Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities

Let B be the unit ball in R-n, n greater than or equal to 3. Let 0 <p < 1 < q less than or equal to (n + 2)/(n - 2). In 1994, Ambrosetti et al. found that the semilinear elliptic Dirichlet problem -Deltau = lambdau(p) + u(q) in B, u > 0 in B, u = 0 on partial derivativeB, admits at least two solutions for small lambda > 0 and no solution for large lambda. In this paper, we prove thta there is a critical number Lambda > 0 such that this problem has exactly two solutions for lambda is an element of (0,Lambda), exactly one solution for lambda = Lambda and no solution for lambda > Lambda.