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 (P λ ) posed in a bounded domain Ω in R 2 (see below). We achieve this by approximating the singular function u −β log(u) by a function l ǫ (u) which pointwisely converges to −u β log(u) as ǫ → 0. Using variational techniques, the perturbed equationthe parameter λ > 0 is small enough. Letting ǫ → 0 and proving a pointwise gradient estimate, we show that the solution u ǫ converges to a nontrivial nonnegative solution of the original problem (P λ ).