Über Gleichungen vom gemischt-zusammengesetzten Typ. (On equations of mixed-composite type) (Q1098013)

By means of functional analytic methods the solvability of the following problems \[ Lu=(\partial /\partial x)(Tu+au_ x)+cu=f(x,y),\quad u|_{\partial D}=0,\quad u_ n|_{\Gamma \quad -_ 0}=0. \] \[ Pu=Lu-u| u|^{\rho}=f(x,y,u),\quad \rho \geq -1/2,\quad u|_{\partial D}=0,\quad u_ n|_{\Gamma \quad -_ 0}=0, \] is considered. Here \(T=K(y)\partial\) 2/\(\partial x\) \(2+\partial\) 2/\(\partial y\) 2, yk(y)\(\geq 0\). D is a domain in \(R^ 2\) with the boundary \(\partial D\); \(n=(n_ 1,n_ 2)\) is the outer normal vector. \(\partial D=\Gamma_ 0\cup \Gamma_-\cup \Gamma_ 1\), where \(\Gamma_ 0\) is a piecewise smooth curve at \(y>0\) with the ends at points A,B of the x-axis. \(\Gamma_-\) is a characteristic curve for T, outgoing from A. \(\Gamma_ 1\) is a noncharacteristic curve corresponding to T, outgoing from B and intersecting \(\Gamma_-\) in C. \(\Gamma\) \(-_ 0\subset \Gamma_ 0\), \(n_ 1(\Gamma\) \(-_ 0)<0.\) The existence of generalized solutions for the problems above is proved.