Hénon-Lane-Emden Conjecture

• Hénon-Lane-Emden conjecture is a hypothesis establishing a precise critical hyperbola that governs the existence or nonexistence of positive solutions in weighted elliptic systems.
• It employs techniques like variational methods, weighted Pohozaev identities, and Hamiltonian ODE reduction to analyze scalar, polyharmonic, and fractional PDE cases.
• The conjecture is fully resolved for radial solutions and distributional cases, while challenges persist for nonradial and high-dimensional extensions.

The Hénon-Lane-Emden conjecture posits a Liouville-type nonexistence result for positive classical (and, in variants, weak or distributional) solutions of nonlinear elliptic systems of Hénon-Lane-Emden type, especially under criticality conditions tied to weighted Sobolev exponents and the system's scaling. In its archetypal form, the conjecture predicts an exact threshold—the “critical hyperbola”—below which no nontrivial positive solutions exist, and on or above which nontrivial solutions emerge. This threshold is governed by a precise weighted balance between nonlinearity, weights, solution structure, and spatial dimension, and underpins a unifying theory encompassing classical, higher-order, and fractional PDE systems, including both scalar and fully coupled settings.

1. Definition of the Hénon-Lane-Emden System and the Critical Hyperbola

The canonical weighted Hénon-Lane-Emden system in $\mathbb{R}^n\setminus\{0\}$ is

$$\left\{ \begin{array}{l} -\Delta u = |x|^a |v|^{p-2}v, \ -\Delta v = |x|^b |u|^{q-2}u, \end{array} \right.$$

with $u,v: \mathbb{R}^n\setminus\{0\}\to\mathbb{R}$, exponents $p,q>1$, and real weights $a,b$. The purely power-law case ($a=b=0$) recovers the classical Lane-Emden system.

The critical threshold for nonexistence is encapsulated by the (weighted) critical hyperbola: $\frac{n+a}{p} + \frac{n+b}{q} = n-2.$ For $1/p + 1/q < 1$ (“anticoercivity”), the conjecture asserts:

• If $\frac{n+a}{p} + \frac{n+b}{q} > n-2$ (subcritical), no positive solution exists.
• If equality holds (critical), positive radial solutions exist via variational methods.
• If $\frac{n+a}{p} + \frac{n+b}{q} < n-2$ (supercritical), infinitely many positive radial solutions exist, vanishing at infinity (Musina et al., 2013).

Extensions to higher-order (polyharmonic) or fractional Laplacians adjust the criticality condition accordingly, e.g. for the polyharmonic case with order $m$,

$\frac{n+a}{p+1} + \frac{n+b}{q+1} = n - 2m,$

and further generalizations include scalar cases, systems on balls or the half-space, and other boundary conditions (Phan, 2014, Dai et al., 2018, Cao et al., 2019).

2. Historical Development and Principal Conjectures

The Hénon-Lane-Emden conjecture emerged as a generalization of Liouville-type results for scalar Lane-Emden equations, incorporating power-type weights and considering both single equations and coupled systems. The conjecture was formulated to account for the interplay between nonlinear source terms of different components and their respective weights. The connection to the critical hyperbola arises from scaling and Pohozaev-type computations, identifying the precise threshold where the existence–nonexistence dichotomy occurs.

For the second-order system, the conjecture states: $$-\Delta u = |x|^a v^p, \quad -\Delta v = |x|^b u^q, \quad x\in\mathbb{R}^N,$$ predicts nonexistence for $(u,v)\geq 0$ if $(N+a)/(p+1) + (N+b)/(q+1) > N-2$ (Fazly et al., 2011, Musina et al., 2013, Huang et al., 18 Dec 2025).

Generalizations in higher order, governed by the polyharmonic operator $(-\Delta)^m$, adjust the threshold to $(N+a)/(p+1) + (N+b)/(q+1) > N-2m$ (Phan, 2014, Fazly, 2013).

Fractional and higher-order analogues utilize generalized exponents and weights with the same principle (Cao et al., 2019).

3. Main Existence and Nonexistence Results

For radial solutions, the conjecture is settled completely:

• No nontrivial, nonnegative radial solution exists if and only if $\frac{n+a}{p} + \frac{n+b}{q} > n-2$ (Musina et al., 2013).
• On the critical hyperbola ($=$), nontrivial radial solutions exist—constructed variationally via Emden-Fowler and Hamiltonian reduction to systems of ODEs (Musina et al., 2013).
• In the supercritical regime ($<$), classical results (Serrin-Zou, Hulshof-van der Vorst, Lions) guarantee infinitely many positive radial solutions vanishing at infinity.

Nonradial cases present far greater technical difficulty:

• For the second-order system in low dimension ($n\leq 4$), or for bounded solutions in $n=3$, the conjecture holds (Fazly et al., 2011).
• For the polyharmonic case $m\geq 1$, the conjecture is fully settled in the radial class for all $n>2m$ and for all classical solutions in $n\leq 2m+1$ (Fazly, 2013). In higher dimensions, only partial results are known and additional technical restrictions are needed (Phan, 2014).

