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: -Delta(gamma)u = f(x, u) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain with smooth boundary in R-N, Omega boolean AND {x(j) = 0} not equal phi for some j, Delta(gamma) is a subelliptic linear 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), N >= 2, where gamma(x) = (gamma(1)(x), gamma(2)(x), ... , gamma(N)(x)) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f(x, xi) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.