Some Stability Properties for Minimal Solutions of $-\Laplacian u=\lambda\,g(u)$

Thierry Cazenave, Miguel Escobedo and M. Assunta Pozio

Abstract: 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 one-dimensional example.

Keywords: nonlinear elliptic problem; branch of minimal solutions; stability; first eigenvalue.

Classification (MSC2000): 35J65, 35P30, 35B30.

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