Two nontrivial solutions of boundary-value problems for semilinear Δγ-differential equations

In this paper, we study the existence of multiple solutions for the boundary-value problem Δγu+f(x,u)=0inΩ,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}$${\Delta _\gamma }u + f\left( {x,u} \right) = 0in\Omega ,u = 0on\partial \Omega ,$$\end{document} where Ω is a bounded domain with smooth boundary in RN (N ≥ 2) and Δγ is the subelliptic operator of the type Δγu=∑j=1N∂xj(γj2∂xju),∂xju=∂u∂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}$${\Delta _\gamma }u = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}u} \right)} ,{\partial _{{x_j}}}u = \frac{{\partial u}}{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \ldots ,{\gamma _N}} \right).$$\end{document} We use the three critical point theorem.