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Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in RN
Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type...
axioms Article Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in R Kisoeb Park Department of IT Convergence Software, Seoul Theological University, Bucheon 14754, Korea; firstname.lastname@example.org; Tel.: +82-1022400353 Abstract: In this paper, we establish the existence of a nontrivial weak solution to Schrödinger-kirchhoff type equations with the fractional magnetic field without Ambrosetti and Rabinowitz condition using mountain pass theorem under a suitable assumption of the external force. Furthermore, we prove the existence of infinitely many large- or small-energy solutions to this problem with Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem by applying the variational methods, that is, the fountain and the dual fountain theorem with Cerami condition. Keywords: Schrödinger-kirchhoff equation; fractional magnetic operators; variational methods MSC: 35A15; 35J60; 35R11; 47G20 1. Introduction The Schrödinger equation plays the role of Newton’s laws and conservation of energy in classical mechanics. The linear Schrödinger equation represents one of the main results of quantum mechanics which is the evolution of a free non-relativistic quantum particle. The Citation: Park, K. Multiplicity Results of Solutions to Non-Local structure of the nonlinear Schrödinger equation is considerably complicated and requires Magnetic Schrödinger–Kirchhoff more sophisticated analysis; see . This equation has been studied extremely according to Type Equations in R . Axioms 2022, the pure or applied mathematical theory, because it stands out as a prototypical system 11, 38. https://doi.org/10.3390/ that has shown to be crucial to model and understand the characteristics of numerous axioms11020038 areas in nonlinear physics. In particular, the signiﬁcant development of the Bose-Einstein condensates revived researches regarding the nonlinear waveforms for the nonlinear Academic Editors: Omar Bazighifan, Schrödinger equations with external potentials and the related nonlinear partial differential Maria Alessandra Ragusa and equations. For further applications and more details we refer the reader to [2–8]. Indeed, Fahd Jarad the mathematical model for the remarkable Bose-Einstein condensate with attractive inter- Received: 28 December 2021 particle interactions under a magnetic trap is a class of nonlinear Schrödinger equations Accepted: 14 January 2022 with external potentials, which is sometimes called the Gross-Pitaevskii equation [9,10]. Published: 19 January 2022 In this regard the present paper is motivated by some works (see [11–21]) concerning the Publisher’s Note: MDPI stays neutral nonlinear Schrödinger equation with regard to jurisdictional claims in ¶y h ¯ published maps and institutional afﬁl- 2 N ih ¯ = (r + i A(x)) y + W(x)y f (x,jyj )y for x 2 R , (1) ¶t 2m iations. N N where h ¯ is Planck constant A(x) = ( A (x), A (x), . . . , A (x)) : R ! R is a real vector 1 2 N N N (magnetic) potential with magnetic ﬁeld B = curl A, and W(x) : R ! R is a scalar electric potential. Particularly, we are interested in the existence of standing wave solutions, Copyright: © 2022 by the authors. that is, solutions of type (1) when h ¯ is sufﬁciently small, where E is a real number and u(x) Licensee MDPI, Basel, Switzerland. is a complex-value function which satisﬁes This article is an open access article distributed under the terms and 2 2 N (r + i A(x)) u(x) + lV(x)u(x) = l f (x, u )u, x 2 R , (2) j j conditions of the Creative Commons