Multiplicity of solutions for a generalizedKadomtsev-Petviashvili equation with potential in R^2

Electronic Journal of Differential Equations

article

Zheng Xie¹  Jing Chen¹ 

[1] School of Mathematics and Computer science Hunan University of Science and Technology Xiangtan

关键词: Kadomtsev-Petviashvili equation;    variational methods;    penalization techniques;    Ljusternik-Schnirelmann theory.;   

DOI  :  10.58997/ejde.2023.48

学科分类：数学（综合）

来源: Texas State University

【 摘 要 】

In this article, we study the generalized Kadomtsev-Petviashvili equation with a potential $$(-u_{xx}+D_{x}^{-2}u_{yy}+V(\varepsilon x,\varepsilon y)u-f(u))_{x}=0 \quad \text{in }\mathbb{R}^2, $$ where \(D_{x}^{-2}h(x,y)=\int_{-\infty }^{x}\int_{-\infty }^{t}h(s,y)\,ds\,dt \), \(f\) is a nonlinearity, \(\varepsilon\) is a small positive parameter, and the potential \(V\) satisfies a local condition. We prove the existence of nontrivial solitary waves for the modified problem by applying penalization techniques. The relationship between the number of positive solutions and the topology of the set where \(V\) attains its minimum is obtained by using Ljusternik-Schnirelmann theory. With the help of Moser's iteration method, we verify that the solutions of the modified problem are indeed solutions of the original problem for \(\varepsilon>0\) small enough.

【 授权许可】

CC BY