The absence of global solutions of the fourth-order Gauss type equation

A. V. Neklyudov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 1, pp. 123-131

Résumé

We consider solutions of a two-dimensional fourth-order equation with a biharmonic operator and a non-linearity exponential with respect to the solution, an counterpart of the classical Gauss–Bieberbach–Rademacher second-order equation, which was previously considered 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. Conditions are obtained under which a solution does not exist in a circle of sufficiently large radius. It is shown that global solutions on the plane can exist only if the coefficient of nonlinearity degenerates to infinity at a rate not less than $\exp\{-|x|^2\ln|x|\}$. It is shown that otherwise the mean value of the solution on a circle of radius $r$ would have to grow to $+\infty$ with an exponential rate as $r\to\infty$. The Pokhozhaev–Mitidieri nonlinear capacity method, based on the choice of appropriate cutting test functions, proves the impossibility of the existence of such a growing global solution. Also, for solutions in ${\mathbb R}^n$ that are periodic in all variables except for one variable $x_1$, the absence of global solutions is obtained by similar methods when the coefficient degenerates under nonlinearity at a rate slower than $\exp\{-x_1 ^3\}$.