Attribution (CC BY) license (https:// 1 h ¯ creativecommons.org/licenses/by/ where l = and V(x) = W(x) E. The transition from quantum mechanics to 2m 4.0/). classical mechanics can be done formally with h ¯ approach 0. Thus the existence of solutions Axioms 2022, 11, 38. https://doi.org/10.3390/axioms11020038 https://www.mdpi.com/journal/axioms Axioms 2022, 11, 38 2 of 14 for h ¯ small, semi-classical solutions, has important physical interest. Very recently, authors in  established the Bourgain-Brezis-Mironescu type result which constructs a bridge between a fractional magnetic operator and the classical theory. Motivated by this paper, nonlocal fractional problems with magnetic ﬁelds has been extensively studied by many researchers; see [23–29] and the references therein. In this regard, the present paper is devoted to the existence of solutions for the following Kirchhoff type equation with the fractional magnetic ﬁeld s p 2 N K(juj )( D) u + V(x)juj u = l f (x,juj)u in R , (3) s,A p,A where x+y Z Z i(x y) A( ) p ju(x) e u(y)j juj = dxdy, s,A N+ ps N N jx yj R R where 0 < s < 1 < p < +¥ and the fractional magnetic operator ( D) is deﬁned as x+y x+y i(x y) A( ) i(x y) A( ) p 2 2 2 jf(x) e f(y)j (f(x) e f(y)) s N ( D) f(x) = 2 lim dy, x 2 R , p,A N+ ps #!0 jx yj R nB (x) ¥ N N N for all f 2 C (R ,C). Here, B (x) denotes a ball in R centered at x 2 R and radius # > 0 N N N and A : R ! R is the magnetic potential. Also, the nonlinear function f : R R ! R will be stated later (see Section 2). When p = 2, the fractional Laplacian ( D) is a p,A fractional Laplacian contains the magnetic ﬁeld. On the other hand, the standard fractional Laplacian ( D) has been a classical topic for a long time and it is applied in various research ﬁelds, such as social sciences, fractional quantum mechanics, materials science, continuum mechanics, phase transition phenomena, image process, game theory, and Lévy process, fractional Sobolev spaces and their corresponding nonlocal equations, see [30–32] and the references therein. Kirchhoff in  ﬁrst introduced a model given by the equation 2 L 2 ¶ u r E ¶u ¶ u r + j jdx = 0, 2 2 ¶t h 2L ¶x ¶x which extends the classical D’Alembert’s wave equation by taking into account the changes in the length of the strings during the vibrations. In this direction, the non-local problem of Kirchhoff type equations have been investigated in [34–37]. Now in order to conﬁrm the existence of solutions to the nonlinear elliptic equations, the following Ambrosetti and Rabinowitz condition ((AR)-condition) given in  has been widely used; (AR) There exists z > p such that 2 N 0 < z F(x, t) f (x, t)t , for x 2 R and t > 0, where F(x, t) = f (x, s)sds. It is well known that (AR)-condition is essential to ensure the compactness condition of the Euler-Lagrange functional which plays a key role in applying the critical point theory. However, this condition is too restrictive and gets rid of many nonlinearities. Thus many researchers have tried to drop the (AR)-condition in the elliptic problem of nonlocal type (see e.g., [20,39–42]). In this respect, we are to prove the existence of a nontrivial solution for problem (3) without (AR)-condition using the mountain pass theorem with Cerami condition under a suitable assumption of the nonlinearity of f . Furthermore, we present the existence of inﬁnitely many large- or small-energy solutions to our problem without (AR)-condition. Especially, following in ( Remark 1.8), there are many examples which are not fulﬁlling the condition on f in a elliptic problem. Thus, inspired by these examples, we investigate the existence and multiplicity of weak solutions to