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

In this paper, we study the existence of multiple solutions for the boundary value problem -Δγu=f(x,u)+g(x,u)inΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{llll} -\Delta _{\gamma } u&{}= f(x,u) + g(x,u) &{} \text{ in } &{} \Omega , \\ u&{}= 0 &{} \text{ on } &{} \partial \Omega , \end{array} \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} is a bounded domain with smooth boundary in RN(N≥2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^N \ (N \ge 2),$$\end{document}f(x,ξ),g(x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x,\xi ), g(x,\xi )$$\end{document} are Carathéodory functions, f(x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x,\xi )$$\end{document} is odd in ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi$$\end{document}, g(x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x,\xi )$$\end{document} is perturbation term and Δγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{\gamma }$$\end{document} is the strongly degenerate elliptic operator of the type Δγ:=∑j=1N∂xjγj2∂xj,∂xj:=∂∂xj,γ:=(γ1,γ2,…,γN).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma : = (\gamma _1, \gamma _2,\ldots , \gamma _N). \end{aligned}$$\end{document}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).