A local Palais-Smale condition and existence of solitary waves for a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

Lirong Huang; Lirong Huang

doi:10.3934/math.2023725

AIMS Mathematics

2023, Volume 8, Issue 6: 14180-14187. doi: 10.3934/math.2023725

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Research article

A local Palais-Smale condition and existence of solitary waves for a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

Lirong Huang ^,

Department of Mathematics and Physics, Fujian Jiangxia University, Fuzhou 350108, China

Received: 10 February 2023 Revised: 27 March 2023 Accepted: 03 April 2023 Published: 17 April 2023
MSC : 35J05, 35J20

This paper is concerned with a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

$ \bigg\{ \begin{array}{rl} & u_t + (|u|^{p-2}u)_x + u_{xxx} +h_x(x-\tau t, y) +\beta \nabla_y v = 0, \\ & v_x = \nabla_y u.\end{array} $

By proving a local Palais-Smale condition, we manage to prove the existence of solitary waves with the help of a variational characterization on the smallest positive constant of an anisotropic Sobolev inequality (Huang and Rocha, J. Inequal. Appl., 2018,163). The novelty is to give an explicit estimate on the sufficient condition of $ h $ to get the existence of solitary waves.
- nonhomogeneous generalized Kadomtsev-Petviashvili equation,
- solitary wave
Citation: Lirong Huang. A local Palais-Smale condition and existence of solitary waves for a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations[J]. AIMS Mathematics, 2023, 8(6): 14180-14187. doi: 10.3934/math.2023725

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Abstract

This paper is concerned with a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

$ \bigg\{ \begin{array}{rl} & u_t + (|u|^{p-2}u)_x + u_{xxx} +h_x(x-\tau t, y) +\beta \nabla_y v = 0, \\ & v_x = \nabla_y u.\end{array} $

By proving a local Palais-Smale condition, we manage to prove the existence of solitary waves with the help of a variational characterization on the smallest positive constant of an anisotropic Sobolev inequality (Huang and Rocha, J. Inequal. Appl., 2018,163). The novelty is to give an explicit estimate on the sufficient condition of $ h $ to get the existence of solitary waves.

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