Electron. J. Differential Equations, Vol. 2023 (2023), No. 48, pp. 117.
Multiplicity of solutions for a generalized KadomtsevPetviashvili equation with potential in R^2
Zheng Xie, Jing Chen
Abstract:
In this article, we study the generalized KadomtsevPetviashvili equation with
a potential
$$
(u_{xx}+D_{x}^{2}u_{yy}+V(\varepsilon x,\varepsilon y)uf(u))_{x}=0
\quad \text{in }\mathbb{R}^2,
$$
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 LjusternikSchnirelmann 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: KadomtsevPetviashvili equation; variational methods;
penalization techniques; LjusternikSchnirelmann theory.
DOI: https://doi.org/10.58997/ejde.2023.48
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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


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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