Uniqueness of standing waves for nonlinear Schrodinger equations

For m > 0 and p is an element of (1, (N + 2)/(N - 2)), we show the uniqueness and a linearized non-degeneracy of solutions for the following problem: Delta u - vertical bar x vertical bar(m)u + u(p) = 0, u > 0 in R(N) and vertical bar x vertical bar-->infinity lim u(x) = 0.