Zheng Xie, Jing Chen
Abstract:
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.
Submitted May 4, 2023. Published July 17, 2023.
Math Subject Classifications: 35A15, 35A18, 35Q53, 58E05, 76B25.
Key Words: Kadomtsev-Petviashvili equation; variational methods;
penalization techniques; Ljusternik-Schnirelmann theory.
DOI: https://doi.org/10.58997/ejde.2023.48
Show me the PDF file (403 KB), TEX file for this article.
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Zheng Xie School of Mathematics and Computer science Hunan University of Science and Technology Xiangtan, 411201 Hunan, China email: xz@mail.hnust.edu.cn |
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Jing Chen School of Mathematics and Computer science Hunan University of Science and Technology Xiangtan, 411201 Hunan, China email: cjhnust@hnust.edu.cn |
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