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arxiv: 2509.01355 · v2 · submitted 2025-09-01 · 🧮 math.AP

Regularizing effect of the natural growth term in quasilinear problems with sign-changing nonlinearities

Jos\'e Carmona Tapia , Paolo Malanchini , Antonio J. Mart\'inez Aparicio , Pedro J. Mart\'inez-Aparicio This is my paper

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classification 🧮 math.AP
keywords quasilinear elliptic equationsp-Laplaciansign-changing nonlinearityregularizing effectgradient termexistence and nonexistenceDirichlet problemarea condition
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The pith

If f(0) is nonnegative, an area condition on f and g is necessary and sufficient for nonnegative solutions with maximum in (α, β] to the p-Laplacian problem with gradient term.

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

The paper establishes that for the Dirichlet problem -Δ_p u + g(u) |∇u|^p = λ f(u), when f is continuous, vanishes at α and β with f positive exactly in (α, β) and f(0) ≥ 0, the existence of a nonnegative solution u whose maximum norm lies in (α, β] is equivalent to an area condition that relates integrals involving f and g. This condition supplies both a sufficient criterion and a necessary obstruction for such bounded solutions. The authors further show that the gradient-dependent term exerts a regularizing effect: making g more negative allows a solution to exist for any fixed λ, independent of the detailed shape of f. A sympathetic reader cares because the result gives an explicit, checkable balance that controls existence for sign-changing nonlinearities in quasilinear equations.

Core claim

If f(0) ≥ 0, an area condition involving f and g is both sufficient and necessary in order to have a pair (λ, u) ∈ R+ × C0^1(Ω¯) with u ≥ 0 and ||u||_C(Ω¯) ∈ (α, β] solving the problem. The more negative g is, the stronger its regularizing effect. Regardless of the shape of f, for any fixed λ there always exists a function g such that the problem admits a nonnegative solution with maximum in (α, β].

What carries the argument

the area condition involving integrals of f and g, which functions as the necessary and sufficient criterion for existence of nonnegative solutions whose maximum lies between the zeros of f.

If this is right

  • When the area condition holds, a solution pair (λ, u) with the required bounds on u exists.
  • Making g sufficiently negative guarantees existence for every fixed λ, no matter how f is shaped outside (α, β).
  • If the area condition is violated, no nonnegative solution with maximum in (α, β] can exist.
  • The regularizing strength increases monotonically as g becomes more negative.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same area-type balance may control existence when the p-Laplacian is replaced by other divergence-structure operators.
  • Reducing the problem radially and integrating the resulting ODE provides a direct numerical test of whether the area condition is satisfied for given f and g.
  • The result suggests that sign changes in f can be used to localize solution maxima even in problems with indefinite weights or competing nonlinearities.

Load-bearing premise

f vanishes at α and β, is positive exactly between them, and satisfies f(0) ≥ 0, while g is merely continuous and solutions are required to lie in C0^1 with maximum norm in (α, β].

What would settle it

For explicit f with f(0) > 0 and zeros at chosen α < β, and for a concrete continuous g, compute the area integral; a solution with maximum exactly β must exist if and only if that integral satisfies the stated relation.

read the original abstract

We investigate the existence and nonexistence of solutions to the Dirichlet problem \begin{equation*} \tag{$P$} \label{pba} \left\{ \begin{alignedat}{2} -\Delta_p u + g(u) |\nabla u|^p &= \lambda f(u) \quad &&\mbox{in} \;\; \Omega, \\ u &= 0 \quad &&\mbox{on} \;\; \partial\Omega, \end{alignedat} \right. \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a smooth bounded domain, $p\in (1,\infty)$, $\lambda>0$ and $g\in C(\mathbb{R})$. Our main assumption is that $:f \mathbb{R}\to \mathbb{R}$ is a continuous function such that $f(s)>0$ for all $s\in (\alpha,\beta)$, where $0<\alpha<\beta$ are two zeros of $f$. If $f(0)\geq 0$, we show that an area condition involving $f$ and $g$ is both sufficient and necessary in order to have a pair $(\lambda,u)\in \mathbb{R}^+\times C_0^1(\overline{\Omega})$, with $u\geq 0$ and $\|u\|_{C(\overline{\Omega})}\in (\alpha,\beta]$, solving~\eqref{pba}. We also study how the presence of the gradient term affects the existence of solution. Roughly speaking, the more negative $g$ is, the stronger its regularizing effect on~\eqref{pba}. We prove that, regardless of the shape of $f$, for any fixed $\lambda$, there always exists a function $g$ such that~\eqref{pba} admits a nonnegative solution with maximum in $(\alpha,\beta]$.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the quasilinear Dirichlet problem (P) involving the p-Laplacian with an additional natural growth term g(u)|∇u|^p on the left-hand side and right-hand side λf(u), where f is continuous, positive precisely on (α,β) for zeros α<β, and f(0)≥0. It claims that an area condition involving integrals of f and g (via the integrating factor exp(−∫g)) is both necessary and sufficient for existence of nonnegative solutions u∈C₀¹(Ω̄) with ||u||_∞∈(α,β]. It further claims that, independently of the detailed shape of f, for any fixed λ>0 there exists a choice of g making such a solution exist.

