Uniqueness of Nonnegative Solutions to Elliptic Differential Inequalities on Finsler Manifolds
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Uniqueness of Nonnegative Solutions to Elliptic Differential Inequalities on Finsler Manifolds Changwei Xiong1 Received: 25 December 2018 / Accepted: 20 September 2019 / © Springer Nature B.V. 2019
Abstract We consider a class of elliptic differential inequalities involving Finsler p-Laplacian and a positive potential function on forward geodesically complete noncompact Finsler measure spaces with finite reversibility. Under various volume growth conditions concerning geodesic balls with a given center and the potential function, we prove that the only nonnegative weak solution of the differential inequalities is identically zero. Keywords Uniqueness of nonnegative solutions · Elliptic differential inequality · Finsler measure space Mathematics Subject Classification (2010) 35R45 · 35J92 · 58J05
1 Introduction One of the most important topics in global analysis, which has been investigated extensively, is the uniqueness of nonnegative solutions to the elliptic differential inequalities, either of the simple form g u + uσ ≤ 0, (1.1) or of its generalized form 1 (1.2) div a(x)|∇g u|p−2 ∇g u + v(x)uσ ≤ 0, a(x) on an n-dimensional complete noncompact Riemannian manifold (M n , g). Here p > 1 and σ > p − 1 are two constants, a(x) ∈ Liploc (M) with a(x) > 0 is a weight function, and v(x) ∈ L1loc (M) with v(x) > 0 a.e. on M is a potential function. Let us first review some history on this topic. This research was supported by Australian Laureate Fellowship FL150100126 of the Australian Research Council. We would like to thank the referee for careful reading of the paper and for valuable suggestions and comments which made this paper better and more readable. We are also grateful to Ben Andrews for his support. Changwei Xiong
[email protected] 1
Mathematical Sciences Institute, Australian National University, Canberra, ACT 2601, Australia
Changwei Xiong
For the problem on Euclidean space Rn , the study may date back to the seminal works by Gidas [10] and Gidas and Spruck [11], where they proved that the only nonnegative weak n solution to Eq. 1.1 with σ > 1 on Rn is identically zero if and only if σ ≤ n−2 when n ≥ 3. For n ≤ 2, it is also known that the same holds for Eq. 1.1 with any σ > 1. After [10] and [11], there appeared a series of papers investigating problems related to Eq. 1.2 on Rn ; see e.g. [18–21] by Mitidieri and Pokhozhaev, [7] by D’Ambrosio and Lucente, [8] by D’Ambrosio and Mitidieri, [22] by Monticelli, [2] by Caristi, D’Ambrosio and Mitidieri, [3] by Caristi and Mitidieri, and [4] by Caristi, Mitidieri and Pokhozhaev. In contrast, the study of the problem on a general complete noncompact Riemannian manifold received adequate attention mainly in the last decade. Motivated by Kurta’s paper [16], Grigor’yan and Kondratiev in [13] proved that the inequality Eq. 1.2 with p = 2 has no nontrivial nonnegative weak solution if there exist positive constants C0 , C, R0 and ε0 , 1 and k ∈ 0, σ −1 such that 1 2σ a(x)v(x)− σ −1 +ε dx ≤ CR σ −1 +C0 ε (ln R)k (1.3) BR (x0 )
for R ≥ R0 and 0
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