Positive solutions of a class of nonlinear eigenvalue problems
The author studies positive solutions of \[ \begin{aligned} \Delta u + \lambda f(u) & = 0\quad\text{in }\Omega,\\ u&= 0\quad\text{in }\partial\Omega.\end{aligned} \] Here \(f(0) > 0\) and \(f\) has \(n\) different zeros for positive values \(a_k\), \(\lambda\) is a positive parameter. An important geometrical assumption in this work is that \(\Omega\) is symmetric and convex in \(N\) orthogonal directions. The author then proves a number of results, mainly concerning the number of solutions as well as bounds for the solutions. An example would be \(f(u) = d + \cos u\), \(0 < d < 1\).
scientific article; zbMATH DE number 2075748