A CRITICAL FRACTIONAL LAPLACE EQUATION IN THE RESONANT CASE

作者：Servadei Raffaella^*

来源：Topological Methods in Nonlinear Analysis, 2014, 43(1): 251-267.

• Critical nonlinearities
• best critical Sobolev constant
• variationaltechniques
• Linking Theorem
• integrodifferentialoperators
• fractional Laplacian

In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities
{(-Delta)(s)u - lambda u = vertical bar u vertical bar(2*-2)u in Omega, u = 0 in R-n \ Omega,
started in the recent papers [13], [17]-[19]. Here s is an element of (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 R-n, n > 2s, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when n not equal 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 [13], [17]-[19], we are able to show that this critical problem admits a nontrivial solution provided
n > 4s and lambda > 0,
n = 4s and lambda > 0 is different from the eigenvalues of (-Delta)(s),
2s < n < 4s and lambda > 0 is sufficiently large.
In this way we extend completely the famous result of Brezis and Nirenberg (see [4], [5], [9], [23]) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.

• 出版日期2014-3