Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces

The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces Arab. J. Math. https://doi.org/10.1007/s40065-023-00429-w Arabian Journal of Mathematics Osvaldo Méndez The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces Received: 24 January 2023 / Accepted: 11 April 2023 © The Author(s) 2023 Abstract Given a Musielak–Orlicz function ϕ(x , s) :  ×[0, ∞) → R on a bounded regular domain ⊂ R and a continuous function M :[0, ∞) → (0, ∞), we show that the eigenvalue problem for the ∂ϕ ∇ u(x ) ∂ϕ u(x ) elliptic Kirchhoff’s equation − M ϕ(x , |∇ u(x )|)dx div (x , |∇ u(x )|) = λ (x , |u(x )|) ∂ s |∇ u(x )| ∂ s |u(x )| 1,ϕ has infinitely many solutions in the Sobolev space W (). No conditions on ϕ are required beyond those that guarantee the compactness of the Sobolev embedding theorem. Keywords Variable exponent p-Laplacian · Sobolev embedding · Modular spaces · Musielak–Orlicz spaces · Variable exponent spaces · Kirchhoff equations · Nonlinear wave equation Mathematics Subject Classification Primary 35A01; Secondary 46A80 1 Introduction In 1883, G. Kirchhoff [13] noted that the vibration of an elastic, variable-length string is modeled by means of the following variant of the classical wave equation: 2 1 2 ∂ u ∂ u ∂ u = M , (1.1) 2 2 ∂ t ∂ x ∂ x where M :[0, ∞) −→ [0, ∞) is a suitable increasing function. Since then, a vast amount of literature was devoted to studying the solvability of various Kirchhoff-type equations [2]. In higher dimensions, (1.1) takes up the form ⎛ ⎞ ∂ u ⎝ ⎠ = M |∇ u( y)| dy u. (1.2) ∂ t The stationary problem M |∇ u( y)| dy u = f (x , u) (1.3) u| = 0 O. Méndez (B) Department of Mathematical Sciences, University of Texas at El Paso, 124 Bell Hall, 500W University Ave., El Paso, TX, USA E-mail: osvaldomendez2007@hotmail.com; osmendez@utep.edu 123 Arab. J. Math. has been extensively studied under different assumptions on M and f , see for example [1,19,20,26,27]and the references therein. Of particular interest is the extension of (1.2) to equations involving the p-Laplacian [10,18]: If 1 < p < ∞ is a real number and M ≥ 0 is a continuous function, the p-Kirchhoff operator is defined as ⎛ ⎞ 1, p 1, p p p−2 ⎝ ⎠ K : W () −→ W () K (u) =− M |∇ u( y)| dy div |∇ u(x )| ∇ u(x ) , (1.4) p p 0 0 1, p which clearly generalizes the right-hand side of (1.1). In (1.4), W () stands for the closure of C () in 0 0 1, p(·) the usual Sobolev space W (). The reader is referred to Sect. 3 for the precise terminology to be used in this work. From the physical point of view, this operator arose from the need of finding a mathematical model for the motion of a vibrating string under less stringent assumptions than those assumed for the classical derivation of the linear wave equation. Specifically, the linear wave equation is obtained under the assumption that the length of the vibrating string remains constant during the motion. By removing this assumption, the nonlinear operator K comes into the play. Various boundary value problems associated to the p-Kirchhoff operator (1.4) have been studied for example in [10,18,19]. The emergence of the variable exponent Lebesgue spaces and the subsequent realization of their role in applications [5] sparked interest in the study of boundary value problems of the type K (u) = f (x , u) (1.5) u| = 0, for a variable exponent p = p(x ). The variability of the exponent opens a new class of highly non-trivial difficulties, mainly related to its modular nature, that is to its direct relation to the functional p(x ) u −→ |u(x )| dx rather than with the norm u −→ u , p(x ) as discussed in [2,21]. A vast amount of literature exists on boundary value problems of the type (1.5), under the assumption of the variability of the exponent p(x ). We refer the reader to some of the most significant from the point of view of the present work, such as [2–4,6–8,21,23]. In this article we observe that the treatment of a wide class of eigenvalue problem for Kirchhoff-type operators, including, but not limited to the variable-exponent case, can be unified by the consideration of Musielak–Orlicz spaces. With this objective in mind, we study the eigenvalue problem for a general Kirchhoff equation in this framework. In fact, given a suitable Musielak–Orlicz (MO)-function ϕ and an appropriate function M (we refer the reader to the next section for a detailed account of the notation and terminology), the generalized Kirchhoff operator is naturally given by ∂ϕ ∇ u(x ) K (u) =− M ρ (|∇ u|) div (x , |∇ u(x )|) . (1.6) ∂ s |∇ u(x )| We provide a characterization of the first eigenvalue for the operator (1.6) via a Musielak–Orlicz Sobolev embedding theorem that has been obtained in [16, Theorem 5.1]. The present work is organized as follows. In the next section we introduce the notation and terminology to be used in the exposition and present a brief survey on the literature. In Section 3 the definition and basic properties of the Musielak–Orlicz spaces needed in the sequel are given. Section 4 is a brief survey on the Sobolev embedding theorems in the context of Musielak–Orlicz spaces. In Section 5 we delve into some natural properties of the Musielak–Orlicz operators to be considered later and the functional analytic stage is set for the treatment of the eigenvalue problems developed in detail in Section 7. Section 8 contains applications to p(x ) the variable exponent case, i.e., to the case ϕ(x , t ) = t . 123 Arab. J. Math. 2 Known results In the sequel,  ⊂ R will denote a bounded domain with a regular boundary (the cone condition will do) and M() will stand for the vector space of all real-valued, Borel-measurable functions defined on . The subset of M consisting of functions p :  −→ [1, ∞) will be denoted by P(). The Lebesgue measure of a subset A ⊂ R will be denoted by | A|. For p ∈ P(), the following notation will be used throughout this work: p := essinf p, p := esssup p. −  + For p ∈ P(), the eigenvalue problem q−2 K (u) = λ|u| u in (2.1) u| = 0 was studied in [2]for M subject to α−1 β−1 m t ≤ M (t ) ≤ m t , (2.2) 1 2 for β ≥ α> 1, m ≥ m > 0 and variable exponents p, q ∈ C () satisfying either [2, Theorem 3.1, Theorem 2 1 3.4, Theorem 3.6] β p < q ≤ q < p , (2.3) + − + 1 < q ≤ q <α p , (2.4) − + − or 1 < q(x)< p(x)< p (x ) (2.5) in .Here np(x ) p (x ) = 1 ( p(x )) +∞1 ( p(x )) (1,n) [n,∞) n − p(x ) and 1 stands for the characteristic function of the set A. An anisotropic variant of (2.1) was considered in [23], whereas [4] deals with the following weighted version of (2.1): q−2 K (u) = λV |u| u (2.6) u| = 0, for 0 ≤ V ∈ L (), p < n and M subject to (2.2)[4, Theorem 1.4]. The generalized version of (1.5)given by |u| K (u) = B f (x , s)ds f (x , u) (2.7) u| = 0 1, p(·) is studied in [12]. Specifically, problem (2.7) is shown to have a solution in W () under the following assumptions [12, Theorem 3.1]: (i) M (s)ds ≥ mt , m > 0, for sufficiently large t, 123 Arab. J. Math. (ii) for some positive constants c , c the Carathéodory function 1 2 f :  × R → R q(x ) ∗ satisfies the bound | f (x , t )|≤ c + c |t | for q ∈ C (),1 < q(x)< p (x ), 1 2 (iii) for some positive constants A , A , 1 2 B(s)ds ≤ A + A t , 1 2 (iv) β q <α p . 1 + 1 − γ −1 For M (t ) = a + bγ t , the study of the solvability of a hyperbolic equation related to the operator K can be found in [3]. A polyharmonic version of (1.5) was studied in [6]. In [8], a discussion of problem 1.5 is presented for a linear function M (t ) = a+bt [8, Theorem 1.1], whereas in [7] the existence of a solution of (1.5) is proved provided, among other conditions, that M (t ) ≥ m > 0for t > 0[7, Theorems 3.1−3.4]. Associated to every Musielak–Orlicz function ϕ, the so-called Matuszewska index of ϕ (see [17]) general- izes the role of the exponent p in the classical Lebesgue spaces; in particular the exponent p is easily verified to be the Matuszewska index of the MO function given by p(x ) (x , t ) −→ t . As is observed in [16], sharp conditions (trivially satisfied by the exponent p for the Sobolev embedding stated in [11,15]) on the Matuszewska index of the MO function ϕ guarantee the compactness of the Sobolev embedding 1,ϕ W () → L () (2.8) for a bounded domain  ⊂ R . Via the compactness of the Sobolev embedding, a natural characterization of the first eigenvalue of the Kirchhoff’s operator can be given, and the results outlined above can be regarded as particular cases of our more general approach, which allows for less stringent conditions than the ones stated in the first part of this Section. 