Existence and multiplicity of nonnegative solutions for semilinear elliptic problems involving nonlinearities indefinite in sign

Let r, s is an element of]1, 2[ and lambda, mu is an element of]0,+infinity[. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 0 boundary data for the semilinear elliptic equation -Delta u = lambda u(s-1) - u(r-1) in Omega subset of R-N, where N >= 2. We prove that there exists a positive constant. such that the above problem has at least two solutions, at least one solution or no solution according to whether lambda > Lambda, lambda = Lambda or lambda < Lambda. In particular, a result by Hernandez, Macebo and Vega is improved and, for the semilinear case, a result by Diaz and Hernandez is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given. (C) 2012 Elsevier Ltd. All rights reserved.