On the asymptotic behavior of some semilinear equations’ solutions on the model riemannian manifolds

T. Kusano, M. Naito [12] studied the positive solutions of the equation (1) in and received some conditions for the existence of positive radially symmetric solutions of the equation, which are cited in the paper (Theorem and Theorem 𝐵). The aim of this work is to study the positive solutions of a semilinear elliptic equation (1) on model Riemannian manifolds. The first result obtained in the study of solutions of (1) is fairly obvious and is based on the properties of manifolds of the parabolic type. Theorem. Let the manifold is such that ∫︁ ∞ 𝑟0 𝑞𝑛−1(𝑡) = ∞. Then any non-negative solution of the equation (1) is identically zero. Next, we consider the manifold of the hyperbolic type, i. e. we assume that ∫︁ ∞ 𝑟0 𝑞𝑛−1(𝑡) 0, the equation (1) has positive radially symmetric solution on 𝑀𝑞, such that 𝑢(0) = 𝛼. The assertion of Theorem can be derived as the corollary of this theorem. In addition, in this paper we find an upper bound for the solutions of the equation (1). Theorem. If the equation (1) has a positive radially symmetric solution, then the following estimate is valid: 𝑢(𝑟) ≤ (︃ 1 𝛼1−𝛾 + (𝛾 − 1) ∫︀ 0 1 𝑞𝑛−1(𝑡) ∫︀ 0 𝑞𝑛−1(𝜉)𝑝(𝜉)𝑑𝜉𝑑𝑡 )︃ 1 𝛾−1. Now we note a condition of positivity of the Φ(𝑟). Lemma. Let the function 𝑞(𝑟) is convex. If (︁ 𝑛+2−𝛾(𝑛−2) 2 (𝑟)𝑝(𝑟) )︁ ≥ 0, then Φ(𝑟) ≥ 0. Also in the paper we obtained the conditions under which the equation (1) has no positive radially symmetric solutions. Theorem. Let the function 𝑞(𝑟) is convex. Assume that (︁ 𝑛+2−𝛾(𝑛−2) 2 (𝑟)𝑝(𝑟) )︁ ≥ 0, and ∫︁ ∞ 𝑞𝑛−1−𝛾(𝑛−2)(𝜉)𝑝(𝜉)𝑑𝜉 = ∞. Then the equation (1) has no positive radially symmetric solution on 𝑀𝑞. Theorem. Let the function 𝑞(𝑟) is convex. Assume that (︁ 𝑛+2−𝛾(𝑛−2) 2 (𝑟)𝑝(𝑟) )︁ ≥ 0, ∫︁ ∞ 𝑞𝑛−1−𝛾(𝑛−2)(𝜉)𝑝(𝜉)𝑑𝜉