On some Critical Problems for the Fractional Laplacian Operator

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.