the fractional p-Laplacian Equation (3) with the external magnetic potential. The strategy of the proof for these results Axioms 2022, 11, 38 3 of 14 is to approach the problem by applying the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition. As far as we are aware, none have reported such multiplicity results for our problem with the external magnetic ﬁeld. This present paper is organized as follows. in Section 2, we state some basic results to deal with this type equation with the fractional magnetic ﬁeld and review well known facts for the fractional Sobolev space. And under certain assumptions of f , we establish the existence of a weak solution of problem (3) using mountain pass theorem. 2. Preliminaries Let the potential function V 2 C(R ,R) be continuous and bounded from below. Assume that 1 N (V) V 2 L (R ), ess inf V(x) > 0 and lim V(x) = +¥. jxj!¥ loc x2R N p 1 N Let L (R ) denote the real valued Lebesgue space with V(x)juj 2 L (R ), equipped with the norm jjujj = V(x)juj dx. p,V s N The fractional Sobolev spaceH (R ) is then deﬁned as for s 2 (0, 1) and p 2 (1, +¥) Z Z ju(x) u(y)j s N N H (R ) = u 2 L (R ) : dxdy < +¥ . V V N+ ps N N jx yj R R s N The space H (R ) is endowed with the norm Z Z ju(x) u(y)j p p p p jjujj := jjujj + [u] with [u] := dxdy. s N s s p,V N+ ps H (R ) N N V jx yj R R For further details on the fractional Sobolev spaces we refer the reader to  and the references therein. We recall the embedding theorem; see e.g., . Lemma 1. Let (V) hold and s 2 (0, 1), p 2 (1, +¥) and let p be the fractional critical Sobolev exponent, that is N p if s p < N, N s p p := +¥ if s p N. s N g N Then, the embedding H (R ) ! L (R ) is continuous for any g 2 [ p, p ] and moreover, the s N g N embedding H (R ) ,! L (R ) is compact for any g 2 [ p, p ). N N p Let L (R ,C) be the Lebesgue space of functions u : R ! C with V(x)juj 2 1 N s N ¥ N L (R ). Deﬁne H (R ,C) as the closure of C (R ,C) with respect to the norm A,V p p p jjujj = (jjujj +juj ), s,A p,V s,A where the magnetic Gagliardo seminorm is given by x+y Z Z i(x y) A( ) p ju(x) e u(y)j juj = dxdy. s,A N+ ps N N jx yj R R In fact, arguing as in ( Proposition 2.1), we can easily show that it is a reﬂexive and separable Banach space as the similar arguments in ([45,47] Appendix). The following Lemmas 2 and 3 can be shown by applying as a general exponent p instead of p = 2 the same argument in ( Lemmas 3.4 and 3.5). Lemma 2. If (V) holds and r 2 [ p, p ], then the embedding s N r N H (R ,C) ,! L (R ,C) A,V Axioms 2022, 11, 38 4 of 14 is continuous. Furthermore, for any compact subset G R and r 2 [1, p ), then the embedding s N s r H (R ,C) ,! H (G,C) ,! L (G,C) A,V V is continuous and the latter is compact, where H (G,C) is endowed with the following norm: Z Z Z ju(x) u(y)j kuk = V(x)juj dx + dxdy . s,V N+ ps jx yj G G G s N Lemma 3. Under the assumption (V), for all bounded sequencefu g inH (R ,C) the sequence A,V r N fju jg admits a subsequence converging strongly to some u in L (R ) for all r 2 [ p, p ). + + For our problem, we suppose that K : R ! R satisﬁes the following conditions: 0 0 (K1) K 2 C(R ) satisﬁes inf + K(t) a > 0, where a > 0 is a constant. 0 t2R (K2) There is a positive constant q 2 [1, ) such that qK(t) = q K(h)dh K(t)t for N ps any t 0. A typical example for K is given by K(t) = b + b t with m > 0, b > 0, and b 0. 0 1 0 1 Now we assume that for 1 < pq < q < p and p 2 (1, +¥), N + (F1) f : R R ! R satisﬁes the Carathéodory condition. N + (F2) f 2 C(R R ,R), and there exist constants c , c > 0 such that 1 2 p 2 q 2 N + j f (x, t)j c t + c t , for all (x, t) 2 R R , q 2 ( pq, p ). 1 2 p 1 N (F3) f (x, t) = o(t ) as t ! 