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

Abstract

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

\({\Delta _\gamma }u + f\left( {x,u} \right) = 0in\Omega ,u = 0on\partial \Omega ,\)

where Ω is a bounded domain with smooth boundary in R^N (N ≥ 2) and Δ_γ is the subelliptic operator of the type

\({\Delta _\gamma }u = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}u} \right)} ,{\partial _{{x_j}}}u = \frac{{\partial u}}{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \ldots ,{\gamma _N}} \right).\)

We use the three critical point theorem.