A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents
Journal
Archive for Rational Mechanics and Analysis
Journal Volume
203
Journal Issue
3
Pages
943-968
Date Issued
2012
Author(s)
Li, Y.Y.
Abstract
In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, where 0 ≦ s 2 < s 1 ≦ 2, 0 ≠ λ ∈ ℝ and 0 ∈ ∂Ω. 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 ∂Ω at 0 H(0) < 0, then (0.1) has a least-energy solution. Therefore, this paper has completed the study of (0.1) for the least-energy solutions. We also prove existence or nonexistence of positive entire solutions of (0.1) with Ω = ℝ + N under different situations of s 1, s 2 and λ. © 2011 Springer-Verlag.
Type
journal article