0 for x 2 R uniformly. F(x,t) (F4) lim = ¥ uniformly for almost all x 2 R , where the number q is given in t!¥ pq (K2), and F(x, t) = f (x, h)h dh. (F5) There exist m > p and r > 0 such that 2 p N f (x, t)t mF(x, t) $t b(x) for all x 2 R and t r, 1 N ¥ N where $ 0 and b 2 L (R )\ L (R ) with b(x) 0. s N The Euler functional corresponding to the problem (3) is J : H (R ,C) ! R A,V deﬁned as follows p p J (u) = (K(juj ) +jjujj ) l F(x,juj) dx. s,A p,V s N The functional J is Fréchet differentiable on H (R ,C), A,V Z Z p 2 ju(x) E(x, y)u(y)j (u(x) E(x, y)u(y)) [v(x) E(x, y)v(y)] hJ (z), vi = R K(juj ) dxdy s,A N+ ps N N jx yj R R Z Z p 2 + V(x)juj uv ¯ dx l f (x,juj)uv ¯ dx N N R R x+y s N i(x y) A( ) for any u, v 2 H (R ,C), where E(x, y) := e and v ¯ denotes complex con- A,V s N 0 jugation of v 2 C. Hereafter, h,i denotes the duality pairing between (H (R ,C)) A,V s N and H (R ,C). Following in , we observe that the critical points of J are exactly A,V the weak solutions of (1.1) and the functional J is weakly lower semi-continuous in s N H (R ,C). A,V The following result is to show that the energy functional J fulﬁlls the geometric conditions. Axioms 2022, 11, 38 5 of 14 Lemma 4. Let s 2 (0, 1), p 2 (1, +¥) and N > ps. Assume that (V), (K1), (K2) and (F1)–(F4) hold. Then the geometric conditions in the mountain pass theorem hold, i.e., (1) u = 0 is a strict local minimum for J . s N (2) J is unbounded from below on H (R ,C). A,V Proof. According to (F2) and (F3), for any # > 0, we can choose a positive constant denoted C(#) such that p 1 q 1 N + j f (x, t)tj #t + C(#)t , for all (x, t) 2 R R . (4) Assume that jjujj < 1. Owing to (K1), (K2) and (4), one has s,A p p J (u) = (K(juj ) +jjujj ) l F(x,juj)dx s,A p,V minf1, aq g le lC(#) p p q jjujj jjujj jjujj p N q N s,A L (R ) L (R ) p p q minf1, aq g leC lCC(#) p p q jjujj jjujj jjujj s,A s,A s,A p p q minf1,aq g for some constant C. Choose e > 0 so small that 0 < leC < . Then 2 p minf1, aq g p q J (u) jjujj C(l, e)Cjjujj . s,A s, A 2 p Since q > p, there is R > 0 small sufﬁciently and d > 0 such that J (u) d > 0 when jjujj = R. Therefore u = 0 is a strict local minimum for J . s,A l Next we prove the condition (2). By the condition (F4), for any C > 0, we can choose a constant d > 0 such that pq F(x, t) Ct (5) for t > d and for almost all x 2 R . Under the assumption (K2), we note that for all x 1, K(x) K(1)(1 + x ). (6) s N Relations (5) and (6) with Lemma 3 imply that for v 2 H (R ,C) A,V p p J (tv) = (K(jtvj ) +jjtvjj ) l F(x,jtvj)dx s,A p,V pq p pq K(1)(1 +jtvj ) +jjtvjj lC jtvj dx s,A p,V fjtvj>dg pq p pq pq pq pq 2K(1)t jvj ) + t jjvjj lt C jvj dx s, A p,V fjtvj>dg pq p pq pq = t 2K(1)jjvjj +jjvjj lC jvj dx s,A p,V fjtvj>dg for t > 0. If C is large sufﬁciently, then we deduce that J (tv) ! ¥ as t ! ¥. Hence the functional J is unbounded from below. The proof is completed. First of all, we introduce the Cerami condition, which was initially provided by Cerami . Deﬁnition 1. Let the functional Y be C and c 2 R. If any sequence fu g satisfying Y(u ) ! c and (1 +jju jj)jjY (u )jj ! 0 n n n Axioms 2022, 11, 38 6 of 14 possesses a convergent subsequence, we say that Y fulﬁls Cerami condition ((C) -condition in short) at the level c. s N Deﬁnition 2. A function u 2 H (R ,C) is called weak solution of problem (3) if u satisﬁes A,V Z Z p 2 ju(x) E(x, y)u(y)j (u(x) E(x, y)u(y)) [f(x) E(x, y)f(y)] R K(juj ) dxdy s,A N+ ps N N jx yj R R Z Z p 2 ¯ ¯ + V(x)juj uf dx = R l f (x,ju(x)j)uf dx R W s N for all f 2 H (R ,C). A,V The following lemma plays a crucial role in establishing the existence of a nontrivial weak solution to the given problem. Lemma 5. Let s 2 (0, 1), p 2 (1, +¥) and N > ps. Assume that (V), (K1), (K2), (F1)–(F2), and (F4)–(F5) hold. Then the functional J satisﬁes the (C) -condition for any l > 0. s N Proof. For c 2 R, let fu g be a (C) -sequence in H (R ,C), that is, n c A,V J (u ) ! c and jjJ (u )jj 0 (1 +jju jj ) ! 