LIOUVILLE THEOREMS FOR STABLE WEAK SOLUTIONS OF ELLIPTIC PROBLEMS INVOLVING GRUSHIN OPERATOR

We consider the boundary value problem {-div(G )(w(1)del(G)u) = w(2)f(u) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded or unbounded C-1 domain of R-N, w(1),w(2) is an element of L-loc(1)(Omega) \ {0} are nonnegative functions, f is an increasing function, del G and div(G) are Grushin gradient and Grushin divergence, respectively. We prove some Liouville theorems for stable weak solutions of the problem under suitable assumptions on Omega, w(1), w(2) and f. We also show the sharpness of our results when Q = R-N and f has power or exponential growth.