On Some Semilinear Elliptic Problems with Singular Potentials Involving Symmetry

Abstract

This paper deals with the existence and multiplicity of solutions for a class of semilinear elliptic problems of the form \begin{equation*} \begin{cases} -\Delta u = \displaystyle \frac{\mu}{|x|^2}u + f(x,u) & \text{ in } & \Omega, \\ u = 0 & \text{ on } & \partial \Omega, \end{cases} \end{equation*} where $\Omega = \Omega_1 \times \Omega_2 \subset \mathbb{R}^N$ ($N \geqq 5$) is a bounded domain having cylindrical symmetry, $\Omega_1 \subset \mathbb{R}^m$ is a bounded regular domain and $\Omega_2$ is a $k-$dimensional ball of radius $R$, centered in the origin and $m+k = N$, and $m \geqq 2$, $k \geqq 3$, $0 \leqq \mu \lt \mu^\star = \left(\frac{N-2}{2}\right)^2$. The proofs rely essentially on the critical point theory tools combined with the Hardy inequality.