Abstract
We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction having the combined effects of a singular term and of a parametric (p-1)-superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter lambda >0 varies. Moreover, we prove the existence of a minimal positive solution u^*_lambda and study the monotonicity and continuity properties of the map lambda rightarrow u^*_lambda .
Highlights
In a recent paper, the authors [15] studied the following singular parametric p-Laplacian Dirichlet problem−Δpu = u−η + λf (x, u) in Ω, u = 0 on ∂Ω, u > 0, λ > 0, 0 < η < 1, 1 < p.They proved a result describing the dependence of the set of positive solutions as the parameter λ > 0 varies, assuming that f (x, ·) is (p − 1)-superlinear.In the present paper, we consider a singular parametric Dirichlet problem driven by the (p, q)-Laplacian, that is, the sum of a p-Laplacian and of a q-Laplacian with 1 < q < p
We consider a singular parametric Dirichlet problem driven by the (p, q)-Laplacian, that is, the sum of a p-Laplacian and of a q-Laplacian with 1 < q < p
Applying variational tools from critical point theory along with suitable truncation and comparison techniques, we prove a bifurcation-type result as in [15], which describes in a precise way the dependence of the set of positive solutions as the parameter λ > 0 changes
Summary
The authors [15] studied the following singular parametric p-Laplacian Dirichlet problem. Applying variational tools from critical point theory along with suitable truncation and comparison techniques, we prove a bifurcation-type result as in [15], which describes in a precise way the dependence of the set of positive solutions as the parameter λ > 0 changes In this direction we mention the recent works of Papageorgiou– Radulescu–Repovs [12] and Papageorgiou–Vetro–Vetro [14] which deal with nonlinear singular parametric Dirichlet problems. In Papageorgiou–Radulescu– Repovs [12] the equation is driven by a nonhomogeneous differential operator and in the reaction we have the competing effects of a parametric singular term and of a (p − 1)-superlinear perturbation. We mention the survey paper of Radulescu [18] on anisotropic (p, q)-equations
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