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

## TL;DR

This paper investigates a Dirichlet problem involving a combined $(p,q)$-Laplacian operator with parametric nonlinearities, establishing bifurcation results and analyzing how the smallest positive solution varies with the parameter.

## Contribution

It provides new bifurcation analysis and studies the monotonicity and continuity of the smallest positive solutions in a $(p,q)$-Laplacian problem with parametric nonlinearities.

## Key findings

- Bifurcation-type results describing solution set changes with parameter.
- Monotonicity of the smallest positive solution map.
- Continuity of the smallest positive solution with respect to the parameter.

## 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.