3 Musielak–Orlicz spaces Throughout this paper  ⊂ R , n ≥ 1 will stand for a bounded, Lipschitz domain. A convex, left-continuous function ϕ :[0, ∞) −→ [0, ∞) with ϕ(0) = 0, lim ϕ(x ) =∞ and lim + ϕ(x ) = 0 will be said to be an Orlicz function. In particular, x →∞ x →0 any Orlicz function is non-decreasing. The term generalized Orlicz function or Musielak–Orlicz (MO) function will refer to a function ϕ :  ×[0, ∞) →[0, ∞) such that ϕ(x , ·) :[0, ∞) →[0, ∞) is an Orlicz function for each fixed x ∈  and ϕ(·, y) :  →[0, ∞) is Lebesgue measurable for each fixed y ∈ R. 123 Arab. J. Math. The Musielak–Orlicz space L (),[24,25], is the real-vector space X of all extended-real valued, Borel- measurable functions u on  for which ϕ(x,λ|u(x )|) dx < ∞ for some λ> 0, furnished with the norm |u(x )| u = inf λ> 0 : ϕ x , ≤ 1 . The functional ρ (u) = ϕ(x , |u(x )|) dx (3.1) is a convex, left-continuous modular on X [9,11,24]. It is well known [9]that L () is a Banach space. Since it will be needed in the sequel, we define the complementary function ϕ of ϕ as ϕ :  ×[0, ∞) −→ [0, ∞) (3.2) ϕ (x , t ) = sup (tu − ϕ(x , u)) . (3.3) u≥0 The complementary function ϕ is itself a MO-function (see [9]) and Hölder’s inequality holds, namely for ϕ ϕ f ∈ L () and g ∈ L (), f (x )g(x ) dx ≤ 2 f  g . (3.4) ϕ ϕ If in addition ϕ(x , t ) dx < ∞ (3.5) for any K ⊂  with Lebesgue measure |K | < ∞ and inf ϕ(x , 1)> 0, (3.6) x ∈ 1,ϕ ϕ the Musielak–Orlicz Sobolev space W () consisting of all functions in L () whose distributional deriva- tives are in L (), is a Banach space when furnished with the norm u =u +|∇ u| , 1,ϕ ϕ ϕ 1,ϕ where ∇ stands for the gradient operator and |·| denotes the Euclidean norm in R . The Sobolev space W () ∞ 1,ϕ is defined to be the closure of C () in W (). 4 Sobolev-type embeddings The central idea of this Section is the Sobolev Embedding Theorem 4.7. In order to facilitate the flow of ideas we present a few definitions. The Matuszewska index of an Orlicz function ϕ was introduced by Matuszewska and Orlicz in [17]. Definition 4.1 For ϕ as above and each x ∈ ,set ϕ(x , tu) M (x , t ) = lim sup . (4.1) ϕ(x , u) u→∞ 123 Arab. J. Math. The Matuszewska index of ϕ is defined to be ln M (x , t ) ln M (x , t ) m(x ) = lim = inf . (4.2) t →∞ ln t t >1 ln t Definition 4.2 The limit (4.1)issaidtobe uniform if for each δ> 0 there exist s > 1and T > 1suchthat, for all (x , t ) ∈  ×[T , ∞), one as ϕ(x , ts ) M (x , t ) − δ< < M (x , t ) + δ. (4.3) ϕ(x , s ) The following examples illustrate the above definition for some well known MO functions: Example 4.3 Let  ⊆ R be a bounded domain and p :  −→ (0, ∞) be Borel-measurable. The MO function ϕ :  ×[0, ∞) −→ [0, ∞) p(x ) ϕ(x , t ) = t (4.4) has Matuszewska index equal to p(x ). In this case, the convergence (4.2) is trivially uniform on  and the limit (4.2) is clearly uniform. Lemma 4.4 Let  ⊂ R be a bounded domain and ϕ an MO function as described above. If the Matuszewska index m is the restriction to  of a continuous function m ˜ on the closure of , i.e., m ˜ :  −→ R, (4.5) and the convergence to the limits (4.1) and (4.2) is uniform, then there exist C > 1,T > 1 and S > 1 such 0 0 that uniformly in  it holds ϕ(x , sT ) ≤ Cϕ(x , s) (4.6) for any s ≥ S . Condition (4.6) will be referred to as the  condition. Proof Fix δ> 0, then for some T > 1 one has for any t ≥ T , by virtue of (4.2) 0 0 m(x )−δ m(x )+δ t < M (x , t)< t (4.7) uniformly in . By definition of M (x , t ) and on account of the uniformity assumption of the infimum (4.1), there exists a positive number N for which, uniformly for t ≥ T and x ∈ , it holds that ϕ(x , st ) m(x )+δ sup < t . (4.8) ϕ(x , s) s≥ N In particular, for all s ≥ N : sup m(x )+δ ϕ(x , sT ) ≤ T ϕ(x , s). Corollary 4.5 There exists S > 1 and a constant C > 1 such that ϕ(x , 2s) ≤ Cϕ(x , s) (4.9) for any x ∈ ,s ≥ S . Lemma 4.6 If the statement of corollary 4.5 holds, then ρ-convergence is equivalent to norm-convergence in L (). 123 Arab. J. Math. Proof It suffices to show that if (u )ρ -converges to 0 and converges a.e. to 0. then it converges to 0 in the j ϕ topology of the norm. This will be automatically implied by the validity of the equality lim ρ (λx ) = 0 (4.10) ϕ n j →∞ for any λ> 0. It is obviously necessary to show (4.10) only for λ> 1. Let N =[log λ]+ 1 ≥ 1. A simple argument shows that for C, S as in Corollary 4.5 N −1 N N ϕ(x,λ|u (x )|) ≤ ϕ(x , 2S ) + C ϕ(x,(λ/2 )|u (x )|). n 0 n Since the second term in the right-hand side tends a.e. to 0 as n →∞, it follows that ρ (λu ) = ϕ(x,λ|u (x )|)dx → 0as n →∞. (4.11) ϕ n n We refer the reader to [16] for the proof of the following theorem. Theorem 4.7 Let  ⊂ R be a bounded domain and ϕ :  ×[0, ∞) −→ R a Musielak–Orlicz function that satisfies condition (3.6) and for which the limits in (4.1) and (4.2) are uniform; assume that the Matuszewska index m is the restriction to  of a continuous function m ˜ defined on the closure of , that 1 < m =: inf m, and that there exists a function β : (0, ∞) −→ (0, ∞) such that, uniformly in  and for t > 0: ϕ(x , t ) ≤ β(t ). (4.12) Then the embedding 1,ϕ W () → L () (4.13) is compact. Corollary 4.8 For ϕ satisfying the conditions of Theorem 4.7, there exists a positive constant C depending 1,ϕ only on n,,ϕ such that for any u ∈ W () u ≤ C |∇ u| . (4.14) ϕ ϕ Proof See [16]. 123 Arab. J. Math. 5  -type Musielak–Orlicz functions From now on we assume that  ⊆ R is a bounded domain satisfying the cone condition and ϕ is a MO- function satisfying the conditions of Theorem 4.7. Denote the conjugate of ϕ by ϕ . Theorem 5.1 Let ϕ be a M O function on ; assume that ϕ satisfies the conditions of Theorem 4.7;in particular, ϕ satisfies the  condition 4.6, i.e., for some K > 0,S > 1 it holds that ϕ(x , 2s) ≤ K ϕ(x , s) for all s ≥ S , x ∈ . (5.1) Then, sup ρ (u) : ρ (|∇ u|) ≤ r < ∞. (5.2) ϕ ϕ Proof It follows from (5.1) that for arbitrary v ∈ L () (recall that ϕ is nonnegative and nondecreasing) ρ (2v) = ϕ(x , 2|v(x )|)dx ⎛ ⎞ ⎜ ⎟ = + ϕ(x , 2|v(x )|)dx ⎝ ⎠ |v|<S |v|≥S 0 0 ⎛ ⎞ ⎜ ⎟ ≤ ϕ(x , 2S )dx + Kρ (v) ⎝ 0 ϕ ⎠ |v|<S ⎛ ⎞ ⎝ ⎠ ≤ ϕ(x , 2S )dx + Kρ (v) . (5.3) 0 ϕ 1,ϕ If r ≥ 1and u ∈ W () with ρ (|∇ u|) ≤ r, (5.4) it is a simple matter to verify that if |∇ u| ≥ 1 |∇ u| 1 1 = ρ ≤ ρ (|∇ u|) ϕ ϕ |∇ u| |∇ u| ϕ ϕ ≤ r ; |∇ u| it is thus clear that if (5.4) holds, then: |∇ u| ≤ r. Therefore, |∇ u| ≤ max{1, r}= b. It follows from the preceding reasoning in conjunction with Poincaré inequality that if ρ (|∇ u|) ≤ r, then, for some C > 0, u ≤ C |∇ u| ≤ Cb. ϕ ϕ Hence, ρ ≤ 1. (5.5) Cb 123 Arab. J. Math. If Cb < 2, (5.5) implies u u ρ (u) = ρ Cb ≤ ρ 2 ϕ ϕ ϕ Cb Cb ≤ ϕ(x , 2S )dx + Kρ 0 ϕ Cb ≤ ϕ(x , 2S )dx + K . (5.6) Otherwise, on account of the iteration of inequality (5.3) ρ (u) = ρ Cb ϕ ϕ Cb [log Cb]+1 ≤ ρ 2 Cb [log Cb] [log Cb]+1 2 2 ≤ 1 + K + ...K ϕ(x , 2S )dx + K . (5.7) In all, (5.6)and (5.7) yield (5.2). An immediate consequence of the preceding theorem is the following functional-analytic result: Lemma 5.2 For ϕ as in Theorem 4.7 and r > 0, the modular ball 1,ϕ B := u ∈ W () : ρ (|∇ u|) ≤ r r ϕ is weakly closed. Proof B is clearly convex. It suffices to show that it is also norm-closed. If (u ) norm-converges to u ∈ r j 1,ϕ W (),then ρ (|∇(u − u)|) → 0as n →∞ and there is no loss of generality in assuming that ∇ u →∇ u ϕ j j a.e. in . Theorem 4.7 guarantees that (u ) can be chosen so that u → u a.e. in .Since j j ρ (|∇ u|) = ϕ(x , lim |∇ u (x )|)dx ≤ lim inf ϕ(x , |∇ u (x )|)dx ≤ r, (5.8) ϕ n n n→∞ n→∞ it follows that u ∈ B , i.e., B is norm-closed (and convex) and hence weakly closed. r r 6 Differentiability properties Aiming at a full description of the Fréchet derivative of the functionals to be introduced momentarily, a further assumption is imposed unto the MO function ϕ at this point, namely, it is from now on required that ϕ be an N function. More precisely: Definition 6.1 An MO functionissaidtobean N -function iff it satisfies the condition ϕ(x , t ) lim = 0 a.e.. (6.1) t →0 It is well known [9]thatif ϕ is an N -function, it can be written as ϕ(x , t ) = φ(x , s) ds, (6.2) where φ(x , ·) is the right t-derivative of ϕ. On the other hand, the conjugate function ϕ can be written as ∗ −1 ϕ (x , t ) = φ (x , s) ds. (6.3) The proof of the following theorem can be found in [14,22]; we include it here in the interest of completeness. 