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: (-Δ)^{α/2}u=λu^q+u^{\frac{N+α}{N-α}}, \quad u>0 &\quad in Ω, u=0&\quad on \partialΩ,$$ where $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $N\ge1$, $λ>0$, $0 0$, at least one if $λ=Λ$, no solution if $λ>Λ$. For $q=1$ we show existence of at least one solution for $0<λ<λ_1$ and nonexistence for $λ\geλ_1$. When $q>1$ the existence is shown for every $λ>0$. Also we prove that the solutions are bounded and regular.