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

We consider solutions of sometwo-dimensional fourth-order equation with a biharmonic operator and exponential nonlinearityof a counterpart of the classicalGauss–Bieberbach–Rademacher second-order equation, which was previously inspectedby many authors in connection with problems of the geometry of surfaces with negative Gaussiancurvature, rarefied gas dynamics, and the theory of automorphic functions. We obtain some conditionsfor the absence of a solution in a disk of sufficiently large radius andshow that global solutions on the plane can exist only if the coefficient ofnonlinearity decays at infinity at the rate at least $ \exp\{-|x|^{2}\log|x|\} $ .Otherwise the mean value of the solution on a circle of radius $ r $ wouldtend to $ +\infty $ with exponential rate as $ r\to\infty $ . ThePokhozhaev–Mitidieri nonlinear capacity method, based on the choice of appropriate cutofftest functions, proves the impossibility of the existence of such global solution.Also, for the solutions in  $ {𝕉}^{n} $ , periodic in all but  $ x_{1} $ variables,the absence of global solutions is obtained by similar methods when the nonlinearitycoefficient decays at rate slower than $ \exp\{-x_{1}^{3}\} $ .