On strictly positive solutions for some semilinear elliptic problems

Abstract

Abstract. Let B be a ball in R N , N ≥ 1, let m be a possibly discontinuous and unbounded function that changes sign in B and let 0 < p < 1. We study existence and nonexistence of strictly positive solutions for semilinear elliptic problems of the form −Δu = m (x) u p in B, u = 0 on ∂B. Mathematics Subject Classification (2000). 35J25, 35J61, 35B09, 35J65.

A curve of positive solutions for an indefinite sublinear Dirichlet problem

“…It was first proved in [13,Theorem 4.4] (see also [14,Theorem 3.7]) that if S(a) ∈ P • then (P a,q ) has a positive solution (which may not belong to P • ). We note that this condition is not sharp, since there exists a such that S(a) < 0 in Ω but (P a,q ) has a positive solution for some q ∈ (0, 1) (see [15,Section 1]). Later on, in [15], the authors studied (P a,q ) in the one-dimensional and radial cases, establishing several sufficient conditions on a (as well as some necessary ones) for the existence of a positive solution of (P a,q ).…”

“…We note that this condition is not sharp, since there exists a such that S(a) < 0 in Ω but (P a,q ) has a positive solution for some q ∈ (0, 1) (see [15,Section 1]). Later on, in [15], the authors studied (P a,q ) in the one-dimensional and radial cases, establishing several sufficient conditions on a (as well as some necessary ones) for the existence of a positive solution of (P a,q ). Some of these results were then extended to the case of a smooth bounded domain in [17].…”

On Dirichlet problems with singular nonlinearity of indefinite sign

“…[15], [7], [18]), but to our knowledge no theorems are known when m is allowed to change sign in Ω. In fact, even when (1.2) is sublinear (that is, γ ∈ (−1, 0)) and one-dimensional, these kind of problems are quite intriguing and involved and, as far as we know, only recently existence of (strictly) positive solutions has started being studied in detail when m changes sign in Ω (see [13], [16], [14] and [10]; and [17] for the p-laplacian). Let us also mention that nonnegative solutions of these semilinear problems have been studied carefully in [2].…”

Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

“…The investigation of (P α ) in the sublinear case has been carried out mostly for α = −∞ [5,9,12,14,15,16,17,19,27] and α = 0 [1,6,12,17,18]. To recall these results, we consider the conditions We also introduce the positivity set A α (a) := {q ∈ (0, 1) : any nontrivial solution of (P α ) lies in P • }.…”