Absence of solutions of elliptic differential inequalities in the neighborhood of conic point of boundary.

*(English. Russian original)*Zbl 1222.35217
Russ. Math. 46, No. 9, 48-57 (2002); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2002, No. 9, 50-59 (2002).

From the introduction: This article is devoted to determination of conditions for absence of nontrivial solutions of semilinear differential inequalities and systems of inequalities of elliptic type in a neighborhood of a conic point of boundary. We also state a result for respective parabolic problem. The absence of
solution is a consequence of the presence of a singularity of the form \(1/| x|^\sigma\), \(\sigma>2\) in the inequalities under consideration.

The theory of linear boundary value problems in conic domains goes back to the classical work by V. A. Kondrat’ev. The study of the semilinear and nonlinear equations is carried out mainly on the basis of known assertions for respective linear problems. In this way, sufficiently fine estimates of the growth of solutions were obtained. A consequence of these estimates is, in particular, the absence of nontrivial solution under some additional conditions.

In this article to prove the absence of a solution we use the method of test functions without use of any information from the linear theory.

The theory of linear boundary value problems in conic domains goes back to the classical work by V. A. Kondrat’ev. The study of the semilinear and nonlinear equations is carried out mainly on the basis of known assertions for respective linear problems. In this way, sufficiently fine estimates of the growth of solutions were obtained. A consequence of these estimates is, in particular, the absence of nontrivial solution under some additional conditions.

In this article to prove the absence of a solution we use the method of test functions without use of any information from the linear theory.