Lane–Emden Systems Overview

• Lane–Emden systems are coupled nonlinear differential equations defined by mutual power-law interactions and critical threshold geometry.
• They provide a framework to analyze existence, uniqueness, stability, symmetry, and concentration phenomena in classical, fractional, and higher-order variants.
• Applications span Liouville-type theorems, Dirichlet blow-up analyses, and nonlocal formulations in MEMS, Hamiltonian, and singular systems.

Lane–Emden systems are coupled nonlinear differential systems whose canonical semilinear elliptic form is

$-\Delta u=v^p,\qquad -\Delta v=u^q,$

posed either in $\mathbb R^N$, in a bounded domain with Dirichlet or Navier conditions, or in nonlocal, quasilinear, and fully nonlinear variants. Their analysis is organized by critical exponent geometry, especially the Sobolev critical hyperbola, together with questions of existence, nonexistence, uniqueness, stability, symmetry, concentration, and asymptotic behavior. In current usage, the subject includes fractional, logarithmic, $p$-Laplacian, Pucci, higher-order, Hamiltonian, singular, and free-boundary analogues, all retaining the characteristic mutual power-law coupling between components.

1. Classical semilinear model and critical structures

The standard Lane–Emden system on $\mathbb R^N$ is

$-\Delta u=v^p,\qquad -\Delta v=u^q,$

with positive exponents and positive solutions. A central structural feature is scale invariance when $pq\neq 1$: if $(u,v)$ is a solution, then

$(R^\alpha u(Rx),\,R^\beta v(Rx))$

is also a solution, where

$$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$

These exponents recur throughout the theory in decay estimates, singular profiles, and dynamical rescalings (Li et al., 2018).

The first major threshold is the Sobolev critical hyperbola

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

In the classical radial theory, the region

$\mathbb R^N$0

is the regime in which singular radial profiles of the form

$\mathbb R^N$1

appear, with $\mathbb R^N$2 chosen so that the pair solves the system away from the origin (Chen et al., 2013). In subcritical Liouville problems, the opposite inequality,

$\mathbb R^N$3

is the natural nonexistence range conjectured for nontrivial nonnegative entire solutions (Li et al., 2018).

The same geometry persists in several extensions. For the fractional Laplacian on a bounded domain, the critical hyperbola becomes

$\mathbb R^N$4

reflecting fractional Sobolev scaling (Leite et al., 2015). For the critical Hamiltonian system on perforated bounded domains, the equality

$\mathbb R^N$5

again marks the loss of compactness and the onset of Coron-type concentration phenomena (Jin et al., 2022). Across these settings, the critical hyperbola functions as the basic organizing threshold for solvability and asymptotics.

2. Entire-space stability and Liouville theory

For entire radial solutions, stability introduces a second layer of criticality beyond mere solvability. In the radial setting, a positive solution $\mathbb R^N$6 is called stable when the linearized system admits a positive supersolution $\mathbb R^N$7, and radial monotonicity gives

$\mathbb R^N$8

The exact stability threshold is the Joseph–Lundgren critical curve

$\mathbb R^N$9

with $p$0 as above. If $p$1 and this inequality holds, stable positive radial solutions exist and satisfy

$p$2

in $p$3; if $p$4, or if $p$5 and the inequality fails, no stable radial solution exists. The same inequality is equivalent to stability of the singular solution $p$6, and its derivation is tied to a sharp weighted Hardy–Rellich inequality with optimal constant

$p$7

(Chen et al., 2013).

Liouville-type nonexistence occupies the complementary side of the subject. The Lane–Emden conjecture asserts that no nontrivial nonnegative classical solution exists in the subcritical regime

$p$8

A proved region is the whole subcritical range with $p$9. Another proved region is given by $\mathbb R^N$0, or by $\mathbb R^N$1 together with

$\mathbb R^N$2

The proof uses Sobolev embeddings on $\mathbb R^N$3, scale invariance, and Pohozaev-type inequalities on spheres (Li et al., 2018).

