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Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian

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Pith's one-line read Every least energy solution of the vectorial p-Laplacian Lane–Emden system is a scalar ground state multiplied by a constant unit vector.

desk verdict Theorem 1.7 is a new and likely true classification, but the written proof overreaches: it treats all minimizers of a scale-invariant quotient as if they solved the system. read the letter →

arxiv 2510.15694 v5 pith:BI2D52Q3 submitted 2025-10-17 math.AP

Annamaria Canino , Simone Mauro This is my paper · ORCID
classification math.AP MSC 35A0135A1535J0535J2035J2535J62
keywords vectorialp-LaplacianquasilinearellipticsystemsleastenergysolutionsLane-EmdenequationsclassificationDirichletboundaryconditionsregularityRayleighquotient
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies systems where a vector-valued unknown u=(u^1,...,u^m) solves a quasilinear elliptic equation driven by the vectorial p-Laplacian. For the power-type Lane–Emden nonlinearity, it proves that every least energy solution has the factored form u=(c^1 ω,...,c^m ω), where c lies on the unit sphere in R^m and ω is a positive solution of the corresponding scalar equation. This matters because it says the multicomponent ground state is completely described by a scalar profile and a fixed direction in the target space; the vector structure does not create genuinely vectorial ground states. The paper also establishes existence of solutions in sublinear and superlinear regimes and a regularity result for weak solutions.

What carries the argument

The scale-invariant quotient Q_{p,q} and its infimum l_{p,q} are the central object. Proposition 5.1 establishes an equivalence between minimizers of l_{p,q} and least energy solutions, including the identity l_{p,q} = (pq/(q-p)c_{p,q})^{(q-p)/q}. The factorization step uses Lagrange's identity on the vectors u_i and ∇u_i to convert the equality |Du|=|∇|u|| into the conclusion that the ratios u_i/|u| are constant a.e., giving u=cω.

What would settle it

Find a least-energy solution of the two-component system in a ball with q just below p* whose two components are not constant multiples of a single positive function; equivalently, exhibit a minimizer u of Q_{p,q} for which |Du|>|∇|u|| on a set of positive measure. Either observation would contradict Theorem 1.7.

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Extended reading notes

Core claim

The central claim is Theorem 1.7: for q in (1,p*) and λ below the first eigenvalue λ1 of the scalar p-Laplacian, the system -Δ_p u = λ|u|^{p-2}u + |u|^{q-2}u has a least energy solution, and every such solution is of the form (c^1 ω,...,c^m ω) with c=(c^1,...,c^m) in S^{m-1} and ω>0 solving -Δ_p ω = λ ω^{p-1} + ω^{q-1} with zero Dirichlet boundary conditions. The proof shows that the least energy level of the vector system is determined by the infimum of the Rayleigh-type quotient Q_{p,q}(u) = (∫|Du|^p - λ∫|u|^p)/(∫|u|^q)^{p/q}, and that any minimizer of this quotient has |Du|=|∇|u||, forcing all components u^i to be constant multiples of one positive scalar function.

Load-bearing premise

Everything rests on Proposition 5.1, the exact equivalence between minimizers of the scale-invariant quotient Q_{p,q} and least-energy solutions; if that equivalence fails for some subcritical q or the minimizer is not achieved at the least-energy scale, the factored form does not follow.

Editorial extensions

If this is right

  • All least energy solutions of the vector Lane–Emden system share the same scalar spatial profile, so no ground state exists whose components vary independently.
  • The ground state energy of the vector system is directly computable from the scalar ground state: c_{p,q} = ((q-p)/(pq)) l_{p,q}^{q/(q-p)}.
  • Some components of a least energy solution may vanish, and signs may differ, since only the unit vector c is constrained, not its entries.
  • Existence is assured for every subcritical q and every λ<λ1, including both p-sublinear and p-superlinear cases.
  • Under p≥2 and subcritical growth, weak solutions are bounded and, on C^{1,α} domains, are C^{1,β}, so the classified ground states are classical.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the classification is correct, it suggests that for a whole family of homogeneous vector problems driven by the vectorial p-Laplacian, the ground-state manifold is a copy of the scalar ground-state set times S^{m-1}, so symmetry breaking appears only through the choice of direction in target space.
  • A natural extension would be to test whether the same factored form persists under small anisotropic perturbations of the nonlinearity; the proof's reliance on exact scale invariance indicates that such perturbations could produce non-polarized ground states.
  • The threshold λ<λ1 is probably where the quotient Q_{p,q} remains bounded below; at or above λ1, the classification likely fails or requires a different normalization.
  • One could numerically minimize Q_{p,q} for a two-component system on a disk and check whether the ratio u^1/u^2 is constant for all computed minimizers, providing a concrete test of the factorized-structure claim.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies variational quasilinear elliptic systems driven by the vectorial p-Laplacian on a bounded domain, with Dirichlet boundary conditions. After establishing differentiability of the energy functional, the authors prove existence of one global minimizer in the p-sublinear case, infinitely many solutions in the p-superlinear case (with and without a linear eigenvalue perturbation), and an L∞/C^{1,β} regularity result for p≥2. The main new contribution is Theorem 1.7, which asserts that for the Lane–Emden type system (P_{p,q}) with q∈(1,p*) and λ<λ1, there exists a least energy solution and every such solution has the factorized form u=(c^1ω,…,c^mω) with c∈S^{m−1} and ω>0 solving the scalar equation. The proof uses a scale-invariant Rayleigh quotient l_{p,q} and a Lagrange identity to reduce the vectorial problem to the scalar one.

