The Neumann problem for the generalized H\'enon equation. Local analysis

Alexander I. Nazarov; Alexandra P. Shcheglova

arxiv: 2604.26286 · v2 · submitted 2026-04-29 · 🧮 math.AP

The Neumann problem for the generalized H\'enon equation. Local analysis

Alexander I. Nazarov , Alexandra P. Shcheglova This is my paper

Pith reviewed 2026-05-12 02:54 UTC · model grok-4.3

classification 🧮 math.AP

keywords generalized Hénon equationNeumann problemradial solutionsecond variationenergy functionallocal minimizerp-Laplacian

0 comments

The pith

For n at least 4 and p close to 2, the second variation at the radial solution of the generalized Hénon Neumann problem is positive for large α.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates positive radial solutions to a Neumann boundary value problem involving the p-Laplacian and a power nonlinearity weighted by |x|^α inside the unit ball. It proves that when the dimension n is at least 4 and p is sufficiently close to 2, the second variation of the associated energy functional at this radial solution remains positive for every exponent q between p and np/(n-p) once α grows large enough. The same positivity is established for any 2 < p < n provided q is sufficiently close to p. A sympathetic reader would care because positivity of the second variation implies the radial solution is at least a local minimizer, giving local stability information even in parameter regimes where the solution is already known not to be a global minimizer.

Core claim

Let n ≥ 4 and let p > 2 be sufficiently close to 2. Then for all p < q < np/(n-p), for sufficiently large α the second variation of the energy functional is positive. The same holds true for all 2 < p < n if q > p is sufficiently close to p.

What carries the argument

The second variation of the energy functional evaluated at the positive radial solution.

Load-bearing premise

The assumption that p lies sufficiently close to 2 or q lies sufficiently close to p, without an explicit quantitative bound on the distance.

What would settle it

An explicit computation showing that the quadratic form of the second variation has a negative direction for some n=4, some p>2 arbitrarily close to 2, some q in (p, np/(n-p)), and some sufficiently large α.

read the original abstract

For the boundary value problem $$\left\{ \begin{array}{rcll} -\Delta_p u+u^{p-1}&=&|x|^{\alpha}u^{q-1}&\mbox{in }\Omega,\\ \frac{\displaystyle\partial u}{\displaystyle\partial{\bf n}}&=&0&\mbox{on }\partial \Omega, \end{array}\right. $$ in the unit ball $\Omega$, we investigate the properties of the positive radial solution. It is known, that for $1<p<n$, $\frac{(n-1)p}{n-p}<q<\frac{np}{n-p}$ and sufficiently large $\alpha$ this solution does not provide global minimum to the corresponding energy functional, see [M. Gazzini, E. Serra, 2008] for $p=2$ and [A.P. Shcheglova, 2018] in general case. Nevertheless, it is shown in [M. Gazzini, E. Serra, 2008] that for $n\ge 4$, $p=2$, $2<q<\frac{2n}{n-2}$ and sufficiently large $\alpha$ the radial solution is at least a local minimizer of the energy functional. We partially generalize this result. Namely, let $n\ge4$ and let $p>2$ be sufficiently close to $2$. Then for all $p<q<\frac{np}{n-p}$, for sufficiently large $\alpha$ the second variation of the energy functional is positive. The same holds true for all $2<p<n$ if $q>p$ is sufficiently close to $p$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends the Gazzini-Serra local minimizer result to p near 2 for the Neumann generalized Hénon problem, but the uniformity of the second-variation estimates across p may be a soft spot.

read the letter

This paper extends the local-minimizer property for the radial solution of the Neumann Hénon problem from the p=2 case to p close to 2, using positivity of the second variation of the energy. They do this by generalizing the estimates from Gazzini-Serra to the p-Laplacian, handling the additional terms that appear when p differs from 2. The result also covers the regime where q is close to p for any p in (2,n). This is a genuine if small step beyond the cited works. The paper is clear on what it assumes from prior results, like existence of the radial solution for large alpha, and it focuses on the local stability via the quadratic form. The abstract lays out the claims without overstatement. The soft spot is the handling of p near 2. The second variation includes p-dependent pieces, and the solution u itself varies with p. For the positivity to carry over, the error terms need to be small uniformly when p is close to 2 and alpha large. The stress-test raises a fair point that without explicit control on the asymptotics or a quantified delta for 'sufficiently close,' the argument rests on an unverified continuity. If the full paper has the estimates, this is minor; otherwise it is the main thing to check. This work is aimed at researchers in nonlinear elliptic PDEs who study variational problems with weights and radial symmetry. It would interest people following the line of Gazzini-Serra and Shcheglova on these Hénon-type equations. I think it deserves peer review. The extension is natural and the techniques look standard, so referees can verify the details and suggest fixes if the uniformity needs strengthening.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the Neumann problem for the generalized Hénon equation -Δ_p u + u^{p-1} = |x|^α u^{q-1} in the unit ball Ω with homogeneous Neumann boundary conditions. It claims that for n ≥ 4 and p > 2 sufficiently close to 2, the positive radial solution u_{p,α} is a local minimizer of the energy functional for all p < q < np/(n-p) and α sufficiently large, by showing positivity of the second variation. An analogous statement is given for 2 < p < n when q is sufficiently close to p. The work partially extends the p=2 local-minimality result of Gazzini-Serra (2008) and references the non-minimality result of Shcheglova (2018) for large α.

