Existence, positivity and boundedness of solutions for systems of quasilinear elliptic equations

Abdelkrim Moussaoui; Jean V\'elin

REVIEW 2 major objections 8 minor 41 references

Existence, positivity and boundedness of solutions for systems of quasilinear elliptic equations

T0 review · 2 major / 8 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Coupled p,q-Laplacian systems are shown to have nontrivial, bounded, positive solutions.

desk verdict Real niche, sensible framework, but Theorem 2 and Theorem 3 rest on false or unverified claims; the paper needs substantive repairs. read the letter →

arxiv 1908.00930 v1 pith:V3H3MZLO submitted 2019-08-02 math.AP

Abdelkrim Moussaoui , Jean Vélin This is my paper · ORCID

classification math.AP MSC 35J6035P3047J1035A1535D30

keywords quasilinearellipticsystemsp-Laplacianq-LaplaciannoncooperativeBesovspacesMoseriterationfiberingmethodboundedness

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

This paper studies a Dirichlet system of two quasilinear equations, one driven by the p-Laplacian and one by the q-Laplacian, with nonlinear right-hand sides that may cross and carry no sign condition. The authors claim that under a coupled subcritical growth assumption with no cooperative structure, the system still admits a nontrivial Hölder-regular solution, that every weak solution is automatically bounded, and that with two extra structural hypotheses the solution can be chosen strictly positive in both components. The significance is that classical variational and sub-supersolution methods fail in this regime, so the proofs combine a fixed-point theorem, compactness measured in Besov spaces, a Moser iteration, and a comparison argument based on the fibering method.

What carries the argument

The load-bearing object is the solution operator $T(u,v)=(z,w)$ that solves $-\Delta_p z=f(x,u,v)$, $-\Delta_q w=g(x,u,v)$ with Dirichlet conditions. Schaefer's fixed-point theorem turns existence into uniform a priori bounds for $\tau T$; those bounds come from testing with scaled components and a weighted first eigenvalue $\lambda_{p,q}$ for the coupled system. Compactness of $T$ is obtained from Besov-space bounds $B^{1+C,p}_\infty\hookrightarrow W^{1,p}_0$, rather than from Sobolev embeddings alone. For boundedness, a Moser iteration with exponents $\delta_k=pC f_k$, $\gamma_k=qC f_k$, $f_k=D(C^k+1/D)$, shows that $\ln\|u\|_{\delta_k}$ grows at most geometrically, hence the $L^\infty$ bounds are finite and explicit. Positivity uses an auxiliary scalar problem and a comparison principle from the fibering method to show the fixed-point solution dominates a positive subsolution.

What would settle it

In a bounded domain of $\mathbb{R}^3$, take a right-hand side whose size is $|x|^{-1/4}$ near a point; its smoothed approximations converge in the relevant $L^{p'}$ space while their sup norms diverge, so Lemma 6's claimed uniform $c_0$ fails for data satisfying (H.1), and the Moser estimate in Lemma 7 that multiplies by $c_0$ does not close.

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Extended reading notes

Core claim

On its own terms the paper establishes: Theorem 1, system (P) has at least one nontrivial solution $(u^*,v^*)$ in $C^{1,\sigma}(\Omega)\times C^{1,\sigma}(\Omega)$ under (H.1)-(H.2); Theorem 2, every weak solution of (P) lies in $L^\infty(\Omega)\times L^\infty(\Omega)$; and Theorem 3, adding (H.3)-(H.4) yields a solution with both components strictly positive. Growth condition (H.2) is coupled: in the paper's notation $\max(|sf|,|tg|) \le k_{p,q}(x)\min(|s|^{\alpha+1}|t|^{\beta+1}, |s|^{pC}+|t|^{qC})$ with $\frac{\alpha+1}{p}+\frac{\beta+1}{q}=1$ and $1<C<\min\{p^*/p,q^*/q\}$, so the system is nonvariational and noncooperative. Existence is obtained through a fixed-point map whose compactness is proved with Besov-space embeddings; boundedness through a Moser iteration with geometric exponent sequences; positivity through a comparison principle built on the fibering method.

Load-bearing premise

The boundedness argument rests on Lemma 6's assertion that right-hand sides can be approximated by smooth functions that are uniformly bounded in $L^\infty$ independently of the approximation parameter; convergence in $L^{p'_C}$ alone does not yield such a bound, and without it the Moser iteration in Lemma 7 lacks a needed estimate.

