# Nonlinear Dirichlet problems with unilateral growth on the reaction

## TL;DR

This paper investigates a nonlinear Dirichlet problem involving the p-Laplace operator with unilateral growth restrictions, establishing multiple solutions including constant sign and nodal solutions, and extending existing multiplicity results.

## Contribution

It introduces two new multiplicity theorems for solutions of the p-Laplace Dirichlet problem under unilateral growth conditions, extending prior results especially in the semilinear case.

## Key findings

- Proved existence of three nontrivial solutions with different signs.
- Established four solutions in the semilinear case using Morse theory.
- Extended multiplicity results for parametric coercive Dirichlet problems.

## Abstract

We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, $p=2$), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multiplicity results existing for a class of parametric, coercive Dirichlet problems.