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On the subcritical Lane-Emden equation on Riemannian models with polynomial volume growth

Alessandra De Luca, Matteo Muratori, Nicola Soave

TL;DR

The paper investigates the Lane-Emden equation in the Sobolev-subcritical range on complete noncompact Riemannian manifolds with polynomial volume growth and nonpositive curvature, revealing new phenomena not present in Euclidean or hyperbolic spaces. It introduces two exponents, $\tilde{2}_{\alpha}$ and $2^*_{\alpha}$, tied to the asymptotic growth rate $\psi(r)\sim r^{\alpha}$ of model manifolds, and partitions the subcritical regime into strongly subcritical, slightly subcritical, and intermediate, with distinct existence and nonexistence behaviors. Key results show sharp nonexistence for $q\le\tilde{2}_{\alpha}$, existence of radial finite-energy solutions for $q$ in the slightly subcritical range, and a delicate intermediate regime where both existence and nonexistence can occur depending on the specific model function $\psi$; nonuniqueness in Dirichlet problems on balls also arises in this regime. The authors develop a blend of variational, Pohozaev-type, and model-manifold constructions, providing explicit psi examples and demonstrating rich geometric-analytic phenomena that interpolate between Euclidean and hyperbolic settings and affect spectral and embedding properties.

Abstract

We focus on the problems of existence and non-existence of positive solutions for the Sobolev-subcritical Lane-Emden equation on certain Riemannian manifolds (mainly models) with asymptotically negative curvature, which, from the viewpoint of the volume growth of geodesic balls, can be regarded as intermediate settings between the Euclidean and the hyperbolic spaces. A number of interesting phenomena arise: the subcritical regime naturally divides into three further ranges, characterized by existence phenomena (slightly subcritical), non-existence phenomena (strongly subcritical), and by a mixed behavior where existence and non-existence strongly depend on additional assumptions on the manifold (intermediate). In the intermediate regime, we further show that the radial homogeneous Dirichlet problem in geodesics balls may admit multiple positive solutions, thereby revealing substantial differences with respect to both the Euclidean and the hyperbolic settings.

On the subcritical Lane-Emden equation on Riemannian models with polynomial volume growth

TL;DR

The paper investigates the Lane-Emden equation in the Sobolev-subcritical range on complete noncompact Riemannian manifolds with polynomial volume growth and nonpositive curvature, revealing new phenomena not present in Euclidean or hyperbolic spaces. It introduces two exponents, and , tied to the asymptotic growth rate of model manifolds, and partitions the subcritical regime into strongly subcritical, slightly subcritical, and intermediate, with distinct existence and nonexistence behaviors. Key results show sharp nonexistence for , existence of radial finite-energy solutions for in the slightly subcritical range, and a delicate intermediate regime where both existence and nonexistence can occur depending on the specific model function ; nonuniqueness in Dirichlet problems on balls also arises in this regime. The authors develop a blend of variational, Pohozaev-type, and model-manifold constructions, providing explicit psi examples and demonstrating rich geometric-analytic phenomena that interpolate between Euclidean and hyperbolic settings and affect spectral and embedding properties.

Abstract

We focus on the problems of existence and non-existence of positive solutions for the Sobolev-subcritical Lane-Emden equation on certain Riemannian manifolds (mainly models) with asymptotically negative curvature, which, from the viewpoint of the volume growth of geodesic balls, can be regarded as intermediate settings between the Euclidean and the hyperbolic spaces. A number of interesting phenomena arise: the subcritical regime naturally divides into three further ranges, characterized by existence phenomena (slightly subcritical), non-existence phenomena (strongly subcritical), and by a mixed behavior where existence and non-existence strongly depend on additional assumptions on the manifold (intermediate). In the intermediate regime, we further show that the radial homogeneous Dirichlet problem in geodesics balls may admit multiple positive solutions, thereby revealing substantial differences with respect to both the Euclidean and the hyperbolic settings.

Paper Structure

This paper contains 12 sections, 23 theorems, 278 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Geometric background and main results
  3. The strongly subcritical regime
  4. Non-existence of nonnegative nontrivial supersolutions for $q \le \tilde{2}_\alpha$
  5. Existence of nonnegative nontrivial supersolutions for $q >\tilde{2}_\alpha$
  6. The slightly subcritical regime
  7. Examples of model functions satisfying the assumptions of Theorems \ref{['da capire']} and \ref{['exi-crit']}
  8. The intermediate regime
  9. Non-existence of solutions under assumption \ref{['hp 4 psi']}
  10. Examples of model functions satisfying the assumptions of Theorem \ref{['thm: non-ex int']}
  11. Existence of solutions on certain Cartan-Hadamard manifolds
  12. Non-uniqueness for the Dirichlet problem in geodesics balls

Key Result

Theorem 1.3

Let $\alpha>1$. Let $\mathbb{M}^n$ be a model manifold associated with a model function $\psi$ satisfying and for some $K,r_0>0$. Then, for every $q \in \left(1,\tilde{2}_\alpha \right]$, the Lane-Emden equation LE does not have nonnegative nontrivial supersolutions; namely, if $u \ge 0$ satisfies in the sense of distributions, then $u \equiv 0$.

Figures (1)

  • Figure 1: A representation of the graphs of $w_1$ (blue) and $w_2$ (red).

Theorems & Definitions (56)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 46 more