Hénon-Lane-Emden System

• The Hénon-Lane-Emden system is a weighted elliptic PDE model with coupled nonlinearities, where the critical hyperbola determines the threshold between existence and nonexistence of solutions.
• It utilizes radial symmetry and the Emden-Fowler transformation to reduce the system to a Hamiltonian ODE framework, facilitating variational analysis.
• Existence and uniqueness results for positive radial solutions are established under stringent conditions like anticoercivity and specific weight exponents, with implications for decay and symmetry properties.

The Hénon-Lane-Emden system is a prototypical weighted elliptic system of the form

$-\Delta u = |x|^a v^p,\qquad -\Delta v = |x|^b u^q$

posed in $\mathbb{R}^N\setminus\{0\}$ or in $\mathbb{R}^N$, where $N\geq2$, $a,b\in\mathbb{R}$, and $p,q>1$. It generalizes both the scalar (weighted) Hénon equation and the Lane-Emden system, introducing a rich mathematical structure through the interplay of weights and nonlinearities. In the radial setting, solutions satisfy $u(x)=u(r)$, $v(x)=v(r)$, $r=|x|$. The critical curve known as the "critical hyperbola"

$\frac{a+N}{p+1} + \frac{b+N}{q+1} = N-2$

plays a central role in both existence and qualitative theory.

1. Problem Setting and Critical Hyperbola

For $N\geq2$, $a,b\in\mathbb{R}$, and $p,q>1$, the system is considered in $\mathbb{R}^N\setminus\{0\}$ under appropriate regularity and decay conditions. One usually imposes

• $u(r),v(r)\to0$ as $r\to\infty$,
• finiteness of certain weighted norms near $r=0$ or continuity at $r=0$ if $a,b>-N$.

The "anticoercivity" condition

$\frac{1}{p}+\frac{1}{q}<1$

ensures the associated variational quotients are finite.

The critical hyperbola,

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

marks the threshold for the existence of nontrivial positive solutions. The system is scaling-invariant under

$$u_\lambda(x)=\lambda^{-\alpha}u(\lambda x),\qquad v_\lambda(x)=\lambda^{-\beta}v(\lambda x),$$

with $\alpha,\beta$ determined by matching the scaling in the system, leading directly to the hyperbola condition (see (Musina et al., 2013)).

This curve separates subcritical ($<$), critical ($=$), and supercritical ($>$) regimes, deeply influencing existence/non-existence and qualitative properties.

2. Variational and Reduction Framework

In the radial case, for $a,b\in\mathbb{R}$, solutions are sought in weighted Sobolev spaces: $$D^{2,p'}(\mathbb{R}^N;|x|^a\,dx),\quad p'=\frac{p}{p-1},$$ with norm

$$\|u\|_{a,p'} = \left(\int_{\mathbb{R}^N} |x|^a |\Delta u|^{p'}\,dx \right)^{1/p'}.$$

These spaces embed continuously (but not compactly in the critical regime) into corresponding weighted Lebesgue spaces.

Via the Emden-Fowler transformation,

$$s=-\ln{r}, \quad u(r)=r^{-\lambda_1}g(s), \quad v(r)=r^{-\lambda_2}f(s),$$

where $\lambda_1 = \frac{b+N}{q}$ and $\lambda_2 = \frac{a+N}{p}$, the system reduces to a conservative Hamiltonian ODE system: $$\begin{cases} -g'' + 2A g' + T g = |f|^{p-2}f, \ -f'' - 2A f' + T f = |g|^{q-2}g, \end{cases}$$ with explicit parameters $A,T$ depending on $(p,q,a,b,N)$.

A "quotient minimization" problem is formulated in $W^{2,p'}(\mathbb{R})$,

$$\inf_{g\not\equiv0} \frac{\int_{\mathbb{R}} |g'' - 2A g' - T g|^{p'} ds}{\left(\int_{\mathbb{R}} |g|^q ds\right)^{p'/q}},$$

whose minimizers yield the required profile functions after inversion of the transform (Musina et al., 2013).

3. Existence and Qualitative Properties of Radial Solutions

Existence Theorem:

If $a+N\neq 0$, $b+N\neq 0$, $p,q>1$ satisfy both the critical hyperbola and the anticoercivity condition, then there exists a nontrivial radial solution,

$$(u,v) \in D^{2,p'}(\mathbb{R}^N; |x|^a\,dx) \times D^{2,q'}(\mathbb{R}^N;|x|^b dx),$$

vanishing at infinity. The explicit decay is,

$$u(r)=O\left(r^{-(a+N)/p}\right),\qquad v(r)=O\left(r^{-(b+N)/q}\right) \text{ as } r\to\infty.$$

Positivity: If $a,b>-N$, any radial $C^2$ solution decaying at infinity is either strictly positive or identically zero.

Uniqueness for $p=2$: By reducing to a fourth-order ODE and relying on works such as Bhakta–Musina, uniqueness (up to scaling, dilation, and sign) is guaranteed for each $(a,b,q)$ on the critical hyperbola when $p=2$, $a,b\neq-N$.

4. Nonexistence, Pohozaev Identity, and Thresholds

Nonexistence Outside the Hyperbola:

If

$\frac{a+N}{p+1} + \frac{b+N}{q+1} < N-2,$

no nontrivial positive radial $C^2$ solution exists [Mitidieri, Serrin–Zou].

Degenerate Weight Case:

If $a \leq -N$ (or $b \leq -N$) with specific limit conditions, any radial solution with $u\geq0,v>0$ must be identically zero. This is proven by analyzing the sign structure in the ODE via energy arguments.

Pohozaev Type Argument:

Although much of the theory in (Musina et al., 2013) is based on ODE/Hamiltonian analysis, nonexistence results can also be re-derived via a suitable Pohozaev identity: multiply each equation by $x\cdot\nabla u$ or $x\cdot\nabla v$, integrate over an annulus, and analyze the limits as radius tends to zero and infinity. Subcriticality forces all sufficiently decaying solutions to vanish.

5. Related Results and Broad Generalizations

• In the non-anticoercive regime or away from the critical hyperbola, variational approaches and fourth-order reductions can be utilized, leading to existence results even in weighted Rellich–Sobolev spaces, including for nonradial ("symmetry-breaking") solutions [(Musina et al., 2013), Appendix].
• The boundary between existence and nonexistence is sharp, coinciding with the critical hyperbola in both classical and very weak solution senses (see e.g. (Carioli et al., 2015) for supersolutions).
• For uniqueness, symmetry, and detailed behavior, reductions to ODEs and analysis of associated critical exponents are essential.

6. Implications and Open Problems

The radial theory for the Hénon-Lane-Emden system on the critical hyperbola is now fully characterized:

• Existence: nontrivial, strictly positive radial solutions exist for exponents on the hyperbola and under $\frac{1}{p}+\frac{1}{q}<1$;
• Qualitative structure: strictly positive, monotone decay, explicit algebraic rates at infinity;
• Uniqueness: established in special cases ($p=2$);
• Nonexistence: proven outside the hyperbola and at degenerate weights via both ODE and integral/Pohozaev methods.

The symmetric setting provides ground states, while suitable spectral conditions on the weight exponents may allow for symmetry-breaking (nonradial) ground states. Open directions include further classification of sign-changing and nonradial solutions, bifurcation analysis in Sobolev-critical or supercritical cases, and sharp thresholds for symmetry-breaking (Musina et al., 2013).