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

We go further in the investigation of the Robin problem $(P_{\alpha})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $u\geq0$ 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 $\Omega$, $u\geq0$ in $\Omega$, $\partial_{\nu}u=0$ on $\partial \Omega$.