On Stable Solutions Of The Weighted Lane-Emden Equation Involving Grushin Operator

In this article, we study the weighted Lane-Emden equationdiv(G)(omega(1)(z)vertical bar del(G)u vertical bar(p-2)del(G)u) = omega(2)(z)vertical bar u vertical bar(q-1)u, z= (x,y) epsilon R-N = R-N1 x R-N2;where N = N-1 + N-2 >= 2; p >= 2 and q > p - 1, while omega(i)(z) epsilon L-loc(1) (R-N) \ {0} g(i = 1; 2) are nonnegative functions satisfying omega(1)(z) <= C parallel to k parallel to(0)(G) and omega(2)(z) >= C parallel to k parallel to(d)(G) for large parallel to z parallel to(G) with d > theta p: Here alpha >= 0 and parallel to z parallel to(G) = (vertical bar x vertical bar(2(1 +alpha)) + vertical bar y vertical bar(2))(1/2(1+alpha)) : divG (resp.,del(G)) is Grushin divergence (resp., Grushin gradient). We prove that stable weak solutions to the equation must be zero under various assumptions on d; theta; p; q and N-alpha = N-1 + (1 + alpha) N-2.