The critical boundary between existence and nonexistence for Lane–Emden systems

← All digests

Family 370 · Partial differential equations
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Can two positive fields sustain one another everywhere in space when each acts as the source of the other’s elliptic equation? The Hénon–Lane–Emden system asks this through

$$-\Delta u=|x|^A v^p, \qquad -\Delta v=|x|^B u^q.$$

The powers $p,q>0$ describe nonlinear coupling. The real numbers $A,B$ make the source strength depend on distance from the origin. Setting $A=B=0$ gives the unweighted Lane–Emden system.

The conjecture predicts that one inequality in the exponents forbids positive solutions on all of $\mathbb R^n$, with no need to assume symmetry, boundedness or decay. Family 370 claims this nonexistence theorem in every dimension $n\ge2$, including singular weights and powers below one.

Together with an earlier radial existence theorem, the claim gives an exact existence–nonexistence classification when $n\ge3$ and $A,B>-2$. The new manuscript supplies the nonexistence half; it does not claim to have newly constructed the complementary radial solutions.

2. Background and essential definitions

Entire positive solutions

The Laplacian is $\Delta u=\sum_{i=1}^n\partial_i^2u$. An equation $-\Delta u=f>0$ says that $u$ is superharmonic: its local curvature is constrained by a positive source. Superharmonic functions need not be constant, and in higher dimensions a source can generate a positive decaying potential. Positivity alone therefore does not settle the coupled problem.

An entire solution is defined throughout the whole space, rather than just a bounded domain. The solution class here is

$$u,v\in C^2(\mathbb R^n\setminus\{0\})\cap C(\mathbb R^n), \qquad u,v>0\quad\text{on }\mathbb R^n.$$

The equations hold classically away from zero. At zero the functions are only required to be continuous and strictly positive. This matters because a negative weight, such as $|x|^{-1}$, is singular there. No differentiability at the origin is initially assumed.

A Liouville-type theorem rules out nontrivial entire solutions in a specified class. Here the conclusion is that no strictly positive pair exists. It does not rule out solutions on bounded regions with boundary conditions, or functions singular at the origin outside this class.

The critical hyperbola

The claimed subcritical condition is

$$\frac{n+A}{p+1}+\frac{n+B}{q+1}>n-2.$$

Equality defines the critical hyperbola in the exponent plane. “Subcritical” refers to the strict inequality above; it does not mean simply that both powers are less than some universal number.

In the unweighted case this becomes

$$\frac1{p+1}+\frac1{q+1}>\frac{n-2}{n}.$$

On the diagonal $p=q=s$, for $n\ge3$, it reduces to $s<(n+2)/(n-2)$, the familiar scalar Sobolev critical exponent. The system has two potentially different powers, however, so a single scalar threshold is not enough.

Scaling reveals the relevant energy

In the nontrivial remaining range $pq>1$ and $A,B>-2$, choose positive numbers $\alpha,\beta$ satisfying

$$p\beta=\alpha+2+A, \qquad q\alpha=\beta+2+B.$$

Then the rescaled functions

$$u_R(x)=R^\alpha u(Rx), \qquad v_R(x)=R^\beta v(Rx)$$

solve the same weighted system. These dilations are centered at the origin; arbitrary translations would move the spatial weights and change the equation.

The two natural energy densities are $|x|^B u^{q+1}$ and $|x|^A v^{p+1}$. Their integrals scale with the same power $R^{\alpha+\beta+2-n}$. Subcriticality makes that exponent positive, which is the eventual source of the contradiction.

3. Development history and previous partial results

The scalar theory supplied the first major model. Gidas and Spruck established subcritical nonexistence for positive scalar solutions, along with local consequences. For systems, Mitidieri developed Rellich identities and obtained early nonexistence regions.

Serrin and Zou proved the unweighted three-dimensional assertion under a growth condition. Poláčik, Quittner and Souplet removed that additional condition, and Souplet settled the four-dimensional Lane–Emden conjecture. Higher dimensions remained more difficult because spherical estimates lose strength as dimension grows.

