Nonlinear elliptic differential equations with nonlocal boundary conditions (Q1175247)

The author proves the existence of weak solutions to the class of boundary value problems of the form: \[ \sum_{|\alpha|\leq1}(- 1)^{|\alpha|}D^ \alpha f_ \alpha(x,u,Du)+g(x,u,Du)=F, \hbox { in } \Omega\subset\mathbb{R}^ n, \] \[ \sum_{|\alpha|=1}f_ \alpha(x,u,Du)v_ \alpha+h_ 1(x,u)+h^*_ 2(x,u(\Phi(x))) =0, \hbox { on } \partial \Omega, \] where \(\alpha\) is a multi-index and \(v_ \alpha\) denotes the coordinates of the exterior normal unit vector on \(\partial\Omega\). Here, the functions \(f_ \alpha,g,h_ 1\), and \(h^*_ 2\) basically satisfy polynomial growth conditions - eleven technical conditions in all. A sufficient condition for uniqueness of the solution and a case of nonexistence are also provided.