0 as n ! ¥, n n n l s, A s,A which means c = J (u ) + o(1) and J (u ), u = o(1), (7) l n n n s N where o(1) ! 0 as n ! ¥. If fu g is bounded in H (R ,C), it follows from the A,V analogous argument as in the proof of Lemma 4.2 in  that sequence fu g converges s N strongly to u inH (R ,C). Hence, it sufﬁces to ensure that the sequencefu g is bounded A,V s N in H (R ,C). We argue by contradiction. Assume that the sequence u is unbounded f g A,V s N in H (R ,C). So then we may assume that A,V jju jj ! ¥, as n ! ¥. s,A Due to the condition (7), we have that p p c = J (u ) + o(1) = (K(ju j ) +jju jj ) l F(x,ju j) dx + o(1). (8) l n n n n p,V s,A Since jju jj ! ¥ as n ! ¥, we assert by (8) that s,A 1 c o(1) p p F(x, u ) dx (K(ju j ) +jju jj ) + n n n p,V s,A pl l l 1 c o(1) minf1, aq gjju jj + ! ¥ as n ! ¥. (9) s,A pl l l s N Define a sequencefw g by w = u /jju jj . Then it is immediate thatfw g H (R ,C) n n n n n s,A A,V andjjw jj = 1. Hence, up to a subsequence, still denoted byfw g, we obtain w * w in n s,A n n s N H (R ,C) as n ! ¥, we have A,V N r N w (x) ! w(x) for a.e. x 2 R and jw j ! jwj in L (R ) as n ! ¥ (10) n n for p r < p . Set S = x 2 R : w(x) 6= 0 . By the convergence (10), we know that u = w jju jj ! ¥ as n ! ¥ j j j j n n n s,A Axioms 2022, 11, 38 7 of 14 for all x 2 S. Then it follows from (K2) and (F3) that for all x 2 S, F(x,ju j) F(x,ju j) n n lim lim p p pq p n!¥ n!¥ K(ju j ) +jju jj n n K(1)(1 +ju j ) +jju jj s, A p,V n n s,A p,V F(x,ju j) lim pq pq n!¥ 2K(1)jju jj +jju jj n n s,A p,V F(x,ju j) lim pq n!¥ (2K(1) + 1)jju jj s,A F(x,ju j) n pq lim jw j pq n!¥ (2K(1) + 1)ju j = ¥, (11) q + where the inequality K(h) K(1)(1 + h ) is used for all h 2 R because if 0 h < 1, then K(h) = K(s) ds K(1), and if h > 1, then K(h) K(1)h . Thus we obtain that jSj = 0, where jj is the Lebesgue measure in R . Indeed, assume that jSj 6= 0. pq N Taking account into (F4) we can choose t > 1 such that F(x, t) > t for all x 2 R and t < t. By means of (F1) and (F2), we derive that there is M > 0 such that jF(x, t)j M for all (x, t) 2 R (0, t ]. Hence there is a M 2 R such that F(x, t) M for all 0 0 0 N + (x, t) 2 R R , and thus F(x,ju j) M n 0 0, (12) p p K(ju j ) +jju jj n n s,A p,V for all x 2 R and for all n 2 N. In accordance with (9), (11), (12) and the Fatou lemma, we infer that F(x,ju j) dx N n = lim inf n!¥ l l F(x,ju j) dx + c o(1) N n 2F(x,ju j) lim inf dx p p n!¥ N R K(ju j ) +jju jj n n s,A p,V Z Z 2F(x,ju j) 2M n 0 = lim inf dx lim sup dx p p p p n!¥ S K(ju j ) +jju jj S K(ju j ) +jju jj n!¥ n n n n s,A p,V s,A p,V 2(F(x,ju j) M ) n 0 = lim inf dx p p n!¥ S K(ju j ) +jju jj n n s,A p,V 2(F(x,ju j) M ) n 0 lim inf dx p p n!¥ K(ju j ) +jju jj n n s,A p,V Z Z 2F(x,ju j) 2M = lim inf dx lim sup dx = ¥, p p p p n!¥ S S K(ju j ) +jju jj n!¥ K(ju j ) +jju jj n n n n s,A p,V s,A p,V which is a contradiction. This means w(x) = 0 for almost all x 2 R . Notice that V(x) ! +¥ as jxj ! ¥, then 1 1 p q jju jj C (ju j +ju j ) dx n 8 n n p,V pq m ju jr 1 1 1 jju jj M , n 0 p,V 2 pq m Axioms 2022, 11, 38 8 of 14 where M is a positive constant. Combining this with (F2) and (F5), one has c + 1 J (u ) J (u ), u l n n n 1 1 1 1 p p p p (K(ju j ) (K(ju j )ju j + jju jj n n n n s,A s,A s,A p,V p m p m + l f (x,ju j)ju j F(x,ju j) dx n n n 1 1 1 1 p p p p (K(ju j ) (K(ju j )ju j + jju jj n n n n s,A s,A s,A p,V p m pq m Z Z 2 p q + l f (x,ju j)ju j F(x,ju j) dx C (ju j +ju j ) dx n n n n n ju j>r ju jr n n 1 1 1 1 1 p p p p p (K(ju j )ju j (K(ju j )ju j + jju jj n n n n n s,A s,A s,A s,A p,V pq m 2 pq m $ju j + b(x) dx M n o 1 1 1 l$ l p p min a, jju jj jju jj jjbjj M , n n 1 N p N 0 s,A L (R ) L (R ) pq m 2 m m which implies l$ 1 n o lim supjjw jj n p L (R ) 1 1 1 n!