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

Abstract

We go further in the investigation of the Robin problem $(P_{\alpha })$: $-\Delta u=a(x)u^q$ in $\Omega$, $u\ge 0$ in $\Omega$, ${\partial }_{\nu }u=\alpha u$ on $\partial \Omega$; on a bounded domain $\Omega \subset {\mathbb{R}}^N$, with $a$ sign-changing and $0<q<1$. Assuming the existence of a positive solution for $\alpha =0$ (which holds if $q$ is close enough to 1), we sharpen the description of the nontrivial solution set of $(P_{\alpha })$ for $\alpha >0$. Moreover, strengthening the assumptions on $a$ and $q$ we provide a global (i.e. for every $\alpha >0$) exactness result on the number of solutions of $(P_{\alpha })$ . Our approach also applies to the problem $(S_{\alpha })$: $-\Delta u=\alpha u+a(x)u^q$ in $\Omega$, $u\ge 0$ in $\Omega$, ${\partial }_{\nu }u=0$ on $\partial \Omega$.