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Towards a unified approach to nonexistence of solutions for a class of differential inequalities. (English) Zbl 1115.35157

An approach for obtaining a priori estimates for solutions of nonlinear partial differential equations and inequalities is suggested. In contrast to most known techniques for proving non-existence of solutions that use comparison methods or the study of the associated energy functionals, a new approach is based on test functions and scaling arguments and applies to wide classes of nonlinear problems for which necessary conditions for solvability can be derived. Estimates for the solutions are obtained by establishing sharp local integral estimates on possible solutions and by the subsequent analysis of the asymptotic behaviour of the estimates with respect to parameters of the problem. The choice of test functions is determined by the nature of the nonlinear problem and the concept of solution employed. Implicit conditions for the nonexistence of the solutions are expressed in the form of certain integral estimates for the solutions formulated in terms of capacitary-type integrals of test functions. The procedure reduces to the optimal choice of the parameters that determine the corresponding test function based on the analysis of an appropriate set of algebraic inequalities.

MSC:

35R45 Partial differential inequalities and systems of partial differential inequalities
35B45 A priori estimates in context of PDEs
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations
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