The weighted subcritical Lane–Emden system

Source and verification

This article explains family 370 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Liouville theorems rule out entire solutions under structural hypotheses. Family 370 claims that the weighted subcritical hyperbola alone excludes positive solutions of the Hénon–Lane–Emden system, without symmetry, boundedness, or prescribed decay at infinity.

🏷️ Preliminaries: a coupled elliptic system

For $n\ge 2$, exponents $p,q>0$, and real weights $A,B$, consider

$$\begin{array}{c}-\Delta u=|x{|}^A{v}^p,\\ -\Delta v=|x{|}^B{u}^q.\end{array}$$

The right sides are positive, so both functions are superharmonic away from the origin. The weights may be singular there. The solution class is therefore $C^2({\mathbb{R}}^n\setminus \{0\})\cap C({\mathbb{R}}^n)$, with strict positivity also at the origin. When $A=B=0$, this is the unweighted Lane–Emden system.

The threshold is measured by

$$\delta =\frac{n+A}{p+1}+\frac{n+B}{q+1}-(n-2).$$

The word subcritical in this article means $\delta >0$. It refers to this coupled weighted relation, not merely to either exponent in isolation.

🏷️ The claimed nonexistence theorem

Family 370, as stated in the manuscript

If $\delta >0$, the system has no strictly positive entire solution in the origin-continuous class above. No radial symmetry, finite-energy, stability, boundedness, or growth hypothesis is required. (OpenAI, 2026)

The theorem is a strict-inequality result. The source combines it with a separate radial existence theorem to obtain an existence criterion when $n\ge 3$ and $A,B>-2$; that broader conclusion uses an additional input. It should not be confused with nonexistence on the equality line.

🏷️ Scaling and a scalar consistency check

Try the dilation $u_{\lambda }(x)={\lambda }^{a}u(\lambda x)$, $v_{\lambda }(x)={\lambda }^{b}v(\lambda x)$. Direct substitution gives the conditions

$$\begin{array}{c}pb-a=2+A,\\ qa-b=2+B.\end{array}$$

For $pq\ne 1$, solving yields

$$\begin{array}{rl}a& =\frac{2+A+p(2+B)}{pq-1},\\ b& =\frac{2+B+q(2+A)}{pq-1}.\end{array}$$

These formulas describe the dilation symmetry; they do not prove nonexistence, and one cannot use them unchanged when $pq=1$. The manuscript separately treats low-exponent regimes.

As a check, put $u=v$, $p=q=s$, and $A=B=0$ with $n\ge 3$. The inequality reduces to

$$\begin{array}{c}\frac{2n}{s+1}>n-2\\ ⟺\\ s<\frac{n+2}{n-2}.\end{array}$$

At the critical exponent, the explicit positive function

$$\begin{array}{c}w(x)=c_n(1+|x{|}^2{)}^{-(n-2)/2},\\ c_n=[n(n-2){]}^{(n-2)/4}\end{array}$$

satisfies $-\Delta w=w^{(n+2)/(n-2)}$, as radial differentiation verifies. Thus even the scalar special case requires the strict threshold.

🏷️ Localized virial estimates and weighted layers

Classical Pohozaev identities compare scaling of gradient energy and nonlinear energy. On the whole space, applying them directly can leave boundary terms at infinity whose decay is unknown. Assuming those terms vanish would introduce an extra hypothesis absent from the stated theorem.

The manuscript instead uses a compactly supported potential-kernel virial identity. Origin-centered dilations control source masses on a fixed ball, while the localized identity retains the exact coefficients of the spatial weights. The goal is an a priori localized-energy bound coming from the positive gap $\delta$.

The weighted problem creates an additional obstacle: translations change $|x{|}^A$ and $|x{|}^B$. A superlevel set of $|x{|}^B{u}^q$ does not have constant $u$ on its endpoints. The proposed interval comparison uses directions near the radial direction and intervals short enough that weight variation stays small. Discarded intervals are estimated by polar integration and a high/low layer decomposition.

The crucial estimate bounds all errors by an arbitrarily small multiple of the energy being estimated, plus a fixed constant. This must be proved before assuming that energy is bounded. Absorbing the small relative error using $\delta >0$ then gives the bound; origin-centered dilation supplies the contradiction.

This order avoids circular reasoning. The threshold supplies an energy estimate only after the weighted interval errors are controlled independently of the energy’s size.

🏷️ Key steps and proof outlines

Use a locally finite interaction measure instead of total energy

In the range $n\ge 3$, $A,B>-2$, $pq>1$, set $f=|x{|}^B{u}^q$ and $g=|x{|}^A{v}^p$. The source first establishes exact Newton-potential representations $u=K\ast g$, $v=K\ast f$, with $K(x)=((n-2)|S^{n-1}|{)}^{-1}|x{|}^{2-n}$. Introduce

$$d\pi (x,y)=f(x)g(y)K(x-y)dxdy.$$

Its marginals are $fudx$ and $gvdy$. Its total mass need not be finite. All subsequent identities use compact cutoffs and prove absolute convergence before exchanging integrations.