123 Arab. J. Math. Theorem 6.2 Let ϕ be an N -function; assume that ∂ϕ ϕ (x , t ) = (x , t ) ∂ t is continuous a.e. x ∈ . Define the operator T as T : M() −→ M() ∂ϕ T (u) = ϕ (x , |u(x )|) = (x , |u(x )|). ϕ t ∂ t Then, from the assumption ϕ ϕ T (L ()) ⊆ L () (6.4) it follows that the operator ϕ ϕ T : L () −→ L () is continuous and bounded. ϕ ϕ Proof Assume (u ) ⊆ L () converges to u ∈ L ().On  ×[0, ∞) define w(x , t ) = ϕ (x , |u(x ) + t |) − ϕ (x , |u(x )|); t t then on account of the assumption on ϕ , w is a Carathéodory function and w(x , 0) = 0. If ϕ ϕ T : L () −→ L () ϕ ∗ is continuous at 0, then T (u − u) −→ 0in L () as n →∞. If ϕ satisfies the  condition, the latter is w n 2 equivalent to ρ (T (u − u)) = ϕ (x , |ϕ (x , |u (x )|) − ϕ (x , |u(x )|)|) → 0as n →∞, ϕ w n t n t that is T (u ) −→ T (u) in L () as n →∞. ϕ n ϕ t t Therefore, it is enough to show that T is continuous at 0 under the assumption that ϕ (x , 0) = 0 a.e. in . ϕ t Assume that T is not continuous at 0; let r > 0and let (u ) be a sequence that converges to 0 in L (),for ϕ n which T (u ) ∗ ≥ r for any n ∈ N. ϕ n ϕ Since norm convergence implies modular convergence, one can, without loss of generality assume that max ρ (u ), u  < , ϕ n n ϕ and hence that ϕ(x , |u (x )|) dx < ∞. Due to the validity of the  condition for ϕ , norm con- n 2 n=1 vergence and modular convergence are equivalent on L (). It follows that there exists (r)> 0 such that ρ ∗ T (u ) ≥ (r ) for any n ∈ N. We next claim the existence of a sequence of real numbers ( ),a ϕ ϕ n k sequence ( ) of subsets of  and a subsequence (u ) of (u ) satisfying the following conditions: k n n (i)  <  , k+1 k (ii) | |≤  , k k ∗ 2 (iii) ϕ (x , |T (u )|) dx > (r ). ϕ n k 3 123 Arab. J. Math. (iv) If E ⊆  is measurable and | E | < 2 ,then k+1 (r ) ϕ (x , |T (u )|) dx < . ϕ n t k Set  = ,  =||, n = 1. We assume that  , n and  are given, then, by assumption, ϕ (·, |u (·)|) ∈ 1 1 1 k k k t n L () and on account of the  condition one has ϕ (x , |ϕ (x , |u (x )|)|) dx < ∞. t n Since the measure A −→ μ( A) = ϕ (x , |ϕ (x , |u (x )|)|) dx t n defined on the Borel σ algebra B of subsets of  is absolutely continuous with respect to the Lebesgue measure, (r ) one can find  such that any X ∈ B with | X | < 2 satisfies ϕ (x , |ϕ (x , |u (x )|)|) dx < . The k+1 k+1 t n assumption  ≤ 2 would contradict (iii ). We now proceed to the construction of  and n . k k+1 k+1 k+1 It is well known [14] that the strong convergence of (u ) in L () implies the convergence in measure of both (T (u )) and (T (T (u ))). Consequently, there exists n ∈ N such that ϕ n ϕ ϕ n k+1 t t (r ) x ∈  : T (T (u )) > < < < . ϕ ϕ n k+1 k t k+1 3|| 2 Define (r ) = x ∈  : T ∗ (T (u )) > . k+1 ϕ ϕ n t k+1 3|| Next, ⎛ ⎞ ⎜ ⎟ ∗ ∗ ϕ (x , |T (u )|) dx = − ϕ (x , |T (u )|) dx ϕ n ⎝ ⎠ ϕ n t k+1 t k+1 k+1 k+1 (r ) 2(r ) >(r ) − = . 3 3 By construction ∞ ∞ ≤  < 2 . (6.5) j j k+1 j =k+1 j =k+1 Set u (x ) if x ∈  \ n k j v(x ) = j =k+1 0 otherwise. It is clear that v ∈ L (). Next, observe that: ∗ ∗ ϕ (x , T (v(x ))) dx ≥ ϕ (x , T (u (x ))) dx ϕ ϕ n t t k k=1 ∞ \∪ k j j =k+1 123 Arab. J. Math. ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = − ϕ (x , T (u (x ))) dx ϕ n t k ⎝ ⎠ k=1 k ∪ j =k+1 2 1 ≥ (r ) − (r ) 3 3 k=1 =∞, which follows by (6.5) and condition (i v). This contradicts assumption (6.4). Hence, T is continuous at 0. We next prepare the ground for the next lemma, which deals with the differentiability properties needed in the sequel. Let M : R −→ [0, ∞) be continuous and write M (s) = M (t ) dt . (6.6) Consider the maps F : L () −→ [0, ∞) F (u) = ρ (u) (6.7) and 1,ϕ H : W () −→ [0, ∞) H (u) = M ρ (|∇ u|) . (6.8) Recall that a MO function ϕ is said to be locally integrable if for any t > 0 and any subset W ⊆  with μ(W)< ∞ one has ϕ(x , t ) dx < ∞. Lemma 6.3 In the terminology of the preceding paragraph, let ϕ be an N function; suppose that ϕ(x , ·) ∂ t is continuous a.e. x . Assume that the complementary function ϕ of ϕ satisfies the  condition and is locally integrable. If the maps ϕ ϕ D : L () −→ L () (6.9) D (u) = ϕ(·, |u(·)|) (6.10) ∂ t and 1,ϕ D : W () −→ L () (6.11) D (u) = ϕ(·, |∇ u(·)|) (6.12) ∂ t are well defined, then the functionals (6.7) and (6.8) are Fréchet differentiable for u = 0.Inthiscase, the derivatives of F and H at u = 0 are given, respectively by ∂ϕ u(x ) F (u), h= (x , |u(x )|) h(x ) dx , (6.13) ∂ s |u(x )| and by ∂ϕ ∇ u(x ) H (u), h=− M ρ (|∇ u) div (x , |∇ u(x )|) , h , (6.14) ∂ s |∇ u(x )| with the understanding that = 0 if x = 0. |x | 123 Arab. J. Math. Proof It suffices to show that under the stipulated conditions (6.13) holds; a straightforward application of the chain rule will yield the full statement of the Lemma. Observe that for u(x ) = 0, h ∈ L () one has |u(x ) + th(x )|−|u(x )| 2u(x )h(x ) + t (h(x )) t |u(x ) + th(x )|+|u(x )| u(x )h(x ) −→ as t −→ 0. |u(x )| Therefore, for some θ ∈ (|u(x )|, |u(x ) + th(x )|), ϕ(x , |u(x ) + th(x )|) − ϕ(x , |u(x )|) |u(x ) + th(x )|−|u(x )| ∂ϕ = (x,θ) t t ∂ t u(x )h(x ) ∂ϕ −→ (x , |u(x )|) as t −→ 0. |u(x )| ∂ t By hypothesis, ∂ϕ h(x ) (x , |u(x )|) dx < ∞, (6.15) ∂ t which in conjunction with the above limit yields: |u(x ) + th(x )|−|u(x )| ∂ϕ lim (x,θ(x ))dx t −→0 t ∂ t {x :u(x )=0} u(x )h(x ) ∂ϕ = (x , |u(x )|)dx . (6.16) |u(x )| ∂ t {x :u(x )=0} On the other hand, one has by assumption, one has a.e. x ∈ : ϕ(x , |th(x )|) −→ 0as t −→ 0 and by convexity, for 0 < t < 1, ϕ(x , |th(x )|) ≤ ϕ(x , |h(x )|). Since h ∈ L (), ϕ(x , |h(x )|) dx < ∞, it is easily derived by way of application of Lebesgue’s theorem that ϕ(x , |u(x ) + th(x )|− ϕ(x , |u(x )|) dx {x :u(x )=0} = ϕ(x , t |h(x )|)/t dx → 0as t → 0. (6.17) {x :u(x )=0} In all, ⎛ ⎞ F (u + th) − F (u) ϕ(x , |u(x ) + th(x )|) − ϕ(x , |u(x )|) ⎜ ⎟ = + dx ⎝ ⎠ t t {x :u(x )=0} {x :u(x )=0} u(x )h(x ) ∂ϕ → (x , |u(x )|)dx as t −→ 0. |u(x )| ∂ t {x :u(x )=0} 123 Arab. J. Math. We conclude that F is Gâteaux differentiable and that its derivative is equal to the right-hand side of (6.13). The proof of the Gâteaux differentiability of G follows along the same lines. Hence, it suffices to prove that under the additional assumptions (6.9)and (6.11), the operators ϕ ϕ ∗ L : L () −→ (L ()) L (u) = F (u) (6.18) 1,ϕ 1,ϕ L : W () −→ W () 0 0 L(u) = G (u) (6.19) are continuous at u = 0. A standard functional-analytic result guarantees that in this case F and G are Fréchet differentiable and that the Fréchet and the Gâteaux derivatives coincide. To this end, consider a convergent ϕ ϕ sequence (u ) ⊂ L (),say u −→ u in L () as j −→ ∞: it is well known that there is no loss of j j generality by assuming that u converges to u almost everywhere in  (see [9,11])). ∂ϕ Since (x , |u(x )|) ∈ L () there exists λ > 0for which ∂ t ∂ϕ ϕ x,λ (x , |u(x )|) dx < ∞. (6.20) ∂ t Notice that for h ≤ 1 one has u (x ) j ∂ϕ u(x ) ∂ϕ (x , |u (x )|) − (x , |u(x )|)) h(x ) dx (6.21) |u (x )| ∂ t |u(x )| ∂ t {x :u (x )=0,u(x )=0} u (x ) ∂ϕ ∂ϕ = (x , |u (x )|) − (x , |u(x )|) h(x ) dx |u (x )| ∂ t ∂ t {x :u (x )=0,u(x )=0} u (x ) j u(x ) ∂ϕ + − (x , |u(x )|)h(x ) dx . (6.22) |u (x )| |u(x )| ∂ t {x :u (x )=0,u(x )=0} Theorem 6.2 guarantees the continuity of the map (6.9); it is apparent from this fact in conjunction with Hölder’s inequality (3.4) that the integral in (6.21) tends to 0 as j tends to infinity. As to the remaining integral, set, for j ∈ N: ∂ϕ (x , |u(x )|) = f (x ) ∂ t u (x ) u(x ) r (x ) = − f (x ). |uj (x )| |u(x )| Recall that in the terminology of Lemma 4.4, ∗ ∗ ϕ (x , sT ) ≤ C ϕ (x , s), (6.23) for any s ≥ S .For λ as in (6.20)and any λ> 2λ let k be defined by the inequalities 0 0 0 k−2 k−1 2 <λ/λ ≤ 2 . Notice that |r |≤ 2. For S as in Lemma 4.4, one readily obtains, for any positive integer m,using the j 0 monotonicity of ϕ in the second variable ∗ m ϕ x , 2 λ f (x ) ∗ m m−1 m−1 = ϕ x , 2 λ f (x ) I 2 λ f (x ) + I 2 λ f (x ) 0 0 0 [S ,∞) [0,S ) 0 0 ∗ m−1 ∗ ≤ C ϕ (x , 2 λ f (x )) + ϕ (x , 2S ). (6.24) 0 0 The iteration of the preceding inequality yields 123 Arab. J. Math. ∗ ∗ k ϕ x,λr ≤ ϕ x , 2 λ f (x ) j 0 ∗ k k−1 k−1 = ϕ x , 2 λ f (x ) I 2 λ f (x ) + I 2 λ f (x ) 0 [S ,∞) 0 [0,S ) 0 0 0 k−1 i ∗ k ∗ ≤ C ϕ (x , 2S ) + C ϕ (x,λ f (x )). 0 0 i =0 A routine application of Lebesgue’s dominated convergence quickly shows that u (x ) u(x ) ∂ϕ ϕ x,λ − (x , |u(x )|) dx −→ 0as j −→ ∞. (6.25) |u (x )| |u(x )| ∂ t {x :u (x )=0,u(x )=0} A similar argument shows that (6.25) holds for 0 <λ ≤ 2λ . It follows from the arbitrariness of λ that " " " " u u ∂ϕ " " − (·, |u|) → 0as j →∞. " " |u | |u| ∂ t j ϕ L () On account of Hölders inequality (3.4), the integral (6.22) is bounded by " " " " u u ∂ϕ " " 2 − (·, |u|) −→ 0as j −→ ∞. (6.26) " " |u | |u| ∂ t ∗ j ϕ L () The proof of the continuity of L follows along the same lines and will be skipped. This concludes the continuity argument and hence F and G are Fréchet differentiable. Lemma 6.4 If M :[0, ∞) −→ [0, ∞) is strictly increasing, the modular M - ball 1,ϕ B = u ∈ W () : M ρ (|∇ u|) ≤ r (6.27) r ϕ 1,ϕ is weakly closed in W (). −1 Proof For any r > 0set s = M (r ); it is then clear from the above assumptions that 1,ϕ B = V = u ∈ W () : ρ (|∇ u|) ≤ s (6.28) r s ϕ r and the latter set is weakly closed (Lemma 5.2). 