Three Nontrivial Solutions of Boundary Value Problems for Semilinear Δγ-Laplace Equation

In this paper, we study the multiplicity of weak solutions to the boundary value problem Delta(gamma)u + f(x, u) = 0 in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain with smooth boundary in R-N (N >= 2) and Delta(gamma) is the subelliptic operator of the type Delta(gamma) := Sigma(N)(j=1) partial derivative(xj) (gamma(2)(j)partial derivative(xj)), partial derivative(xj) := partial derivative/partial derivative(xj), gamma = (gamma(1), gamma(2), ..., gamma(N)), the nonlinearity f(x, xi) is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.