Significance. If the claims hold, the work provides a sharp necessary-and-sufficient criterion for existence in the presence of sign-changing nonlinearities, emphasizing the regularizing role of the gradient term. The result that g can always be chosen to guarantee existence for fixed λ is a notable flexibility statement for natural-growth problems. These findings would contribute to the literature on quasilinear elliptic equations with p-growth and gradient nonlinearities.

major comments (1)
  1. [§3 (necessity direction)] §3 (necessity direction): after introducing the integrating factor φ(u)=exp(−∫_0^u g(s)ds) and rewriting the PDE in divergence form, integration over Ω and the divergence theorem produce the identity −∫_∂Ω φ(0)|∇u|^{p−2}(∇u·ν)dσ=λ∫_Ω f(u)φ(u)dx. The boundary integrand has definite sign, yet the manuscript does not supply a co-area-level-set argument, boundary-gradient estimate, or vanishing argument that would allow this term to be absorbed or controlled so as to force the sign of the one-dimensional area ∫_α^β f(s)φ(s)ds. Without such control the necessity implication does not follow for arbitrary continuous g and p>1.
minor comments (2)
  1. [Introduction] The precise statement of the area condition (the integral that must be positive) is referenced but not displayed in the introduction or abstract; placing the explicit formula early would improve readability.
  2. [Preliminaries] Notation for the space C₀¹(Ω̄) is used consistently, but the precise regularity needed for the integrating factor and the application of the divergence theorem could be stated once in the preliminaries.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the recognition of the potential importance of the necessary-and-sufficient area condition and the flexibility result for the choice of g. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3 (necessity direction)] §3 (necessity direction): after introducing the integrating factor φ(u)=exp(−∫_0^u g(s)ds) and rewriting the PDE in divergence form, integration over Ω and the divergence theorem produce the identity −∫_∂Ω φ(0)|∇u|^{p−2}(∇u·ν)dσ=λ∫_Ω f(u)φ(u)dx. The boundary integrand has definite sign, yet the manuscript does not supply a co-area-level-set argument, boundary-gradient estimate, or vanishing argument that would allow this term to be absorbed or controlled so as to force the sign of the one-dimensional area ∫_α^β f(s)φ(s)ds. Without such control the necessity implication does not follow for arbitrary continuous g and p>1.

    Authors: We agree that the necessity direction in Section 3 requires a more explicit justification to connect the integrated identity to the sign of the one-dimensional area integral ∫_α^β f(s)φ(s) ds. While the left-hand side is nonnegative (and typically positive) because u vanishes on ∂Ω with ∇u·ν ≤ 0 and φ(0) > 0, the manuscript's argument that this forces the area condition when ||u||_∞ ∈ (α,β] was not spelled out with sufficient detail. In the revised manuscript we will insert a dedicated paragraph (or short subsection) that applies the co-area formula to the level sets of u. Since u is C^1, nonnegative, vanishes on the boundary and attains a maximum M ∈ (α,β], the range of u is the interval [0,M]. We will show that ∫_Ω f(u)φ(u) dx can be rewritten, via the co-area formula and the change of variables along the values of u, as a positive multiple of ∫_0^M f(s)φ(s) ds (with the weight coming from the surface measures of the level sets). Because f>0 on (α,β) and M ∈ (α,β], the sign of the volume integral is then the same as the sign of the area integral over [α,β]. We will also add a brief boundary-gradient estimate (using the C^1 regularity and the Hopf-type lemma for the p-Laplacian) to guarantee that the boundary integral is strictly positive whenever a nontrivial solution exists. These additions will make the necessity implication rigorous for arbitrary continuous g and p>1. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rely on direct PDE analysis without reduction to inputs by construction

full rationale

The paper establishes sufficiency and necessity of an area condition on f and g for existence of nonnegative solutions with max-norm in (α,β] when f(0)≥0. This is obtained by introducing the integrating factor φ(u)=exp(−∫g) to rewrite the equation, integrating over Ω, and applying the divergence theorem to relate the volume integral of f(u)φ(u) to boundary flux. The resulting condition is a direct consequence of the PDE and boundary conditions under the stated continuity assumptions, not a re-derivation or fit of the same quantity. No steps reduce by construction to prior fitted values, self-definitions, or load-bearing self-citations. The additional claim that a suitable g exists for any fixed λ and arbitrary f is likewise obtained by explicit construction of g to enforce the area condition. The skeptic's concern about uncontrolled boundary flux affects proof validity (correctness risk) but does not indicate circularity, as the paper's chain remains independent of its own outputs. The work is self-contained against external benchmarks and uses standard quasilinear techniques without renaming or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper relies on standard background results in elliptic PDE theory rather than new axioms or parameters. No free parameters, invented entities, or ad-hoc assumptions beyond the stated continuity and positivity conditions on f and g.

axioms (2)
  • standard math Standard maximum principles and regularity theory for the p-Laplacian operator
    Implicitly used to obtain C^1 solutions and control the maximum norm
  • domain assumption Continuity of f and g together with the sign condition on f in (α,β)
    Explicitly stated as the main assumption on f

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