¥ m min a, pq m 2 l$ n o = jjwjj . (13) p N L (R ) 1 1 1 m min a, pq m 2 Hence, it follows from (13) that w 6= 0. Thus, we can conclude a contradiction. Therefore, s N fu g is bounded in H (R ,C). This complete the proof. A,V Using Lemma 5, we prove the existence of a nontrivial weak solution to our problem. Theorem 1. Under the same assumptions of Lemma 5, the problem (3) has a nontrivial weak solution for all l > 0. Proof. Note that J (0) = 0. By Lemma 4, the mountain pass geometric conditions are satisﬁed. From Lemma 5, J fulﬁls the (C) -condition for any l > 0. Subsequently, l c problem (3) admits a nontrivial weak solution for any l > 0 by Lemmas 4 and 5. Next, applying the fountain theorem in ( Theorem 3.6), we indicate inﬁnitely many weak solutions for problem (3). To do this, we refer to the following lemma. Lemma 6 (). Let E be a reﬂexive and separable Banach space. Then there exist fe g E and f f g E such that E = spanfe : n = 1, 2,g, E = spanf f : n = 1, 2,g, and 1 if i = j f , e = 0 if i 6= j. L L k ¥ Let us denote E = spanfe g, Y = E , and Z = E . In order to obtain n n n n k k n=1 n=k the existence result, we use the following Fountain theorem. Axioms 2022, 11, 38 9 of 14 Lemma 7 ([49,50]). Let E be a real Banach space, I 2 C (E,R) satisﬁes the (C) -condition for any c > 0 and I is even. If for each sufﬁciently large k 2 N, there exist $ > s > 0 such that the k k following conditions hold: (1) b := inffI(u) : z 2 Z ,jjujj = s g ! ¥ as k ! ¥; k k E k (2) a := maxfI(u) : u 2 Y ,jjujj = $ g 0. k k k Then the functional I has an unbounded sequence of critical values, i.e., there exists a sequence fu g E such that I (u ) = 0 and I(u ) ! +¥ as n ! +¥. n n n Theorem 2. Let s 2 (0, 1), p 2 (1, +¥) and N > ps. Assume that (V), (K1), (K2) and (F1)–(F4) hold. Then for any l > 0, problem (3) has an unbounded sequence of nontrivial weak solutions s N fu g in H (R ,C) such that J (u ) ! ¥ as n ! ¥. n l n A,V Proof. The proof follows the lines of that of Lemma 3.2 in . To apply Lemma 7, let us s N denote E := H (R ,C) and I := J . Plainly, J is an even functional and ensures the l l A,V (C) -condition. It sufﬁces to show that there exist $ > s > 0 with the conditions (1) and k k (2) in Lemma 7. Let us denote V = sup jjzjj q N . k L (R ) jjujj =1,z2Z s,A k Then, it is obvious to verify that V ! 0 as k ! ¥. For any z 2 Z , assume that jjujj > 1. k k s,A minf1,aq g Choose e > 0 so small that 0 < leC < . Then it follows from (4) that 2 p p p J (u) = (K([u] ) +jjujj ) l F(x,juj)dx s,A p,V minf1, aq g jjujj l F(x,juj)dx s,A minf1, aq g le lC(e) p p q jjujj jjujj jjujj p N q N s,A L (R ) L (R ) p p q minf1, aq g p q q jjujj lC(e)V jjujj s,A k s, A 2 p minf1, aq g q q p p = lC(e)V jjujj jjujj (14) k s,A s,A 2 p h i q p q 2 plC(e) Choose s = V . Since p < q, p 2 (1, +¥) and V ! 0 as k ! ¥, we infer k 1 k minf1,aq g k s ! ¥ as k ! ¥. Hence, if u 2 Z and jjujj = s , then we deduce that k k s,A k minf1, aq g J (u) s ! ¥ as k ! ¥, 2 p which implies (1). Now we prove condition (2). To do this, we claim that J (u) ! ¥ as jjujj ! ¥ l s,A for all u 2 Y . Let us assume that this is false for some k. Then we can choose a sequence s N fu g in H (R ,C) such that A,V jju jj ! ¥ as n ! ¥ and J (u ) M. n s,A l n Let w = u /jju jj . Then it is obvious that jjw jj = 1. Since dimY < ¥, there is n n n s,A n s,A k w 2 Y nf0g such that up to a subsequence, jjw wjj ! 