On some critical problems for the fractional Laplacian operator

DOI10.1016/J.JDE.2012.02.023zbMATH Open1245.35034arXiv1106.6081OpenAlexW1976659984MaRDI QIDQ424439

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Publication date: 1 June 2012

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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 partialOmega,${\displaystyle }$ where $Omegasubsetmathbb{R}^N$ is a smooth bounded domain, $Nge1$, $lambda>0$, $0<q<fracN+alphaN-alpha$, $0<alpha<minN,2$. 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 $0<lambda<lambd{a}_1$ and nonexistence for $lambdagelambd{a}_1$. When $q>1$ the existence is shown for every $lambda>0$. Also we prove that the solutions are bounded and regular.

Full work available at URL: https://arxiv.org/abs/1106.6081

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