Affiliation:
1. School of Mathematics and Statistics & Key Laboratory of Nonlinear Analysis and Applications Central China Normal University Wuhan China
2. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences Central China Normal University Wuhan China
Abstract
AbstractIn this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem
where , , , , , is the outward unit normal vector at the boundary , is the usual critical exponent for the Sobolev embedding and is the critical exponent for the Sobolev trace embedding . By establishing an improved Pohozaev identity, we show that problem () has no nontrivial solution if . Applying the mountain pass theorem without the condition and the delicate estimates for the mountain pass level, we obtain the existence of a positive solution for all and the different values of the parameters and . Particularly, for , , , we prove that problem () has a positive solution if and only if . Moreover, the existence of multiple solutions for () is also obtained by dual variational principle for all and suitable .
Funder
National Natural Science Foundation of China