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

ISSN： 1473-7124

Source： Proceedings Section A: Mathematics - Royal Society of Edinburgh, Vol.133, Iss.3, 2003-06, pp. : 705-717

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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-Δu = λu^p + u^q in B,u > 0 in B,u = 0 on ∂B,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.

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