Radially Symmetric Solutions in PDEs

• Radially symmetric solutions are defined by their dependence solely on the radius, reducing multi-dimensional PDEs to one-dimensional forms.
• The use of Emden–Fowler transformation and variational methods underpins the methodology, establishing existence, uniqueness, and sharp constant estimates.
• These solutions play a pivotal role in analyzing the critical weighted Hénon–Lane–Emden system, influencing applications in nonlinear analysis and astrophysics.

Radially symmetric solutions constitute a distinguished class of solutions to elliptic, parabolic, and hyperbolic partial differential equations and systems, as well as geometric variational problems, in which the dependence on spatial variables reduces to the radius $r=|x|$. Such solutions underpin sharp existence, symmetry, classification, and blow-up results in nonlinear analysis. Their structure is pivotal in quantifying best constants in functional inequalities, establishing uniqueness criteria, and constructing special localized patterns, including solitons and breathers. This entry systematically presents their formulation, variational underpinnings, existence on the critical Hénon–Lane–Emden hyperbola, regularity and qualitative features, as well as nonexistence phenomena and methodological consequences.

1. Weighted Hénon–Lane–Emden System and the Critical Hyperbola

The Hénon–Lane–Emden system generalizes the classical Lane–Emden equation to two mutually interacting components with spatial weights: $$\begin{cases} -\Delta u = |x|^a\,v^p, \ -\Delta v = |x|^b\,u^q, \end{cases} \quad x\in\R^n, \quad a,b\in\R,\; p,q>1.$$ The parameters $(p,q)$ are said to be on the weighted critical hyperbola if

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

a threshold ensuring scale invariance and sharp embedding properties for the associated Sobolev spaces. The anticoercivity condition

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

guarantees integrability of the nonlinearity and compactness of minimizing sequences.

Radial symmetry imposes $u(x) = u(r)$ and $v(x) = v(r)$, which facilitates reduction to one-dimensional problems and explicit consideration of scaling phenomena relevant to the critical hyperbola.

2. Variational and Emden–Fowler Reduction Framework

For radially symmetric $(u,v)$, the Emden–Fowler transform is employed: $$s = -\ln|x|, \quad u(x) = |x|^{-\lambda_1}\,g(s), \quad v(x) = |x|^{-\lambda_2}\,f(s),$$ with exponents $\lambda_1 = \frac{b+n}{q+1}$, $\lambda_2 = \frac{a+n}{p+1}$, obeying $\lambda_1+\lambda_2 = n-2$ on the critical hyperbola.

This leads to a coupled Hamiltonian ODE system on $\R$: $$\begin{cases} -g'' + 2A\,g' + T\,g = |f|^{p-1}f, \ -f'' -2A\,f' + T\,f = |g|^{q-1}g, \end{cases}$$ with $A = \frac12(n-2-\lambda_1+\lambda_2)$, $T = \lambda_1\lambda_2$, and solution spaces given by weighted Sobolev classes $W^{2,p'}$, $W^{2,q'}$.

The problem is variational: one minimizes Rayleigh quotient functionals

$$I_{p',q}(A,T) = \inf_{g\neq0} \frac{\int_{\R}|(-\partial^2+2A\partial +T)g(s)|^{q'}\,ds}{\left(\int_{\R}|g(s)|^q\,ds\right)^{q'/q}},$$

with analogous expression for $I_{q',p}(-A,T)$. Existence of minimizers is proved by direct methods, underpinning existence of nontrivial radially symmetric solutions.

3. Existence and Uniqueness of Radial Solutions on the Critical Hyperbola

Theorem (Musina–Sreenadh (Musina et al., 2013)): Let $n\ge2$, $a,b\in\R\setminus\{-n\}$, $p,q>1$, and $(p,q)$ on the critical hyperbola. If $(u,v)$ are as above and the anticoercivity condition holds, then there exists a nontrivial radially symmetric solution

$u \in \D^{2,p'}(\R^n;|x|^{-a}\,dx), \quad v \in \D^{2,q'}(\R^n;|x|^{-b}\,dx)$

with prescribed decay rates

$$\lim_{|x|\to\infty} |x|^{\lambda_1} u(x) = \lim_{|x|\to\infty} |x|^{\lambda_2} v(x) = 0,$$

and positivity $u,v>0$ iff $a>-n,\, b>-n$.

When $p=2<q$, there is uniqueness: the reduction leads to a single fourth-order ODE for $g$, and, up to translation, inversion, and sign, only one nontrivial solution exists in the radial class.

4. Qualitative Properties: Positivity, Monotonicity, Decay

For $a,b>-n$, the system admits strictly positive, strictly radially decreasing solutions: $$\frac{d}{dr} u(r) < 0, \quad \frac{d}{dr} v(r) < 0, \quad r>0.$$ The solutions possess prescribed asymptotic decay governed by exponents $\lambda_1, \lambda_2$,

$$|\nabla^k u(x)| = O(|x|^{-\lambda_1-k}), \quad |\nabla^k v(x)| = O(|x|^{-\lambda_2-k}), \quad k=0,1.$$

Near the origin $r\to0$, solutions are regular, and the Emden–Fowler variables produce $g(s), f(s)\to 0$ as $s\to\pm\infty$ at an exponential rate determined by the linearization of their ODE system.

If either $a\leq -n$ or $b\leq -n$, any nonnegative radial solution satisfying finite limit conditions at zero or infinity must vanish identically.

5. Nonexistence Results and Threshold Phenomena

No nontrivial positive radial $C^2(\R^n)$ solution exists continuous at $0$ or $\infty$ if

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

i.e., below the critical hyperbola. This strengthens the original Lane–Emden conjecture and extends it to weighted cases. The result is obtained by various techniques including Pohožaev-type identities, shooting arguments in the Emden–Fowler formulation, and energy estimates.

Moreover, Musina–Sreenadh establish that if $a\leq -n$ or $b\leq -n$, the only nonnegative radial solution under appropriate finite limit conditions is the zero solution.

6. Methodological Consequences and Applications

• The reduction to Hamiltonian ODE systems via the Emden–Fowler transform is fundamental in the analysis of radial symmetry for coupled PDEs.
• Weighted Sobolev spaces tailored to the system's scaling yield sharp constants for embedding and variational inequalities.
• Radial solutions characterize minimal—often ground state—energy configurations and provide explicit best constants in critical inequalities.
• Nonexistence results delineate the precise region in parameter space where positive solutions may or may not exist—this is crucial for applications in astrophysics, nonlinear analysis, and study of singularities.

7. Connections to Broader Topics

Radially symmetric solutions, their classification, and criticality phenomena directly inform related areas such as:

• Lane–Emden systems with fractional Laplacian, nonlocal nonlinearities, and measure sources (Silva, 2024).
• Stability and symmetry breaking in Euler and Navier–Stokes equations (Fan et al., 2024, Hashimoto et al., 6 Jul 2025).
• Blow-up and ground-state analysis in nonlinear Schrödinger-type systems (Hespanha et al., 18 Mar 2025, Inui et al., 2018).
• Rigidity theorems and inverse problems in elliptic equations and overdetermined boundary value problems (Ciraolo et al., 2013).

These themes highlight the centrality of radially symmetric solutions in the contemporary analysis of nonlinear PDEs, geometric variational problems, and the calculus of variations.