On some Critical Problems for the Fractional Laplacian Operator

Details On some Critical Problems for the Fractional Laplacian Operator On some Critical Problems for the Fractional Laplacian Operator

2011-06-29

arxiv.org/abs/1106.6081v2

Mathematics

Analysis of PDEs

Scientific paper

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, $N\ge1$, $\lambda>0$, $00$, 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.

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