A recent partial result by Kui Li, Mingxiang Li and Juncheng Wei covers, for $n\ge5$ and $p,q\ge1$, a region satisfying

$$\frac1{p+1}+\frac1{q+1} \ge1-\frac2n+\frac4{n^2}.$$

This lies strictly inside the full conjectured subcritical range. Family 370 claims the remaining region as well, including exponents below one.

For the weighted problem, Bidaut-Véron and Giacomini identified the critical hyperbola in the regular radial theory. Phan formulated the unrestricted weighted question for $A,B>-2$ and proved bounded, low-dimensional and conditional cases.

Fazly and Ghoussoub obtained results under boundedness and stability assumptions for nonnegative weights. Li and Zhang proved the weighted three-dimensional conjecture without boundedness. Other work covered restricted exponent regions in dimensions four and five. Huang and Zou formulated the broader origin-continuous, all-real-weight assertion and proved cases with stability, decay or energy hypotheses.

The new claim removes those extra restrictions in the nonradial higher-dimensional problem. Its method continues the tradition of scaling and Rellich–Pohozaev identities, but obtains the needed local energy bound without first assuming finite total energy or a decay rate.

4. What the new manuscript claims

For every integer $n\ge2$, every $p,q>0$, and every real $A,B$, the manuscript claims that

$$\frac{n+A}{p+1}+\frac{n+B}{q+1}>n-2$$

excludes all strictly positive pairs in the origin-continuous solution class defined above.

The conclusion requires no radial symmetry, global boundedness, integrability, finite energy, stability, or prescribed growth or decay at infinity. With zero weights it yields the usual subcritical Lane–Emden nonexistence statement for positive classical entire solutions.

The source also makes preliminary exclusions independent of the subcritical inequality: a pair in this solution class would necessarily have $n\ge3$, $A,B>-2$, and $pq>1$. Continuity and positivity at zero are what make the weight restriction unavoidable; the two-dimensional exclusion follows from spherical averages and positivity at large radius.

For $n\ge3$ and $A,B>-2$, combining the new result with the cited radial existence theorem gives the exact classification:

$$\text{a positive entire pair exists} \quad\Longleftrightarrow\quad \frac{n+A}{p+1}+\frac{n+B}{q+1}\le n-2.$$

In the existence range a pair can be chosen radial, including equality on the critical hyperbola. This is an existence statement, not a claim that all positive solutions are radial or unique.

The broader all-real-weight formulation belongs to the nonexistence theorem. The complementary existence classification is specifically restricted to $n\ge3$ and $A,B>-2$.

5. Why the advance matters

Importance: 5/5. If correct, the manuscript resolves the full subcritical Lane–Emden conjecture and its weighted extension, central nonexistence questions for nonlinear elliptic systems.

Liouville theorems have consequences beyond whole-space equations. In a blow-up argument, a sequence of increasingly large solutions on bounded domains can be rescaled around its peaks. If a nontrivial entire limit would violate a Liouville theorem, such blow-up is impossible. This mechanism underlies a priori bounds and singularity estimates, although each application has its own hypotheses.

The weighted result also adds information that cannot be obtained by merely inserting the same powers into the unweighted theorem. Spatial weights alter the critical inequality, the scaling and the geometry of the source layers.

6. The main idea

Express each field through the other’s source

For $n\ge3$, let

$$K(x)=c_n|x|^{2-n}, \qquad c_n=\frac1{(n-2)|S^{n-1}|},$$

be the Newton kernel, the fundamental solution of $-\Delta$. Define source densities

$$f=|x|^B u^q, \qquad g=|x|^A v^p.$$

After the preliminary reductions, the manuscript obtains

$$u=K*g,\qquad v=K*f.$$

Convolution means integration against the translated kernel. Initially nonnegative additive constants may occur; source-mass estimates rule them out. These same estimates control source masses on origin-centered balls and distant contributions to potentials on a fixed ball.

The key issue is that source-mass control is weaker than energy control. Integrals of $f$ and $g$ do not immediately bound integrals of $fu$ and $gv$, because the potential may be large where its source concentrates.

Localize the scaling identity

The interaction between the two sources can be written as a nonnegative measure

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

Integrating out one variable produces $fu$ or $gv$. Thus the two energy densities are the two marginals of the same interaction.

