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

The boundary value problem is examined for the system of elliptic equations of from $-\mathrm{\Delta }u+A(x)u=0\text{in}\mathrm{\Omega },$ 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$Itisassumedthat$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.