Infinitely many solutions for quasilinear Schrödinger equation with concave-convex nonlinearities

IF 1.7 4区 数学 Q1 Mathematics
Lijuan Chen, Caisheng Chen, Qiang Chen, Yunfeng Wei
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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 $\lambda \in [0,\lambda _{0}]$ .
具有凹凸非线性的准线性薛定谔方程的无限多解
在这项工作中,我们研究了以下具有参数 α 和凹凸非线性的准线性薛定谔方程的无穷多个解的存在性: 0.1 $$\begin{aligned}& -\Delta _{p}u+V(x) \vert u \vert ^{p-2}u-\Delta _{p}\bigl( \vert u \vert ^{2\alpha}\bigr) \vert u \vert ^{2\alpha -2}u= \lambda h_{1}(x) \vert u \vert ^{m-2}u+h_{2}(x) \vert u \vert ^{q-2}u、\& \quad x\in {\mathbb{R}}^{N}, \end{aligned}$$ 其中 $\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ , $1< p< N$ , $\lambda \ge 0$ , and $1< m< p0$ such that Eq.(0.1) 在$W^{1,p}({\mathbb{R}}^{N})$ 中有无限多的高能解,前提是$\lambda \ in [0,\lambda _{0}]$。
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来源期刊
Boundary Value Problems
Boundary Value Problems MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.00
自引率
5.90%
发文量
83
审稿时长
4 months
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.
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