A compactly supported radial cutoff gives a localized virial identity, a version of the Rellich–Pohozaev scaling identity. It retains the exact coefficients involving $n+A$ and $n+B$, together with terms generated near the cutoff’s edge.

Using a compact cutoff avoids assuming that boundary terms vanish at infinity. But it creates an obligation: those cutoff terms must be controlled without already knowing the energy bound being sought.

Compare intervals in weighted source layers

The manuscript decomposes each source into superlevel sets. This is the layer-cake idea: a nonnegative function is the integral of indicators of the sets where it exceeds a level.

Intersecting these sets with lines produces intervals. An elementary comparison of pairs of intervals relates their Newtonian interaction to endpoint contributions and a directional pressure term. The comparison is organized with nonnegative quantities, avoiding undefined subtraction when an interaction integral over an unbounded interval diverges.

Weights make the endpoints subtle. If $|x|^Bu(x)^q=t$, then

$$u(x)=\left(\frac{t}{|x|^B}\right)^{1/q},$$

which varies with the endpoint location. It is not the constant $t^{1/q}$ available in the unweighted problem.

The argument therefore uses short intervals and directions close to the local radial direction. On each retained interval the weight changes only slightly. Long or poorly positioned intervals are discarded with estimates based on polar integration, source masses and a high/low decomposition of source levels.

The resulting errors have the form $\varepsilon E+C_\varepsilon$, where $E$ is the localized energy and $C_\varepsilon$ is independent of the particular solution. Obtaining this estimate without assuming an upper bound on $E$ is the central claimed advance.

Absorb errors, then dilate

Let

$$\gamma=\frac{n+A}{p+1}+\frac{n+B}{q+1}-(n-2)>0.$$

The localized identity, pressure estimate and comparison of the two energy marginals lead to an inequality of the schematic form

$$\frac\gamma2 E\le C\varepsilon E+C_\varepsilon.$$

Choosing $\varepsilon$ small compared with the fixed positive gap $\gamma$ absorbs the energy term on the right. This yields a universal bound for the energy on the unit ball.

Apply that same bound to every rescaled solution $(u_R,v_R)$. The exact scaling relation is

$$\begin{aligned} &\int_{B_1}\bigl(|x|^B u_R^{q+1}+|x|^A v_R^{p+1}\bigr)\,dx\\ &\qquad=R^{\alpha+\beta+2-n} \int_{B_R}\bigl(|x|^B u^{q+1}+|x|^A v^{p+1}\bigr)\,dx. \end{aligned}$$

The exponent is positive under subcriticality. For $R\ge1$, the last integral is at least its fixed strictly positive value on $B_1$. The right side therefore grows without bound, contradicting the uniform estimate for the left side.

7. Scope, limitations, and open questions

Strict positivity includes the origin, and continuity there is part of the solution class. Singular profiles that blow up at zero, or different notions of weak solution, require separate analysis. Sign-changing solutions are also outside the theorem.

The subcritical inequality is strict. On the critical hyperbola the absorption gap disappears, and the cited radial theory supplies positive solutions in the admissible existence range. The nonexistence conclusion cannot be extended there unchanged.

The result concerns the second-order Laplacian with radial power weights. Higher-order, fractional or nonradial-weight systems require their own estimates. The existence classification does not describe every solution, its uniqueness or its detailed asymptotic behavior.

Applications to bounded-domain estimates likewise need explicit rescaling and compactness hypotheses. The manuscript supplies a strong entire-solution obstruction, rather than every possible application of that obstruction.

8. Sources and further reading

The Gidas–Spruck and Souplet papers linked above explain the scalar and low-dimensional precedents. Phan’s paper formulates the weighted classification, while Bidaut-Véron–Giacomini supply the radial existence theorem used for its complementary half.

For optional background, the Laplace operator and fundamental solutions explain the differential and potential formulations. The manuscript below is the source for the unrestricted subcritical nonexistence claim.

Source manuscripts

• M1. OpenAI, The Subcritical Hénon–Lane–Emden Conjecture (September 24, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.