LIOUVILLE TYPE THEOREM FOR STABLE SOLUTIONS TO ELLIPTIC EQUATIONS INVOLVING THE GRUSHIN OPERATOR

Tóm tắt

We study a Liouville type theorem for stable solutions of the following
semilinear equation involving Grushin operators -(∆xu + a2|x|2α∆yu) =
|u|p-1u, (x, y) ∈ RN = RN1×RN2, where p > 1, α > 0 and a 6= 0. Basing on the
technique of Farina [1], we establish the nonexistence of nontrivial stable solutions
under the range p < pc(Nα) where Nα = N1 +(1+α)N2, and pc(Nα) is a certain
(explicitly given) positive constant depending on Nα.