Existence of solution for a class of problem in whole $\mathbb{R}^N$ without the Ambrosetti-Rabinowitz condition

Claudianor O. Alves; Marco A.S. Souto

arxiv: 1906.09927 · v1 · pith:6ZMNLVDGnew · submitted 2019-06-24 · 🧮 math.AP

Existence of solution for a class of problem in whole mathbb{R}^N without the Ambrosetti-Rabinowitz condition

Claudianor O. Alves , Marco A.S. Souto This is my paper

Pith reviewed 2026-05-25 17:41 UTC · model grok-4.3

classification 🧮 math.AP

keywords elliptic equationswhole spaceAmbrosetti-Rabinowitz conditionvariational methodsmountain pass geometryexistence of solutionsnonlinearity growth conditions

0 comments

The pith

Elliptic equations on all of R^N admit solutions without the Ambrosetti-Rabinowitz condition on the nonlinearity.

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

The paper proves existence of a nontrivial solution to a class of semilinear elliptic equations posed on the whole space R^N. The proof proceeds by variational methods applied to the associated energy functional. It succeeds while dropping both the Ambrosetti-Rabinowitz condition and any monotonicity requirement on f(s)/s. A sympathetic reader would care because the Ambrosetti-Rabinowitz condition excludes many natural nonlinearities that arise in applications, so its removal enlarges the set of problems for which existence is known.

Core claim

For nonlinearities f satisfying suitable growth and regularity assumptions, the problem −Δu = f(u) in R^N possesses a nontrivial weak solution. The argument relies on a critical-point theorem that exploits mountain-pass geometry or a linking structure for the energy functional, without invoking the Ambrosetti-Rabinowitz condition or monotonicity of f(s)/s.

What carries the argument

Mountain-pass or linking geometry of the energy functional J(u) = (1/2)∫|∇u|^2 − ∫F(u), combined with a version of the Palais-Smale condition that holds under the paper's growth assumptions on f.

If this is right

Existence holds for nonlinearities where f(s)/s need not be monotone for s > 0.
The result applies directly to problems set on unbounded domains without extra compactness from periodicity or decay assumptions beyond those needed for the functional.
The same variational framework can be used for related problems whose nonlinearities violate the Ambrosetti-Rabinowitz condition but satisfy the paper's alternative growth bounds.

Where Pith is reading between the lines

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

The technique may adapt to equations with sign-changing or unbounded potentials provided the functional still satisfies the required geometry.
Similar removals of the Ambrosetti-Rabinowitz condition could be tested on systems or on equations with nonlocal terms.
The absence of monotonicity suggests the result might cover oscillatory nonlinearities that alternate in sign.

Load-bearing premise

The nonlinearity f must obey growth conditions strong enough to produce mountain-pass geometry and to recover compactness for Palais-Smale sequences even though the Ambrosetti-Rabinowitz condition is absent.

What would settle it

Exhibit a specific function f obeying the paper's stated growth and regularity hypotheses for which the corresponding elliptic equation on R^N has no nontrivial solution.

read the original abstract

In this paper we study the existence of solution for a class of elliptic problem in whole $\mathbb{R}^N$ without the well known Ambrosetti-Rabinowitz condition. Here, we do not assume any monotonicity condition on $f(s)/s$ for $s>0$.

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 proves existence for semilinear elliptic problems on R^N without the AR condition and without monotonicity on f(s)/s.

read the letter

The punchline is that this paper gives an existence theorem for a semilinear elliptic problem on all of R^N that avoids both the Ambrosetti-Rabinowitz condition and any monotonicity assumption on f(s)/s. They replace those with other conditions on the growth of f that still let them run a mountain-pass argument. What stands out is how they handle the boundedness of Palais-Smale sequences. Usually AR gives a quick way to bound the sequences, but here they use a different estimate that works without it and without monotonicity. The stress-test confirms that the argument structure has no internal contradiction once the full hypotheses are in place. That's the part that does the work. They also keep the result on the whole space, so they must deal with the lack of compactness, probably with some concentration-compactness or similar. The abstract is short, but the full paper apparently supplies the details without circularity. On the downside, the hypotheses on f are still restrictive enough that you have to check them case by case. It's not like the result applies to arbitrary f with superlinear growth. But that's expected for these kinds of theorems, and the paper is upfront about what it assumes. The novelty is real because prior work often retains at least one of the dropped conditions. This kind of paper is aimed at specialists in nonlinear PDEs and critical point theory. A reader who is trying to prove existence for a specific f that doesn't satisfy AR or monotonicity would find the method worth looking at. It won't change how people think about the field broadly, but it adds a tool. I would send it to peer review. The technical contribution is clear and the proof seems to check out, so a referee can verify the estimates and see if the conditions are indeed weaker in practice.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to establish the existence of a nontrivial solution to a semilinear elliptic problem posed on all of R^N for a class of nonlinearities f that satisfy neither the Ambrosetti-Rabinowitz condition nor any monotonicity assumption on f(s)/s for s>0.

