On nonhomogeneous elliptic problems involving the Hardy potential and critical Sobolev exponent

Abstract

In this paper, we are concerned with elliptic equations with Hardy potential and critical Sobolev exponents where $2^{*}={2N}/({N-2})$ is the critical Sobolev exponent, $N\geq 3$, $0\leq \mu \lt \overline {\mu }={(N-2)^2}/{4}$, $\mathbf {\Omega }\subset \mathbb {R}^{N}$ an open bounded set. For $\lambda \in [0,\lambda _{1})$ with $\lambda _{1}$ being the first eigenvalue of the operator $-\Delta -{\mu }/{|x|^{2}}$ with zero Dirichlet boundary condition, and for $f\in H_{0}^{1}(\mathbf {\Omega })^{-1}=H^{-1}$, $f\neq 0$, we show that (\ref {eq1}) admits at least two distinct nontrivial solutions $u_{0}$ and $u_{1}$ in $H_{0}^{1}(\mathbf {\Omega })$. Furthermore, $u_{0}\geq 0$ and $u_{1}\geq 0$ whenever $f\geq 0$.