0 and w (x) ! w(x) n n s,A Axioms 2022, 11, 38 10 of 14 for almost all x 2 R as n ! ¥. Thus we have (14) that, 1 M 1 J (u ) p p p p p p K(ju j ) +jju jj K(ju j ) +jju jj n n n n s,A p,V s,A p,V F(x,ju j) = l dx p p K(ju j ) +jju jj n n s,A p,V F(x,ju j) l dx. (15) pq fw (x)6=0g (2K(1) + 1)jju jj s,A If we follow the analogous argument as in the proof of Lemma 5, we derive by (12), (15), (F4) and Fatou’s lemma that Z Z 1 F(x,ju j) M n 0 lim inf dx lim sup dx pq pq pl n!¥ fw (x)6=0g fw (x)6=0g n n!¥ n (2K(1) + 1)jju jj (2K(1) + 1)jju jj n n s,A s,A Z Z F(x,ju j) M F(x,ju j) M n 0 n 0 = lim inf dx lim inf dx pq pq n!¥ n!¥ fw (x)6=0g fw (x)6=0g n n (2K(1) + 1)jju jj (2K(1) + 1)jju jj n n s,A s,A Z Z F(x,ju j) M n 0 = lim inf dx lim sup dx pq pq n!¥ fw (x)6=0g fw (x)6=0g n n n!¥ (2K(1) + 1)jju jj (2K(1) + 1)jju jj n n s,A s,A 1 F(x,ju j) pq lim inf jw j dx = ¥, pq n!¥ 2K(1) + 1 fw (x)6=0g n u j j where M was given in the proof of Lemma 5. This is impossible. Thus, J (u) ! ¥ as 0 l jjujj ! ¥ for all u 2 Y . Choose $ > s > 0 large sufﬁciently and let jjujj = $ , we s,A k k k s,A k ﬁnally obtain a = maxfJ (u) : u 2 Y ,jjujj = $ g 0. k l k s,A k This completes the proof. Deﬁnition 3. Let E be a real separable and reﬂexive Banach space. We say that I satisﬁes the (C) -condition (with respect to Y ) if any sequence fu g E for which u 2 Y , n n n n c n2N for any n 2 N, I(u ) ! c and jj(Ij ) (u )jj (1 +jju jj ) ! 0 as n ! ¥, n Y n E n E contains a subsequence converging to a critical point of E. Lemma 8 (Dual Fountain Theorem ( Theorem 3.11)). Assume that E is a real Banach space, I 2 C (E,R) is an even functional. If there is k > 0 so that, for each k k , there are 0 0 $ > s > 0 such that k k (A1) inffI(u) : u 2 Z ,jjujj = $ g 0. k E k (A2) b := maxfI(u) : u 2 Y ,jjujj = s g < 0. k k E k (A3) g := inffI(u) : u 2 Z ,jjujj $ g ! 0 as k ! ¥. k k k (A4) I satisﬁes the (C) -condition for every c 2 [d , 0). c k Then I has a sequence of negative critical values c < 0 satisfying c ! 0 as n ! ¥. n n Lemma 9. Let s 2 (0, 1), p 2 (1, +¥) and N > ps. Assume that (V), (K1), (K2) and (F1)–(F5) hold. Then the functional J satisﬁes the (C) -condition. l c Proof. The proof is carried out by the analogous argument as in . With the help of Lemmas 8 and 9 we are ready to demonstrate our second assertion. Axioms 2022, 11, 38 11 of 14 Theorem 3. Let s 2 (0, 1), p 2 (1, +¥) and N > ps. Assume that (V), (K1), (K2) and (F1)–(F5) s N hold. Then the problem (3) has a sequence of nontrivial weak solutions fu g in H (R ,C) such A,V that J (u ) ! 0 as n ! ¥ for any l > 0. l n Proof. Invoking Lemma 9, we get that J is even and satisﬁes the (C) -condition for all c 2 R. Now it remains to show that conditions (A1), (A2) and (A3) of Lemma 8 are satisﬁed. (A1): Let us denote q = sup jjujj , q = sup jjujj . p N q N 1,k 2,k L (R ) L (R ) jjujj =1,u2Z jjujj =1,u2Z s, A k s, A k Then, it is immediate to verify that q ! 0 and q ! 0 as k ! ¥. Set J = maxfq , q g. 1,k 2,k k 1,k 2,k Then it follows that p p J (u) = (K(juj ) +jjujj ) l F(x,juj)dx s,A p,V minf1, aq g lc lc p p 2 q jjujj jjujj jjujj p q N N s,A L (R ) L (R ) p p q minf1, aq g lc lc p p p q q 1 2 jjujj q jjujj q jjujj s,A 1,k s,A 2,k s, A p p q minf1, aq g c c p p q 1 2 jjujj l + J jjujj s,A s,A p p q for sufﬁciently large k and jjujj 1. Choose s,A p 2q 2 pl c c 1 2 $ = + J . 1 k minf1, aq g p q Let u 2 Z with jjujj = $ > 1 for k large enough. Then, there exists k 2 N such that k s,A k 0 minf1, aq g c c p p 2q 1 2 J (u) jjujj l + J jjujj s,A s,A p p q minf1, aq g = $ 0 2 p for all k 2 N with k k , because minf1, aq g lim $ = ¥. 