# Existence of Solutions to Nonlinear Legendre Boundary Value Problems

## TL;DR

This paper investigates conditions under which solutions exist for nonlinear Legendre boundary value problems, providing criteria based on the nonlinearities' size and behavior, and offering qualitative insights into solutions depending on parameters.

## Contribution

It establishes new sufficient conditions for the existence of solutions to nonlinear Legendre boundary value problems, including qualitative descriptions based on parameters.

## Key findings

- Sufficient conditions for solution existence are derived.
- Results depend on the size and limiting behavior of nonlinearities.
- Qualitative descriptions of solutions are provided depending on parameters.

## Abstract

In this paper, we consider nonlinearly perturbed Legendre differential equations subject to the usual boundary conditions. For such problems we establish sufficient conditions for the existence of solutions and in some cases we provide a qualitative description of solutions depending on a parameter. The results presented depend on the size and limiting behavior of the nonlinearities.