Nontrivial stationary solutions to GKP equation in bounded domain

Abstract

In this article, using the Mountain Pass Lemma due to Ambrosetti and Rabinowitz, we obtain the existence of nontrivial stationary solutions of Generalized Kadomtsev–Petviashvili (GKP) equation in a bounded domain with smooth boundary and for superlinear nonlinear term f (u) which satisfies some growth condition. Based on a Pohozaev type variational identity for cylindrical symmetric solution, we obtain the nonexistence of the nontrivial cylindrical symmetric solution for super-critical nonlinearity, i.e. f(u)=|u| ^{p −2} u where .

Keywords:

• GKP equation
• Stationary solution
• Cylindrical symmetry
• Mathematics Subject Classification: 35J60

Acknowledgment

The author was supported by Grant 10071080 and 10101024 from the NNSF of China.