Abstract:
In this paper, we consider the generalized solutions of the inequality
−div(A(x,u,∇u)∇u)⩾F(x,u,∇u)uq,q>1,
on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville's theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality −Δu⩾uq, q>1, on spherically symmetric noncompact Riemannian manifolds.
Letter to the Editor A. G. Losev, Yu. S. Fedorenko Mat. Zametki, 2009, 85:3, 480
This publication is cited in the following 9 articles:
S. S. Vikharev, A. G. Losev, “Triviality of Bounded Solutions of the Stationary Ginzburg–Landau Equation on Spherically Symmetric Manifolds”, Math. Notes, 101:2 (2017), 208–218
Losev A.G., “Solvability of the Dirichlet Problem For the Poisson Equation on Some Noncompact Riemannian Manifolds”, Differ. Equ., 53:12 (2017), 1595–1604
S. S. Vikharev, “O nekotorykh liuvillevykh teoremakh dlya statsionarnogo uravneniya Ginzburga–Landau na kvazimodelnykh rimanovykh mnogoobraziyakh”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 15:2 (2015), 127–135
E. A. Mazepa, “The positive solutions to quasilinear elliptic inequalities on model Riemannian manifolds”, Russian Math. (Iz. VUZ), 59:9 (2015), 18–25
A. P. Sazonov, “Polozhitelnye resheniya ellipticheskikh uravnenii na rimanovykh mnogoobraziyakh spetsialnogo vida”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2015, no. 3(28), 6–18
E. A. Mazepa, “Polozhitelnye resheniya kvazilineinykh ellipticheskikh neravenstv na rimanovykh proizvedeniyakh”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2015, no. 6(31), 6–16
A. G. Losev, E. A. Mazepa, “On asymptotic behavior of positive solutions of some quasilinear inequalities on model Riemannian manifolds”, Ufa Math. J., 5:1 (2013), 83–89