Positive Solution for a Class of Degenerate Quasilinear Elliptic Equations in R-N

Abstract

We establish a result on the existence of a positive solution for the following class of degenerate quasilinear elliptic problems:(P) {-Delta(up)u + V(x)vertical bar x vertical bar-(ap+)vertical bar u vertical bar(p-2)u = K(x)f(x, u), in R-N, u > 0, in R-N, u epsilon D-u(1,p)(R-N),where -Delta(ap)u = -div(vertical bar x vertical bar(-ap)vertical bar del u vertical bar(p-2)del u), 1 < p < N, -infinity < a < N-p/p, a <= e <= a + 1, d = 1 + a - e, and p* := p*(a, e) = Np/N-dp denotes the Hardy-Sobolev's , and denotes the Hardy-Sobolev's critical exponent, V and K are bounded nonnegative continuous potentials, K vanishes at infinity, and f has a subcritical growth at infinity. The technique used here is the variational approach.