A related half-space Liouville theorem excludes positive Dirichlet solutions in

$\mathbb R^N$4

for every $\mathbb R^N$5, under the weaker assumption of boundedness on finite strips

$\mathbb R^N$6

The argument combines moving planes, stripwise gradient bounds, convexity in the normal direction, and a final contradiction showing that no such positive classical solution can exist (Li et al., 2024). This extends earlier half-space nonexistence results that required either global boundedness or exponent restrictions.

3. Bounded-domain Dirichlet problems, blow-up, and concentration

On bounded smooth domains, Lane–Emden systems display a delicate interplay between compactness and concentration. For minimal-energy solutions of

$\mathbb R^N$7

with $\mathbb R^N$8 approaching the critical hyperbola from below through

$\mathbb R^N$9

the blow-up point remains strictly interior even on nonconvex domains. This removes the convexity assumption used in earlier work. The proof replaces moving-plane boundary control with a contradiction argument based on local Pohozaev identities and sharp Green-function asymptotics near the boundary (Choi, 2015).

In dimension two, the large-exponent regime is governed less by the classical critical hyperbola than by balance between the two exponents. For positive classical solutions in a smooth bounded planar domain, if

$-\Delta u=v^p,\qquad -\Delta v=u^q,$0

and the exponents are comparable, for example $-\Delta u=v^p,\qquad -\Delta v=u^q,$1 or $-\Delta u=v^p,\qquad -\Delta v=u^q,$2, then $-\Delta u=v^p,\qquad -\Delta v=u^q,$3 bounds are uniform in $-\Delta u=v^p,\qquad -\Delta v=u^q,$4. When

$-\Delta u=v^p,\qquad -\Delta v=u^q,$5

both components are uniformly bounded, and in star-shaped domains the energy

$-\Delta u=v^p,\qquad -\Delta v=u^q,$6

is also uniformly bounded. The comparability hypothesis is essential: in the disk with $-\Delta u=v^p,\qquad -\Delta v=u^q,$7 and $-\Delta u=v^p,\qquad -\Delta v=u^q,$8, one has $-\Delta u=v^p,\qquad -\Delta v=u^q,$9 but

$pq\neq 1$0

so uniform boundedness fails for the second component (Kamburov et al., 2022).

At the critical hyperbola, perforated geometry restores solvability. In

$pq\neq 1$1

with $pq\neq 1$2,

$pq\neq 1$3

there exists a family of positive solutions concentrating around the center of the hole as $pq\neq 1$4. The approximate profile is built from the entire-space ground state, projected onto the punctured domain and corrected by Lyapunov–Schmidt reduction. The reduced energy has a nondegenerate saddle point, producing what is described as the first existence result for the critical Lane–Emden system on a bounded domain (Jin et al., 2022).

Beyond criticality, suitable domains admit supercritical boundary layers. For smooth bounded $pq\neq 1$5, $pq\neq 1$6, and

$pq\neq 1$7

there are positive solutions concentrating along one or several $pq\neq 1$8-dimensional submanifolds of $pq\neq 1$9 as the exponent pair approaches a lower-dimensional critical hyperbola. The analysis proceeds by symmetry reduction to a weighted $(u,v)$0-dimensional problem, $(u,v)$1, followed by blow-up analysis and finite-dimensional reduction (Guo et al., 2023).

4. Nonlocal and fractional systems

The fractional Lane–Emden system on a smooth bounded domain $(u,v)$2 is

$(u,v)$3

with $(u,v)$4. In this nonlocal setting, the natural Dirichlet condition is prescribed on the whole complement, and standard local tools must be replaced by fractional regularity, maximum principles, and Pohozaev-type identities. The sharp trichotomy is as follows: if $(u,v)$5, there is a unique positive viscosity solution; if $(u,v)$6 and

$(u,v)$7

there exists at least one positive viscosity solution; and if $(u,v)$8 is star-shaped with

$(u,v)$9

then no positive viscosity solution exists. The nonexistence proof is based on a fractional Rellich identity, while existence is obtained variationally after reducing the system to a scalar nonlocal functional problem (Leite et al., 2015).

