含参数拟线性非齐次椭圆型方程的多重解

数学物理学报 ›› 2019, Vol. 39 ›› Issue (2): 286-296.

含参数拟线性非齐次椭圆型方程的多重解

宋洪雪^1,^2,*(),魏云峰^2,³

¹ 南京邮电大学理学院南京 210023
² 河海大学理学院南京 210098
³ 南京审计大学统计与数学学院南京 211815

收稿日期:2018-01-15 出版日期:2019-04-26 发布日期:2019-05-05
通讯作者: 宋洪雪 E-mail:songhx@njupt.edu.cn
基金资助:
国家自然科学基金(61503198);中国博士后科学基金面上项目(2017M611664);南京邮电大学校级科研基金(NY217092);南京邮电大学校级科研基金(NY218076)

Multiple Solutions for Quasilinear Nonhomogeneous Elliptic Equations with a Parameter

Hongxue Song^1,^2,*(),Yunfeng Wei^2,³

¹ School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023
² College of Science, Hohai University, Nanjing 210098
³ School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815

Received:2018-01-15 Online:2019-04-26 Published:2019-05-05
Contact: Hongxue Song E-mail:songhx@njupt.edu.cn
Supported by:
the NSFC(61503198);the China Postdoctoral Science Foundations(2017M611664);the NUPTSF(NY217092);the NUPTSF(NY218076)

摘要/Abstract

摘要：

该文研究如下形式的拟线性非齐次椭圆型方程

$ \begin{eqnarray*}\label{eq:0a1} -\triangle_{p}u-\triangle_{p}(|u|^{2\alpha})|u|^{2\alpha -2}u + V(x)|u|^{p-2}u=h(u)+g(x), ~~~x\in \mathbb{R}^{N}, \end{eqnarray*} $

其中$1 <p\leq N$ ($N\geq 3$), $\frac{1}{2} <\alpha\leq 1 $, $V\in C(\mathbb{R}^{N}, \mathbb{R})$, $h \in C(\mathbb{R}, \mathbb{R})$,而且扰动项$g\in L^{p'}(\mathbb{R}^{N})$,这里$p'=\frac{p}{p-1}$.利用变量代换结合极小极大方法可以证明该问题存在多重解.

关键词: 拟线性椭圆型方程, Ekeland变分原理, P-S序列

Abstract:

In this paper, we study the following quasilinear nonhomogeneous elliptic equations of the form

$-\triangle_{p}u-\triangle_{p}(|u|^{2\alpha})|u|^{2\alpha -2}u + V(x)|u|^{p-2}u=h(u)+g(x), ~~~x\in \mathbb{R}^{N}, $

where $1 <p\leq N$ ($N\geq 3$), $\frac{1}{2} <\alpha\leq 1 $, $V\in C(\mathbb{R}^{N}, \mathbb{R})$, $h \in C(\mathbb{R}, \mathbb{R})$ and $g\in L^{p'}(\mathbb{R}^{N})$, where $p'=\frac{p}{p-1}$, is a disturbance term. Using a variable replacement and minimax method, we show the existence and multiplicity of solutions to this problem.

Key words: Quasilinear elliptic equations, Ekeland's variational principle, Palais-Smale sequences

中图分类号:

O175.25

宋洪雪,魏云峰. 含参数拟线性非齐次椭圆型方程的多重解[J]. 数学物理学报, 2019, 39(2): 286-296.

Hongxue Song,Yunfeng Wei. Multiple Solutions for Quasilinear Nonhomogeneous Elliptic Equations with a Parameter[J]. Acta mathematica scientia,Series A, 2019, 39(2): 286-296.

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