Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

Abstract

Let $\Omega \subset {\mathbb {R}}^{N}$ ($N\ge 1$) be a smooth bounded domain, $a\in C({\overline{\Omega }})$ a sign-changing function, and $0\le q<1$. We investigate the Robin problem

where $\alpha \in [-\infty ,\infty )$ and $\nu$ is the unit outward normal to $\partial \Omega$. Due to the lack of strong maximum principle structure, this problem may have dead core solutions. However, for a large class of weights a we recover a positivity property when q is close to 1, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of q: $(P_{\alpha })$ has exactly one nontrivial solution for $\alpha \le 0$, exactly two nontrivial solutions for $\alpha >0$ small, and no such solution for $\alpha >0$ large. Assuming some further conditions on a, we show that these solutions lie in a subcontinuum. These results partially rely on (and extend) our previous work (Kaufmann et al. in J Differ Equ 263:4481–4502, 2017), where the cases $\alpha =-\infty$ (Dirichlet) and $\alpha =0$ (Neumann) have been considered. We also obtain some results for arbitrary $q\in \left[ 0,1\right)$. Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and a priori lower and upper bounds.