A critical fractional Laplace equation in the resonant case
Keywords
Critical nonlinearities, best critical Sobolev constant, variationaltechniques, Linking Theorem, integrodifferentialoperators, fractional LaplacianAbstract
In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities $$ \cases (-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\text{in }} \Omega,\\ u=0 & {\text{in }} \mathbb{R}^n\setminus \Omega, \endcases $$ started in the recent papers \cite{13}, \cite{17}-\cite{19}. Here $s\in (0,1)$ is a fixed parameter, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive constant, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\RR^n$, $n> 2s$, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and $\lambda$ is an eigenvalue of the operator $(-\Delta)^s$ with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a non-trivial solution, so that with the results obtained in \cite{13}, \cite{17}-\cite{19}, we are able to show that this critical problem admits a nontrivial solution provided \roster \item"$\bullet$" $n> 4s$ and $\lambda> 0$, \item"$\bullet$" $n=4s$ and $\lambda> 0$ is different from the eigenvalues of $(-\Delta)^s$, \item"$\bullet$" $2s< n< 4s$ and $\lambda> 0$ is sufficiently large. \endroster In this way we extend completely the famous result of Brezis and Nirenberg (see \cite{4}, \cite{5}, \cite{9}, \cite{23}) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.Downloads
Published
2016-04-12
How to Cite
1.
SERVADEI, Raffaella. A critical fractional Laplace equation in the resonant case. Topological Methods in Nonlinear Analysis. Online. 12 April 2016. Vol. 43, no. 1, pp. 251 - 267. [Accessed 29 March 2024].
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