Significance. If rigorously established, the result would be a useful extension of local analysis for weighted p-Laplace problems near the semilinear case p=2, clarifying the parameter regime where the radial solution remains locally stable before losing global minimality as α grows. It correctly builds on the second-variation techniques from Gazzini-Serra while handling the additional nonlinear terms in the p-Laplacian linearization.

major comments (2)

[Proof of the main theorem (following the statement for n≥4)] The proof that the second variation remains positive for p sufficiently close to 2 (the central claim) relies on a continuity argument from the p=2 case but provides no uniform-in-p estimates on the radial solution u_{p,α} or on the remainder terms in the quadratic form as α → ∞. In particular, the extra term (p-2)∫ |∇u|^{p-4} (∇u · ∇v)^2 together with the p-dependent coefficients in the linearized operator and the weights u^{p-2}, u^{q-2} are not shown to be o(1) uniformly near p=2; this leaves the 'sufficiently close' condition unquantified and the extension non-rigorous.
[Section on second variation analysis] No explicit δ>0 or modulus of continuity is derived for the second-variation quadratic form Q_p(v) near p=2; the manuscript assumes the radial solution exists and is positive but does not control how its boundary-layer profile (which depends on p) affects the positivity inherited from the p=2 case of Gazzini-Serra.

minor comments (2)

[Abstract and Introduction] The abstract and introduction could explicitly recall the precise form of the energy functional J_p whose second variation is analyzed.
[Introduction] Notation for the radial solution u_{p,α} and the range of q should be introduced consistently before the main statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The concerns raised about the rigor of the continuity argument in p are valid, and we will strengthen the proof by adding the missing uniform estimates and modulus of continuity. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

Referee: [Proof of the main theorem (following the statement for n≥4)] The proof that the second variation remains positive for p sufficiently close to 2 (the central claim) relies on a continuity argument from the p=2 case but provides no uniform-in-p estimates on the radial solution u_{p,α} or on the remainder terms in the quadratic form as α → ∞. In particular, the extra term (p-2)∫ |∇u|^{p-4} (∇u · ∇v)^2 together with the p-dependent coefficients in the linearized operator and the weights u^{p-2}, u^{q-2} are not shown to be o(1) uniformly near p=2; this leaves the 'sufficiently close' condition unquantified and the extension non-rigorous.

Authors: We agree that the current presentation of the continuity argument lacks explicit uniform-in-p estimates, rendering the 'sufficiently close' condition unquantified. To address this, we will add a dedicated subsection deriving uniform bounds on the radial solution u_{p,α} as p → 2 (for fixed large α), leveraging the known convergence of solutions to the p-Laplace equation to the semilinear case in C^{1,β} norms. We will then control the remainder terms in the second variation, including the (p-2) integral term, by showing they are absorbed into the positive definite quadratic form from the p=2 case of Gazzini-Serra, using the positivity margin for large α. This will yield an explicit (though possibly small) δ > 0 depending on n, q, and α. The revised proof will thus be fully rigorous. revision: yes
Referee: [Section on second variation analysis] No explicit δ>0 or modulus of continuity is derived for the second-variation quadratic form Q_p(v) near p=2; the manuscript assumes the radial solution exists and is positive but does not control how its boundary-layer profile (which depends on p) affects the positivity inherited from the p=2 case of Gazzini-Serra.

Authors: We acknowledge that no explicit modulus of continuity for Q_p(v) is currently derived, and the dependence of the boundary-layer profile on p is not quantified. In the revision, we will establish continuity of the quadratic form with respect to p by analyzing the linearized operator and weights via asymptotic expansions of the radial solution near the boundary (using the large-α concentration). Standard comparison principles and elliptic regularity will control the p-dependence of the layer, ensuring that the positivity inherited from Gazzini-Serra persists for p sufficiently close to 2. An explicit δ > 0 (modulo the large-α regime) will be provided, together with the corresponding modulus. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim rests on independent analysis

full rationale

The manuscript states its main result as a direct generalization of the p=2 local-minimizer property established in the external reference Gazzini-Serra 2008. The single self-citation (Shcheglova 2018) appears only in the background sentence establishing that the radial solution fails to be a global minimizer; that fact is not invoked in the derivation of second-variation positivity. No equation or step in the provided abstract reduces the claimed positivity to a fitted parameter, a self-referential definition, or an unverified self-citation chain. The phrase 'sufficiently close to 2' is presented as the domain on which a continuity argument from the p=2 case is expected to hold, without any definitional circularity. The derivation chain is therefore self-contained against the stated assumptions and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard variational calculus and radial symmetry assumptions drawn from prior literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

domain assumption Existence of a positive radial solution to the boundary-value problem
Invoked implicitly when discussing its second variation; existence is taken from earlier works.
standard math Standard Sobolev embeddings and trace theorems for radial functions in the unit ball
Required for the energy functional to be well-defined and differentiable.

