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

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.