Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation

Abstract

In this paper, we study the following generalized Kadomtsev-Petviashvili equation ut+uxxx+(h(u))x=Dx−1Δyu,{u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where (t,x,y)∈R+×R×RN−1\left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1}, N≥2N\ge 2, Dx−1f(x,y)=∫−∞xf(s,y)ds{D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s, ft=∂f∂t{f}_{t}=\frac{\partial f}{\partial t}, fx=∂f∂x{f}_{x}=\frac{\partial f}{\partial x} and Δy=∑i=1N−1∂2∂yi2{\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}}. We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in RN{{\mathbb{R}}}^{N}.

Keywords

• generalized kadomtsev-petviashvili equation
• ground state solutions
• multiplicity of solutions
• 35j60
• 35j20