Variants for very weak (distributional) solutions yield a completely sharp threshold for existence/nonexistence, with extensions to more general inequalities and inhomogeneous cases (Carioli et al., 2015).

In fractional or higher-order settings, nonexistence results (Liouville theorems) are obtained under analogous criticality conditions using nonlocal integral and scaling sphere methods. The critical hyperbola is explicitly characterized, and scaling sphere techniques yield both decay estimates and nonexistence in the subcritical regime (Cao et al., 2019).

4. Methodologies: Proof Techniques and Analytical Tools

The principal methodologies include:

• Weighted Pohozaev and Rellich-Pohozaev Identities: Extended to encompass higher order and weighted systems, these identities allow derivation of algebraic constraints from integrating bulk terms against well-chosen test functions and analyzing boundary decay (Fazly, 2013, Phan, 2014).
• ODE Hamiltonian Reduction: For radial solutions, the Emden–Fowler transform reduces the PDE system to a Hamiltonian system of second-order ODEs in logarithmic variables, admitting conserved energies under the critical hyperbola and enabling direct variational construction (Musina et al., 2013).
• Decay and Integral Estimates: Global integral inequalities and rescaling arguments (Morrey-type, measure–feedback) yield decay estimates for solutions and their derivatives, leading to contradiction in the nonexistence regime (Phan, 2014, Cao et al., 2019).
• Stability and Morse Index Arguments: For solutions stable outside compact sets or with finite Morse index, additional Liouville theorems are found, often via multiplier techniques for the linearized operator and iteration of stability inequalities (Fazly et al., 2011, Huang et al., 18 Dec 2025).
• Scaling Sphere Method: In fractional and nonlocal contexts, the scaling sphere (Kelvin transform) method is used to obtain monotonicity and lower bound estimates, ultimately forcing nonexistence under subcriticality via bootstrapping arguments (Cao et al., 2019, Dai et al., 2018).
• Variational Construction: On the critical hyperbola, minimization of functionals in weighted Sobolev spaces, exploiting compactness and the symmetric structure, leads to ground state radial solutions (Musina et al., 2013).

5. Sharp Thresholds, Distributional Solutions, and Extensions

For very weak (distributional) positive supersolutions, the precise set of admissible weights $(a,b)$, given $(p,q)$, is captured by regions $E_{p,q}$ classified by coercivity:

• In the homogeneous case (critical hyperbola), explicit power-type solutions appear.
• The existence region is sharp: outside it, no nontrivial positive supersolutions exist, proved via Green’s function and nonlinear integral estimates (Carioli et al., 2015).

For fractional and higher-order polyharmonic systems, the adaptation of these techniques remains effective, and critical hyperbolae specify the exact Liouville threshold for system-wide nonexistence (Cao et al., 2019).

Boundary-value problems on domains and half-spaces connect the conjecture to a priori estimates and existence of bounded solutions, with nonexistence in unbounded space implying uniform bounds for solution families in bounded geometries (Dai et al., 2018).

6. Open Problems, Challenges, and Future Directions

While the conjecture is completely proved for radial solutions and for nonradial bounded solutions in low dimensions, significant open cases remain in high dimensions for unbounded or slowly decaying nonradial solutions without stability or boundedness constraints. Principal technical hurdles include:

• Lack of suitable Sobolev embedding for high-order and high-dimensional cases, impeding control of boundary terms (Phan, 2014).
• Absence of maximum principles for systems or higher-order operators, preventing direct comparison of $u$ and $v$ components.
• Complexity of the “measure–feedback” schemes for controlling disparate homogeneities introduced by weights $|x|^a$, $|x|^b$.

Further future directions include:

• Classification of nonradial and symmetry-breaking solutions on the critical hyperbola, especially via bifurcations in weighted Sobolev inequalities.
• Development of new trace-type or monotonicity formulas/identities appropriate for higher-order and system settings.
• A finer analysis of stability, uniqueness, Morse index, and multi-peak or sign-changing solutions in the critical regime.
• Complete resolution in the general case for all dimensions by relaxing technical boundedness or stability requirements (Huang et al., 18 Dec 2025).

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Key References (arXiv IDs):

• (Musina et al., 2013): Radial existence and nonexistence; ODE reduction.
• (Fazly, 2013): Polyharmonic conjecture—low and high dimension results.
• (Phan, 2014): Advanced Liouville-type theorems for polyharmonic systems.
• (Fazly et al., 2011): Liouville theorems in second and fourth order; stability and Morse index.
• (Carioli et al., 2015): Very weak (distributional) supersolutions, sharp threshold.
• (Cao et al., 2019): Fractional and higher-order extensions; scaling sphere method.
• (Dai et al., 2018): Critical order on the half-space.
• (Huang et al., 18 Dec 2025): Liouville-type theorems for stable solutions; recent advances.