Concentration on curves for a Neumann Ambrosetti-Prodi-type problem in two-dimensional domains
Given a smooth, bounded domain \(\Omega\subset\mathbb{R}^2\), the authors consider the following Neumann Ambrosetti-Prodi type problem \[ -\Delta u=g(u)-s\psi \text{ in } \Omega, \qquad \frac{\partial u}{\partial \nu}=0 \text{ on } \partial\Omega, \] where \(p>1\), \(s>0\) is a parameter, \(\psi\) is a positive function, \(\nu\) is the outward normal of \(\partial\Omega\) and \(g\) is a function satisfying certain conditions, which is taken to be \(|u|^p\). It is assumed that \(\Gamma\) is a curve intersecting \(\partial\Omega\) orthogonally at exactly two points and dividing \(\Omega\) into two parts. Furthermore, \(\Gamma\) is assumed to satisfy a stationary and non-stationary conditions with respect to the functional \(\int_{\Gamma}\psi^{\sigma},\) where \(\sigma=\frac{p+3}{2p}\). Under these conditions and provided \(s\) is large and away from certain critical numbers, the authors prove the existence of a solution \(u_s\) which concentrates along the whole \(\Gamma\), exponentially small in \(s\) at any positive distance from \(\Gamma\). In more detail, firstly the authors introduce modified Fermi coordinates in a tubular neighborhood of \(\Gamma\) in \(\overline{\Omega}\) and expand the Laplace-Beltrami operator in these coordinates. They then reach a key step in their proof: the construction of an accurate approximate solution in a neighborhood of the curve \(\Gamma_{\varepsilon}=\varepsilon^{-1}\Gamma\), and the construction of exact solutions by linearizing the equation around this approximation. The associated linear operator happens to be invertible, with an inverse that is controlled in a suitable norm by a negative power of \(\varepsilon\), as long as \(\varepsilon\) stays away from critical values where resonance occurs. Finally, the relationship between the size of the error and that of the inverse of the linearization leads to a fixed point scheme.
scientific article; zbMATH DE number 6982495
scientific article