Multiplicity of solutions for Neumann problems resonant at any eigenvalue (Q2017067)

The authors consider the following semilinear Neumann problem \[ \begin{cases} -\Delta u(z)=f(z,u(z)) & \text{in}\;\Omega,\\ \dfrac{\partial u}{\partial n}=0 & \text{on}\;\partial\Omega, \end{cases} \] where \(\Omega\subset {\mathbb R}^N\) is a bounded domain with a \(C^2\)-smooth boundary. The reaction term \(f(z,u)\) is supposed to be resonant both at \(\pm\infty\) and at zero with respect to any eigenvalue of the negative Neumann Laplacian. Using the reduction method and Morse theory, the authors show that the problem considered has at least two nontrivial smooth solutions.