A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in ℝ^N

2015/04/05 by Zhong, Xuexiu, Zou, Wenming
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1504.01133

Abstract

In this paper, we will study the following PDE in ℝ^N involving multiple Hardy-Sobolev critical exponents: \begincases Δu+∑_i=1^lλ_i \fracu^{2^*(s_i)-1}|x|^s_i+u^{2^*-1}=0 \hboxin ℝ^N, u∈ D₀^1,2(ℝ^N), \endcases where 00 for 1≤ i≤ k; λ_i<0 for k+1≤ i≤ l. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence and non-existence of the positive ground state solution. The regularity of the least-energy solution are also investigated.

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