PORTUGALIAE MATHEMATICA Vol. 59, No. 4, pp. 373391 (2002) 

Some Stability Properties for Minimal Solutions of $\Laplacian u=\lambda\,g(u)$Thierry Cazenave, Miguel Escobedo and M. Assunta PozioAbstract: We study the stability of the branch of minimal solutions $(u_\lambda)_{0<\lambda <\lambda^*}$ of $\Laplacian u =\lambda\,g(u)$ for a nonlinearity $g$ which is neither concave nor convex. We show that it is related to the regularity of the map $\lambda\mapsto u_\lambda $. We then show that in dimensions $N=1$ and $N=2$, discontinuities in the branch of minimal solutions can be produced by arbitrarilly small perturbations of the nonlinearity $g$. In dimensions $N\ge 3$ the perturbation has to be large enough. We also study in detail a specific onedimensional example. Keywords: nonlinear elliptic problem; branch of minimal solutions; stability; first eigenvalue. Classification (MSC2000): 35J65, 35P30, 35B30. Full text of the article:
Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.
© 2002 Sociedade Portuguesa de Matemática
