A GENERAL APPROACH TO A PRIORI ESTIMATES OFSOLUTIONS TO NONLINEAR EQUATIONS AND ORDINARYDERIVATIVE INEQUALITIES

This article outlines a general approach to a priori estimates of solutions to nonlinear equations and ordinary derivative inequalities, based on the method of trial functions.This approach covers a fairly wide class of nonlinear problems, for which we study the problem of the absence of nontrivial solutions (see [1]).Our approach is based on a priori estimates.First, we obtain an a priori estimate for the solution of the nonlinear problem under consideration.Then we obtain the asymptotics of this a priori estimate.The proof of the absence of a solution is carried out by contradiction.The derivation of an a priori estimate is based on the trial function method.The optimal choice of the trial function leads to a minimax nonlinear problem, which generates a nonlinear capacitance.To analyze the absence problem, it is enough to obtain an exact estimate of the first term of the asymptotics of this capacity.