Some existence and uniqueness results for logistic Choquard equation

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions for a nonlocal fractional p-Laplacian logistic Choquard equation, including subdiffusive and superdiffusive cases, with applications to Brezis-Nirenberg type problems.

Contribution

It provides new existence and uniqueness results for a class of doubly nonlocal logistic equations involving fractional p-Laplacian and Choquard terms, extending previous work to broader nonlinearities and boundary conditions.

Findings

Proved existence and nonexistence results under general conditions.

Established regularity of weak solutions.

Identified conditions for multiple solutions, including nodal solutions.

Abstract

We consider the following doubly nonlocal nonlinear logistic problem driven by the fractional $p$-Laplacian \begin{equation*} \pl u = f(x,u) -\cq ~\text{in}~ \O, ~u=0 ~\text{in}~ \Rn\setminus\O. \end{equation*} Here $ \O \subset \Rn (N\geq2)$ is a bounded domain with $ C^{1,1}$ boundary $\partial \O$, $ s \in (0,1) $, $p \in (1,\infty)$ are such that $ps < N$. Also $p_{s,\a}^\#\leq r<\infty$ , where $p_{s,\a}^\#=(2N-\a)/2N$. Under suitable and general assumptions on the nonlinearity $f$, we study the existence, nonexistence, uniqueness, and regularity of weak solutions. As for applications, we treat cases of subdiffusive type logistic Choquard problem. We also consider in the superdiffusive case the Brezis-Nirenberg type problem with logistic Choquard and show the existence of a nontrivial solution for a suitable choice of $\l$. Finally for a particular choice of $f$ viz. $f(x,t)=\l t^{q-1}$ with $1<p<2r<q$, we show the existence of at least one energy nodal solution.