Existence of solutions to nonlocal and singular elliptic problems via Galerkin method

We study the existence of solutions to the nonlocal elliptic equation $$ -M(|u|^2)Delta u = f(x,u) $$ with zero Dirichlet boundary conditions on a bounded and smooth domain of $mathbb{R}^n$. We consider the $M$-linear case with $fin H^{-1}(Omega )$, and the sub-linear case $f(u)=u^{alpha}$, $0 Keywords Nonlocal elliptic problems --- Galerkin Method.