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

We go further in the investigation of the Robin problem $(P_α)$: $-Δu=a(x)u^{q}$ in $Ω$, $u\geq0$ in $Ω$, $\partial_νu=αu$ on $\partial Ω$; on a bounded domain $Ω\subset\mathbb{R}^{N}$, 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 $(S_α)$: $-Δu=αu + a(x)u^{q}$ in $Ω$, $u\geq0$ in $Ω$, $\partial_νu=0$ on $\partial Ω$.

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