pith-pipeline@v0.9.0 · 5608 in / 1327 out tokens · 42265 ms · 2026-05-12T02:54:52.719032+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We partially generalize this result. Namely, let n≥4 and let p>2 be sufficiently close to 2. Then for all p<q<np/(n-p), for sufficiently large α the second variation of the energy functional is positive.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

D²Q_{p,q,α}(v_α;h,h) = D²Q(v_α;h₁,h₁) + p F_{p,q,α}(g) with F containing the extra (p-2) term |∇v|^{p-4}(∇v·∇g)²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Gazzini, E

M. Gazzini, E. Serra. The Neumann problem for the H´ enon equation, trace inequalities and Steklov eigenvalues. Ann. Inst. Henri Poincar´ e, Analyse Non Lineaire25, 281–302 (2008)

work page 2008
[2]

M. H´ enon. Numerical experiments on the stability of spherical stellar systems. Astronomy and Astrophysics24, 229–238 (1973)

work page 1973
[3]

Kolonitskii, A.I

S.B. Kolonitskii, A.I. Nazarov. Multiplicity of solutions to the Dirichlet problem for gen- eralized H´ enon equation. J. Math. Sci.144(6), 4624–4644 (2007). Transl. from Problemy Mat. Analiza35, 91–110 (2007)

work page 2007
[4]

Lami Dozo, O

E. Lami Dozo, O. Torn´ e. Symmetry and symmetry breaking for minimizers in the trace inequality. Comm. Contemp. Math.7(6), 727–746 (2005)

work page 2005
[5]

Mart´ ınez, J.D

S. Mart´ ınez, J.D. Rossi. Isolation and simplicity for the first eigenvalue of thep-Laplacian with a nonlinear boundary condition. Abstr. and Appl. Anal.7(5), 287–293 (2002)

work page 2002
[6]

A.I. Nazarov. On the symmetry of extremals in the weight embedding theorem. J. Math. Sci.107(3), 3841–3859 (2001). Transl. from Problemy Mat. Analiza23, 50–75 (2001)

work page 2001
[7]

W.-M. Ni. A nonlinear Dirichlet problem on a unit ball and its applications. Indiana Univ. Math. J.31(6), 801–807 (1982)

work page 1982
[8]

Shcheglova

A.P. Shcheglova. The Neumann problem for the generalized H´ enon equation. J. Math. Sci. 235(3), 360–373 (2018). Transl. from Problemy Mat. Analiza95, 3–19 (2018)

work page 2018
[9]

Smets, M

D. Smets, M. Willem, J. Su. Non-radial ground states for the H´ enon equation. Commun. Contemp. Math.4(3), 467–480 (2002). 18

work page 2002

[1] [1]

Gazzini, E

M. Gazzini, E. Serra. The Neumann problem for the H´ enon equation, trace inequalities and Steklov eigenvalues. Ann. Inst. Henri Poincar´ e, Analyse Non Lineaire25, 281–302 (2008)

work page 2008

[2] [2]

M. H´ enon. Numerical experiments on the stability of spherical stellar systems. Astronomy and Astrophysics24, 229–238 (1973)

work page 1973

[3] [3]

Kolonitskii, A.I

S.B. Kolonitskii, A.I. Nazarov. Multiplicity of solutions to the Dirichlet problem for gen- eralized H´ enon equation. J. Math. Sci.144(6), 4624–4644 (2007). Transl. from Problemy Mat. Analiza35, 91–110 (2007)

work page 2007

[4] [4]

Lami Dozo, O

E. Lami Dozo, O. Torn´ e. Symmetry and symmetry breaking for minimizers in the trace inequality. Comm. Contemp. Math.7(6), 727–746 (2005)

work page 2005

[5] [5]

Mart´ ınez, J.D

S. Mart´ ınez, J.D. Rossi. Isolation and simplicity for the first eigenvalue of thep-Laplacian with a nonlinear boundary condition. Abstr. and Appl. Anal.7(5), 287–293 (2002)

work page 2002

[6] [6]

A.I. Nazarov. On the symmetry of extremals in the weight embedding theorem. J. Math. Sci.107(3), 3841–3859 (2001). Transl. from Problemy Mat. Analiza23, 50–75 (2001)

work page 2001

[7] [7]

W.-M. Ni. A nonlinear Dirichlet problem on a unit ball and its applications. Indiana Univ. Math. J.31(6), 801–807 (1982)

work page 1982

[8] [8]

Shcheglova

A.P. Shcheglova. The Neumann problem for the generalized H´ enon equation. J. Math. Sci. 235(3), 360–373 (2018). Transl. from Problemy Mat. Analiza95, 3–19 (2018)

work page 2018

[9] [9]

Smets, M

D. Smets, M. Willem, J. Su. Non-radial ground states for the H´ enon equation. Commun. Contemp. Math.4(3), 467–480 (2002). 18

work page 2002