7 Kirchhoff-type eigenvalue problem We are now ready to prove the main result of this work. Theorem 7.1 Let M ∈ C ((0, ∞), (0, ∞)).Set M (t ) = M (s)ds. Then, for any r > 0, there exists a 1,ϕ solution (u,λ) ∈ W () × (0, ∞) to the equation ∂ϕ ∇ u(x ) ∂ϕ u(x ) − M ρ (|∇ u|) div (x , |∇ u(x )|) = λ (x , |u(x )|) , (7.1) ∂ s |∇ u(x )| ∂ s |u(x )| satisfying M (ρ (|∇ u|)) = r (7.2) 123 Arab. J. Math. Proof Theorem 5.1 guarantees that for any r > 0, 0 < sup ρ (u) : u ∈ B = S < ∞. (7.3) ϕ r r 1,ϕ We next observe that Theorem 4.7 implies the existence of a sequence (u ) ⊂ V with u  u in W () j s j 0 r 0 and u −→ u in L () such that j 0 ρ (u) −→ S = ρ (u ) . (7.4) ϕ r ϕ 0 To see this, we notice that a.e. in , ϕ(x , |u (x )|) −→ ϕ(x , |u(x )|) and that on account of convexity, for any n ∈ N: 1 1 ϕ(x , |u (x )|) ≤ ϕ(x , 2|u (x ) − u(x )|) + ϕ(x , 2|u(x )|). (7.5) n n 2 2 Select n large enough so that 2u − u  < 1; for such n, it holds, by way of the convexity of ϕ(x , ·), n ϕ 1 1 2|u (x ) − u(x )|2u − u n n ϕ ϕ(x , 2|u (x ) − u(x )|)dx = ϕ x , dx 2 2 2u − u n ϕ |u (x ) − u(x )| ≤u − u  ϕ x , dx (7.6) n ϕ u − u n ϕ Denote the left-hand side and the right-hand side of (7.5)by v and w respectively. Then the following n n conditions hold: (i) v (x ) → v(x ) = ϕ(x , |u(x )|) ∈ L () a.e. in (ii) w (x ) → w(x ) = ϕ(x , 2|u(x )|) ∈ L () a.e. in (iii) v ,w ∈ L () for any n ∈ N n n 1 1 (iv) w dx → ϕ(x , 2|u|)dx = ρ (2u). n ϕ 2 2 Since w − v ≥ 0 a.e in , Fatou’s Lemma leads to: (w − v)dx ≤ w dx + lim inf (−v ) dx = w dx − lim sup v dx and (w + v)dx ≤ w dx + lim inf v dx . The two last statements yield lim ϕ(x , |u (x )|)dx = ϕ(x , |u(x )|)dx n→∞ or, equivalently ρ (u ) −→ ρ (u) as n →∞. (7.7) ϕ n ϕ By construction ρ (u ) −→ S ;(7.7) is therefore the desired result. ϕ n r 123 Arab. J. Math. Lemma 6.4 yields M ρ (|∇ u |) ≤ r. Furthermore, the continuity and monotonicity of the modular ρ ϕ 0 ϕ immediately yield M (ρ (|∇ u |)) = r. ϕ 0 As is apparent from the above, u is a solution of the constrained maximization problem of the type max F (v) with G(v) = r. (7.8) It is a routine matter to show, via the Implicit Function Theorem, that the preceding statement implies that kerG (u ) = ker F (u ), (7.9) 0 0 from which it is clear (since neither functional is null) that u satisfies equation (7.1)for some λ> 0. This concludes the proof of the claim. 8 Applications A particular instance of Theorem 7.1 deserves to be stressed, namely its implication in the consideration of variable exponent Lebesgue spaces. More precisely, if  ⊂ R is bounded then ϕ :  ×[0, ∞) −→ [0, ∞) (8.1) p(x ) ϕ(x , t ) = (8.2) p(x ) satisfies the conditions of Theorem 4.7 iff p is the restriction to  of a function p ˜ ∈ C (R , R) and 1 < p = inf p(x ) ≤ p = sup p(x)< ∞. (8.3) − + x ∈ x ∈ The conjugate function is clearly given by p(x ) p(x ) − 1 p(x )−1 ϕ (x , t ) = t . (8.4) p(x ) It is straightforward to verify the conditions of Lemma 6.3 for this case. It is customary to write, in this case p(x ) |u(x )| ρ (u) = dx . (8.5) p(x ) Therefore, Theorem 7.1 yields the following result: Theorem 8.1 If  ⊂ R is bounded, M satisfies the conditions of Theorem 7.1 and p :  −→ (1, ∞) is a variable exponent satisfying the assumptions (8.3) then for each r > 0 there exists a solution (u ,λ) to the eigenvalue problem p(x )−2 p(x )−2 M ρ (|∇ u|) di v |∇ u| ∇ u = λ|u| u (8.6) with M ρ (|∇ u |) = r. p 0 With the aid of the following compactness theorem, the techniques used in Sections 6 and 7, one can derive Theorems 8.3 and 8.4, particular cases of which were obtained in [2]and [4], respectively, via different methods. 123 Arab. J. Math. Theorem 8.2 [9,15] Let  ⊂ R ,n > 1 be a bounded domain, p ∈ C () with 1 < p ≤ p < n. (8.7) − + For 0 <ε < and q ∈ P() such that n−1 np(x ) q(x)< − , (8.8) n − p(x ) the embedding 1, p(·) q(·) W () → L () (8.9) is compact. Theorem 8.3 Givenafunction M ∈ C ((0, ∞), [0, ∞)),M (t)> 0 for t > 0. Under the hypotheses of 1, p(·) Theorem 8.2, for any r > 0, there exists a solution (u,λ) ∈ W () × (0, ∞) to the equation p−2 q−2 − M ρ (|∇ u|) div |∇ u| ∇ u = λ|u| u, (8.10) satisfying M ρ (|∇ u|) = r. (8.11) p(x ) Proof The proof follows from Theorem 7.1 and Theorem 8.3 by considering ϕ(x , t ) = and by way of p(x ) the bound ⎧ ⎫ ⎨ ⎬ |u(x )| sup : ρ (|∇ u|) ≤ r < ∞, (8.12) ⎩ q(x ) ⎭ which is easy derived as in Theorem 5.1 via the compactness result of Theorem 8.2. Theorem 8.4 Under the assumptions of Theorem 8.2, for any 0 ≤ V ∈ L () and r > 0, there exists a 1, p(·) solution (u,λ) ∈ W () × (0, ∞) to the equation |∇ u| p−2 q−2 − M ρ div |∇ u| ∇ u = λV |u| u, (8.13) satisfying |∇ u| M ρ = r. (8.14) Proof The proof follows along the same lines as those of Theorem 7.1 by observing that the functional V (x ) p(·) q(x ) L ()  u → T (u) = |u(x )| dx q(x ) is Fréchet differentiable for u = 0 and that, for h ∈ C (), q(x )−2 T (u), h= V (x )|u(x )| u(x )h(x )dx . If B , r > 0, is defined as in Lemma 6.4, it follows as in the proof of Theorem 7.1 that there exists u ∈ B r 0 r p(x ) ∇ u with M dx = r such that p(x ) ⎧ ⎫ ⎨ ⎬ V (x ) V (x ) q(x ) q(x ) |u (x )| dx = max |u(x )| dx , u ∈ B . 0 r ⎩ ⎭ q(x ) q(x ) Reasoning mutatis mutandis as in the proof of Theorem 7.1 it can be shown that u is in fact a sought-for solution to Problem 8.13. 123 Arab. J. Math. Acknowledgements The author is pleased to express his gratitude to the referees for their careful revision of the manuscript and their valuable suggestions. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Funding No funding was provided or used for the research leading to this article. Declarations Conflict of interest The author declares having no conflict of, competing or financial interest in regards to the research presented in this work. References 1. Alves, G.; Corrêa, F.: Positive solutions for a quasilinear elliptic equation of Kirchhoffs type. Comput. Math. Appl. 49, 85–93 (2005) 2. Arouzi, G.; Mirzapour, M.: Eigenvalue problems for p(x )-Kirchhoff-type equations. EJDE 2013(253), 1–10 (2013) 3. Autuori, G.; Pucci, P.; Salvatori, M.C.: Global nonexistence for nonlinear-Kirchhoff systems. Arch. Rat. Mech. Anal. 196, 489–516 (2010) 4. Chung, N.: On some p(x )-Kirchhoff type equations with weights. J. Appl. Math. Inform. 32(1–2), 113–128 (2014) 5. Chen, Y.; Levine, S.; Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006) (electronic). 6. Colasuonno, F.; Pucci, P.: Multiplicity of solutions for p(x )-polyharmonic Kirchhoff equations. Nonlinear Anal. 74, 5962– 5974 (2011) 7. Dai, G.; Hao, R.: Existence of solutions for a p(x )-Kirchhoff-type equation. J. Math. Anal. Appl. 79, 275–284 (2009) 8. Dai, G.; Liu, D.: Infinitely many positive solutions for a p(x )-Kirchhoff-type equation. J. Math. Anal. Appl. 359, 704–710 (2009) 9. Diening, L.; Harjulehto, P.; Hästö, P.; Ruži ˚ cka, ˇ M.: Lebesgue and Sobolev spaces with variable exponents, vol. 2017. Lecture notes in Mathematics. Springer, Heidelberg (2011) 10. Dreher, M.: The Kirchhoff equation for the p-Laplacian. Rend. Semin. Mat. Univ. Polit. Torino 64(2), 217–238 (2006) 11. D.E. Edmunds, J. Lang, O. Méndez, Differential operators on spaces of variable integrability, World Scientific (2015) 12. Fan, X.: On non-local p(x )-Laplacian Dirichlet problems. Nonlinear Anal. Theor. 72, 3314–3323 (2010) 13. G. Kirchoff Mechaniks, Teubner, Leipzig (1883) 14. Krasnoselskii, M.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, Elsford (1964) p(x ) k, p(x ) 15. O. Kovácik, ˘ J. Rákosník, On spaces L () and W (), Czechoslovak Math. J. 41 (1991), 592–618. 16. Lang, J.; Méndez, O.: Sharp conditions for the compactness of the Sobolev embedding theorem on Musielak–Orlicz spaces. Math. Nachr. 292(2), 377–388 (2019) 17. Matuszewska, W.; Orlicz, W.: On certain properties of ϕ-functions. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 8, 430–443 (1960) 18. Li, H.: Blow-up of solutions to a p-Kirchhoff-type parabolic equation with general nonlinearity. J. Dyn Control Syst 26, 383–392 (2020) 19. Li, L.; Sun, J.: Existence and multiplicity solutions for a non local equation of Kirchhoff type. Differ. Equ. Appl. 10(4), 369–386 (2018) 20. Liang, Z.; Li, F.; Shi, J.: Positive solutions to Kirchhofs type equations with nonlinearity having prescribed asymptotic behavior. Ann. Inst. H. Poincaré Anal. Non Linéaire 31(1), 155–167 (2014) 21. O. Méndez, On the eigenvalue problem for a class of Kirchhoff-type equations. J. Math. Anal. Appl. 494(2) (2021). 22. O. Méndez, J. Lang, Analysis on function spaces of Musielak–Orlicz type. Chapman and Hall/CRC Monographs and Research Notes in Mathematics (2018) 23. Mihailescu, M.; Radulescu, V.; Pucci, P.: Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent. J. Math. Anal. Appl. 340(1), 687–698 (2008) 24. J. Musielak, Orlicz Spaces and Modular Spaces, Lect. Notes in Math. 1034, 1983. 25. Nakano, H.: Modulared Semi-ordered Linear Spaces. Maruzen, Tokyo (1950) 26. Sun, J.; Tang, V.: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal. theo. 74(4), 1212–1222 (2011) 27. Sun, J.; Wu, T.: Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains. Proc. R. Soc. Edinburgh: Sect. A Math. 146(2), 435–448 (2016) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arabian Journal of Mathematics Springer Journals

The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces

Méndez, Osvaldo
Arabian Journal of Mathematics , Volume 12 (3) – Dec 1, 2023
Download PDF
19 pages

Loading next page...
