Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent

Jing  Zhang; Shiwang Ma

doi:10.3934/dcdsb.2016033

Article Contents

2016, Volume 21, Issue 6: 1999-2009. Doi: 10.3934/dcdsb.2016033

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Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent

Jing Zhang^1, and
Shiwang Ma^2,

1.
Mathematics Science College, Inner Mongolia Normal University, Hohhot 010022
2.
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071

Received: December 31, 2013

Revised: March 31, 2016

Published: July 2016

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Abstract

In this paper, we are concerned with the following nonlinear Schrödinger equations with hardy potential and critical Sobolev exponent \begin{equation}\label{eq0.1} \left\{\begin{array}{ll} -\Delta u+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^{*}-2}u,& \textrm{in}\, \mathbb{R}^N, \\ u>0, & \textrm{in}\,\mathcal{D}^{1,2}(\mathbb{R}^N), （1） \end{array} \right. \end{equation} where $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent, $0\leq \mu<\overline{\mu}=\frac{(N-2)^2}{4}$, $a(x)\in C(\mathbb{R}^N)$. We first use an abstract perturbation method in critical point theory to obtain the existence of positive solutions of （1） for small value of $|\lambda|$. Secondly, we focus on an anisotropic elliptic equation of the form \begin{equation}\label{eq0.2} -{\rm div}(B_\lambda(x)\nabla u)+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^*-2}u, x\in\mathbb{R}^N. （2） \end{equation} The same abstract method is used to yield existence result of positive solutions of （2） for small value of $|\lambda|$.

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Mathematics Subject Classification: Primary: 35B33, 35B20; Secondary: 35B09.

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