Boundary value problems for semilinear elliptic equations of arbitrary order in unbounded domains (Q1826027)

Let \(\Omega \subset {\mathbb{R}}^ n\) be a (possibly unbounded) domain. The author studies the \(H^{2m,p}(\Omega)\) solvability of a boundary value problem corresponding to the equation \(Au=f(x,u)\), where A is a uniformly elliptic operator of order 2m and f fulfills the growth assumption \(| f(x,u)| \leq \sum^{\infty}_{k=1}V_ k(x)| u|^{b_ k}\), where \((1-2mp/n)b_ k<1\) and \(V_ k\) belong to certain spaces of functions depending on n,m,p and \(b_ k\) (e.g. the growth \(| f(x,u)| \leq V(x)e^{C| u|}\) is allowed provided \(n<2mp\) and \(V\in L^ p(\Omega))\). The main tool is the Schauder fixed point theorem and a theorem of Krasnosel'skij.