2 p k!¥ Therefore, inffJ (u) : u 2 Z ,jjujj = $ g 0. l k s,A k (A2): Observe that jjjj , jjjj and jjjj are equivalent on Y . Then there p N pq N s,A k L (R ) L (R ) exist positive constants V and V such that 1,k 2,k jjujj p N V jjujj and jjujj V jjujj pq N 1,k s,A s,A 2,k L (R ) L (R ) for any u 2 Y . From (F2)–(F4), for anyM > 0 there are positive constants C (M) such that k 7 pq pq p F(x, t) MV t C (M)t 2,k Axioms 2022, 11, 38 12 of 14 N + q for almost all (x, t) 2 R R . Since K(h) K(1)(1 + h ) for all h 2 R , it follows that p p J (u) = (K(juj ) +jjujj ) l F(x,juj)dx p,V s,A Z Z pq p pq pq p K(1)(1 +juj ) +jjujj lMV u dx + lC (M) juj dx s,A p,V 2,k N N R R Z Z pq pq pq pq p 2K(1)jjujj +jjujj lMV u dx + lC (M) juj dx s,A s,A 2,k N N R R pq pq p p (2K(1) + 1)jjujj lMjjujj + lC (M)V jjujj s, A s,A s,A 1,k pq pq p for any u 2 Y with jjujj 1. Let f (t) = (2K(1) + 1)t lMt + lC (M)V t . k s, A 7 p 1,k If M is large thoroughly, then lim f (t) = ¥, and thus there is t 2 (1,¥) such that t!¥ 0 f (t) < 0 for all t 2 [t ,¥). Hence J (u) < 0 for all u 2 Y with jjujj = t . Choosing 0 l k s,A 0 s = t for all k 2 N, one has b := maxfJ (u) : u 2 Y ,jjujj = s g < 0. s,A k l k k If necessary, we can change k to a large value, so that $ > s > 0 for all k k . 0 0 k k (A3): Because Y \Z 6= f and 0 < s < $ , we have g b < 0 for all k k . For k k k k k k 0 any u 2 Z with jjujj = 1 and 0 < t < $ , one has k s,A k minf1, aq g lc lc p p q 1 2 J (tu) jjtujj jjtujj jjtujj l p N q N s,A L (R ) L (R ) p p q lc lc 1 p 2 q p q t jjujj t jjujj p q N N L (R ) L (R ) p q lc lc p p q q 1 2 $ J $ J k k k k p q for large enough k. Hence, it follows from the deﬁnition of $ that lc lc p p q q 1 2 g $ J $ J k k k k p q p q ( p q) p ( p q)q p 2q p 2q lc 2 pl c c lc 2 pl c c 1 1 2 p 2q 2 1 2 p 2q = + J + J . k k 1 1 p minf1, aq g p q q minf1, aq g p q Because p < q and J ! 0 as k ! ¥, we derive that lim g = 0. k k!¥ k Hence all conditions of Lemma 8 are required. Consequently, we assert that problem (3) s N has a sequence of nontrivial weak solutions fu g in H (R ,C) such that J (u ) ! 0 as n l n A,P n ! ¥ for any l > 0. 3. Conclusions In this paper, we investigate the existence and multiplicity of weak solutions to the fractional p-Laplacian Equation (3) with the external magnetic potential. The strategy of the proof for these results is to approach the problem variationally by applying the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition. As far as we are aware, the present paper is the ﬁrst attempt to study the multiplicity of nontrivial weak solutions to Schrödinger-Kirchhoff-type problems with the external magnetic potential in these circumstances. We point out that with a similar analysis, our main consequences continue to hold when ( D) v in (3) is changed into any non-local p,A integro-differential operator L , deﬁned as follows; x+y x+y i(x y) A( ) p 2 i(x y) A( ) N 2 2 L v(x) = 2 jv(x) e v(y)j (v(x) e v(y)) M(x y)dy for all x 2 R , (16) R Axioms 2022, 11, 38 13 of 14 where M : R nf0g ! (0, +¥) is a kernel function satisfying properties that 1 N p (M1) m M 2 L (R ), where m(x) = minfjxj , 1g; (N+ ps) N (M2) there exists q > 0, such that M(x) qjxj for all x 2 R nf0g; (K3) M(x) = M( x) for all x 2 R nf0g. Funding: This work was supported by the Seoul Theological University Research Fund of 2022 and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-202161F1). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. 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Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in RN
, Volume 11 (2) –
Jan 19, 2022
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