In this paper we complete the study of the a non-local fractional equation involving critical nonlinearities depending on a real parameter, started in some recent papers by the same authors (joint with other coauthors). Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and the parameter is an eigenvalue of the fractional Laplace opertaor with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a nontrivial solution, so that with the results obtained in some recent papers, we are able to show that this critical problem admits a nontrivial solutionunder suitable assumptions on the dimension of the space and on the parameter appearing in the equation. In this way we extend completely the famous result of Brezis and Nirenberg for the critical Laplace equation to the non-local setting of the fractional Laplace equation.
A critical fractional Laplace equation in the resonant case
SERVADEI, RAFFAELLA
2014
Abstract
In this paper we complete the study of the a non-local fractional equation involving critical nonlinearities depending on a real parameter, started in some recent papers by the same authors (joint with other coauthors). Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and the parameter is an eigenvalue of the fractional Laplace opertaor with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a nontrivial solution, so that with the results obtained in some recent papers, we are able to show that this critical problem admits a nontrivial solutionunder suitable assumptions on the dimension of the space and on the parameter appearing in the equation. In this way we extend completely the famous result of Brezis and Nirenberg for the critical Laplace equation to the non-local setting of the fractional Laplace equation.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
