Solutions with sign information for nonlinear nonhomogeneous elliptic equations
The paper deals with the existence of multiple nontrivial solutions to the Dirichlet problem \[ \begin{cases} -\text{div\,}a(Du(z))=f(x,u(z)) & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^N\) is a bounded and \(C^2\)-smooth domain, the map \(a:\;\mathbb{R}^N\to \mathbb{R}^N\) is strictly monotone and \(f\) is a Carathéodory function. The authors show the existence of at least three nontrivial solutions, all with sign information, by means of variational methods combined with truncation and comparison techniques. In the particular case of \((p,2)\)-equations, tools from Morse theory are employed to show the existence of four nontrivial solutions. Finally, for a special class of parametric equations, multiplicity theorems are obtained that generalize earlier results on the subject.
scientific article; zbMATH DE number 6712491