On the solvability of a nonselfadjoint quasilinear elliptic boundary value problem. (Q1394479)

Let \(\Omega \) be a bounded open subset of \(\mathbb R^N \) with sufficiently smooth boundary \(\partial \Omega ,~A=\sum _{| \alpha | \leq 2m} a_{\alpha }(x) D^{\alpha} \) be a linear elliptic partial differential operator of order \(2m\) acting on the Sobolev space \(H^{2m} (\Omega),\) and \(B\) be a system of \(m\) linear boundary operators of order less than \(2m-1\) defined on \(\partial \Omega .\)The following boundary value problem: \[ (1)~~Au(x)= f(x, D^{\alpha_1}u(x),\dots,D^{\alpha_k} u(x)),~x\in{\Omega};~Bu(x)=0,~x\in{\partial \Omega}, \] is considered.Each \(\alpha _i\) is a multiindex with \(| \alpha _i| \leq{2m} \) and \(f:\Omega \times \mathbb R^k \to \mathbb R \) has the form \(f(x, t_1, \dots, t_k)=\sum _{i=1}^k f_i(x,t_i);\) each \(f_i\) is a Caratheodory function and \(f\) may grow sublinearly. Let \(X= \{u \in H^{2m} (\Omega) | Bu=0\} .\) The operator \(L:X \to L^2(\Omega),\) defined by \(Lu= A(u),\) is supposed to be a Fredholm operator which has a non-trivial kernel. The author states additional conditions on \(L\) and \(f\) which guarantee the existence of at least one solution of (1) in \(X.\) The proof is based on techniques of nonlinear functional analysis. Two concrete examples are presented.