Existence and Non-existence of Solutions for Semilinear bi-Δγ-Laplace Equation

In this paper, we study existence and non-existence of weak solutions for semilinear bi-Delta(gamma)-Laplace equation Delta(2)(gamma)u = f (x, u) in Omega, u = partial derivative(v)u = 0 on partial derivative Omega, where Omega is a bounded domain with smooth boundary in R-N (N >= 2), f (x, xi) is a Caratheodory function 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)), Delta(2)(gamma) := Delta(gamma) (Delta(gamma)).