tailieunhanh - Liouville-type theorems for a quasilinear elliptic equation of the H´enon-type

In the paper "Liouville-type theorems for a quasilinear elliptic equation of the H´enon-type", We consider the H´enon-type quasilinear elliptic equation −Δmu = |x| aup where Δmu = div(|∇u| m−2∇u), m > 1, p>m − 1 and a ≥ 0. We are concerned with the Liouville property, . the nonexistence of positive solutions in the whole space RN . We prove the optimal Liouville-type theorem for dimension N | Nonlinear Differ. Equ. Appl. c 2015 Springer Basel Nonlinear Differential Equations DOI s00030-015-0345-y and Applications NoDEA Liouville-type theorems for a quasilinear elliptic equation of the H enon-type Quoc Hung Phan and Anh Tuan Duong Abstract. We consider the H enon-type quasilinear elliptic equation Δm u x a up where Δm u div u m 2 u m gt 1 p gt m 1 and a 0. We are concerned with the Liouville property . the nonex- istence of positive solutions in the whole space RN . We prove the optimal Liouville-type theorem for dimension N lt m 1 and give partial results for higher dimensions. Mathematics Subject Classification. Primary 35B53 35J62 Secondary 35K57 35B33. Keywords. Quasilinear Liouville-type theorem H enon-type equation. 1. Introduction This article is devoted to the study of positive solutions of the following elliptic equation Δm u x a up x Ω where Δm u div u m 2 u denotes the m-Laplace operator Ω is a domain of RN . We assume throughout the paper that 1 lt m lt p 1 and a 0. The interest of Eq. started from the case of classical Laplacian Δu x a up which is called the H enon equation. Since the pioneering work of H enon 12 in 1973 on the studying of rotating stellar structures a variety of results on the qualitative properties of the solutions to problem have been established. In particular the results on the existence and nonexistence the multiplic- ity the symmetry-breaking properties and blow-up profile of solutions were obtained see 2 4 17 23 26 27 . Q. H. Phan and A. T. Duong NoDEA The problem for m 2 arises in the theory of quasi-regular and quasi-conformal mappings and in mathematical modelling of non-Newtonian fluids. Media with m gt 2 and m lt 2 are called dilatant fluids and pseudo- plastics respectively see the references in 25 for a discussion of the physical background . In general case of m the existence and nonexistence results were widely studied. Among others Clement et al. 7 applied the mountain pass .