Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity

Let Omega subset of R-N (N >= 1) be a bounded and smooth domain and a : Omega -> R be a sign-changing weight satisfying integral(Omega) a < 0. We prove the existence of a positive solution u(q) for the problem (P-a,P-q) {-Delta u = a(x)u(q) in Omega, partial derivative u/partial derivative nu = 0 on partial derivative Omega, if q(0) < q < 1, for some q(0) = q(0)(a) > 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 Omega is a ball and a is radial, we give some explicit conditions on q and a ensuring the existence of a positive solution of (P-a,P-q). We also obtain some properties of the set of q's such that (P-a,P-q) admits a solution which is positive on (Omega) over bar. Finally, we present some results on non-negative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method.