Positive solutions for the Robin $p$-Laplacian plus an indefinite potential

Abstract

We consider a nonlinear elliptic equation driven by the Robin $p$-Laplacian plus an indefinite potential. In the reaction we have the competing effects of a strictly $(p-1)$-sublinear parametric term and of a $(p-1)$-linear and nonuniformly nonresonant term. We study the set of positive solutions as the parameter $\lambda >0$ varies. We prove a bifurcation-type result for large values of the positive parameter $\lambda$. Also, we show that for all admissible $\lambda >0$, the problem has a smallest positive solution ${\overline{u}}_{\lambda }$ and we study the monotonicity and continuity properties of the map $\lambda \mapsto {\overline{u}}_{\lambda }$.