Multiple solutions for asymptotically linear resonant elliptic problems
The author considers the nonlinear elliptic problem \[ \begin{cases} -\Delta u= g(x,u)\quad &\text{in }\Omega,\\ u= 0\quad\text{on }\partial\Omega,\end{cases}\tag{P} \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\), \(g:\Omega\times \mathbb R\to \mathbb R\) is a function of class \(C^1\) such that \(g(x, 0)= 0\) and which is asymptotically linear at infinity. He defines critical groups starting with homological groups with coefficients in \(\mathbb Z\). He uses some theorems in \textit{K. C. Chang} [Infinite-dimensional Morse theory and multiple solution problems, Birkhäuser, Boston (1993; Zbl 0779.58005)] to compute explicitely these groups in order to prove existence of multiple solutions of (P). He studies resonant and nonresonant cases.
scientific article; zbMATH DE number 1997216