The Absence of Global Solutions of a Fourth-Order Gauss Type Equation

We consider solutions of some two-dimensional fourth-order equation with a biharmonic operator and exponential nonlinearity of a counterpart of the classical Gauss-Bieberbach-Rademacher second-order equation, which was previously inspected by many authors in connection with problems of the geometry of surfaces with negative Gaussian curvature, rarefied gas dynamics, and the theory of automorphic functions. We obtain some conditions for the absence of a solution in a disk of sufficiently large radius and show that global solutions on the plane can exist only if the coefficient of nonlinearity decays at infinity at the rate at least exp{-|x|2 ln |x|}. Otherwise the mean value of the solution on a circle of radius r would tend to +8 with exponential rate as r.8. The Pokhozhaev-Mitidieri non-linear capacity method, based on the choice of appropriate cutoff test functions, proves the impossibility of the existence of such global solution. Also, for the solutions in Rn, periodic in all but x1 variables, the absence of global solutions is obtained by similar methods when the nonlinearity coefficient decays at rate slower than exp{-x31}.