A broader Hamiltonian version replaces the pure powers by

$(R^\alpha u(Rx),\,R^\beta v(Rx))$0

with $(R^\alpha u(Rx),\,R^\beta v(Rx))$1. Under the growth conditions

$(R^\alpha u(Rx),\,R^\beta v(Rx))$2

and additional subcritical restrictions when $(R^\alpha u(Rx),\,R^\beta v(Rx))$3, there exists a weak solution in a mixed fractional space

$(R^\alpha u(Rx),\,R^\beta v(Rx))$4

for a suitable $(R^\alpha u(Rx),\,R^\beta v(Rx))$5. The functional is strongly indefinite, and the proof adapts the De Figueiredo–Felmer method to the fractional setting through spectral calculus for the Dirichlet fractional Laplacian and a linking argument on the splitting $(R^\alpha u(Rx),\,R^\beta v(Rx))$6 (Dussel et al., 20 Jan 2025).

Another nonlocal extension replaces $(R^\alpha u(Rx),\,R^\beta v(Rx))$7 by the logarithmic Laplacian $(R^\alpha u(Rx),\,R^\beta v(Rx))$8, the formal derivative of $(R^\alpha u(Rx),\,R^\beta v(Rx))$9 at $$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$0. For

$$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$1

every positive solution with algebraic decay at infinity is radially symmetric and monotone decreasing about some point. The proof requires a direct moving-plane theory specifically adapted to $$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$2, including a maximum principle for antisymmetric functions, a narrow region principle, and a decay-at-infinity principle (Zhang et al., 2022).

5. Higher-order, quasilinear, and fully nonlinear generalizations

The Lane–Emden paradigm extends to critical-order and polyharmonic systems. In bounded, smooth, strictly convex domains $$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$3, $$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$4, the higher critical-order system

$$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$5

with Navier conditions admits uniform $$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$6 a priori estimates provided $$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$7 and one exponent is controlled by the other, for example $$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$8 or $$\alpha=\frac{2(p+1)}{pq-1},\qquad \beta=\frac{2(q+1)}{pq-1}.$$9. The proof combines a local moving-planes method near the boundary, boundary monotonicity of all intermediate Laplacian iterates, Harnack inequalities, and precise relations among the maxima of $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$0, $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$1, $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$2, and $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$3 (Dai et al., 2022).

In quasilinear measure-data problems, the prototype becomes

$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$4

with $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$5, $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$6, and $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$7 a nonnegative Radon measure. Here the natural framework is nonlinear potential theory. Existence is characterized by iterated Wolff-potential conditions, and solutions satisfy sharp global pointwise estimates of Brezis–Kamin type in terms of $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$8. The same approach also yields corresponding results for fractional Laplacian systems, showing that in the sub-natural regime the correct threshold is potential-theoretic rather than variational (Silva et al., 2024).

Fully nonlinear Lane–Emden systems replace $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$9 by operators $\mathbb R^N$00 satisfying ellipticity, homogeneity, and regularity assumptions. In the critical homogeneity case $\mathbb R^N$01, the system exhibits two principal half-eigenvalues and hence two principal spectral curves

$\mathbb R^N$02

in the $\mathbb R^N$03-plane, corresponding to positive and negative eigenfunction pairs. A third spectral curve associated with a second eigenvalue may also appear, together with an anti-maximum principle. In the genuinely sublinear regime $\mathbb R^N$04, the inhomogeneous Dirichlet problem has a unique positive viscosity solution for every $\mathbb R^N$05 (Santos et al., 2020).

