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

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

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

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.