On Stable Solutions to Weighted Quasilinear Problems of Gelfand Type

Let p >= 2 and w, f is an element of L-loc(1)(R-N ) be nonnegative functions such that w(x) <= C-l vertical bar x vertical bar(a); and f (x) >= C-2 vertical bar x vertical bar(b) for large vertical bar x vertical bar. We prove the Liouville type theorem for stable W-loc(1,p) solutions of weighted quasilinear problem -div(w(x)vertical bar del(u)vertical bar(P-2) del u) = f (x)e(u) in R-N . The result holds true for N < (p - a) (p + 3) + 4b /p - 1 and is sharp in the case that w and f are Hardy-Henon potentials. We also prove the full classification of solutions which are stable outside a compact set to Gelfand equation -Delta N-u = e(u )in R-N .