UNIQUENESS AND NONDEGENERACY OF SIGN-CHANGING RADIAL SOLUTIONS TO AN ALMOST CRITICAL ELLIPTIC PROBLEM

We study uniqueness of sign-changing radial solutions for the following semi-linear elliptic equation Delta u - u + vertical bar u vertical bar(p-1) u = 0 in R-N, u is an element of H-1 (R-N), where 1 < p < N+2/N -2, N >= 3. It is well-known that this equation has a unique positive radial solution. The existence of sign -changing radial solutions with exactly k nodes is also known. However, the uniqueness of such solutions is open. In this paper, we show that such sign -changing radial solution is unique when p is close to ii,+q. Moreover, those solutions are non -degenerate, i.e., the kernel of the linearized operator is exactly N -dimensional.