Positive Solutions for Nonlinear Robin Problems with Concave Terms

We consider a parametric Robin problem driven by the

-Laplacian plus a potential. In the reaction we have the combined effects of a parametric concave term and of a

(p \!-\! 1) ( p  ⁣ −  ⁣ 1 )

-linear perturbation. We consider the case of uniform nonresonance with respect to the principal eigenvalue

\widehat{\lambda}_1>0 λ ^ 1 > 0

and the case of nonuniform nonresonance with respect to

\widehat{\lambda}_1>0 λ ^ 1 > 0

. For both cases we prove a bifurcation-type theorem describing the dependence on the parameter

\lambda>0 λ > 0

of the set of positive solutions. We also establish the existence of a smallest positive solution

\widetilde{u}^*_{\lambda} u ~ λ ∗

for every admissible parameter

\lambda>0 λ > 0

and determine the monotonicity and continuity properties of the map

\lambda\longmapsto\widetilde{u}_{\lambda}^* λ ⟼ u ~ λ ∗

.