On nonexistence of solutions to some nonlinear functional differential inequalities

We consider nonexistence of nontrivial solutions for several classes of nonlinear functional differential inequalities. In particular, we obtain sufficient conditions for nonexistence of such solutions for the following types of inequalities: semilinear elliptic inequalities with a transformed argument in the nonlinear term, including higher order ones; quasilinear elliptic inequalities with a transformed argument in the nonlinear term dependent on the absolute value of the gradient of the solution; elliptic inequalities with the principal part of the p-Laplacian type with similar transformations in the lower order terms; parabolic partial differential inequalities with a transformed temporal argument in the nonlinear term. In the case of the untransformed argument these results coincide with the well-known optimal results of Mitidieri and Pohozaev, but in the general case they depend on the character of the transformation of the argument. The results apply to different types of transformations of the argument, such as dilatations, rotations, contractions, and shifts. © 2018, Springer International Publishing AG, part of Springer Nature.