On a Dirichlet problem with $(p,q)$-Laplacian and parametric concave-convex nonlinearity

Abstract

A homogeneous Dirichlet problem with $(p,q)$-Laplace differential operator and reaction given by a parametric $p$-convex term plus a $q$-concave one is investigated. A bifurcation-type result, describing changes in the set of positive solutions as the parameter $\lambda>0$ varies, is proven. Since for every admissible $\lambda$ the problem has a smallest positive solution $\bar u_{\lambda}$, both monotonicity and continuity of the map $ \lambda \mapsto \bar u_{\lambda}$ are studied.