Significance. If the classification theorem were correct, it would give a complete and elegant description of least energy ground states for the vectorial Lane–Emden system: all ground states are obtained from a scalar ground state by a constant unit vector of coefficients. This would extend the scalar and cooperative-system results of Saldaña–Tavares and Correia–Oliveira–Tavares to the vectorial p-Laplacian setting. The existence and regularity results are largely standard adaptations, but the paper collects them in a convenient unified framework. The main novelty is the classification, and the proof strategy—passing through the quotient l_{p,q} and using the identity of Hynd–Kawohl–Lindqvist—is natural and potentially effective. However, as written, the proof of the classification contains a gap that is load-bearing for the theorem, and the theorem statement itself appears to be false in the included case q=p. There is also a gap in the Palais–Smale boundedness argument for Theorem 1.4 when λ>0. These issues are fixable, but they require substantive revision.

major comments (3)
  1. [Proof of Theorem 1.4, Step 3] The boundedness of Palais–Smale sequences for λ>0 is not established. The proof uses the weighted Young inequality to write ∫(|Du|^p − λ|u|^p) ≥ ‖u‖^p − Cδ − Cδ^{q/p}∫|u|^q, and then claims that for δ small enough this is ≥ c‖u‖^p − C_2(δ). This absorption is invalid because the term Cδ^{q/p}∫|u|^q is not controlled by ‖u‖^p when q>p: for a Palais–Smale sequence, ∫|u|^q may grow like ‖u‖^q, and no a priori bound is available. Thus the conclusion that C(1+‖u_n‖) ≥ (1/p−1/μ)‖u_n‖^p − C_2(δ) does not follow. This gap also affects Step 1 of Theorem 1.7 for q>p, which explicitly relies on “Step 3 of proof of Theorem 1.4” to obtain a converging subsequence of a minimizing sequence.
  2. [Proof of Theorem 1.7, Step 2 (p. 16–17)] The final inference is invalid. The proof starts with an arbitrary minimizer u of the quotient l_{p,q}. Such a minimizer satisfies the constrained Euler equation −Δ_p u − λ|u|^{p−2}u = l_{p,q}|u|^{q−2}u, not the unconstrained system (P_{p,q}). Therefore, after showing u=cω, the statement “ω solves (5.2) by Proposition 5.2” does not follow, because Proposition 5.2 applies only to solutions of (P_{p,q}). The theorem remains plausibly true, but the proof must start from a least energy solution w, decompose the normalized function v=w/|w|_q into cη, and then use that w=c(|w|_q η) solves (P_{p,q}) to conclude that |w|_q η solves (5.2). The Harnack positivity assertion for ω at this stage is also unsupported, since u is not yet known to solve an elliptic equation.
  3. [Theorem 1.7 statement (p. 4)] The statement includes q=p, but for q=p the problem reduces to the eigenvalue equation −Δ_p u = (λ+1)|u|^{p−2}u. Nontrivial solutions exist only if λ+1 is an eigenvalue of the Dirichlet p-Laplacian. For generic λ<λ1, e.g. λ=0 when λ1≠1, there is no nontrivial solution, so the asserted existence of a least energy solution is false. The theorem should exclude q=p or impose an additional condition such as λ+1=λ1. The proof itself only treats q<p and q>p, consistent with this correction.
minor comments (4)
  1. [Proposition 5.2 (p. 16)] In the converse direction, the displayed equation has a typo: the left-hand side should be ∫|∇ω|^{p−2}∇ω·∇φ_j, not ∫|ω|^{p−2}ω φ_j. The intended equation is clear from context, but the typo should be corrected.
  2. [Proof of Theorem 1.7, Step 1] For q∈(1,p), the coercivity bound explicitly uses λ<λ1. It would be helpful to state that λ<λ1 is used here; the proof is correct but the role of the eigenvalue is implicit.
  3. [Proof of Theorem 1.7, Step 2] The notation “t:=|v|^{-1}_q” is ambiguous; it should read t=1/|v|_q. Also, after substituting tv, the denominator (∫|v|^q)^{p/q} should be written explicitly to avoid confusion with the definition of Q_{p,q}(tv).
  4. [Proposition 5.1] The statement says “if and only if”, but the proof only shows one direction and then an equality chain. The logical structure is correct, but the wording could be clarified to indicate that the converse follows from the equality l_{p,q}=(pq/(q−p)c_{p,q})^{(q−p)/q}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the least-energy classification is obtained from independent variational identities and an elementary algebraic lemma, not from its own conclusion.

