On the role played by the Fučík spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist (Q1599975)

The authors study the semilinear elliptic boundary value problem \[ \begin{cases} -\Delta u=f(x,u) \quad &\text{in }\Omega,\\ u=0\quad &\text{on } \partial \Omega,\end{cases} \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial \Omega\), and \(f\) is a given function. The main goal of the authors is to explore the role played by the Fučík spectrum of \(-\Delta\) in this problem.