Linear elliptic system with nonlinear boundary conditions without Landesman-Lazer conditions

TL;DR

This paper investigates a boundary value problem for a linear elliptic system with nonlinear boundary conditions, relaxing Landesman-Lazer conditions, using Leray-Schauder degree theory to establish existence results.

Contribution

It introduces a new approach to analyze elliptic systems with nonlinear boundary conditions without relying on Landesman-Lazer conditions.

Findings

Existence of solutions under relaxed boundary conditions

Application of Leray-Schauder degree methods to nonlinear boundary problems

Extension of classical results to systems with bounded nonlinearities

Abstract

The boundary value problem is examined for the system of elliptic equations of from $-\Delta u + A(x)u = 0 \quad\text{in} \Omega,$ where $A(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$ and $\frac{\partial u}{\partial \nu}+g(u)=h(x) \quad\text{on} \partial\Omega$ It is assumed that $g\in C(\mathbb{R}^{k},\mathbb{R}^{k})$ is a bounded function which may vanish at infinity. The proofs are based on Leray-Schauder degree methods.