Nonnegative Solutions of an Indefinite Sublinear Robin Problem I: Positivity, Exact Multiplicity, and Existence of a Subcontinuum - Ramos-Quoirin, H (Springer, 2020)
Let Ω ⊂ RN (N≥ 1) be a smooth bounded domain, a∈ C(Ω ¯) a sign-changing function, and 0 ≤ q< 1. We investigate the Robin problem [Figure not available: see fulltext.]where 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 in a subcontinuum. These results partially rely on (and extend) our previous work (Kaufmann et al. in J Differ Equ 263:4481–4502, 2017), 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. © 2020, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.