Stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain - Jiafa Xu; Zhongli Wei; Youzheng Ding (Electronic Journal of Qualitative Theory of Differential Equations, 2012)
In this work, we are mainly concerned with the existence of stationary solutions for the generalized Kadomtsev-Petviashvili equation in bounded domain in $\mathbb{R}^n$ \[\left\{\begin{aligned} &\frac{\partial^3}{\partial x^3}u(x,y)+\frac{\partial}{\partial x}f(u(x,y))=D_x^{-1}\Delta_yu(x,y),\ \text{in}\ \Omega,\\ &D_x^{-1}u|_{\partial\Omega}=0,\ u|_{\partial\Omega}=0, \end{aligned}\right.\] where $\Omega\in \mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$. We utilize critical point theory to establish our main results.