A Semilinear Elliptic Problem with Critical Exponent and Potential Terms

TL;DR

This paper investigates the existence, non-existence, and multiplicity of solutions for a semilinear elliptic PDE with critical Sobolev exponent and nonlocal potential terms, extending classical methods to nonlocal eigenvalue problems.

Contribution

It introduces new existence and multiplicity results for a class of nonlocal elliptic equations with critical exponents, combining Brezis-Nirenberg techniques with potential regularity analysis.

Findings

Established conditions for existence and non-existence of solutions.

Proved multiplicity results under certain parameter regimes.

Analyzed a nonlocal eigenvalue problem related to the main PDE.

Abstract

This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega ,\nonumber u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation} Here, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $N\geq3$, $2^*=\frac{2N}{N-2}$, $\lambda\in\mathbb{R}$, $\lambda\in(0,N)$, $I_\alpha$ is the Riesz potential and \begin{align} I_\alpha*_\Omega u(x):=\int_\Omega \frac{\Gamma(\frac{N-\alpha}{2})}{\Gamma(\frac{\alpha}{2})\pi^\frac{N}{2}2^\alpha|x-y|^{N-\alpha}} u(y)dy. \nonumber \end{align} We study the non-existence, existence and multiplicity results. Our argument combines Brezis-Nirenberg's method with the regularity results involving potential terms. Especially, we study the following nonlocal eigenvalue problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u\mbox{ in }\Omega ,\nonumber \lambda\in\mathbb{R},\,u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation}