Positive solutions of elliptic equations on riemannian manifolds of a special type

In this paper we study the asymptotic behavior of positive solutions of elliptic equations Δ𝑢 + 𝑝(𝑟)𝑢𝛾 = 0 and div (𝜎(𝑟)∇𝑢) + 𝑝(𝑟)𝑢𝛾 = 0 on complete Riemannian manifolds. The conditions of existence and nonexistence of positive solutions of the equations studied on such manifolds. Let - complete Riemannian manifold can be represented as a union of = ∪ 𝐷, where - a compact and isometric to the direct product of [0;∞) × 𝑆, where - compact Riemannian manifold with metric 𝑑𝑠2 = ℎ2(𝑟)𝑑𝑟2 + 𝑞2(𝑟)𝑑𝜃2. Where ℎ(𝑟) and 𝑞(𝑟) - a positive, smooth on [0;∞) functions, and - the standard Riemannian metric on the sphere 𝑆. The following assertions. Theorem 1. Let the manifold is such that ∫︁ ∞ 1 ℎ(𝑡)𝑑𝑡 𝑞𝑛-1(𝑡) = ∞. Then every non-negative solution (1) is identically zero. Theorem 2. Let the manifold is such that ∫︁ ∞ 1 ℎ(𝑡)𝑑𝑡 𝑞𝑛-1(𝑡) = 0 the equation (1) is on a positive radially symmetric solution such that 𝑢(0) = 𝛼. Theorem 3. Let the manifold is such that ∫︁ ∞ 1 ℎ(𝑡)𝑑𝑡 𝜎(𝑟)𝑞𝑛-1(𝑡) = ∞. Then every non-negative solution (2) is identically zero. Theorem 4. Let the manifold is such that ∫︁ ∞ 1 ℎ(𝑡)𝑑𝑡 𝜎(𝑟)𝑞𝑛-1(𝑡) 0 the equation (2) is on a positive radially symmetric solution such that 𝑢(0) = 𝛼. In addition, the found conditions under which the equations (1) and (2) haven’t a positive radially symmetric solutions.