Nontrivial solutions to boundary value problems for semilinear Δγ-differential equations

In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: −Δγu=f(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}$$ - {\Delta _\gamma }u = f(x,u)\;{\rm{in}}\;\Omega ,\;\;\;\;\;u = 0\;{\rm{on}}\;\partial \Omega ,$$\end{document} where Ω is a bounded domain with smooth boundary in ℝN,Ω∩{xj=0}≠∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^N},\;\Omega \cap \left\{ {{x_j} = 0} \right\} \ne \emptyset $$\end{document} for some j, Δγ is a subelliptic linear operator of the type Δγ:=∑j=1N∂xj(γj2∂xj),∂xj:=∂∂xj,N≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _\gamma }: = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}(\gamma _j^2{\partial _{{x_j}}}),\;\;\;\;\;{\partial _{{x_j}}}: = {\partial \over {\partial {x_j}}}} ,\;\;\;\;\;N \ge 2,$$\end{document} where γ(x) = (γ1(x), γ2(x), …, γN(x)) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f(x, ξ) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.