A global compact result for a semilinear elliptic problem with Hardy potential and critical non-linearities on ℝ N

Abstract

In this paper, we are concerned with the following class of elliptic problems: $$ \left\{ \begin{gathered} - \Delta u - \mu \frac{u} {{|x|^2 }} + a(x)u = |u|^{2* - 2} u + k(x)|u|^{q - 2} u, \hfill \\ u \in H^1 (\mathbb{R}^N ), \hfill \\ \end{gathered} \right. $$ where 2* = 2N/(N - 2) is the critical Sobolev exponent, 2 < q < 2*, \( 0 \leqslant \mu < \bar \mu \triangleq \frac{{(N - 2)^2 }} {4} \), a(x), k(x) ∈ C(ℝN). Through a compactness analysis of the functional corresponding to the problems (*), we obtain the existence of positive solutions for this problem under certain assumptions on a(x) and k(x).