UNIQUENESS OF SOLUTIONS FOR A NONLOCAL ELLIPTIC EIGENVALUE PROBLEM

We examine equations of the form {(-Delta)(1/2) u = lambda g(x) f(u) in Omega u = 0 on partial derivative Omega, where lambda > 0 is a parameter and Omega is a smooth bounded domain in R-N, N >= 2. Here g is a positive function and f is an increasing, convex function with f(0) = 1 and either f blows up at 1 or f is superlinear at infinity. We show that the extremal solution u* associated with the extremal parameter lambda* is the unique solution. We also show that when f is suitably supercritical and Omega satisfies certain geometrical conditions then there is a unique solution for small positive. lambda