Numerical continuation and the Gelfand problem
This paper is concerned with the Gelfand problem \[ \Delta u+\lambda e^u= 0\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega, \] where \(\lambda>0\), and where \(\Omega\in\mathbb{R}^3\) denotes a bounded domain. The author is interested in finding the number of positive solutions in terms of the `bifurcation diagram' where the sup norm of the solutions \(u\) is plotted against \(\lambda\). To calculate the diagram a numerical approach is proposed which is based on finite differences together with an arclength method. Results are presented for a number of special domains as cube and punctured cubes where smaller cubes are deleted.
scientific article; zbMATH DE number 1183199
scientific article