Existence of solutions for a class of degenerate quasilinear elliptic equation in R-N with vanishing potentials

Abstract

We establish the existence of positive solution for the following class of degenerate quasilinear elliptic problem(P) {-Lu-ap + V(x)vertical bar x vertical bar(-ap*)vertical bar u vertical bar(p-2)u = f(u) in R-N, u > 0 in R-N; u is an element of D-a(1,p) (R-N)where -Lu-ap = -div(vertical bar x vertical bar(-ap)vertical bar del vertical bar(p-2)del u), 1 < N, -infinity < a < N-p/p, a <= e <= a + 1, d = 1 + a - e, and p* = p* (a, e) = Np/N-dp denote the Hardy-Sobolev's critical exponent, V is a bounded nonnegative vanishing potential and f has a subcritical growth at infinity. The technique used here is a truncation argument together with the variational approach.