Large Volume Nonexistence Results

• The paper establishes explicit critical thresholds beyond which minimizers or solutions fail to exist in variational problems involving nonlocal energies.
• It employs techniques such as geometric slicing, splitting arguments, and nonlocal Pohozaev identities to rigorously quantify these limits.
• The results illustrate how factors like hyperbolic geometry, Coulomb repulsion, and fractional operators drive the nonexistence phenomena in large-volume settings.

Nonexistence results for large volumes are foundational in the study of nonlocal variational problems, partial differential equations (PDEs), and geometric analysis. Such results assert that, above a certain threshold for the domain size or prescribed mass, no minimizer or nontrivial solution exists for the associated energy, PDE, or variational problem. These obstructions, depending on the interplay of local and nonlocal effects, have been sharply quantified in recent literature, with explicit bounds and rigorous proof strategies.

1. Formulation of Large-Volume Nonexistence in Variational Problems

Nonexistence results arise in several models where energetic or dynamical constraints become incompatible with large domain size or volume. Consider representative functionals:

• Nonlocal isoperimetric problem on $\mathbb{H}^n$: For $E\subset\mathbb{H}^n$ measurable with $|E|=m$, energy

$$\mathcal{F}_\alpha(E) = P_{\mathbb{H}^n}(E) + \gamma \iint_{E\times E} \frac{dx\,dy}{[d_{\mathbb{H}^n}(x,y)]^\alpha},$$

with perimeter $P_{\mathbb{H}^n}(E)$ and exponent $0<\alpha<2$ (Li et al., 14 Dec 2025).

• Liquid drop model: For $\Omega\subset\mathbb{R}^3$ with $|\Omega|=A$,

$$E(\Omega) = \mathrm{Per}(\Omega) + \frac{1}{2}\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|},$$

minimizing over sets of fixed volume (Frank et al., 2016).

• Integro-differential PDEs: For $L$ of order $2s$ (fractional Laplacian or similar), and bounded domain $\Omega$,

$$\begin{cases} Lu = f(x,u) & \text{in }\Omega, \ u=0 & \text{in }\mathbb{R}^n\setminus\Omega, \end{cases}$$

where nonexistence for large $|\Omega|$ follows for certain supercritical power nonlinearities and eigenvalue problems (Ros-Oton, 2017).

The central result in each setting is the existence of a critical mass/volume or parameter beyond which minimizers or solutions cannot exist.

2. Mechanisms and Geometric Effects Leading to Nonexistence

Underlying these results are geometric and analytic mechanisms that render solution existence for large volumes impossible:

• Splitting and Escape-to-Infinity: In hyperbolic space $\mathbb{H}^n$, negative curvature accelerates the "mass escape" effect, as geodesic distances expand exponentially; splitting a set along horospheres incurs vanishingly small nonlocal energy cost for large $|E|$. For $\alpha<2$, this effect dominates, prohibiting minimizers at high volume (Li et al., 14 Dec 2025).
• Coulomb Energy in Liquid Drop Model: The competition between perimeter (surface tension) and nonlocal Coulomb repulsion grows in favor of the latter as the nucleus size increases, resulting in fragmentation or nonexistence above the explicit threshold $A^*\leq 8$ (Frank et al., 2016).
• Pohozaev Identity and Fractional Trace Inequalities: For fractional PDEs, integral identities combine with sharp trace inequalities to show that, for supercritical exponents, or domains beyond explicit volume bounds (depending on the eigenvalue), only the trivial solution is possible on star-shaped sets (Ros-Oton, 2017).

3. Precise Statements of Main Nonexistence Theorems

The quantitative nature of these results allows explicit thresholds:

Model	Nonexistence Threshold	Reference
Nonlocal isoperimetric on $\mathbb{H}^n$	$m \geq m_0(n,\alpha,\gamma)$ for $0<\alpha<2$	(Li et al., 14 Dec 2025)
Liquid drop model	$A > 8$	(Frank et al., 2016)
Fractional Laplacian eigenvalue	$|\Omega| > V_{*}(\lambda)$ (explicit in parameters)	(Ros-Oton, 2017)
Supercritical power $p\geq p_*(n,s)$	Any $|\Omega|$ for star-shaped $\Omega$	(Ros-Oton, 2017)

In each case, the threshold is determined by the dimensional constants, interaction exponents, and underlying geometric features.

4. Methods of Proof and Analytical Tools

The proof strategies exploit convexity, comparison principles, and sharp interpolation or slicing arguments:

• Slicing and Splitting: Both (Frank et al., 2016) and (Li et al., 14 Dec 2025) employ a splitting of sets (using hyperplanes in $\mathbb{R}^3$ or horospheres in $\mathbb{H}^n$) and comparison of interface energies to cross-term nonlocal energies. Averaging over directions and applying the coarea or Fubini-type identities facilitate sharp global inequalities.
• Differential Inequalities: In hyperbolic nonlocal isoperimetric problems, differential inequalities for mass functions $U(t)$, arising from geometric decompositions and the coarea formula, are integrated to produce contradictions for large volume (Li et al., 14 Dec 2025).
• Nonlocal Pohozaev Identities: For integro-differential operators, the nonlocal Pohozaev identity equates volume integrals to boundary terms. By exploiting sharp boundary regularity results $u/d^s\in C^{s-\varepsilon}(\overline{\Omega})$ and trace inequalities, one deduces impossibility of nontrivial solutions above critical exponents or volumes (Ros-Oton, 2017).

5. Sharpness, Open Problems, and Comparison Across Settings

The thresholds established are either conjecturally sharp or the best quantitative bounds currently available:

• In the liquid drop model, $A^*\leq8$ is the first explicit constant, within a factor of $2.3$ of the conjectured critical volume $A_c\approx3.518$, but closing this gap remains open (Frank et al., 2016).
• For the nonlocal isoperimetric problem on $\mathbb{H}^n$, existence/nonexistence is resolved for $0<\alpha<2$. The borderline case $\alpha=2$ is open as the method hinges on tail decay and diameter estimates failing at the endpoint (Li et al., 14 Dec 2025).
• For fractional Laplacian Dirichlet problems, thresholds in both the eigenvalue and supercritical power setting are known to be essentially sharp. In the supercritical regime, any star-shaped domain yields nonexistence, while in the linear case, the threshold matches the scaling of the first eigenvalue for balls (Ros-Oton, 2017).

A plausible implication is potential for similar techniques in further classes of nonlocal PDEs and multi-phase variational problems, pending refinement of slicing or interpolation arguments.

6. Related Problems and Extensions

Analogous large-volume nonexistence results hold in other geometric and analytic frameworks:

• In Euclidean space, similar competition between local and nonlocal terms leads to nonexistence for certain energy functionals, but the decay mechanisms differ, involving algebraic (rather than exponential) tail energies (Li et al., 14 Dec 2025).
• Extensions to more general nonlinearities, including sign-changing or mixed terms, are possible by combining the Pohozaev-type integral identities with sharper functional inequalities, as in (Ros-Oton, 2017).
• Loss of compactness phenomena—such as fragmentation into multiple droplets or mass escaping to infinity—describe the failure mode at or beyond the critical threshold, an area of ongoing research (Frank et al., 2016).

These results collectively illustrate the rigorous constraints imposed by geometry, nonlocality, and scaling on the existence of minimizers and solutions in large domains.