Semilinear Robin problems resonant at both zero and infinity (Q1689406)

Given a bounded domain \(\Omega \subseteq \mathbb{R}^N\) with a \(C^2\)-boundary \(\partial \Omega\), the authors consider the following semilinear Robin problem \[ -\Delta u(z)=f(z,u(z)) \quad \text{in }\Omega, \qquad \frac{\partial u}{\partial n}+\beta(z)u=0 \quad\text{on }\partial \Omega,\tag{1} \] where the nonlinearity \(f:\Omega\times \mathbb{R}\to\mathbb{R}\) is measurable in the first argument, continuously differentiable with respect to the second variable and the boundary function \(\beta\) belongs to \(W^{1,\infty}(\partial \Omega)\) such that \(\beta(z) \geq 0\) for all \(z \in \partial \Omega\). Applying variational methods and critical groups, the authors prove the existence of at least two nontrivial smooth solutions of problem (1) under certain resonance conditions at \(\pm\infty\) and at zero.