Boundary layer solutions to functional elliptic equations
Let \(\Omega\) be a bounded open set of \(\mathbb{R}^n\), \(\mathbb{A}:\Omega\times L^p(\Omega)\to \mathbb{R}\), \(p\geq 1\). Assume that for any \(u\in L^p(\Omega)\), \(\mathbb{A}\) is measurable; for a.e. \(x\in \Omega\), \(\mathbb{A}\) is continuous; and there exist two constants \(a_0,a_{\infty}\) such that \[ 0<a_0\leq \mathbb{A}(x,u)\leq a_{\infty} \text{ for a.e. }x\in \Omega, \forall u\in L^p(\Omega). \] The authors consider the following system \[ \begin{cases} -\mathbb{A}(x,u)\Delta u=\lambda u \text{ in }\Omega,\\ u=0\text{ on }\partial\Omega, \end{cases}\tag{1} \] where \(f\in C^1\), \(f(0)=0\), and \(\lambda\) is a positive parameter. They construct a nontrivial solution and then study the asymptotic behavior when the diffusion coefficient goes to zero.
scientific article; zbMATH DE number 5617956
scientific article