arxiv: 2604.23339 · v1 · submitted 2026-04-25 · 🧮 math.AP

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Blowing-up Solutions with Residual Mass in a Slightly Subcritical Dirichlet Problem

Rufaidah Alharbi , Mohamed Ben Ayed , Khalil El Mehdi

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Pith reviewed 2026-05-08 07:37 UTC · model grok-4.3

classification 🧮 math.AP

keywords blow-up solutionsresidual masssubcritical elliptic problemDirichlet problemasymptotic analysiscritical Sobolev exponentpotential function

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The pith

In dimensions 4 and 5, single blow-up points cannot coexist with residual mass in slightly subcritical Dirichlet problems.

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

The paper examines positive solutions to a slightly subcritical elliptic equation with Dirichlet boundary conditions that blow up at points while possibly converging weakly to a positive limit elsewhere. It establishes that in dimensions four and five, a single blow-up point is incompatible with the presence of residual mass. The analysis uses asymptotic expansions to show this incompatibility and then examines how the boundary behavior of the potential V influences whether blow-up occurs on the boundary or in the interior. Constructions are given for both interior and boundary cases under appropriate conditions on V.

Core claim

In dimensions n=4 and n=5, single blow-up point cannot coexist with residual mass. If the normal derivative of V is positive, any single blow-up solution with residual mass must occur in the interior; if negative at some boundary point, boundary blow-up solutions with residual mass can be constructed. Both simple and non-simple interior blow-up solutions exhibiting residual mass can be constructed without any assumption on the sign of the normal derivative of V.

What carries the argument

Delicate asymptotic expansions of the gradient of the functional associated to the variational problem, used to derive contradictions or conditions for the coexistence of single blow-up and residual mass.

If this is right

Interior bubbling solutions with nonzero weak limit cannot occur in dimensions 4 and 5.
Single blow-up with residual mass requires either multiple blow-up points or specific boundary conditions on V.
Positive normal derivative of V at boundary forces interior location for such solutions.
Negative normal derivative allows construction of boundary blow-up solutions with residual mass.
Simple and non-simple blow-up profiles with residual mass exist in the interior independently of the sign of the normal derivative of V.

Where Pith is reading between the lines

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

These results highlight a dimension-dependent threshold where residual mass forces multiple blow-up points in low dimensions.
The constructions suggest that similar techniques might apply to slightly supercritical problems or other nonlinearities.
The interaction between potential V and domain geometry may determine the number of blow-up points more generally.

Load-bearing premise

That the asymptotic expansions of the gradient of the associated functional are accurate enough to rule out the coexistence of single blow-up and residual mass without additional hidden restrictions on the domain or the potential V.

What would settle it

An explicit construction or numerical evidence of a single interior blow-up solution with positive residual mass in dimension 4 would contradict the main non-existence result.

read the original abstract

In this paper, we study the Dirichlet elliptic problem $(\mathcal{P}_\varepsilon)$: $-\Delta u +V\,u = u^{p-\varepsilon}$, $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega\subset \R^n$ ( $n\geq 3$) is a bounded domain, $V$ is a smooth positive function on $\overline{\Omega}$, $p+1= 2n/(n-2)$ is the critical Sobolev exponent, and $\varepsilon >0$ is a small parameter. First, we show that, unlike the case of weak convergence to zero, interior bubbling solutions with a nonzero weak limit cannot occur in low dimensions. We then treat the general setting by removing the restriction that blow-up points are confined to the interior. Using delicate asymptotic expansions of the gradient of the associated functional, we prove that in dimensions $n=4$ and $n=5$, single blow-up point cannot coexist with residual mass.\\ We further elucidate the role of the sign of the normal derivative of the potential $V$ on the boundary: if it is positive, any single blow-up solution with residual mass must occur in the interior; if it is negative at some boundary point, boundary blow-up solutions with residual mass can be constructed. Finally, we construct both simple and non-simple interior blow-up solutions exhibiting residual mass, without any assumption on the sign of the normal derivative of $V$. These results provide new insights into the interaction between the potential, the geometry of the domain, and the critical nonlinearity.

