Linear second order elliptic partial differential equations with nonlinear boundary conditions at resonance without landesman-lazer conditions

TL;DR

This paper investigates the solvability of linear second order elliptic PDEs with nonlinear boundary conditions at resonance, removing the need for Landesman-Lazer conditions and allowing unbounded nonlinearities.

Contribution

It introduces a new approach to solve elliptic PDEs at resonance without Landesman-Lazer conditions, using topological degree and a priori estimates.

Findings

Established solvability under broader nonlinear boundary conditions.

Extended the theory to include unbounded nonlinearities.

Linked boundary nonlinearity with Steklov spectrum.

Abstract

We are concerned with the solvability of linear second order elliptic partial differential equations with nonlinear boundary conditions at resonance, in which the nonlinear boundary conditions perturbation is not necessarily required to satisfy Landesman-Lazer conditions or the monotonicity assumption. The nonlinearity may be unbounded. The nonlinearity interact, in some sense with the Steklov spectrum on boundary nonlinearity. The proofs are based on a priori estimates for possible solutions to a homotopy on suitable trace and topological degree arguments.