 
/lp/springer-journals/the-eigenvalue-problem-for-kirchhoff-type-operators-in-musielak-orlicz-pIoxtxflKI

References (37)

Publisher
Springer Journals
Copyright
Copyright © The Author(s) 2023
ISSN
2193-5343
eISSN
2193-5351
DOI
10.1007/s40065-023-00429-w
Publisher site
See Article on Publisher Site

Abstract

Arab. J. Math. https://doi.org/10.1007/s40065-023-00429-w Arabian Journal of Mathematics Osvaldo Méndez The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces Received: 24 January 2023 / Accepted: 11 April 2023 © The Author(s) 2023 Abstract Given a Musielak–Orlicz function ϕ(x , s) :  ×[0, ∞) → R on a bounded regular domain ⊂ R and a continuous function M :[0, ∞) → (0, ∞), we show that the eigenvalue problem for the ∂ϕ ∇ u(x ) ∂ϕ u(x ) elliptic Kirchhoff’s equation − M ϕ(x , |∇ u(x )|)dx div (x , |∇ u(x )|) = λ (x , |u(x )|) ∂ s |∇ u(x )| ∂ s |u(x )| 1,ϕ has infinitely many solutions in the Sobolev space W (). No conditions on ϕ are required beyond those that guarantee the compactness of the Sobolev embedding theorem. Keywords Variable exponent p-Laplacian · Sobolev embedding · Modular spaces · Musielak–Orlicz spaces · Variable exponent spaces · Kirchhoff equations · Nonlinear wave equation Mathematics Subject Classification Primary 35A01; Secondary 46A80 1 Introduction In 1883, G. Kirchhoff [13] noted that the vibration of an elastic, variable-length string is modeled by means of the following variant of the classical wave equation: 2 1 2 ∂ u ∂ u ∂ u = M , (1.1) 2 2 ∂ t ∂ x ∂ x where M :[0, ∞) −→ [0, ∞) is a suitable increasing function. Since then, a vast amount of literature was devoted to studying the solvability of various Kirchhoff-type equations [2]. In higher dimensions, (1.1) takes up the form ⎛ ⎞ ∂ u ⎝ ⎠ = M |∇ u( y)| dy u. (1.2) ∂ t The stationary problem M |∇ u( y)| dy u = f (x , u) (1.3) u| = 0 O. Méndez (B) Department of Mathematical Sciences, University of Texas at El Paso, 124 Bell Hall, 500W University Ave., El Paso, TX, USA E-mail: osvaldomendez2007@hotmail.com; osmendez@utep.edu 123 Arab. J. Math. has been extensively studied under different assumptions on M and f , see for example [1,19,20,26,27]and the references therein. Of particular interest is the extension of (1.2) to equations involving the p-Laplacian [10,18]: If 1 < p < ∞ is a real number and M ≥ 0 is a continuous function, the p-Kirchhoff operator is defined as ⎛ ⎞ 1, p 1, p p p−2 ⎝ ⎠ K : W () −→ W () K (u) =− M |∇ u( y)| dy div |∇ u(x )| ∇ u(x ) , (1.4) p p 0 0 1, p which clearly generalizes the right-hand side of (1.1). In (1.4), W () stands for the closure of C () in 0 0 1, p(·) the usual Sobolev space W (). The reader is referred to Sect. 3 for the precise terminology to be used in this work. From the physical point of view, this operator arose from the need of finding a mathematical model for the motion of a vibrating string under less stringent assumptions than those assumed for the classical derivation of the linear wave equation. Specifically, the linear wave equation is obtained under the assumption that the length of the vibrating string remains constant during the motion. By removing this assumption, the nonlinear operator K comes into the play. Various boundary value problems associated to the p-Kirchhoff operator (1.4) have been studied for example in [10,18,19]. The emergence of the variable exponent Lebesgue spaces and the subsequent realization of their role in applications [5] sparked interest in the study of boundary value problems of the type K (u) = f (x , u) (1.5) u| = 0, for a variable exponent p = p(x ). The variability of the exponent opens a new class of highly non-trivial difficulties, mainly related to its modular nature, that is to its direct relation to the functional p(x ) u −→ |u(x )| dx rather than with the norm u −→ u , p(x ) as discussed in [2,21]. A vast amount of literature exists on boundary value problems of the type (1.5), under the assumption of the variability of the exponent p(x ). We refer the reader to some of the most significant from the point of view of the present work, such as [2–4,6–8,21,23]. In this article we observe that the treatment of a wide class of eigenvalue problem for Kirchhoff-type operators, including, but not limited to the variable-exponent case, can be unified by the consideration of Musielak–Orlicz spaces. With this objective in mind, we study the eigenvalue problem for a general Kirchhoff equation in this framework. In fact, given a suitable Musielak–Orlicz (MO)-function ϕ and an appropriate function M (we refer the reader to the next section for a detailed account of the notation and terminology), the generalized Kirchhoff operator is naturally given by ∂ϕ ∇ u(x ) K (u) =− M ρ (|∇ u|) div (x , |∇ u(x )|) . (1.6) ∂ s |∇ u(x )| We provide a characterization of the first eigenvalue for the operator (1.6) via a Musielak–Orlicz Sobolev embedding theorem that has been obtained in [16, Theorem 5.1]. The present work is organized as follows. In the next section we introduce the notation and terminology to be used in the exposition and present a brief survey on the literature. In Section 3 the definition and basic properties of the Musielak–Orlicz spaces needed in the sequel are given. Section 4 is a brief survey on the Sobolev embedding theorems in the context of Musielak–Orlicz spaces. In Section 5 we delve into some natural properties of the Musielak–Orlicz operators to be considered later and the functional analytic stage is set for the treatment of the eigenvalue problems developed in detail in Section 7. Section 8 contains applications to p(x ) the variable exponent case, i.e., to the case ϕ(x , t ) = t . 123 Arab. J. Math. 2 Known results In the sequel,  ⊂ R will denote a bounded domain with a regular boundary (the cone condition will do) and M() will stand for the vector space of all real-valued, Borel-measurable functions defined on . The subset of M consisting of functions p :  −→ [1, ∞) will be denoted by P(). The Lebesgue measure of a subset A ⊂ R will be denoted by | A|. For p ∈ P(), the following notation will be used throughout this work: p := essinf p, p := esssup p. −  + For p ∈ P(), the eigenvalue problem q−2 K (u) = λ|u| u in (2.1) u| = 0 was studied in [2]for M subject to α−1 β−1 m t ≤ M (t ) ≤ m t , (2.2) 1 2 for β ≥ α> 1, m ≥ m > 0 and variable exponents p, q ∈ C () satisfying either [2, Theorem 3.1, Theorem 2 1 3.4, Theorem 3.6] β p < q ≤ q < p , (2.3) + − + 1 < q ≤ q <α p , (2.4) − + − or 1 < q(x)< p(x)< p (x ) (2.5) in .Here np(x ) p (x ) = 1 ( p(x )) +∞1 ( p(x )) (1,n) [n,∞) n − p(x ) and 1 stands for the characteristic function of the set A. An anisotropic variant of (2.1) was considered in [23], whereas [4] deals with the following weighted version of (2.1): q−2 K (u) = λV |u| u (2.6) u| = 0, for 0 ≤ V ∈ L (), p < n and M subject to (2.2)[4, Theorem 1.4]. The generalized version of (1.5)given by |u| K (u) = B f (x , s)ds f (x , u) (2.7) u| = 0 1, p(·) is studied in [12]. Specifically, problem (2.7) is shown to have a solution in W () under the following assumptions [12, Theorem 3.1]: (i) M (s)ds ≥ mt , m > 0, for sufficiently large t, 123 Arab. J. Math. (ii) for some positive constants c , c the Carathéodory function 1 2 f :  × R → R q(x ) ∗ satisfies the bound | f (x , t )|≤ c + c |t | for q ∈ C (),1 < q(x)< p (x ), 1 2 (iii) for some positive constants A , A , 1 2 B(s)ds ≤ A + A t , 1 2 (iv) β q <α p . 1 + 1 − γ −1 For M (t ) = a + bγ t , the study of the solvability of a hyperbolic equation related to the operator K can be found in [3]. A polyharmonic version of (1.5) was studied in [6]. In [8], a discussion of problem 1.5 is presented for a linear function M (t ) = a+bt [8, Theorem 1.1], whereas in [7] the existence of a solution of (1.5) is proved provided, among other conditions, that M (t ) ≥ m > 0for t > 0[7, Theorems 3.1−3.4]. Associated to every Musielak–Orlicz function ϕ, the so-called Matuszewska index of ϕ (see [17]) general- izes the role of the exponent p in the classical Lebesgue spaces; in particular the exponent p is easily verified to be the Matuszewska index of the MO function given by p(x ) (x , t ) −→ t . As is observed in [16], sharp conditions (trivially satisfied by the exponent p for the Sobolev embedding stated in [11,15]) on the Matuszewska index of the MO function ϕ guarantee the compactness of the Sobolev embedding 1,ϕ W () → L () (2.8) for a bounded domain  ⊂ R . Via the compactness of the Sobolev embedding, a natural characterization of the first eigenvalue of the Kirchhoff’s operator can be given, and the results outlined above can be regarded as particular cases of our more general approach, which allows for less stringent conditions than the ones stated in the first part of this Section. 