# Multiplicity of solutions for a p-Schrödinger–Kirchhoff-type integro-differential equation

Multiplicity of solutions for a p-Schrödinger–Kirchhoff-type integro-differential equation We consider the integro-differential problem (P): -a+b∫RN|∇u|pdxp-1Δpu+V(x)|u|p-2u=f(x,u),x∈RN,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} -\left( a+b\left( \int _{\mathbb {R}^{N}}|\nabla u|^{p} \textrm{d} x\right) ^{p-1}\right) \Delta _{p} u +V(x)|u|^{p-2} u =f(x,u), \quad x\in \mathbb {R}^{N}, \end{aligned}\end{document}with |u(x)|⟶0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|u(x)| \longrightarrow 0$$\end{document}, as |x|⟶+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|x| \longrightarrow +\infty$$\end{document}. We assume that a,b>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a, b>0$$\end{document}, N≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N \ge 2$$\end{document}, 1<p<N<+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1<p< N < +\infty$$\end{document}, V∈C(RN)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V\in \textrm{C}^{}(\mathbb {R}^{N})$$\end{document} with inf(V)>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\inf (V)>0$$\end{document}, and that f:RN×R⟶R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f:\mathbb {R}^{N}\times \mathbb {R}\longrightarrow \mathbb {R}$$\end{document} verifies conditions introduced by Duan and Huang. We prove the existence of a non-trivial ground state solution and, by a Ljusternik–Schnirelman scheme, the existence of infinitely many non-trivial solutions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Functional Analysis Springer Journals

# Multiplicity of solutions for a p-Schrödinger–Kirchhoff-type integro-differential equation

, Volume 14 (2) – Apr 1, 2023
19 pages

/lp/springer-journals/multiplicity-of-solutions-for-a-p-schr-dinger-kirchhoff-type-integro-h0mSxlr03E
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Springer Journals
Copyright © Tusi Mathematical Research Group (TMRG) 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
2639-7390
eISSN
2008-8752
DOI
10.1007/s43034-023-00257-1
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### Abstract

We consider the integro-differential problem (P): -a+b∫RN|∇u|pdxp-1Δpu+V(x)|u|p-2u=f(x,u),x∈RN,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} -\left( a+b\left( \int _{\mathbb {R}^{N}}|\nabla u|^{p} \textrm{d} x\right) ^{p-1}\right) \Delta _{p} u +V(x)|u|^{p-2} u =f(x,u), \quad x\in \mathbb {R}^{N}, \end{aligned}\end{document}with |u(x)|⟶0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|u(x)| \longrightarrow 0$$\end{document}, as |x|⟶+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|x| \longrightarrow +\infty$$\end{document}. We assume that a,b>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a, b>0$$\end{document}, N≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N \ge 2$$\end{document}, 1<p<N<+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1<p< N < +\infty$$\end{document}, V∈C(RN)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V\in \textrm{C}^{}(\mathbb {R}^{N})$$\end{document} with inf(V)>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\inf (V)>0$$\end{document}, and that f:RN×R⟶R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f:\mathbb {R}^{N}\times \mathbb {R}\longrightarrow \mathbb {R}$$\end{document} verifies conditions introduced by Duan and Huang. We prove the existence of a non-trivial ground state solution and, by a Ljusternik–Schnirelman scheme, the existence of infinitely many non-trivial solutions.

### Journal

Annals of Functional AnalysisSpringer Journals

Published: Apr 1, 2023

Keywords: p-Schrödinger–Kirchhoff-type equation; Ljusternik–Schnirelman theory; Critical point theory; 45K05; 35J60

### References

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