Editorial extensions

If this is right

Under (H.1)-(H.2), existence of a nontrivial solution does not require $f$ and $g$ to be cooperative, sign-definite, or variational; the same argument covers any Carathéodory nonlinearities satisfying the coupled growth bound.
Every weak solution of (P) is in $L^\infty(\Omega)\times L^\infty(\Omega)$, so the $C^{1,\sigma}$ regularity machinery can be applied to any solution, not just the one constructed by fixed point.
The a priori bounds used in Schaefer's theorem are explicit: solutions of the $\tau$-family stay between $\varepsilon_0/2$ and $2\Theta$ in $W^{1,p}_0\times W^{1,q}_0$, giving a quantitative nontriviality statement.
With (H.3)-(H.4), the produced solution has both components strictly positive, so the result includes sign information rather than mere existence.

Reading between the lines

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

The uniform $L^\infty$ bound for smooth approximants asserted in Lemma 6 is not a consequence of $L^{p'_C}$-convergence; a mildly singular right-hand side already shows the bound can fail. The existence and boundedness results might survive with a truncation-based approximation, but the paper does not provide that repair.
Theorem 3's comparison argument requires the fixed parameter $\lambda=1$ to lie inside the eigenvalue window allowed by Lemma 5; since (H.1)-(H.4) do not state such a spectral condition, positivity is only guaranteed for systems for which that window contains 1.
The Besov-space compactness device is transferable: it suggests the same existence scheme applies to systems with more than two equations or to higher-order quasilinear operators, provided the corresponding compact Besov-to-Sobolev embeddings hold.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper tackles the complementary growth case for non-cooperative (p,q)-Laplacian systems, and the setup is genuinely a corner of the literature: the exponents satisfy (α+1)/p + (β+1)/q = 1 with p_C < p*, q_C < q*, and no sign or cooperativity conditions are assumed. The Schaefer fixed-point framework with Carathéodory nonlinearities assumed only in L^{p'_C} is a sensible way to attack this, and the Besov regularity argument in Lemma 3 is a real feature even if it is terse.

But the main theorems are not proved as written. Lemma 6 asserts that because f lies in L^{p'_C}, one can approximate it by C∞_0 functions that are uniformly bounded in L∞. That is false: an L^p function that is not essentially bounded cannot be approximated in L^p by a sequence with bounded sup norm. This c0 is then used in Lemma 7 to control the right-hand side of the Moser iteration, and without it the exponent bookkeeping does not close. So Theorem 2, and with it the C^{1,σ} conclusion in Theorem 1, is unproved. This looks repairable—one could add an explicit L∞ assumption on the composed nonlinearity or replace the approximation with a truncation—but the current argument has a hole.

The positivity claim has a separate gap. Theorem 3 applies Lemma 5 with λ=1, but (H.1)-(H.4) do not ensure that 1 lies in the intervals where the fibering method works, i.e., λ < λ_{b_p}, or λ = λ_{b_p} or λ in (λ_{b_p}, λ_{b_p}+ε) with a sign condition. Without such an assumption the comparison argument has no starting point. That is a missing hypothesis, not a minor rewording.

The paper is not a mess. The overall strategy is coherent, the writing is readable, and the citations look appropriate. But the two flaws are load-bearing: one breaks the boundedness theorem, the other breaks the positivity theorem. I would not publish it in this form. If the authors can fix the approximation issue and add the needed eigenvalue condition, the paper would be a legitimate contribution for specialists in quasilinear elliptic systems. A serious referee should examine it, but the verdict should be reject pending those repairs.

Referee Report

2 major / 8 minor

Summary. The paper studies the quasilinear elliptic system (P) of p-Laplacian and q-Laplacian equations with Carathéodory nonlinearities. Under growth assumptions (H.1)-(H.2), it claims existence of a nontrivial weak solution in C^{1,σ}×C^{1,σ} (Theorem 1), L∞-boundedness of all weak solutions (Theorem 2), and, under additional monotonicity and pointwise lower-bound assumptions (H.3)-(H.4), existence of a positive solution (Theorem 3). The proofs are organized around Schaefer's fixed point theorem with Besov-space compactness, a Pohozaev fibering comparison argument for positivity, and a Moser iteration for L∞ bounds.

