No abstract

We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum principle does not apply to. Our approach is based on a continuity argument combined with variational techniques, the sub and supersolutions method and some a priori bounds. Both Dirichlet and Neumann homogeneous boundary conditions are considered. As a byproduct, we deduce some existence and uniqueness results. Finally, as an application, we derive some positivity results for indefinite concaveconvex type problems.

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.

Let Ω ⊂ R N (N ≥ 1) be a bounded and smooth domain and a : Ω → R be a sign-changing weight satisfying Ω a < 0. We prove the existence of a positive solution uq for the problemIn doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of uq as q → 1 − . When Ω is a ball and a is radial, we give some explicit conditions on q and a ensuring the existence of a positive solution of (Pa,q). We also obtain some properties of the set of q's such that (Pa,q) admits a solution which is positive on Ω. Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the subsupersolution method. Several methods and results apply as well to the Dirichlet counterpart of (Pa,q). * 2010 Mathematics Subject Classification. 35J25, 35J61. †

Let Ω ⊂ R N (N ≥ 1) be a smooth bounded domain, a ∈ C(Ω) a sign-changing function, and 0 ≤ q < 1. We investigate the Robin problemwhere α ∈ [−∞, ∞) and ν is the unit outward normal to ∂Ω. Due to the lack of strong maximum principle structure, this problem may have dead core solutions. However, for a large class of weights a we recover a positivity property when q is close to 1, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of q: (Pα) has exactly one nontrivial solution for α ≤ 0, exactly two nontrivial solutions for α > 0 small, and no such solution for α > 0 large. Assuming some further conditions on a, we show that these solutions lie on a subcontinuum. These results rely partially on (and extend) our previous work [17], where the cases α = −∞ (Dirichlet) and α = 0 (Neumann) have been considered. We also obtain some results for arbitrary q ∈ [0, 1). Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds. * 2010 Mathematics Subject Classification. 35J15, 35J25, 35J61. †