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

TL;DR

This paper proves the existence of nonnegative solutions to a singular Choquard equation in two dimensions by approximation and variational methods, demonstrating convergence of solutions as the approximation parameter tends to zero.

Contribution

It introduces a novel approximation approach for the singular nonlinearity and establishes existence and convergence results for solutions of the Choquard equation with singular absorption.

Findings

Existence of nonnegative solutions for small parameter  in the singular Choquard problem.

Successful approximation of the singular nonlinearity with convergence of solutions.

Establishment of a pointwise gradient estimate ensuring solution convergence.

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 $(\mc P_\la)$.