Nonnegative solutions of an indefinite sublinear Robin problem II: local and global exactness results

Abstract

We proceed further in the investigation of the Robin problem

$$({\text{P}_\alpha })\;\;\;\;\;\left\{ {\begin{array}{*{20}{c}} { - \vartriangle u = a(x){u^q}}&{\text{in}}&\Omega \\ {u \geqslant 0\;\;\;\;\;\;\;\;}&{\text{in}}&\Omega \\ {{\partial _v}u = \alpha u\;\;\;\;}&{\text{on}}&{\partial \Omega }, \end{array}} \right.$$

on a smooth bounded domain Ω ⊂ ℝ^N, with a sign-changing and 0 < q < 1. Assuming the existence of a positive solution for α = 0 (which holds if q is close enough to 1), we sharpen the description of the nontrivial solution set of (P_α)for α > 0. Moreover, strengthening the assumptions on a and q we provide a global (i.e., for every α > 0) exactness result on the number of solutions of (P_α). Our approach also applies to the problem

$$({\text{S}_\alpha })\;\;\;\;\;\left\{ {\begin{array}{*{20}{c}} { - \vartriangle u = \alpha u + a(x){u^q}}&{\text{in}}&\Omega \\ {u \geqslant 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{\text{in}}&\Omega \\ {{\partial _v}u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{\text{on}}&{\partial \Omega .} \end{array}} \right.$$