A Nonlinear Elliptic PDE with Two Sobolev–Hardy Critical Exponents

By Yanyan Li, Changshou Lin. 2011

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AbstractIn this paper, we consider the following PDE involving two Sobolev–Hardy critical exponents, $$ \label{0.1}\left\{\begin{aligned}& \Delta u + \lambda\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \quad \rm {in}\,\,\Omega,\quad\quad\quad(0.1)\\ & u=0 \quad {\rm on }\,\,\Omega, \end{aligned} \right.$$where 0 ≦ s2 < s1 ≦ 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 s1 = 0 or s2 = 0. In this paper, we prove that if 0 < s2 <  s1 < 2 and the mean curvature of $${\partial \Omega}$$ 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 $${\Omega =\mathbb {R}_{+}^{N}}$$ under different situations of s1, s2 and λ.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/165ed324244a2ccc4d9e91d76338268790a86e00