A Choquard type equation with a singular absorption nonlinearity in two dimension

Abstract

In this article, we show the existence of a nonnegative solution to the singular problem $(\mc P_\la)$ posed in a bounded domain $\Omega$ in $\mb R^2$ (see below). We achieve this by approximating the singular function $u^{-\beta}\log(u)$ by a function $l_\e(u)$ which pointwisely converges to $-u^\beta\log(u)$ as $\e \ra 0$. Using variational techniques, the perturbed equation $-\De u+l_\e(u)=\ds\la \left(\I{\Om}\frac{F(u(y))}{|x-y|^\mu}dy\right)f(u(x))$ is shown to have a solution $u_\e \in H_0^{1}(\Om)$ when the parameter $\la >0$ is small enough. Letting $\e \ra 0$ and proving a pointwise gradient estimate, we show that the solution $u_\e$ converges to a nontrivial nonnegative solution of the original problem $(\mc P_\la)$.