3 Musielak–Orlicz spaces Throughout this paper  ⊂ R , n ≥ 1 will stand for a bounded, Lipschitz domain. A convex, left-continuous function ϕ :[0, ∞) −→ [0, ∞) with ϕ(0) = 0, lim ϕ(x ) =∞ and lim + ϕ(x ) = 0 will be said to be an Orlicz function. In particular, x →∞ x →0 any Orlicz function is non-decreasing. The term generalized Orlicz function or Musielak–Orlicz (MO) function will refer to a function ϕ :  ×[0, ∞) →[0, ∞) such that ϕ(x , ·) :[0, ∞) →[0, ∞) is an Orlicz function for each fixed x ∈  and ϕ(·, y) :  →[0, ∞) is Lebesgue measurable for each fixed y ∈ R. 123 Arab. J. Math. The Musielak–Orlicz space L (),[24,25], is the real-vector space X of all extended-real valued, Borel- measurable functions u on  for which ϕ(x,λ|u(x )|) dx < ∞ for some λ> 0, furnished with the norm |u(x )| u = inf λ> 0 : ϕ x , ≤ 1 . The functional ρ (u) = ϕ(x , |u(x )|) dx (3.1) is a convex, left-continuous modular on X [9,11,24]. It is well known [9]that L () is a Banach space. Since it will be needed in the sequel, we define the complementary function ϕ of ϕ as ϕ :  ×[0, ∞) −→ [0, ∞) (3.2) ϕ (x , t ) = sup (tu − ϕ(x , u)) . (3.3) u≥0 The complementary function ϕ is itself a MO-function (see [9]) and Hölder’s inequality holds, namely for ϕ ϕ f ∈ L () and g ∈ L (), f (x )g(x ) dx ≤ 2 f  g . (3.4) ϕ ϕ If in addition ϕ(x , t ) dx < ∞ (3.5) for any K ⊂  with Lebesgue measure |K | < ∞ and inf ϕ(x , 1)> 0, (3.6) x ∈ 1,ϕ ϕ the Musielak–Orlicz Sobolev space W () consisting of all functions in L () whose distributional deriva- tives are in L (), is a Banach space when furnished with the norm u =u +|∇ u| , 1,ϕ ϕ ϕ 1,ϕ where ∇ stands for the gradient operator and |·| denotes the Euclidean norm in R . The Sobolev space W () ∞ 1,ϕ is defined to be the closure of C () in W (). 4 Sobolev-type embeddings The central idea of this Section is the Sobolev Embedding Theorem 4.7. In order to facilitate the flow of ideas we present a few definitions. The Matuszewska index of an Orlicz function ϕ was introduced by Matuszewska and Orlicz in [17]. Definition 4.1 For ϕ as above and each x ∈ ,set ϕ(x , tu) M (x , t ) = lim sup . (4.1) ϕ(x , u) u→∞ 123 Arab. J. Math. The Matuszewska index of ϕ is defined to be ln M (x , t ) ln M (x , t ) m(x ) = lim = inf . (4.2) t →∞ ln t t >1 ln t Definition 4.2 The limit (4.1)issaidtobe uniform if for each δ> 0 there exist s > 1and T > 1suchthat, for all (x , t ) ∈  ×[T , ∞), one as ϕ(x , ts ) M (x , t ) − δ< < M (x , t ) + δ. (4.3) ϕ(x , s ) The following examples illustrate the above definition for some well known MO functions: Example 4.3 Let  ⊆ R be a bounded domain and p :  −→ (0, ∞) be Borel-measurable. The MO function ϕ :  ×[0, ∞) −→ [0, ∞) p(x ) ϕ(x , t ) = t (4.4) has Matuszewska index equal to p(x ). In this case, the convergence (4.2) is trivially uniform on  and the limit (4.2) is clearly uniform. Lemma 4.4 Let  ⊂ R be a bounded domain and ϕ an MO function as described above. If the Matuszewska index m is the restriction to  of a continuous function m ˜ on the closure of , i.e., m ˜ :  −→ R, (4.5) and the convergence to the limits (4.1) and (4.2) is uniform, then there exist C > 1,T > 1 and S > 1 such 0 0 that uniformly in  it holds ϕ(x , sT ) ≤ Cϕ(x , s) (4.6) for any s ≥ S . Condition (4.6) will be referred to as the  condition. Proof Fix δ> 0, then for some T > 1 one has for any t ≥ T , by virtue of (4.2) 0 0 m(x )−δ m(x )+δ t < M (x , t)< t (4.7) uniformly in . By definition of M (x , t ) and on account of the uniformity assumption of the infimum (4.1), there exists a positive number N for which, uniformly for t ≥ T and x ∈ , it holds that ϕ(x , st ) m(x )+δ sup < t . (4.8) ϕ(x , s) s≥ N In particular, for all s ≥ N : sup m(x )+δ ϕ(x , sT ) ≤ T ϕ(x , s). Corollary 4.5 There exists S > 1 and a constant C > 1 such that ϕ(x , 2s) ≤ Cϕ(x , s) (4.9) for any x ∈ ,s ≥ S . Lemma 4.6 If the statement of corollary 4.5 holds, then ρ-convergence is equivalent to norm-convergence in L (). 123 Arab. J. Math. Proof It suffices to show that if (u )ρ -converges to 0 and converges a.e. to 0. then it converges to 0 in the j ϕ topology of the norm. This will be automatically implied by the validity of the equality lim ρ (λx ) = 0 (4.10) ϕ n j →∞ for any λ> 0. It is obviously necessary to show (4.10) only for λ> 1. Let N =[log λ]+ 1 ≥ 1. A simple argument shows that for C, S as in Corollary 4.5 N −1 N N ϕ(x,λ|u (x )|) ≤ ϕ(x , 2S ) + C ϕ(x,(λ/2 )|u (x )|). n 0 n Since the second term in the right-hand side tends a.e. to 0 as n →∞, it follows that ρ (λu ) = ϕ(x,λ|u (x )|)dx → 0as n →∞. (4.11) ϕ n n We refer the reader to [16] for the proof of the following theorem. Theorem 4.7 Let  ⊂ R be a bounded domain and ϕ :  ×[0, ∞) −→ R a Musielak–Orlicz function that satisfies condition (3.6) and for which the limits in (4.1) and (4.2) are uniform; assume that the Matuszewska index m is the restriction to  of a continuous function m ˜ defined on the closure of , that 1 < m =: inf m, and that there exists a function β : (0, ∞) −→ (0, ∞) such that, uniformly in  and for t > 0: ϕ(x , t ) ≤ β(t ). (4.12) Then the embedding 1,ϕ W () → L () (4.13) is compact. Corollary 4.8 For ϕ satisfying the conditions of Theorem 4.7, there exists a positive constant C depending 1,ϕ only on n,,ϕ such that for any u ∈ W () u ≤ C |∇ u| . (4.14) ϕ ϕ Proof See [16]. 123 Arab. J. Math. 5  -type Musielak–Orlicz functions From now on we assume that  ⊆ R is a bounded domain satisfying the cone condition and ϕ is a MO- function satisfying the conditions of Theorem 4.7. Denote the conjugate of ϕ by ϕ . Theorem 5.1 Let ϕ be a M O function on ; assume that ϕ satisfies the conditions of Theorem 4.7;in particular, ϕ satisfies the  condition 4.6, i.e., for some K > 0,S > 1 it holds that ϕ(x , 2s) ≤ K ϕ(x , s) for all s ≥ S , x ∈ . (5.1) Then, sup ρ (u) : ρ (|∇ u|) ≤ r < ∞. (5.2) ϕ ϕ Proof It follows from (5.1) that for arbitrary v ∈ L () (recall that ϕ is nonnegative and nondecreasing) ρ (2v) = ϕ(x , 2|v(x )|)dx ⎛ ⎞ ⎜ ⎟ = + ϕ(x , 2|v(x )|)dx ⎝ ⎠ |v|<S |v|≥S 0 0 ⎛ ⎞ ⎜ ⎟ ≤ ϕ(x , 2S )dx + Kρ (v) ⎝ 0 ϕ ⎠ |v|<S ⎛ ⎞ ⎝ ⎠ ≤ ϕ(x , 2S )dx + Kρ (v) . (5.3) 0 ϕ 1,ϕ If r ≥ 1and u ∈ W () with ρ (|∇ u|) ≤ r, (5.4) it is a simple matter to verify that if |∇ u| ≥ 1 |∇ u| 1 1 = ρ ≤ ρ (|∇ u|) ϕ ϕ |∇ u| |∇ u| ϕ ϕ ≤ r ; |∇ u| it is thus clear that if (5.4) holds, then: |∇ u| ≤ r. Therefore, |∇ u| ≤ max{1, r}= b. It follows from the preceding reasoning in conjunction with Poincaré inequality that if ρ (|∇ u|) ≤ r, then, for some C > 0, u ≤ C |∇ u| ≤ Cb. ϕ ϕ Hence, ρ ≤ 1. (5.5) Cb 123 Arab. J. Math. If Cb < 2, (5.5) implies u u ρ (u) = ρ Cb ≤ ρ 2 ϕ ϕ ϕ Cb Cb ≤ ϕ(x , 2S )dx + Kρ 0 ϕ Cb ≤ ϕ(x , 2S )dx + K . (5.6) Otherwise, on account of the iteration of inequality (5.3) ρ (u) = ρ Cb ϕ ϕ Cb [log Cb]+1 ≤ ρ 2 Cb [log Cb] [log Cb]+1 2 2 ≤ 1 + K + ...K ϕ(x , 2S )dx + K . (5.7) In all, (5.6)and (5.7) yield (5.2). An immediate consequence of the preceding theorem is the following functional-analytic result: Lemma 5.2 For ϕ as in Theorem 4.7 and r > 0, the modular ball 1,ϕ B := u ∈ W () : ρ (|∇ u|) ≤ r r ϕ is weakly closed. Proof B is clearly convex. It suffices to show that it is also norm-closed. If (u ) norm-converges to u ∈ r j 1,ϕ W (),then ρ (|∇(u − u)|) → 0as n →∞ and there is no loss of generality in assuming that ∇ u →∇ u ϕ j j a.e. in . Theorem 4.7 guarantees that (u ) can be chosen so that u → u a.e. in .Since j j ρ (|∇ u|) = ϕ(x , lim |∇ u (x )|)dx ≤ lim inf ϕ(x , |∇ u (x )|)dx ≤ r, (5.8) ϕ n n n→∞ n→∞ it follows that u ∈ B , i.e., B is norm-closed (and convex) and hence weakly closed. r r 6 Differentiability properties Aiming at a full description of the Fréchet derivative of the functionals to be introduced momentarily, a further assumption is imposed unto the MO function ϕ at this point, namely, it is from now on required that ϕ be an N function. More precisely: Definition 6.1 An MO functionissaidtobean N -function iff it satisfies the condition ϕ(x , t ) lim = 0 a.e.. (6.1) t →0 It is well known [9]thatif ϕ is an N -function, it can be written as ϕ(x , t ) = φ(x , s) ds, (6.2) where φ(x , ·) is the right t-derivative of ϕ. On the other hand, the conjugate function ϕ can be written as ∗ −1 ϕ (x , t ) = φ (x , s) ds. (6.3) The proof of the following theorem can be found in [14,22]; we include it here in the interest of completeness. 