Generalized eigenproblem and nonlinear elliptic system with nonlinear boundary conditions

TL;DR

This paper investigates the solvability of nonlinear elliptic systems with nonlinear boundary conditions, focusing on the generalized Steklov Robin eigensystem and establishing solution existence via Leray-Schauder degree methods.

Contribution

It introduces a new analysis of nonlinear elliptic systems with boundary nonlinearities involving the generalized Steklov Robin spectrum, extending prior spectral theory applications.

Findings

Existence of solutions for nonlinear elliptic systems with boundary nonlinearities.

Analysis of the generalized Steklov Robin eigensystem with matrix weights.

Application of Leray-Schauder degree to establish solvability.

Abstract

We will study solvability of nonlinear second-order elliptic system of partial differential equations with nonlinear boundary conditions. We study the generalized Steklov Robin eigensystem (with possibly matrices weights) in which the spectral parameter is both in the system and on the boundary. We prove the existence of solutions for nonlinear system when both nonlinearities in the differential system and on the boundary interact, in some sense, with the generalized spectrum of Steklov Robin. The method of proof makes use of Leray-Schauder degree.