Positive solutions of semilinear elliptic eigenvalue problems with concave nonlinearities. (English) Zbl 1180.35198

The existence (and non-existence) of positive solutions of the Neumann problem
\[ \begin{aligned} \Delta u+\lambda|u|^{p-1}u= \lambda m(x)&\quad\text{in }\Omega,\\ \frac{\partial u}{\partial n}= \lambda(\sigma(x)-c|u|^{q-1})u &\quad\text{on }\partial\Omega\end{aligned} \]
is studied in a smooth bounded domain in \(\mathbb R^n\). A central concept is the Nehari manifold, which here is studied with the fibering method.


35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations