# Nonlinear nonhomogeneous boundary value problems with competition phenomena

## TL;DR

This paper investigates a nonlinear boundary value problem with competing nonlinearities, demonstrating the existence of multiple solutions, including constant sign and nodal solutions, using variational and topological methods.

## Contribution

It establishes the existence of at least five solutions for small parameters and analyzes their properties, introducing new results for problems with nonhomogeneous operators and competing nonlinearities.

## Key findings

- At least five solutions exist for small parameter values.
- Four solutions have constant sign, one is nodal.
- Extremal solutions are monotonic and continuous in the parameter.

## Abstract

We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values $\lambda>0$, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter $\lambda>0$ varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory.