Nonlinear nonhomogeneous singular problems

Abstract

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $(p-1)$ near $+\infty$ and with a reaction which has the competing effects of a parametric singular term and a $(p-1)$-superlinear perturbation which does not satisfy the usual Ambrosetti-Rabinowitz condition. Using variational tools, together with suitable truncation and strong comparison techniques, we prove a "bifurcation-type" theorem that describes the set of positive solutions as the parameter $\lambda$ moves on the positive semiaxis. We also show that for every $\lambda >0$, the problem has a smallest positive solution $u_{\lambda }^{\ast }$ and we demonstrate the monotonicity and continuity properties of the map $\lambda \mapsto u_{\lambda }^{\ast }$.