123 Arab. J. Math. Theorem 6.2 Let ϕ be an N -function; assume that ∂ϕ ϕ (x , t ) = (x , t ) ∂ t is continuous a.e. x ∈ . Define the operator T as T : M() −→ M() ∂ϕ T (u) = ϕ (x , |u(x )|) = (x , |u(x )|). ϕ t ∂ t Then, from the assumption ϕ ϕ T (L ()) ⊆ L () (6.4) it follows that the operator ϕ ϕ T : L () −→ L () is continuous and bounded. ϕ ϕ Proof Assume (u ) ⊆ L () converges to u ∈ L ().On  ×[0, ∞) define w(x , t ) = ϕ (x , |u(x ) + t |) − ϕ (x , |u(x )|); t t then on account of the assumption on ϕ , w is a Carathéodory function and w(x , 0) = 0. If ϕ ϕ T : L () −→ L () ϕ ∗ is continuous at 0, then T (u − u) −→ 0in L () as n →∞. If ϕ satisfies the  condition, the latter is w n 2 equivalent to ρ (T (u − u)) = ϕ (x , |ϕ (x , |u (x )|) − ϕ (x , |u(x )|)|) → 0as n →∞, ϕ w n t n t that is T (u ) −→ T (u) in L () as n →∞. ϕ n ϕ t t Therefore, it is enough to show that T is continuous at 0 under the assumption that ϕ (x , 0) = 0 a.e. in . ϕ t Assume that T is not continuous at 0; let r > 0and let (u ) be a sequence that converges to 0 in L (),for ϕ n which T (u ) ∗ ≥ r for any n ∈ N. ϕ n ϕ Since norm convergence implies modular convergence, one can, without loss of generality assume that max ρ (u ), u  < , ϕ n n ϕ and hence that ϕ(x , |u (x )|) dx < ∞. Due to the validity of the  condition for ϕ , norm con- n 2 n=1 vergence and modular convergence are equivalent on L (). It follows that there exists (r)> 0 such that ρ ∗ T (u ) ≥ (r ) for any n ∈ N. We next claim the existence of a sequence of real numbers ( ),a ϕ ϕ n k sequence ( ) of subsets of  and a subsequence (u ) of (u ) satisfying the following conditions: k n n (i)  <  , k+1 k (ii) | |≤  , k k ∗ 2 (iii) ϕ (x , |T (u )|) dx > (r ). ϕ n k 3 123 Arab. J. Math. (iv) If E ⊆  is measurable and | E | < 2 ,then k+1 (r ) ϕ (x , |T (u )|) dx < . ϕ n t k Set  = ,  =||, n = 1. We assume that  , n and  are given, then, by assumption, ϕ (·, |u (·)|) ∈ 1 1 1 k k k t n L () and on account of the  condition one has ϕ (x , |ϕ (x , |u (x )|)|) dx < ∞. t n Since the measure A −→ μ( A) = ϕ (x , |ϕ (x , |u (x )|)|) dx t n defined on the Borel σ algebra B of subsets of  is absolutely continuous with respect to the Lebesgue measure, (r ) one can find  such that any X ∈ B with | X | < 2 satisfies ϕ (x , |ϕ (x , |u (x )|)|) dx < . The k+1 k+1 t n assumption  ≤ 2 would contradict (iii ). We now proceed to the construction of  and n . k k+1 k+1 k+1 It is well known [14] that the strong convergence of (u ) in L () implies the convergence in measure of both (T (u )) and (T (T (u ))). Consequently, there exists n ∈ N such that ϕ n ϕ ϕ n k+1 t t (r ) x ∈  : T (T (u )) > < < < . ϕ ϕ n k+1 k t k+1 3|| 2 Define (r ) = x ∈  : T ∗ (T (u )) > . k+1 ϕ ϕ n t k+1 3|| Next, ⎛ ⎞ ⎜ ⎟ ∗ ∗ ϕ (x , |T (u )|) dx = − ϕ (x , |T (u )|) dx ϕ n ⎝ ⎠ ϕ n t k+1 t k+1 k+1 k+1 (r ) 2(r ) >(r ) − = . 3 3 By construction ∞ ∞ ≤  < 2 . (6.5) j j k+1 j =k+1 j =k+1 Set u (x ) if x ∈  \ n k j v(x ) = j =k+1 0 otherwise. It is clear that v ∈ L (). Next, observe that: ∗ ∗ ϕ (x , T (v(x ))) dx ≥ ϕ (x , T (u (x ))) dx ϕ ϕ n t t k k=1 ∞ \∪ k j j =k+1 123 Arab. J. Math. ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = − ϕ (x , T (u (x ))) dx ϕ n t k ⎝ ⎠ k=1 k ∪ j =k+1 2 1 ≥ (r ) − (r ) 3 3 k=1 =∞, which follows by (6.5) and condition (i v). This contradicts assumption (6.4). Hence, T is continuous at 0. We next prepare the ground for the next lemma, which deals with the differentiability properties needed in the sequel. Let M : R −→ [0, ∞) be continuous and write M (s) = M (t ) dt . (6.6) Consider the maps F : L () −→ [0, ∞) F (u) = ρ (u) (6.7) and 1,ϕ H : W () −→ [0, ∞) H (u) = M ρ (|∇ u|) . (6.8) Recall that a MO function ϕ is said to be locally integrable if for any t > 0 and any subset W ⊆  with μ(W)< ∞ one has ϕ(x , t ) dx < ∞. Lemma 6.3 In the terminology of the preceding paragraph, let ϕ be an N function; suppose that ϕ(x , ·) ∂ t is continuous a.e. x . Assume that the complementary function ϕ of ϕ satisfies the  condition and is locally integrable. If the maps ϕ ϕ D : L () −→ L () (6.9) D (u) = ϕ(·, |u(·)|) (6.10) ∂ t and 1,ϕ D : W () −→ L () (6.11) D (u) = ϕ(·, |∇ u(·)|) (6.12) ∂ t are well defined, then the functionals (6.7) and (6.8) are Fréchet differentiable for u = 0.Inthiscase, the derivatives of F and H at u = 0 are given, respectively by ∂ϕ u(x ) F (u), h= (x , |u(x )|) h(x ) dx , (6.13) ∂ s |u(x )| and by ∂ϕ ∇ u(x ) H (u), h=− M ρ (|∇ u) div (x , |∇ u(x )|) , h , (6.14) ∂ s |∇ u(x )| with the understanding that = 0 if x = 0. |x | 123 Arab. J. Math. Proof It suffices to show that under the stipulated conditions (6.13) holds; a straightforward application of the chain rule will yield the full statement of the Lemma. Observe that for u(x ) = 0, h ∈ L () one has |u(x ) + th(x )|−|u(x )| 2u(x )h(x ) + t (h(x )) t |u(x ) + th(x )|+|u(x )| u(x )h(x ) −→ as t −→ 0. |u(x )| Therefore, for some θ ∈ (|u(x )|, |u(x ) + th(x )|), ϕ(x , |u(x ) + th(x )|) − ϕ(x , |u(x )|) |u(x ) + th(x )|−|u(x )| ∂ϕ = (x,θ) t t ∂ t u(x )h(x ) ∂ϕ −→ (x , |u(x )|) as t −→ 0. |u(x )| ∂ t By hypothesis, ∂ϕ h(x ) (x , |u(x )|) dx < ∞, (6.15) ∂ t which in conjunction with the above limit yields: |u(x ) + th(x )|−|u(x )| ∂ϕ lim (x,θ(x ))dx t −→0 t ∂ t {x :u(x )=0} u(x )h(x ) ∂ϕ = (x , |u(x )|)dx . (6.16) |u(x )| ∂ t {x :u(x )=0} On the other hand, one has by assumption, one has a.e. x ∈ : ϕ(x , |th(x )|) −→ 0as t −→ 0 and by convexity, for 0 < t < 1, ϕ(x , |th(x )|) ≤ ϕ(x , |h(x )|). Since h ∈ L (), ϕ(x , |h(x )|) dx < ∞, it is easily derived by way of application of Lebesgue’s theorem that ϕ(x , |u(x ) + th(x )|− ϕ(x , |u(x )|) dx {x :u(x )=0} = ϕ(x , t |h(x )|)/t dx → 0as t → 0. (6.17) {x :u(x )=0} In all, ⎛ ⎞ F (u + th) − F (u) ϕ(x , |u(x ) + th(x )|) − ϕ(x , |u(x )|) ⎜ ⎟ = + dx ⎝ ⎠ t t {x :u(x )=0} {x :u(x )=0} u(x )h(x ) ∂ϕ → (x , |u(x )|)dx as t −→ 0. |u(x )| ∂ t {x :u(x )=0} 123 Arab. J. Math. We conclude that F is Gâteaux differentiable and that its derivative is equal to the right-hand side of (6.13). The proof of the Gâteaux differentiability of G follows along the same lines. Hence, it suffices to prove that under the additional assumptions (6.9)and (6.11), the operators ϕ ϕ ∗ L : L () −→ (L ()) L (u) = F (u) (6.18) 1,ϕ 1,ϕ L : W () −→ W () 0 0 L(u) = G (u) (6.19) are continuous at u = 0. A standard functional-analytic result guarantees that in this case F and G are Fréchet differentiable and that the Fréchet and the Gâteaux derivatives coincide. To this end, consider a convergent ϕ ϕ sequence (u ) ⊂ L (),say u −→ u in L () as j −→ ∞: it is well known that there is no loss of j j generality by assuming that u converges to u almost everywhere in  (see [9,11])). ∂ϕ Since (x , |u(x )|) ∈ L () there exists λ > 0for which ∂ t ∂ϕ ϕ x,λ (x , |u(x )|) dx < ∞. (6.20) ∂ t Notice that for h ≤ 1 one has u (x ) j ∂ϕ u(x ) ∂ϕ (x , |u (x )|) − (x , |u(x )|)) h(x ) dx (6.21) |u (x )| ∂ t |u(x )| ∂ t {x :u (x )=0,u(x )=0} u (x ) ∂ϕ ∂ϕ = (x , |u (x )|) − (x , |u(x )|) h(x ) dx |u (x )| ∂ t ∂ t {x :u (x )=0,u(x )=0} u (x ) j u(x ) ∂ϕ + − (x , |u(x )|)h(x ) dx . (6.22) |u (x )| |u(x )| ∂ t {x :u (x )=0,u(x )=0} Theorem 6.2 guarantees the continuity of the map (6.9); it is apparent from this fact in conjunction with Hölder’s inequality (3.4) that the integral in (6.21) tends to 0 as j tends to infinity. As to the remaining integral, set, for j ∈ N: ∂ϕ (x , |u(x )|) = f (x ) ∂ t u (x ) u(x ) r (x ) = − f (x ). |uj (x )| |u(x )| Recall that in the terminology of Lemma 4.4, ∗ ∗ ϕ (x , sT ) ≤ C ϕ (x , s), (6.23) for any s ≥ S .For λ as in (6.20)and any λ> 2λ let k be defined by the inequalities 0 0 0 k−2 k−1 2 <λ/λ ≤ 2 . Notice that |r |≤ 2. For S as in Lemma 4.4, one readily obtains, for any positive integer m,using the j 0 monotonicity of ϕ in the second variable ∗ m ϕ x , 2 λ f (x ) ∗ m m−1 m−1 = ϕ x , 2 λ f (x ) I 2 λ f (x ) + I 2 λ f (x ) 0 0 0 [S ,∞) [0,S ) 0 0 ∗ m−1 ∗ ≤ C ϕ (x , 2 λ f (x )) + ϕ (x , 2S ). (6.24) 0 0 The iteration of the preceding inequality yields 123 Arab. J. Math. ∗ ∗ k ϕ x,λr ≤ ϕ x , 2 λ f (x ) j 0 ∗ k k−1 k−1 = ϕ x , 2 λ f (x ) I 2 λ f (x ) + I 2 λ f (x ) 0 [S ,∞) 0 [0,S ) 0 0 0 k−1 i ∗ k ∗ ≤ C ϕ (x , 2S ) + C ϕ (x,λ f (x )). 0 0 i =0 A routine application of Lebesgue’s dominated convergence quickly shows that u (x ) u(x ) ∂ϕ ϕ x,λ − (x , |u(x )|) dx −→ 0as j −→ ∞. (6.25) |u (x )| |u(x )| ∂ t {x :u (x )=0,u(x )=0} A similar argument shows that (6.25) holds for 0 <λ ≤ 2λ . It follows from the arbitrariness of λ that " " " " u u ∂ϕ " " − (·, |u|) → 0as j →∞. " " |u | |u| ∂ t j ϕ L () On account of Hölders inequality (3.4), the integral (6.22) is bounded by " " " " u u ∂ϕ " " 2 − (·, |u|) −→ 0as j −→ ∞. (6.26) " " |u | |u| ∂ t ∗ j ϕ L () The proof of the continuity of L follows along the same lines and will be skipped. This concludes the continuity argument and hence F and G are Fréchet differentiable. Lemma 6.4 If M :[0, ∞) −→ [0, ∞) is strictly increasing, the modular M - ball 1,ϕ B = u ∈ W () : M ρ (|∇ u|) ≤ r (6.27) r ϕ 1,ϕ is weakly closed in W (). −1 Proof For any r > 0set s = M (r ); it is then clear from the above assumptions that 1,ϕ B = V = u ∈ W () : ρ (|∇ u|) ≤ s (6.28) r s ϕ r and the latter set is weakly closed (Lemma 5.2). 