Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities

Technical Summary: Existence and Non-existence of Nonradial Solutions to an Elliptic Equation on Hyperbolic Space

Problem Statement
This article investigates the semilinear elliptic equation on the $N$-dimensional hyperbolic space $\mathbb{B}^N$ (modeled as the Poincaré ball):
$$-\Delta_{\mathbb{B}^N} u - \lambda u = |u|^{p-1}u, \quad u \in H^1(\mathbb{B}^N),$$
where $N \geq 3$, $\lambda > \frac{N(N-2)}{4}$, and $1 < p \leq 2^* - 1$ (with $2^* = \frac{2N}{N-2}$ being the critical Sobolev exponent). The study addresses two primary questions:

1. Existence: Does the equation possess nonradial, sign-changing solutions in the critical case ($p = 2^* - 1$) when the spectral parameter $\lambda$ lies in the range $\frac{N(N-2)}{4} < \lambda < \lambda_1$, where $\lambda_1 = \frac{(N-1)^2}{4}$ is the bottom of the $L^2$-spectrum of $-\Delta_{\mathbb{B}^N}$?
2. Non-existence: Can one establish the non-existence of non-trivial solutions for spectral values $\lambda > \lambda_1$ under specific symmetry constraints?

Methodology
The authors employ a combination of variational methods, conformal geometry, group theory, and spectral analysis.

• Conformal Transformation: The problem on the hyperbolic space is transformed into a weighted Euclidean Dirichlet problem on the unit ball $B \subset \mathbb{R}^N$. Using the conformal factor $h(x) = \frac{2}{1-|x|^2}$, the equation $(Eq_\lambda)$ is mapped to:
  $$-\Delta v - \tilde{\lambda} h^2 v = |v|^{2^*-2}v, \quad v \in H^1_0(B),$$
  where $\tilde{\lambda} = \lambda - \frac{N(N-2)}{4}$. This allows the application of Euclidean variational techniques.

• Symmetry and Equivariant Subspaces: To construct nonradial solutions, the authors restrict the variational problem to $\phi$-equivariant subspaces of $H^1_0(B)$. They utilize a closed subgroup $\Gamma \subset O(N)$ and a homomorphism $\phi: \Gamma \to \mathbb{Z}_2$. Functions $u$ in these subspaces satisfy $u(gx) = \phi(g)u(x)$, forcing sign changes and breaking radial symmetry. The existence of such subspaces relies on the construction of $n$ distinct equivariant subspaces with trivial intersections, based on the decomposition $N = 4n + m$.

• Existence Proof (Critical Case):

  • The authors define an equivariant Sobolev quotient $S_\phi$ and compare it to the corresponding Euclidean quotient $S_\phi^\infty$.
  • By constructing a test function based on the Aubin-Talenti bubbles (solutions to the Euclidean critical equation) and utilizing the improved decay estimates of these bubbles (derived via Kelvin transformation and equivariance), they prove the strict inequality $S_\phi < S_\phi^\infty$.
  • This strict inequality, combined with the concentration-compactness principle and the fact that the fixed point set of the group action is only the origin, ensures the compactness of minimizing sequences, yielding a non-trivial minimizer.

• Non-existence Proof ($\lambda > \lambda_1$):

  • For $\lambda > \lambda_1$, the equation is viewed as a Schrödinger-type eigenvalue problem $-(\Delta_{\mathbb{B}^N} + V)u = \lambda u$ with potential $V = -|u|^{p-1}$.
  • The authors invoke a result by Borthwick and Marzuola stating that if the potential $V$ decays sufficiently fast ($V = o(\rho^{-1})$), no $L^2$-eigenfunctions exist for $\lambda > \lambda_1$.
  • To verify this decay condition for sign-changing solutions, they introduce a Quantitative Orbit Packing (QOP) condition. This condition measures how many points can be packed into an orbit of a subgroup $\Gamma$ on the unit sphere.
  • They prove that if $|u|$ is $\Gamma$-invariant and $\Gamma$ satisfies the (QOP) condition, the solution $u$ must decay at a rate sufficient to satisfy the hypothesis of the non-existence theorem, thereby forcing $u \equiv 0$.

Key Contributions and Results

1. Existence of Nonradial Sign-Changing Solutions (Theorem 1.1):
   For dimensions $N = 4n + m$ ($n \geq 1, m \in \{0,1,2,3\}$) and parameters $\frac{N(N-2)}{4} < \lambda < \frac{(N-1)^2}{4}$ with critical exponent $p = 2^* - 1$, the equation possesses at least $n$ distinct nonradial sign-changing solutions. This result extends previous work on radial solutions and improves multiplicity results for lower dimensions.

2. Symmetry-Restricted Non-existence (Theorem 1.2):
   For $\lambda > \lambda_1$ and $1 < p \leq 2^* - 1$, if a solution $u$ satisfies $|u| \in H^1(\mathbb{B}^N)$ and is invariant under a subgroup $\Gamma \subset O(N)$ that satisfies the (QOP) condition, then $u \equiv 0$.

   • This result generalizes known non-existence theorems for radial solutions to a broad class of nonradial symmetric solutions.
   • It identifies specific group structures (e.g., products of orthogonal groups) that guarantee non-existence, while noting that groups with finite orbits do not satisfy the necessary packing condition.

Significance and Claims
The paper claims to be the first to address these specific directions regarding nonradial sign-changing solutions in the critical hyperbolic setting and symmetry-based non-existence for $\lambda > \lambda_1$.

• Improvement on Existing Literature: The existence result improves upon prior findings by Mancini, Sandeep, and others, who established existence for positive radial solutions or radial sign-changing solutions under different constraints. This work specifically targets the nonradial and sign-changing regime in the critical case.
• Novel Non-existence Mechanism: The non-existence result is significant because it moves beyond the standard Pohozaev identity (which fails for sign-changing solutions due to nodal sets) and comparison principles. Instead, it utilizes spectral theory and a geometric "orbit packing" condition to derive decay estimates, proving non-existence for a large class of symmetric functions, including an infinite-dimensional subspace of nonradial functions.

The authors maintain a modest tone, framing their work as establishing existence and multiplicity for specific spectral ranges and providing a partial non-existence result contingent on the symmetry group's geometric properties, rather than claiming a complete classification of all possible solutions.