Semilinear Neumann equations with indefinite and unbounded potential (Q2828641)

The paper deals with a semilinear Neumann problem with an indefinite and unbounded potential and a Carathéodory reaction term, namely NEWLINE\[NEWLINE \begin{cases} -\Delta u(z)+\beta(z)u(z)=f(z,u(z)) &\quad \text{in}\;\Omega,\\ \dfrac{\partial u}{\partial n}=0 &\quad \text{on}\;\partial\Omega, \end{cases} NEWLINE\]NEWLINE where \(\Omega\subset\mathbb{R}^N,\) \(N\geq3,\) is a bounded and \(C^2\)-smooth domain, \(\beta\in L^s(\Omega),\) \(s>N,\) is a sign changing and bounded from below, and \(f\) is a Carathéodory function.NEWLINENEWLINEUnder suitable asymptotic conditions on the reaction term \(f\) that make coercive the corresponding energy functional, the authors prove multiplicity theorems producing three or four solutions with sign information on them. The approach adopted combines variational methods based on the critical point theory with suitable perturbation and truncation techniques, and with Morse theory.