Significance. If valid, the results would extend existence and qualitative theory for non-cooperative, non-variational (p,q)-Laplacian systems to a subcritical growth window with no sign conditions on the nonlinearities. The combination of Schaefer fixed point theory with Besov regularity and de Thélin's eigenvalue is a plausible and potentially useful strategy, and the nontriviality estimates in Proposition 1 are clearly structured. However, the paper does not supply machine-checked proofs or numerical validation, and its value rests entirely on the analytic arguments. The present version has two independent load-bearing gaps that affect Theorems 2 and 3 and, through Theorem 2, also the regularity claim in Theorem 1.

major comments (2)

[Section 4, Lemma 6 and Lemma 7] The assertion that the L^{p'_C}(Ω)-approximants f_ε can be chosen with a uniform L∞ bound c_0 independent of ε is false. Convergence in L^{p'_C}(Ω) does not imply uniform boundedness: for an unbounded f ∈ L^{p'_C}(Ω), every sequence of smooth functions converging to f in L^{p'_C} has L∞-norms tending to infinity, since a uniformly L∞-bounded subsequence would possess an L∞ weak-* limit equal to f. This invalidates the Hölder estimate immediately after (4.10), where Lemma 7 uses ∫ |u|^{a_k+1}|f_ε| dx ≤ c_0 |Ω|^{1/r_k}(∫|u|^{δ_k} dx)^{(a_k+1)/δ_k}. With only the L^{p'_C} norm available, that estimate becomes ||f||_{p'_C} ||u||^{a_k+1}_{p_C(a_k+1)}, and since p_C(a_k+1) > δ_{k+1} for p>1, the induction from δ_k to δ_{k+1} in Lemma 7 does not close. Consequently Theorem 2 is not proved; Theorem 1's C^{1,σ} conclusion invokes Theorem 2, so it is also unsupported.
[Section 3, proof of Theorem 3] The comparison argument for positivity applies Lemma 5 with λ=1, but the hypotheses (H.1)-(H.4) impose no condition on the weighted first eigenvalue λ_{b_p} that would place λ=1 in one of the admissible ranges of Lemma 5 (namely λ < λ_{b_p}, or λ = λ_{b_p} or λ_{b_p} < λ < λ_{b_p}+ε_p together with ∫ a_p u_p^{α̂+1}|v_*|^{β̂+1} dx < 0). Without such a condition, the existence of the positive supersolution U from Lemma 5 is not guaranteed, and the conclusion u_* > 0 in Theorem 3 does not follow from (H.1)-(H.4).

minor comments (8)

[Section 2, proof of Lemma 2] There is no Lemma 2.3 in the manuscript; 'Thanks to Lemma 2.3' should refer to the preceding Hölder/embedding estimate.
[Section 2, Proposition 1] The interval '[0,1[' in the sentence 'for any τ ∈ [0,1[' should presumably be '(0,1]' to match the Schaefer parameter range.
[Section 4, formulas (4.1)-(4.12)] The notation 'p−2/2' should be '(p−2)/2' (similarly for q); the current typography is ambiguous.
[Section 4, proof of Theorem 2] The heading 'Proof of Theorem 3' before the discussion following Lemma 7 should read 'Proof of Theorem 2'.
[Section 4, Lemma 6] The final sentence 'a quite similar argument produces that ˜v = u⋆ in Ω' should read '˜v = v⋆ in Ω'.
[Section 2, Lemma 3] The claim that one can pass to a subsequence with |f_nk|, |g_nk| ≤ const. a.e. is not justified by convergence in L^{p'_C}; the compactness argument should be rewritten using only the L^{p'_C} bounds and the Besov estimates.
[Assumption (H.2)] The symbols ∧ and ∨ are assigned meanings opposite to the standard convention (max/min versus min/max); please use explicit max/min notation to avoid confusion.
[Section 2, equation (2.5)] For non-integer α+1 and β+1, expressions such as u^{α+1}_τ v^{β+1}_τ should be written with absolute values, |u_τ|^{α+1}|v_τ|^{β+1}, as in (H.2).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the derivation is deductive and relies on independent external benchmarks; the serious weaknesses are correctness gaps, not circular reductions.

full rationale

The paper's central claims are proved through Schaefer's fixed point theorem, Besov-space compactness, Moser iteration, and Pohozaev's fibering method. No parameter is fitted and no result is obtained by renaming its own hypotheses. The load-bearing external inputs (Minty-Browder, Schaefer, Simon's Besov estimates, Tolksdorf regularity, de Thelin's eigenvalue, and Pohozaev's comparison propositions) are independent of the authors' own results and do not contain the target conclusions. The self-citations ([16], [17], [19], [28]) appear only in the historical/literature discussion and are not used as premises in Lemmas 1-7 or Theorems 1-3. Theorem 1's invocation of Theorem 2 is legitimate because existence of the weak solution is already obtained from Schaefer before Theorem 2 is used for regularity. The genuine defects in the paper, such as Lemma 6's asserted uniform L-infinity bound c0 for L^{p'_C} approximants, the lower-bound derivation in Proposition 1, and Theorem 3's unverified compatibility of lambda=1 with Lemma 5, are mathematical gaps or false assertions, not circular reductions of conclusions to inputs. The paper is therefore not circular in the sense of deriving its predictions from its assumptions by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

