The existence of a nontrivial solution to an elliptic equation with critical Sobolev exponent and a general potential well

The purpose of this paper is to examine a class of elliptic problems that involve negative potentials $ a\in L^{\frac{N}{2} }(\Omega) $ and critical nonlinearities. To discuss this, the well-known eigenvalue problem $ -\Delta- a $ is considered. Under some mild assumptions, an existence result is obtained, which extends the existing results to the critical case.