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.