| Abstract: |
| We investigate existence and non-existence of positive radial solutions for the subcritical Lane-Emden equation $-\Delta_{\mathbb{M}^n} u = u^q $ on a class of non-compact Riemannian models $ \mathbb{M}^n $. A number of interesting phenomena arise: depending on the volume growth, which is required to be polynomial, the subcritical regime divides into three ranges, characterized by existence (slightly subcritical), non-existence (strongly subcritical), and by a mixed behavior where existence and non-existence depend in a very sensitive way on the underlying manifold (intermediate). As a byproduct of our methods of proof we also show that, in some cases, the radial homogeneous Dirichlet problem in geodesics balls can have multiple positive solutions, in contrast with both the Euclidean and the hyperbolic space. |
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