Semilinear elliptic singular perturbation problems with nonuniform interior behavior
Let \(\epsilon^ 2\Delta u+f(x,u)=0\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(\epsilon\) is a small positive parameter and \(\Omega\) is a bounded domain in \(R^ N\) with a smooth boundary \(\partial \Omega\). The authors investigate certain intermediate solutions for the above Dirichlet problem in case f satisfies suitable conditions. The main result is the following. Let f be a \(C^ 1({\bar \Omega}\times R)\) function such that: (A) there are three \(C^ 2({\bar \Omega})\) functions \(z_ i(x)\) \((i=1,2,3)\) so that \(z_ 1(x)<z_ 2(x)<z_ 3(x)\), \(z_ 2(x)>0\), \(f(x,z_ i(x))=0\) \((i=1,2,3)\), \(f_ u(x,z_ i(x))<0\) \((i=1,3)\) and \(z_ 2(x)\) is the unique zero of f between \(z_ 1\) and \(z_ 3\) (x in \({\bar \Omega}\)); (B) the integral of f(x,u) with respect to u, from \(\vartheta\) to \(z_ 1(x)\) is positive for x in \(\partial \Omega\) and \(\vartheta\) between 0 and \(z_ 1;\) (C) there is a nonempty open set \(W\subset \Omega\) so that the integral of f(x,u) with respect to u from \(z_ 1(x)\) to \(z_ 3(x)\) is positive for \(x\in W.\) Let \(u_ 0(x)\) be the unique solution of the equation: ``integral of f(x,u) with respect to u from \(z_ 1(x)\) to \(u_ 0(x)=0''\) for \(x\in W\). Under the above conditions, if \(\gamma\) is a small constant and \(\epsilon\) is sufficiently small then for each ball \(B\subset W\) the previous Dirichlet problem has a solution u(x,\(\epsilon\)) so that (a) \(z_ 1(x)-\gamma <u(x,\epsilon)\) if \(dist(x,\partial \Omega)\gg \epsilon\); (b) \(u(x,\epsilon)<z_ 3(x)+\gamma\) for all \(x\in {\bar \Omega}\); (c) \(\exists x\in B\) so that \(u(x,\epsilon)<u_ 0(x)+\gamma.\) Furthermore, there is a computable \(\delta >0\), independent of \(\epsilon\) so that (d) max[u(x,\(\epsilon\)): \(x\in \Omega]\equiv u(x_ 0,\epsilon)+\delta\).
scientific article; zbMATH DE number 4157050
scientific article