Existence and multiplicity of positive solutions for parametric nonlinear nonhomogeneous singular Robin problems

Abstract

We consider nonlinear Robin problems driven by a nonhomogeneous differential operator and with a reaction that has a singular term and a parametric $$(p-1)$$-superlinear perturbation which need not satisfy the Ambrosetti–Rabinowitz condition. We are looking for positive solutions. Using variational arguments and a suitable truncation and comparison techniques, we prove a bifurcation-type theorem which describes the set of positive solutions as the parameter $$\lambda > 0$$ varies. Also we show the for every admissible value of the parameter $$\lambda >0$$, the problem has a smallest solution $${\bar{u}}_{\lambda }$$ and we determine the monotonicity and continuity properties of the map $$\lambda \rightarrow {\bar{u}}_{\lambda }$$.