A minimization problem associated with elliptic systems of FitzHugh--Nagumo type
The authors deal with the following problem: \[ \begin{cases} -\varepsilon^2\Delta u=f(u)-v\quad & \text{in }\Omega\\ -\Delta v+ \gamma v=\delta u\quad &\text{in }\Omega\\ u=v=0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\varepsilon\) is a parameter, \(\gamma\) and \(\delta\) are nonnegative constants, and \(f\) is a \(C^1\)-function satisfying some natural conditions. The aim of the authors is to prove that for suitably large \(\delta>0\), (1) has solutions, which either oscillate around a constant in a compact subset of \(\Omega\), or have a sharp interior layer. These solutions are local minimum of the corresponding functional.
scientific article; zbMATH DE number 2117903
scientific article