On some Critical Problems for the Fractional Laplacian Operator (1106.6081v2)

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.