A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents

TL;DR

This paper investigates the existence of least-energy and positive entire solutions for a nonlinear elliptic PDE involving two Sobolev-Hardy critical exponents, extending previous results and establishing new existence criteria based on boundary curvature.

Contribution

It proves the existence of least-energy solutions under specific conditions on exponents and boundary curvature, completing the analysis for this class of PDEs.

Findings

Existence of least-energy solutions when $0< s_2 < s_1 <2$ and boundary curvature $H(0)<0$.

Complete characterization of least-energy solutions for the PDE.

Conditions for existence or nonexistence of positive entire solutions in $ n$.

Abstract

In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, \label{0.1} {& \Delta u + \lambda\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \text{in} \Omega, & u=0 \qquad \text{on} \Omega, where $0 \le s_2 < s_1 \le 2$, $0 \ne \lambda \in \Bbb R$ and $0 \in \partial \Omega$. The existence (or nonexistence) for least-energy solutions has been extensively studied when $s_1=0$ or $s_2=0$. In this paper, we prove that if $0< s_2 < s_1 <2$ and the mean curvature of $\partial \Omega$ at 0 $H(0)<0$, then \eqref{0.1} has a least-energy solution. Therefore, this paper has completed the study of \eqref{0.1} for the least-energy solutions. We also prove existence or nonexistence of positive entire solutions of \eqref{0.1} with $\Omega =\rn$ under different situations of $s_1, s_2$ and $\lambda$.