Positive Solutions of an Elliptic Neumann Problem with a Sublinear Indefinite Nonlinearity

Let Ω ⊂ RN (N≥ 1) be a bounded and smooth domain and a: Ω → R be a sign-changing weight satisfying ∫ ΩaOpenSPiltSPi 0. We prove the existence of a positive solution uq for the problem [Equation not available: see fulltext.]if q0OpenSPiltSPi qOpenSPiltSPi 1 , for some q0= q0(a) CloseSPigtSPi 0. In doing so, we improve the existence result previously established in Kaufmann et al. (J Differ Equ 263:4481–4502, 2017). 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 sub-supersolution method. © 2018, Springer International Publishing AG, part of Springer Nature.