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

Abstract

Let B be the unit ball in Rⁿ, n ≥ 3. Let 0 < p < 1 < q ≤ (n + 2)/(n − 2). In 1994, Ambrosetti et al. found that the semilinear elliptic Dirichlet problem admits at least two solutions for small λ > 0 and no solution for large λ. In this paper, we prove that there is a critical number Λ > 0 such that this problem has exactly two solutions for λ ∈ (0, Λ), exactly one solution for λ = Λ and no solution for λ > Λ.