Existence of Infinitely Many Solutions for Δγ-Laplace Problems

Abstract

In this article, we study the existence of infinitelymany solutions for the boundary–value problem

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

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

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

. Our main tools are the local linking and the symmetric mountain pass theorem in critical point theory.