Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity

TL;DR

This paper investigates the existence of multiple solutions for a fractional elliptic problem involving asymmetric critical Choquard nonlinearity, using variational methods like Mountain pass and Linking theorems.

Contribution

It introduces three solution approaches for a fractional Choquard problem with asymmetric nonlinearity, extending variational techniques to this class of problems.

Findings

At least three nontrivial solutions are established.

The problem involves critical Hardy-Littlewood-Sobolev exponent.

Variational methods successfully applied to fractional Choquard equations.

Abstract

In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{equation*} \begin{array}{rl} (-\Delta)^s u &= \displaystyle-\lambda|u|^{q-2}u + au + b\left( \int\limits_{\Omega} \frac{(u^{+}(y))^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy\right) (u^{+})^{2^{*}_{\mu ,s}-2}u \quad\text{in} \; \Omega, u &= 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega, \end{array} \end{equation*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. Here $(-\Delta)^s$ is the fractional Laplace operator, $\lambda > 0$ is a real parameter, $q \in (1, 2)$, $a > 0$ and $b> 0$ are given constants, and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation $u^{+} = \max \{u, 0\}$. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.