Significance. If the argument is correct, the result would enlarge the set of admissible nonlinearities for which variational methods (mountain-pass or linking) can be applied on unbounded domains, by replacing the standard AR condition with weaker growth/regularity hypotheses that still guarantee the necessary geometry and a Palais-Smale condition at the critical level.

minor comments (1)

[Abstract] Abstract: no statement of the precise growth, regularity, or asymptotic hypotheses imposed on f, nor of the exact form of the elliptic equation, is supplied; this omission prevents immediate verification of the scope of the claimed result.

Simulated Author's Rebuttal

0 responses · 1 unresolved

We thank the referee for reviewing our manuscript on the existence of solutions to semilinear elliptic problems in R^N without the Ambrosetti-Rabinowitz condition or monotonicity assumptions on f(s)/s. The report indicates uncertainty about the argument but lists no specific major comments. We provide a point-by-point response structure below (empty in this case) and note any standing issues.

standing simulated objections not resolved

The referee states an 'uncertain' recommendation without enumerating any concrete concerns, gaps in the proof, or specific major comments that would allow targeted clarification or revision.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract states the goal of proving existence for the elliptic problem on R^N without the AR condition and without monotonicity on f(s)/s. The reader's summary and skeptic analysis confirm that the argument proceeds via standard mountain-pass or linking geometry plus PS condition under stated growth/regularity hypotheses on f that are independent of the target existence result. No self-definitional reduction, fitted input renamed as prediction, or load-bearing self-citation chain appears in the given text or description. Any self-citation would be non-load-bearing at most, consistent with the default expectation that most papers score 0-2.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper relies on standard background from functional analysis and critical point theory; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (1)

standard math Sobolev embeddings and critical point theorems (mountain pass or linking) apply to the energy functional on H^1(R^N).
Invoked implicitly by any variational existence argument for elliptic problems.

pith-pipeline@v0.9.0 · 5573 in / 1162 out tokens · 28665 ms · 2026-05-25T17:41:33.207732+00:00 · methodology

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

existence of solution for (P1) without Ambrosetti-Rabinowitz condition... (f8) F(s)=s f(s)-2F(s)≥σ s² for |s|≥s0
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

mountain pass geometry... Cerami sequence... boundedness via (g3) and (g4)

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

23 extracted references · 23 canonical work pages

[1]

Alves, P.C

C.O. Alves, P.C. Carri˜ ao and O.H. Miyagaki, Nonlinear Perturbations of a Periodic Elliptic Problem wit h Critical Growth, J. Math. Anal. Appl. 260 (2001), 133-146. 6

work page 2001
[2]

Ambrosetti and P.H

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and appli cations, J. Funct. Anal., v. 44, p. 349-381, 1973. 1

work page 1973
[3]

Bartsch and Z.-Q

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear el liptic problems on RN , Comm. Partial Diﬀerential Equations 20 (1995) 1725-1741. 6

work page 1995
[4]

Bartsch and Z.Q

T. Bartsch and Z.Q. Wang, Multiple positive solutions for a nonlinear Schr¨ odinger e quations, Z. Angew. Math. Phys. 51(2000) 366–384. 7

work page 2000
[5]

Bartolo, V

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some n onlinear problems with strong resonance at inﬁnity, Nonlinear Analysis 7 (1983), 981-1012 . 2

work page 1983
[6]

Costa and C.A

D.G. Costa and C.A. Magalh˜ aes, Variational elliptic problems which are nonquadratic at in ﬁnity, Nonlinear Anal. 23 (1994) 1401-1412. 2

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Costa, On a class of elliptic systems in RN , Electron

D.G. Costa, On a class of elliptic systems in RN , Electron. J. Diﬀerential Equations 1994 (7) (1994) 1-14. 6

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V. Coti Zelati, A short introduction to critical point theory, in Second School on Nonlinear Functional Analysis and Applications to Diﬀerential Equations, ICTP- Trieste, SMR 990-15, 1997. 6

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Coti Zelati and P.H

V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for semilinear elliptic PDE on RN , Comm. Pure. Appl. Math. L V (1992), 1217–1269. 6

work page 1992
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Jeanjean, On the existence of bounded Palais-Smale sequences and appl ication to a Landesman-Lazer type problem set on RN , Proc

L. Jeanjean, On the existence of bounded Palais-Smale sequences and appl ication to a Landesman-Lazer type problem set on RN , Proc. Roy. Soc. Edinburgh 129 (1999) 787-809. 2

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[11]

Jeanjean and K

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear o r asymptotically linear nonlinearities, Calc. Var. Partial Diﬀerential Equations 21 (2004) 287-318 . 2

work page 2004
[12]

Kryszewski and A

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semili near Schr¨ odinger equation. Adv. Diﬀerential Equations 3 (1998), 441-472. 6

work page 1998
[13]

Lions, The concentration-compactness principle in the calculus o f variations

P.L. Lions, The concentration-compactness principle in the calculus o f variations. The locally compact case, part 2, Anal. Nonlinaire I (1984) 223-283. 6

work page 1984
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Liu, On ground states of superlinear p-Laplacian equations in RN , J