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

The paper proves non-existence of single blow-up with residual mass in dimensions 4 and 5, plus constructions that depend on the sign of the normal derivative of V.

read the letter

The main point is that single blow-up cannot coexist with positive residual mass in n=4 and n=5 for this slightly subcritical problem, and the sign of V's normal derivative on the boundary decides whether boundary blow-up examples exist. Interior constructions work without that sign condition. This is a direct extension of earlier bubbling work to the case with nonzero weak limit, using gradient expansions on the associated functional to reach contradictions when residual mass is assumed present. The constructions appear explicit and tied to the potential's boundary behavior, which fills in some specific open cases in low dimensions. The approach follows standard lines for these problems but applies them to remove the interior-only restriction and handle the residual mass term carefully. The expansions are the core technical step, and the abstract indicates they are carried out to sufficient order to control the critical-point condition. If the error estimates hold without extra restrictions on domain geometry or V beyond smoothness, the non-existence statements look reliable. The constructions for both simple and non-simple interior solutions add concrete examples that do not require sign assumptions on the derivative. The main limitation is that everything rests on the accuracy of those asymptotics; a referee would need to check the remainder terms and the handling of the boundary interaction when blow-up occurs near the boundary. This is targeted at researchers working on blow-up analysis for critical and subcritical elliptic problems. Readers already familiar with the literature on residual mass and bubbling will find the dimension-specific non-existence and the role of V_n useful. The paper has enough new statements and technical content to merit sending it to referees rather than rejecting it outright. I would recommend peer review, with the main request being a close look at the gradient expansions and the boundary cases.

Referee Report

0 major / 3 minor

Summary. The manuscript examines the slightly subcritical Dirichlet problem −Δu + V u = u^{p−ε} (u > 0 in Ω, u = 0 on ∂Ω) for small ε > 0, where p + 1 is the critical Sobolev exponent. It establishes that interior bubbling solutions with nonzero weak limit cannot occur in low dimensions, proves via asymptotic expansions of the gradient of the associated functional that single blow-up points cannot coexist with residual mass in dimensions n = 4 and 5, analyzes the role of the sign of the normal derivative of V in permitting or precluding boundary blow-up with residual mass, and constructs both simple and non-simple interior blow-up solutions with residual mass without sign restrictions on V.

Significance. If the expansions are correct, the results clarify the interaction between blow-up, residual mass, dimension, domain geometry, and the potential V in critical elliptic problems. The non-existence statement in dimensions 4 and 5, the sign-dependent boundary constructions, and the explicit interior constructions (both simple and non-simple) constitute concrete advances that extend prior work on bubbling without residual mass.

minor comments (3)

[Abstract] The abstract refers to 'delicate asymptotic expansions of the gradient of the associated functional' without indicating the leading-order terms or the remainder estimates; adding a short outline of the expansion orders in the introduction would improve accessibility.
[Introduction] Notation for the energy functional J_ε and its gradient ∇J_ε is introduced gradually; an early dedicated subsection collecting all definitions and the precise form of the critical-point equation would reduce cross-referencing.
[Section 3] The statement that 'single blow-up point cannot coexist with residual mass' in n=4,5 is central; a brief remark on whether the argument extends verbatim to multiple isolated blow-up points (or why it does not) would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work, as well as the recommendation for minor revision. We appreciate the recognition of the advances concerning non-existence in low dimensions, the role of the normal derivative of V, and the constructions of interior solutions with residual mass.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes non-existence of single-blow-up solutions with residual mass in dimensions 4 and 5 via direct asymptotic expansions of the gradient of the associated functional, yielding contradictions with the Euler-Lagrange equation. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the boundary and interior constructions are independent of the non-existence argument and rely on explicit sign conditions on the normal derivative of V. The chain is externally falsifiable through the stated expansions and does not rename known results or smuggle ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard tools from elliptic PDE theory such as variational methods, Sobolev embeddings, and regularity theory, with no free parameters or invented entities visible in the abstract.

axioms (2)

standard math Sobolev embedding theorem holds for the critical exponent in bounded domains
Required to set up the functional space and energy functional for the problem.
standard math Elliptic regularity and maximum principle apply to positive solutions
Used to control the behavior of solutions away from blow-up points.

pith-pipeline@v0.9.0 · 5595 in / 1235 out tokens · 61357 ms · 2026-05-08T07:37:18.469609+00:00 · methodology

discussion (0)

Reference graph

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