A perturbed nonlinear elliptic PDE with two Hardy-Sobolev critical exponents

TL;DR

This paper investigates the existence of positive solutions for a perturbed nonlinear elliptic PDE with two Hardy-Sobolev critical exponents in a domain with negative mean curvature at the boundary, using variational methods.

Contribution

It introduces new existence results for ground state solutions of a perturbed PDE involving two Hardy-Sobolev critical exponents, employing concentration compactness and perturbation techniques.

Findings

Existence of ground state solutions under various conditions.

Application of concentration compactness principle.

Use of Nehari manifold and perturbation methods.

Abstract

Let $\Omega$ be a $C^1$ open bounded domain in $\R^N$ ($N\geq 3$) with $0\in \partial \Omega$. Suppose that $\partial\Omega$ is $C^2$ at $0$ and the mean curvature of $\partial\Omega$ at $0$ is negative. Consider the following perturbed PDE involving two Hardy-Sobolev critical exponents: $$ \begin{cases} &\Delta u+\lambda_1 \frac{u^{2^*(s_1)-1}}{|x|^{s_1}}+\lambda_2\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}+\lambda_3\frac{u^p}{|x|^{s_3}}=0\;\quad \hbox{in}\;\Omega,\\ &u(x)>0\;\hbox{in}\;\Omega,\;\, u(x)=0\;\hbox{on}\;\partial\Omega, \end{cases} $$ where $0<s_2<s_1<2, 0\leq s_3<2, 2^*(s_i):=\frac{2(N-s_i)}{N-2}, 0\neq \lambda_i\in \R, \lambda_2>0, 1< p\leq 2^*(s_3)-1$. The existence of ground state solution is studied under different assumptions via the concentration compactness principle and the Nehari manifold method. We also apply a perturbation method to study the existence of positive solution.