full rationale

The derivation chain is self-contained with respect to circularity. The Rayleigh quotient l_{p,q} and the least-energy level c_{p,q} are defined independently, and Proposition 5.1, cited from [16] and proved in the text, relates them by direct scaling: from a solution u one forms v=u/|u|_q, and from a normalized minimizer v one forms l^{1/(q-p)}v; neither direction presupposes the target factorization u=(c^1ω,\dots,c^mω). The factorization in Step 2 follows from equality in the minimizing quotient, which forces |Du|=|∇ω| a.e., together with Lagrange identity (5.3) quoted from [9]; that identity is an elementary algebraic fact independent of the classification being proved, and [9] is not authored by the present authors. Proposition 5.2 is a direct computation, not a hidden assumption. There are no fitted parameters, no parameter renamed as a prediction, and no load-bearing self-citation. One non-circular concern exists as written: Step 2 classifies arbitrary minimizers of l_{p,q}, and the final line “ω solves (5.2) by Proposition 5.2” would require the additional scaling bridge between a least-energy solution and a minimizer; this is a possible proof gap, not a circular reduction. Hence the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

All inputs are standard PDE assumptions (growth conditions, Dirichlet domain, λ<λ1) and cited theorems. No fitted constants, no ad hoc scaling, and no new postulated entities appear. The coefficient vector c∈S^{m-1} is a consequence, not an input.

assumptions (5)
  • standard math Symmetric mountain pass theorem (Ambrosetti–Rabinowitz) as stated in Theorem 2.3
    Used to prove Theorems 1.3 and 1.4; quoted from [1,15].
  • standard math Sobolev embedding W^{1,p}_0(Ω)↪L^q(Ω) for q<p* and spectral decomposition of the vectorial p-Laplacian
    Invoked throughout, e.g., first eigenvalue λ1 and eigenfunctions φk taken from [9].
  • domain assumption Lemma 3.1 from Vannella [21] on uniform L^q smallness over W^{1,p} balls
    Used in the regularity proof of Theorem 1.5; not proved in the paper.
  • domain assumption Harnack inequality for nonnegative weak solutions of scalar quasilinear equations
    Used to assert ω>0 in Step 2 of Theorem 1.7; cited to [20].
  • domain assumption Boundary C^{1,β} regularity theory for scalar p-Laplace equations
    Used to pass from L∞ bounds to C^{1,β} regularity in Theorem 1.5; cited to [4].

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Pith. "Pith review of Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian." pith.science (2026). https://pith.science/paper/BI2D52Q3

@misc{pith2026251015694,
  author       = {Pith},
  title        = {Pith review of: Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BI2D52Q3}},
  note         = {Machine review of arXiv:2510.15694}
}
abstract

We prove existence and regularity results for the following elliptic system: \[ \begin{cases} -\textbf{div}(|D\boldsymbol{u}|^{p-2}D\boldsymbol{u})=\boldsymbol{f}(x,\boldsymbol{u}) & \text{in } \Omega \\ \boldsymbol{u}=0 & \text{on } \partial\Omega, \end{cases} \] where $\boldsymbol{u}=(u^1,\dots,u^m)$, $p>1$, and $\Omega\subset\mathbb{R}^N$ is a bounded domain. We also consider the special case \[\boldsymbol{f}(x,\boldsymbol{u})=\lambda|\boldsymbol{u}|^{p-2}\boldsymbol{u}+|\boldsymbol{u}|^{q-2}\boldsymbol{u},\] and we prove a classification result. In particular, we show that any least energy solution is of the form $(c^1\omega,\dots,c^m\omega)$, where $\boldsymbol{c}=(c^1,\dots,c^m)\in S^{m-1}$ (the $(m-1)$-sphere in $\mathbb R^m$) and $\omega$ is a positive solution of the corresponding scalar equation.

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