A multiplicity result to Schr\"{o}dinger Equation with singular points

Abstract

In this paper, using variational method, we study the existence and mutiplicity of the solutions for the following multi-singular critical elliptic problem 
\begin{eqnarray*}
\begin{cases}\begin{array}{cc}
-\Delta{u}-\displaystyle
\sum_{i=1}^k\frac{\mu_i}{|x-a_i|^2}u=f_\lambda \left( x,u \right)
& x \in{\Omega{\backslash}}\{a_1,...,a_k\},\\
u(x)>0 & x \in{\Omega}{\backslash}\{a_1,...,a_k \},\\
u(x)=0 &x \in{\partial{\Omega}}.
\end{array}
\end{cases}
\end{eqnarray*}
where $\Omega{\subset}\mathbb{R}^N(N\geq3)$ is a smooth bounded
domain such that $a_i\in{\Omega},i=1,2,...,k,$ for $k\geq2$ are
different points, $0\leq{\mu_i} \in \mathbb{R}$.
In this class of nonlinear elliptic Dirichlet boundary value problems the combination effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results.