Linear Elliptic equations with nonlinear boundary conditions under strong resonance conditions

TL;DR

This paper proves the existence and multiplicity of solutions for elliptic equations with nonlinear boundary conditions under strong resonance, using variational methods and Morse theory, especially at the first Steklov eigenvalue.

Contribution

It introduces new results on elliptic problems with nonlinear boundary conditions at strong resonance, focusing on the first Steklov eigenvalue and employing advanced variational techniques.

Findings

Existence of solutions under strong resonance conditions.

Multiple solutions are established for the problem.

Application of Morse theory and critical groups in analysis.

Abstract

In this work we establish existence and multiplicity of solutions for elliptic problem with nonlinear boundary conditions under strong resonance conditions at infinity. The nonlinearity is resonance at infinity and the reso- nance phenomena occurs precisely in the first Steklov eigenvalue problem. In all results we use Variational Methods, Critical Groups and the Morse Theory.