Generalized eigenvalue problems of nonhomogeneous elliptic operators and their application (Q374433)

The following generalized eigenvalue problem is considered NEWLINE\[NEWLINE\begin{aligned} -\text{div\,}a(x,|\nabla u|)\,\nabla u= \lambda|u|^{p-2} u\quad & \text{in }\Omega,\\ u= o\quad &\text{on }\partial\Omega,\end{aligned}NEWLINE\]NEWLINE for a strictly monoton (in the second variable) function \(a\) which fulfils some regularity conditions. The authors prove that there exist values \(\lambda\in\mathbb{R}\) such that the problem has non-trivial solutions. The \(p\)-Laplace equation in a bounded domain \(\Omega\) appears as special case. Using variational methods even the existence of a positive solution can be proved under certain conditions on a general right-hand side \(f(x,u)\). It is shown that the spectrum of the basis operator (left-hand side) is not discrete in a corresponding Sobolev space.