Existence results for Brezis–Nirenberg problems with Hardy potential and singular coefficients

In this paper, we consider the existence of non-trivial solutions to semi-linear Brezis–Nirenberg type problems with Hardy potential and singular coefficients. First, we shall study the corresponding eigenvalue problem, and obtain some basic properties of eigenvalues and asymptotic estimates of the eigenfunctions and approximating eigenfunctions. Secondly, we consider the extremal functions of the best embedding constant, and get some crucial estimates for the cut-off function of the extremal functions. Thirdly, applying different variational theorems for distinct cases of those parameters appearing in the equation, we obtain two existence results for non-trivial solutions to semi-linear Brezis–Nirenberg type problems. Our existence results are divided into non-resonant and resonant cases.