Choose a smooth radial cutoff $\eta$, nonincreasing in $|x|$, equal to one on $B_1$ and vanishing outside $B_2$, with the boundary decay specified below; set $X=x\eta$, $w=-x\cdot \nabla \eta \ge 0$, and write $E_f=\int \eta fu$, $E_g=\int \eta gv$, $E=E_f+E_g$. To avoid confusing reciprocal exponents with the dilation exponents above, put $a_0=1/(q+1)$ and $b_0=1/(p+1)$. Integration by parts in the two potential formulas gives

$$\begin{array}{r}{a}_0(n+B)E_f+b_0(n+A)E_g-a_0{W}_f-b_0{W}_g\\ =(n-2)\int \Phi d\pi ,\end{array}$$

where $W_f=\int wfu$, $W_g=\int wgv$, and $\Phi =(X(x)-X(y))\cdot (x-y)/|x-y{|}^2$. The powers of the cutoff are chosen so that localization errors near its outer boundary remain integrable with constants independent of the solution.

Close pairs use Taylor expansion; distant pairs use potential tails

The derivative of $X$ is $\eta I-w{\omega }_x\otimes {\omega }_x$, where ${\omega }_x=x/|x|$. Thus its leading interaction is $\eta (x)-w(x)D(x,y)$, with $D=((x-y)\cdot {\omega }_x{)}^2/|x-y{|}^2$.

Set $d(x)=\min \{1,(2-|x|{)}_{+}\}$ and use the local comparison length $\ell (x)=\delta d(x{)}^{n+3}$. For close pairs, $d$ stays comparable along their segment and Taylor’s remainder is at most a small multiple of the cutoff. The marginal identities charge its integral to $C\delta E$. For distant pairs, the lower bound $|x-y|>\ell (x)$ permits use of the source’s uniform potential-tail estimate. Sufficient vanishing of $\eta$ makes the resulting powers of $d$ positive. This proves

$$|E_f-E_g|+\int |\Phi -\eta (x)+w(x)D|d\pi \le \epsilon E+C_{\epsilon }.$$

It is a relative estimate, not an assumed bound on $E$.

The interval comparison retains the midpoint displacement

On two parallel lines, write their separation as $\lambda >0$ and longitudinal difference as $h=s-s^{\prime }$. The restricted Newton kernel is proportional to $({\lambda }^2+h^2{)}^{(2-n)/2}$. For bounded intervals $I,J$ with midpoint difference $d_0$, two integrations by parts express their endpoint-average defect through

$$(n-2){\int }_{I\times J}\frac{h^2-d_{0}h}{{\lambda }^2+h^2}\mathcal{K}(h)dsd{s}^{\prime }.$$

The pushforward density of $h$ is the overlap length $|I\cap (J+h)|$, symmetric and nonincreasing away from $d_0$. Pairing $h$ with $-h$ shows that the integral of the $d_{0}h$ term is nonnegative. Dropping it therefore gives the required upper bound by the directional pressure term. Unequal nonnegative interval weights produce the additional error $|c_I-c_J|\mathcal{K}$. The source keeps this comparison in nonnegative form until global finiteness is established, especially because unbounded line integrals can diverge in dimension three.

Weighted source layers recover the correct nonlinear coefficient

For each direction, decompose $\{f>t\}$ into intervals. Retain only short bounded components and weight each by the infimum of a localized angular cutoff. Directions are restricted to a cap near ${\omega }_x$ of radius proportional to $d(x)$, with exact average weight $w(x)$. At a retained endpoint, $f=t$ gives $u=(t/|x{|}^B{)}^{1/q}$, up to the controlled variation of $|x{|}^B$ along that short interval.

The exact layer identity is

$${\int }_0^{f(x)}\left[u(x)-(t/|x{|}^B{)}^{1/q}\right]dt=a_{0}f(x)u(x).$$

Long intervals force a whole radial segment inside a high source level; polar integration bounds their directional probability. Splitting high and low levels makes the lost weight at most $(C_{\kappa }\delta +C_{\kappa ,\delta }/P)E+CP$. Choose cap width $\kappa$, then interval scale $\delta$, then level threshold $P$. This produces the pressure estimate with arbitrarily small relative error.

Combining pressure and localization leaves $\delta E/2$ on the left, where $\delta >0$ is the subcritical gap already defined above. Absorption gives a universal local energy bound. Applying it to every origin-centered dilation multiplies the energy on $B_R$ by $R^{a+b+2-n}$, with $a+b+2-n>0$. Its positive mass on $B_1$ then contradicts that uniform bound as $R\to \infty$. No unproved decay at infinity is inserted. (OpenAI, 2026 s “A localized virial identity,” “An interval-pair inequality,” “Pressure from weighted source layers,” and “The energy bound and the scaling contradiction”)

See Also

• Family 375 is another entire-solution classification in which an energy-growth bound must be derived.
• Family 371 explains how scaling distinguishes energy control from control of a stronger regularity norm.

🏷️ Manuscripts and verification scope

• The Subcritical Hénon–Lane–Emden Conjecture — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page describes the origin-continuous weighted nonexistence theorem with the strict subcritical inequality. The critical equality case is outside its selected statement. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The Subcritical Hénon–Lane–Emden Conjecture.