{"title":"Infinitely many solutions for quasilinear Schrödinger equation with concave-convex nonlinearities","authors":"Lijuan Chen, Caisheng Chen, Qiang Chen, Yunfeng Wei","doi":"10.1186/s13661-023-01805-3","DOIUrl":null,"url":null,"abstract":"In this work, we study the existence of infinitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity: 0.1 $$\\begin{aligned}& -\\Delta _{p}u+V(x) \\vert u \\vert ^{p-2}u-\\Delta _{p}\\bigl( \\vert u \\vert ^{2\\alpha}\\bigr) \\vert u \\vert ^{2\\alpha -2}u= \\lambda h_{1}(x) \\vert u \\vert ^{m-2}u+h_{2}(x) \\vert u \\vert ^{q-2}u, \\\\& \\quad x\\in {\\mathbb{R}}^{N}, \\end{aligned}$$ where $\\Delta _{p}u=\\operatorname{div}(|\\nabla u|^{p-2}\\nabla u)$ , $1< p< N$ , $\\lambda \\ge 0$ , and $1< m< p<2\\alpha p<q<2\\alpha p^{*}=\\frac{2\\alpha pN}{N-p}$ . The functions $V(x)$ , $h_{1}(x)$ , and $h_{2}(x)$ satisfy some suitable conditions. Using variational methods and some special techniques, we prove that there exists $\\lambda _{0}>0$ such that Eq. (0.1) admits infinitely many high energy solutions in $W^{1,p}({\\mathbb{R}}^{N})$ provided that $\\lambda \\in [0,\\lambda _{0}]$ .","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Boundary Value Problems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1186/s13661-023-01805-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
In this work, we study the existence of infinitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity: 0.1 $$\begin{aligned}& -\Delta _{p}u+V(x) \vert u \vert ^{p-2}u-\Delta _{p}\bigl( \vert u \vert ^{2\alpha}\bigr) \vert u \vert ^{2\alpha -2}u= \lambda h_{1}(x) \vert u \vert ^{m-2}u+h_{2}(x) \vert u \vert ^{q-2}u, \\& \quad x\in {\mathbb{R}}^{N}, \end{aligned}$$ where $\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ , $1< p< N$ , $\lambda \ge 0$ , and $1< m< p<2\alpha p0$ such that Eq. (0.1) admits infinitely many high energy solutions in $W^{1,p}({\mathbb{R}}^{N})$ provided that $\lambda \in [0,\lambda _{0}]$ .
期刊介绍:
The main aim of Boundary Value Problems is to provide a forum to promote, encourage, and bring together various disciplines which use the theory, methods, and applications of boundary value problems. Boundary Value Problems will publish very high quality research articles on boundary value problems for ordinary, functional, difference, elliptic, parabolic, and hyperbolic differential equations. Articles on singular, free, and ill-posed boundary value problems, and other areas of abstract and concrete analysis are welcome. In addition to regular research articles, Boundary Value Problems will publish review articles.