Multiple solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations

Abstract

In this paper, we study the existence of multiple solutions for the boundary value problem

$$\begin{aligned} \begin{array}{llll} -\Delta _{\gamma } u&{}= f(x,u) + g(x,u) &{} \text{ in } &{} \Omega , \\ u&{}= 0 &{} \text{ on } &{} \partial \Omega , \end{array} \end{aligned}$$

where $\Omega$ is a bounded domain with smooth boundary in $\mathbb {R}^N \ (N \ge 2),$ $f(x,\xi ), g(x,\xi )$ are Carathéodory functions, $f(x,\xi )$ is odd in $\xi$, $g(x,\xi )$ is perturbation term and $\Delta _{\gamma }$ is the strongly degenerate elliptic operator of the type

$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma : = (\gamma _1, \gamma _2,\ldots , \gamma _N). \end{aligned}$$

We use the minimax method and Rabinowitz’s perturbation method. This result is a generalization of that of Luyen and Tri (Complex Var Elliptic Equ 64(6):1050–1066, 2019).