7 Kirchhoff-type eigenvalue problem We are now ready to prove the main result of this work. Theorem 7.1 Let M ∈ C ((0, ∞), (0, ∞)).Set M (t ) = M (s)ds. Then, for any r > 0, there exists a 1,ϕ solution (u,λ) ∈ W () × (0, ∞) to the equation ∂ϕ ∇ u(x ) ∂ϕ u(x ) − M ρ (|∇ u|) div (x , |∇ u(x )|) = λ (x , |u(x )|) , (7.1) ∂ s |∇ u(x )| ∂ s |u(x )| satisfying M (ρ (|∇ u|)) = r (7.2) 123 Arab. J. Math. Proof Theorem 5.1 guarantees that for any r > 0, 0 < sup ρ (u) : u ∈ B = S < ∞. (7.3) ϕ r r 1,ϕ We next observe that Theorem 4.7 implies the existence of a sequence (u ) ⊂ V with u  u in W () j s j 0 r 0 and u −→ u in L () such that j 0 ρ (u) −→ S = ρ (u ) . (7.4) ϕ r ϕ 0 To see this, we notice that a.e. in , ϕ(x , |u (x )|) −→ ϕ(x , |u(x )|) and that on account of convexity, for any n ∈ N: 1 1 ϕ(x , |u (x )|) ≤ ϕ(x , 2|u (x ) − u(x )|) + ϕ(x , 2|u(x )|). (7.5) n n 2 2 Select n large enough so that 2u − u  < 1; for such n, it holds, by way of the convexity of ϕ(x , ·), n ϕ 1 1 2|u (x ) − u(x )|2u − u n n ϕ ϕ(x , 2|u (x ) − u(x )|)dx = ϕ x , dx 2 2 2u − u n ϕ |u (x ) − u(x )| ≤u − u  ϕ x , dx (7.6) n ϕ u − u n ϕ Denote the left-hand side and the right-hand side of (7.5)by v and w respectively. Then the following n n conditions hold: (i) v (x ) → v(x ) = ϕ(x , |u(x )|) ∈ L () a.e. in (ii) w (x ) → w(x ) = ϕ(x , 2|u(x )|) ∈ L () a.e. in (iii) v ,w ∈ L () for any n ∈ N n n 1 1 (iv) w dx → ϕ(x , 2|u|)dx = ρ (2u). n ϕ 2 2 Since w − v ≥ 0 a.e in , Fatou’s Lemma leads to: (w − v)dx ≤ w dx + lim inf (−v ) dx = w dx − lim sup v dx and (w + v)dx ≤ w dx + lim inf v dx . The two last statements yield lim ϕ(x , |u (x )|)dx = ϕ(x , |u(x )|)dx n→∞ or, equivalently ρ (u ) −→ ρ (u) as n →∞. (7.7) ϕ n ϕ By construction ρ (u ) −→ S ;(7.7) is therefore the desired result. ϕ n r 123 Arab. J. Math. Lemma 6.4 yields M ρ (|∇ u |) ≤ r. Furthermore, the continuity and monotonicity of the modular ρ ϕ 0 ϕ immediately yield M (ρ (|∇ u |)) = r. ϕ 0 As is apparent from the above, u is a solution of the constrained maximization problem of the type max F (v) with G(v) = r. (7.8) It is a routine matter to show, via the Implicit Function Theorem, that the preceding statement implies that kerG (u ) = ker F (u ), (7.9) 0 0 from which it is clear (since neither functional is null) that u satisfies equation (7.1)for some λ> 0. This concludes the proof of the claim. 8 Applications A particular instance of Theorem 7.1 deserves to be stressed, namely its implication in the consideration of variable exponent Lebesgue spaces. More precisely, if  ⊂ R is bounded then ϕ :  ×[0, ∞) −→ [0, ∞) (8.1) p(x ) ϕ(x , t ) = (8.2) p(x ) satisfies the conditions of Theorem 4.7 iff p is the restriction to  of a function p ˜ ∈ C (R , R) and 1 < p = inf p(x ) ≤ p = sup p(x)< ∞. (8.3) − + x ∈ x ∈ The conjugate function is clearly given by p(x ) p(x ) − 1 p(x )−1 ϕ (x , t ) = t . (8.4) p(x ) It is straightforward to verify the conditions of Lemma 6.3 for this case. It is customary to write, in this case p(x ) |u(x )| ρ (u) = dx . (8.5) p(x ) Therefore, Theorem 7.1 yields the following result: Theorem 8.1 If  ⊂ R is bounded, M satisfies the conditions of Theorem 7.1 and p :  −→ (1, ∞) is a variable exponent satisfying the assumptions (8.3) then for each r > 0 there exists a solution (u ,λ) to the eigenvalue problem p(x )−2 p(x )−2 M ρ (|∇ u|) di v |∇ u| ∇ u = λ|u| u (8.6) with M ρ (|∇ u |) = r. p 0 With the aid of the following compactness theorem, the techniques used in Sections 6 and 7, one can derive Theorems 8.3 and 8.4, particular cases of which were obtained in [2]and [4], respectively, via different methods. 123 Arab. J. Math. Theorem 8.2 [9,15] Let  ⊂ R ,n > 1 be a bounded domain, p ∈ C () with 1 < p ≤ p < n. (8.7) − + For 0 <ε < and q ∈ P() such that n−1 np(x ) q(x)< − , (8.8) n − p(x ) the embedding 1, p(·) q(·) W () → L () (8.9) is compact. Theorem 8.3 Givenafunction M ∈ C ((0, ∞), [0, ∞)),M (t)> 0 for t > 0. Under the hypotheses of 1, p(·) Theorem 8.2, for any r > 0, there exists a solution (u,λ) ∈ W () × (0, ∞) to the equation p−2 q−2 − M ρ (|∇ u|) div |∇ u| ∇ u = λ|u| u, (8.10) satisfying M ρ (|∇ u|) = r. (8.11) p(x ) Proof The proof follows from Theorem 7.1 and Theorem 8.3 by considering ϕ(x , t ) = and by way of p(x ) the bound ⎧ ⎫ ⎨ ⎬ |u(x )| sup : ρ (|∇ u|) ≤ r < ∞, (8.12) ⎩ q(x ) ⎭ which is easy derived as in Theorem 5.1 via the compactness result of Theorem 8.2. Theorem 8.4 Under the assumptions of Theorem 8.2, for any 0 ≤ V ∈ L () and r > 0, there exists a 1, p(·) solution (u,λ) ∈ W () × (0, ∞) to the equation |∇ u| p−2 q−2 − M ρ div |∇ u| ∇ u = λV |u| u, (8.13) satisfying |∇ u| M ρ = r. (8.14) Proof The proof follows along the same lines as those of Theorem 7.1 by observing that the functional V (x ) p(·) q(x ) L ()  u → T (u) = |u(x )| dx q(x ) is Fréchet differentiable for u = 0 and that, for h ∈ C (), q(x )−2 T (u), h= V (x )|u(x )| u(x )h(x )dx . If B , r > 0, is defined as in Lemma 6.4, it follows as in the proof of Theorem 7.1 that there exists u ∈ B r 0 r p(x ) ∇ u with M dx = r such that p(x ) ⎧ ⎫ ⎨ ⎬ V (x ) V (x ) q(x ) q(x ) |u (x )| dx = max |u(x )| dx , u ∈ B . 0 r ⎩ ⎭ q(x ) q(x ) Reasoning mutatis mutandis as in the proof of Theorem 7.1 it can be shown that u is in fact a sought-for solution to Problem 8.13. 123 Arab. J. Math. Acknowledgements The author is pleased to express his gratitude to the referees for their careful revision of the manuscript and their valuable suggestions. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Funding No funding was provided or used for the research leading to this article. Declarations Conflict of interest The author declares having no conflict of, competing or financial interest in regards to the research presented in this work. References 1. Alves, G.; Corrêa, F.: Positive solutions for a quasilinear elliptic equation of Kirchhoffs type. Comput. Math. Appl. 49, 85–93 (2005) 2. Arouzi, G.; Mirzapour, M.: Eigenvalue problems for p(x )-Kirchhoff-type equations. EJDE 2013(253), 1–10 (2013) 3. Autuori, G.; Pucci, P.; Salvatori, M.C.: Global nonexistence for nonlinear-Kirchhoff systems. Arch. Rat. Mech. Anal. 196, 489–516 (2010) 4. Chung, N.: On some p(x )-Kirchhoff type equations with weights. J. Appl. Math. Inform. 32(1–2), 113–128 (2014) 5. Chen, Y.; Levine, S.; Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006) (electronic). 6. Colasuonno, F.; Pucci, P.: Multiplicity of solutions for p(x )-polyharmonic Kirchhoff equations. Nonlinear Anal. 74, 5962– 5974 (2011) 7. Dai, G.; Hao, R.: Existence of solutions for a p(x )-Kirchhoff-type equation. J. Math. Anal. Appl. 79, 275–284 (2009) 8. Dai, G.; Liu, D.: Infinitely many positive solutions for a p(x )-Kirchhoff-type equation. J. Math. Anal. Appl. 359, 704–710 (2009) 9. Diening, L.; Harjulehto, P.; Hästö, P.; Ruži ˚ cka, ˇ M.: Lebesgue and Sobolev spaces with variable exponents, vol. 2017. Lecture notes in Mathematics. Springer, Heidelberg (2011) 10. Dreher, M.: The Kirchhoff equation for the p-Laplacian. Rend. Semin. Mat. Univ. Polit. Torino 64(2), 217–238 (2006) 11. D.E. Edmunds, J. Lang, O. Méndez, Differential operators on spaces of variable integrability, World Scientific (2015) 12. Fan, X.: On non-local p(x )-Laplacian Dirichlet problems. Nonlinear Anal. Theor. 72, 3314–3323 (2010) 13. G. Kirchoff Mechaniks, Teubner, Leipzig (1883) 14. Krasnoselskii, M.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, Elsford (1964) p(x ) k, p(x ) 15. O. Kovácik, ˘ J. Rákosník, On spaces L () and W (), Czechoslovak Math. J. 41 (1991), 592–618. 16. Lang, J.; Méndez, O.: Sharp conditions for the compactness of the Sobolev embedding theorem on Musielak–Orlicz spaces. Math. Nachr. 292(2), 377–388 (2019) 17. Matuszewska, W.; Orlicz, W.: On certain properties of ϕ-functions. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 8, 430–443 (1960) 18. Li, H.: Blow-up of solutions to a p-Kirchhoff-type parabolic equation with general nonlinearity. J. Dyn Control Syst 26, 383–392 (2020) 19. Li, L.; Sun, J.: Existence and multiplicity solutions for a non local equation of Kirchhoff type. Differ. Equ. Appl. 10(4), 369–386 (2018) 20. Liang, Z.; Li, F.; Shi, J.: Positive solutions to Kirchhofs type equations with nonlinearity having prescribed asymptotic behavior. Ann. Inst. H. Poincaré Anal. Non Linéaire 31(1), 155–167 (2014) 21. O. Méndez, On the eigenvalue problem for a class of Kirchhoff-type equations. J. Math. Anal. Appl. 494(2) (2021). 22. O. Méndez, J. Lang, Analysis on function spaces of Musielak–Orlicz type. Chapman and Hall/CRC Monographs and Research Notes in Mathematics (2018) 23. Mihailescu, M.; Radulescu, V.; Pucci, P.: Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent. J. Math. Anal. Appl. 340(1), 687–698 (2008) 24. J. Musielak, Orlicz Spaces and Modular Spaces, Lect. Notes in Math. 1034, 1983. 25. Nakano, H.: Modulared Semi-ordered Linear Spaces. Maruzen, Tokyo (1950) 26. Sun, J.; Tang, V.: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal. theo. 74(4), 1212–1222 (2011) 27. Sun, J.; Wu, T.: Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains. Proc. R. Soc. Edinburgh: Sect. A Math. 146(2), 435–448 (2016) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Journal

Arabian Journal of Mathematics – Springer Journals

Published: Dec 1, 2023

Keywords: Variable exponent p-Laplacian; Sobolev embedding; Modular spaces; Musielak–Orlicz spaces; Variable exponent spaces; Kirchhoff equations; Nonlinear wave equation; Primary 35A01; Secondary 46A80

There are no references for this article.