Radial fully nonlinear systems driven by the Pucci extremal operators admit a complementary dynamical-systems description. After transforming radial solutions into a four-dimensional quadratic autonomous flow, one obtains uniqueness of positive radial regular solutions up to scaling in $\mathbb R^N$06, uniqueness in balls, exclusion principles separating ball solutions from entire solutions, and explicit regions of existence and nonexistence. In the Laplacian case, the classical critical hyperbola is also the threshold for existence and nonexistence of radial exterior-domain solutions with Neumann boundary conditions (Maia et al., 2021).

6. Singular nonlinearities and free-boundary formulations

Lane–Emden coupling also appears with singular nonlinearities. A MEMS-motivated system is

$\mathbb R^N$07

in a smooth bounded domain. There exists a critical curve $\mathbb R^N$08 in the positive $\mathbb R^N$09-quadrant separating an existence region $\mathbb R^N$10, where the system has a positive classical minimal solution, from a nonexistence region $\mathbb R^N$11, where no solution exists. The extremal solution on the critical curve is smooth when $\mathbb R^N$12 and $\mathbb R^N$13. The curve therefore plays the role of a pull-in threshold set for the coupled MEMS system (Ó et al., 2019).

A different constrained Hamiltonian variant produces a Lane–Emden system of free-boundary type: $\mathbb R^N$14 supplemented by integral normalizations and constant boundary traces. Under the subcritical Souto condition

$\mathbb R^N$15

there exists at least one solution for every boundary parameter $\mathbb R^N$16. Moreover, for

$\mathbb R^N$17

the positive branch is unique and real analytic, while the associated free energy and energy satisfy

$\mathbb R^N$18

For $\mathbb R^N$19, nonnegative solutions cease to exist under the stated dimensional and convexity assumptions, so a free boundary must appear (Bartolucci et al., 3 Aug 2025).

These singular and constrained variants show that Lane–Emden structure is compatible with parameter-plane critical curves, extremal branches, and monotonicity formulas even when the nonlinearity is not polynomial. A plausible implication is that the classical power-law system is only one representative of a broader cooperative framework in which threshold geometry survives substantial alterations of the reaction terms.

7. Exterior domains and geometric settings

Exterior-domain problems reveal another aspect of the theory. In the two-dimensional exterior domain

$\mathbb R^N$20

the sublinear Lane–Emden–Fowler system

$\mathbb R^N$21

admits bounded positive radial solutions when $\mathbb R^N$22 and

$\mathbb R^N$23

The proof uses the Liouville transformation $\mathbb R^N$24, which converts the radial PDE into a coupled ODE system on $\mathbb R^N$25, followed by Schauder fixed-point arguments and sub/supersolution methods. The constructed solutions satisfy

$\mathbb R^N$26

for prescribed $\mathbb R^N$27 (Covei, 18 Nov 2025).

On Cartan–Hadamard model manifolds $\mathbb R^N$28, radial solutions of

$\mathbb R^N$29

are controlled by the geometry through

$\mathbb R^N$30

In the critical or supercritical regime

$\mathbb R^N$31

there is at least a one-parameter family of globally positive radial solutions. If the manifold is stochastically complete, equivalently $\mathbb R^N$32, the admissible initial data form a one-dimensional shooting curve and both components must vanish at infinity. If it is stochastically incomplete, equivalently $\mathbb R^N$33, the global positivity set becomes genuinely two-dimensional and at least one component has a positive limit at infinity. Finite-energy radial solutions exist if and only if the manifold is isometric to $\mathbb R^N$34 and the exponents are critical (Muratori et al., 2023).

Taken together, these results show that exterior geometry, topology, and curvature do not merely perturb the Euclidean Lane–Emden picture. They can change the dimensionality of the admissible shooting set, alter asymptotic states at infinity, and replace Euclidean scaling by domain- or manifold-dependent mechanisms while preserving the defining cooperative coupling.