Calculating critical groups of solutions for elliptic problem with jumping nonlinearity
The authors consider a semilinear elliptic equation of the form \[ \begin{cases} \Delta u+g(x,u)= 0\quad & \text{in }\Omega,\\ u=0\quad & \text{on } \partial\Omega, \end{cases}\tag{1} \] where \(\Omega\) is a bounded domain, \(g(x,u)\) is a given function. They calculate the critical groups for the trivial critical point. Existence of nontrivial solutions is obtained as a consequence through the Morse relation.
scientific article; zbMATH DE number 1761020