A semilinear elliptic system with vanishing nonlinearities (Q1422580)

Let \(\mathbb R^m\) be the real \(m\)-dimensional space, \(m\geq 1\) and \(\Omega\subset\mathbb R^m\) be a bounded open, connected set with a smooth boundary. Let \(g\in C^1(\mathbb R^N, \mathbb R^N)\) and \(f\in C(\overline\Omega, \mathbb R^N)\). The authors consider the Neumann BVP \[ \begin{gathered} \Delta u+ g(u)= f(x),\quad x\in\Omega,\\ {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega.\end{gathered}\tag{1} \] The main interest of the authors is the resonance case, so they assume that \(g\) will always be a bounded function. To study the existence of solutions to (1) the authors use Leray-Schauder degree arguments.