On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents

Abstract

Suppose Ω ⊂ ℝ^N(N ≥ 3) is a smooth bounded domain, $\xi _i \in \Omega , 0 < a_i < \sqrt {\bar \mu } , \bar \mu : = \left( {\frac{{N - 2}} {2}} \right)^2 ,0 \leqslant \mu _i < \left( {\sqrt {\bar \mu } - a_i } \right)^2 , a_i < b_i < a_i + 1$ and $p_i : = \frac{{2N}} {{N - 2(1 + a_i - b_i )}}$ are the weighted critical Hardy-Sobolev exponents, i = 1, 2,…, k, k ≥ 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem

$$ \sum\limits_{i = 1}^k {\left( { - div(|x - \xi _i |^{ - 2a_i } \nabla u) - \frac{{\mu _i u}} {{|x - \xi _i |^{2(1 + a_i )} }} - \frac{{u^{p_i - 1} }} {{|x - \xi _i |^{b_i p_i } }}} \right)} = 0 $$

with Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters a _i , b _i and μ _i , i = 1, 2,…, k.