On some Critical Problems for the Fractional Laplacian Operator

TL;DR

This paper investigates the existence and nonexistence of positive solutions to a fractional Laplacian elliptic problem with lower order perturbations, revealing conditions under which solutions exist, are bounded, or do not exist.

Contribution

It provides new existence and nonexistence results for positive solutions to a fractional Laplacian problem with lower order terms, depending on parameters and the nonlinearity exponent.

Findings

Multiple solutions for sublinear case when q<1

Existence of solutions for q=1 under certain lambda

Solutions are bounded and regular

Abstract

We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: (-\Delta)^{\alpha/2}u=\lambda u^q+u^{\frac{N+\alpha}{N-\alpha}}, \quad u>0 &\quad in \Omega, u=0&\quad on \partial\Omega,$$ where $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, $N\ge1$, $\lambda>0$, $0<q<\frac{N+\alpha}{N-\alpha}$, $0<\alpha<\min\{N,2\}$. For suitable conditions on $\alpha$ depending on $q$, we prove: In the case $q<1$, there exist at least two solutions for every $0<\lambda<\Lambda$ and some $\Lambda>0$, at least one if $\lambda=\Lambda$, no solution if $\lambda>\Lambda$. For $q=1$ we show existence of at least one solution for $0<\lambda<\lambda_1$ and nonexistence for $\lambda\ge\lambda_1$. When $q>1$ the existence is shown for every $\lambda>0$. Also we prove that the solutions are bounded and regular.