Some remarks on the stability of sign changing solutions
This paper deals with the study of the stability properties of the changing sign solutions of the problems: \[ - \Delta u= h(u)\quad\text{in}\quad D,\quad u= 0\quad\text{on}\quad \partial D,\quad\text{or}\quad {\partial u\over \partial n}= 0\quad\text{on}\quad \partial D, \] \[ - \varepsilon^2 \Delta u= h(u)\quad\text{in}\quad D,\quad u= 0\quad\text{on}\quad \partial D,\quad\text{or}\quad {\partial u\over \partial n}= 0\quad\text{on}\quad \partial D, \] where \(D\) is a bounded domain in \(\mathbb{R}^n\) \((n\geq 2)\) with regular boundary, \(\varepsilon> 0\) and \(h: \mathbb{R}^1\to \mathbb{R}^1\) is defined by \[ h(u)= \begin{cases} \alpha u- \alpha u^2\quad & \text{if}\quad u\geq 0\\ du+ u^2\quad & \text{if} \quad u\leq 0\end{cases} \] where \(\alpha> 0\). By stability is meant stability for the natural corresponding parabolic problem. The instability of many sign changing solutions is proved. A number of methods are found for obtaining stable changing sign solutions. Some of these methods involve singular perturbations.
scientific article; zbMATH DE number 794314
scientific article