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

Abstract

<abstract><p>This paper is concerned with a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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} $\end{document} </tex-math></disp-formula></p> <p>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 <bold>explicit estimate</bold> on the sufficient condition of $ h $ to get the existence of solitary waves.</p></abstract>