A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents

Abstract

In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, \label{0.1} {& Δu + λ\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \text{in} Ω, & u=0 \qquad \text{on} Ω, where $0\le s_2<s_1\le 2$, $0\ne \lambda \in \mathbb{R}$ 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 $Ω=\rn$ under different situations of $s_1,s_2$ and $\lambda$.

References in corpus (1)

• Symmetry and Asymmetry: The Method of Moving Spheres

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