Positive Solutions for Nonlinear Robin Problems with Concave Terms

We consider a parametric Robin problem driven by the $p$-Laplacian plus a potential. In the reaction we have the combined effects of a parametric concave term and of a $(p \!-\! 1)$-linear perturbation. We consider the case of uniform nonresonance with respect to the principal eigenvalue $\widehat{\lambda}_1>0$ and the case of nonuniform nonresonance with respect to $\widehat{\lambda}_1>0$. For both cases we prove a bifurcation-type theorem describing the dependence on the parameter $\lambda>0$ of the set of positive solutions. We also establish the existence of a smallest positive solution $\widehat{u}^*_{\lambda}$ for every admissible parameter $\lambda>0$ and determine the monotonicity and continuity properties of the map $\lambda\longmapsto\widehat{u}_{\lambda}^*$.