Pure proof paper: no numbers are fitted to data. Constants C (1.1) and D (4.8) are existential choices made within the proof, and ε0, Θ in Proposition 1 are derived rather than tuned. λ_{b_p} is determined by the given weight b_p, not chosen. No new entities are postulated; (u⋆,v⋆) is constructed via fixed-point and comparison arguments.

assumptions (6)

standard math Schaefer's fixed point theorem (Theorem 4) applies to the operator T defined by (P_{z,w}).
Section 2, proof of Theorem 1; requires T continuous and compact, which Lemmas 2-3 attempt to show.
standard math Minty-Browder and Sobolev/Poincaré embeddings on W_0^{1,p} × W_0^{1,q} give unique solvability of the decoupled system (P_{z,w}).
Section 2, after (P_{z,w}); relies on (H.1) placing f,g in L^{p'_C}, L^{q'_C}.
domain assumption Pohozaev's fibering results (Proposition 2) for -Δp u = λb|u|^{p-2}u + a|u|^{q-2}u transfer to problem (3.8) with exponents p and α̂+1.
Section 3, Lemma 5 and proof of Theorem 3; the paper does not verify that (3.8) meets Proposition 2's hypothesis set, including the ordering of exponents needed for the fibering maximization.
ad hoc to paper The weighted first eigenvalue λ_{b_p} satisfies either 1 < λ_{b_p} or the alternative window condition with ∫ a_p u_p^{α̂+1}|v⋆|^{β̂+1} < 0 when λ=1.
Section 3, proof of Theorem 3, eq. (3.9); Lemma 5 requires this condition for λ=1, but (H.1)-(H.4) do not imply it.
domain assumption Tolksdorf regularity [38] upgrades weak solutions to C^{1,σ}.
Section 2, end of the proof of Theorem 1; also invoked in Section 4.
domain assumption Simon's Besov estimates [36] give boundedness in B^{1+1/(p-1),p}_∞ used in Lemma 3, and Lemma 1's compact Besov-to-Sobolev embeddings hold as stated.
Section 2, Lemma 1 and Lemma 3; technical assertions are carried by references [27] and [36] without precise theorem numbers for [36].

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Pith. "Pith review of Existence, positivity and boundedness of solutions for systems of quasilinear elliptic equations." pith.science (2026). https://pith.science/paper/V3H3MZLO

@misc{pith2026190800930,
  author       = {Pith},
  title        = {Pith review of: Existence, positivity and boundedness of solutions for systems of quasilinear elliptic equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3H3MZLO}},
  note         = {Machine review of arXiv:1908.00930}
}

read the original abstract

This article sets forth results on the existence, positivity and boundedness of solutions for quasilinear elliptic systems involving p-Laplacian and q-Laplacian operators. The approach combines Schaefer's fixed point, comparison principle as well as Moser's iteration procedure.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

R. A. Adams, Sobolev spaces, Academic Press, New York, London, Toronto, Sydney, San Francisco, 1975

work page 1975

[2] [2]

Adriouch, On quasilinear and anisotropic elliptic systems with Sobol ev critical exponents, Thesis, University of La Rochelle, 2007

K. Adriouch, On quasilinear and anisotropic elliptic systems with Sobol ev critical exponents, Thesis, University of La Rochelle, 2007

work page 2007

[3] [3]

Aghajani and J

A. Aghajani and J. Shamshiri, Multilplcity of positive solutions for quasilinear ellipt ic p-Laplacians systems. Elec. J. Diﬀ. Eqts. (111) (2012), 1-16

work page 2012

[4] [4]

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Azizieh, Ph

C. Azizieh, Ph. Cl´ ement and E. Mitidieri, Existence and a priori estimates for positive solutions for a p-Laplace systems. J. Diﬀ. Eqts. 184 (2) (2002), 422-442. SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS 21

work page 2002

[6] [6]

Br´ ezis, Analyse fonctionnelle, Th´ eorie et applications, Masson, Paris, 1983

H. Br´ ezis, Analyse fonctionnelle, Th´ eorie et applications, Masson, Paris, 1983

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Choi and P.J

Y.S. Choi and P.J. McKenna, A singular gierer-meinhardt system of elliptic equa- tions. Ann. Inst. H. Poincar´ e Anal. Non-lin´ eaire 17 (4) (2000), 503-522

work page 2000

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Ph. Cl´ ement, J. Fleckinger, E. Mitidieri and F. de Th ´ el in, Existence of positive solutions for a non variational quasilinear elliptic syste m, J. Diﬀ. Eqts. 166 (2) (2000), 455-477

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