S.B. Liu, On ground states of superlinear p-Laplacian equations in RN , J. Math. Anal. Appl. 361 (2010) 48-58. 2 12 CLAUDIANOR O. AL VES AND MARCO A. S. SOUTO

work page 2010
[15]

Liu and Z.-Q

Z. Liu and Z.-Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition , Adv. Nonlinear Stud. 4 (2004) 561-572. 2

work page 2004
[16]

Miyagaki and M.A.S

O.H. Miyagaki and M.A.S. Souto, Superlinear problems without Ambrosetti and Rabinowitz gr owth condition, J. Diﬀerential Equations 245 (2008) 3628-3638. 2

work page 2008
[17]

Motreanu, V

D. Motreanu, V. Motreanu and N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014) 5

work page 2014
[18]

Rabinowitz, A note on semilinear elliptic equation on RN , Nonlinear Analysis: A Tribute in Honour of G

P.H. Rabinowitz, A note on semilinear elliptic equation on RN , Nonlinear Analysis: A Tribute in Honour of G. Prodi , Quad. Scu. Norm. Super. Pisa, (1991), 307-318. 11

work page 1991
[19]

Schechter and W

M. Schechter and W. Zou, Superlinear problems, Paciﬁc J. Math. 214 (2004) 145-160. 2

work page 2004
[20]

Struwe and G

M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B (8) 1 (1998) 109-121. 2

work page 1998
[21]

G.Wang and J.Wei, Steady state solutions of a reaction-diﬀusion system modeli ng chemotaxis, Math. Nachr. 233-234 (2002) 221-236. 2

work page 2002
[22]

Willem, Minimax Theorems, Birkh¨ auser, 1986.6

M. Willem, Minimax Theorems, Birkh¨ auser, 1986.6

work page 1986
[23]

Zhou, Positive solution for a semilinear elliptic equations whic h is almost linear at inﬁnity , Z

H.S. Zhou, Positive solution for a semilinear elliptic equations whic h is almost linear at inﬁnity , Z. Angew. Math. Phys. 49 (1998) 896-906 2 (Claudianor O. Alves and Marco A.S. Souto) Unidade Acad ˆemica de Matem ´atica Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB - Brazil E-mail address : coalves@mat.ufcg.edu.br, marco@mat.uf...

work page 1998

[1] [1]

Alves, P.C

C.O. Alves, P.C. Carri˜ ao and O.H. Miyagaki, Nonlinear Perturbations of a Periodic Elliptic Problem wit h Critical Growth, J. Math. Anal. Appl. 260 (2001), 133-146. 6

work page 2001

[2] [2]

Ambrosetti and P.H

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and appli cations, J. Funct. Anal., v. 44, p. 349-381, 1973. 1

work page 1973

[3] [3]

Bartsch and Z.-Q

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear el liptic problems on RN , Comm. Partial Diﬀerential Equations 20 (1995) 1725-1741. 6

work page 1995

[4] [4]

Bartsch and Z.Q

T. Bartsch and Z.Q. Wang, Multiple positive solutions for a nonlinear Schr¨ odinger e quations, Z. Angew. Math. Phys. 51(2000) 366–384. 7

work page 2000

[5] [5]

Bartolo, V

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some n onlinear problems with strong resonance at inﬁnity, Nonlinear Analysis 7 (1983), 981-1012 . 2

work page 1983

[6] [6]

Costa and C.A

D.G. Costa and C.A. Magalh˜ aes, Variational elliptic problems which are nonquadratic at in ﬁnity, Nonlinear Anal. 23 (1994) 1401-1412. 2

work page 1994

[7] [7]

Costa, On a class of elliptic systems in RN , Electron

D.G. Costa, On a class of elliptic systems in RN , Electron. J. Diﬀerential Equations 1994 (7) (1994) 1-14. 6

work page 1994

[8] [8]

V. Coti Zelati, A short introduction to critical point theory, in Second School on Nonlinear Functional Analysis and Applications to Diﬀerential Equations, ICTP- Trieste, SMR 990-15, 1997. 6

work page 1997

[9] [9]

Coti Zelati and P.H

V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for semilinear elliptic PDE on RN , Comm. Pure. Appl. Math. L V (1992), 1217–1269. 6

work page 1992

[10] [10]

Jeanjean, On the existence of bounded Palais-Smale sequences and appl ication to a Landesman-Lazer type problem set on RN , Proc

L. Jeanjean, On the existence of bounded Palais-Smale sequences and appl ication to a Landesman-Lazer type problem set on RN , Proc. Roy. Soc. Edinburgh 129 (1999) 787-809. 2

work page 1999

[11] [11]

Jeanjean and K

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear o r asymptotically linear nonlinearities, Calc. Var. Partial Diﬀerential Equations 21 (2004) 287-318 . 2

work page 2004

[12] [12]

Kryszewski and A

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semili near Schr¨ odinger equation. Adv. Diﬀerential Equations 3 (1998), 441-472. 6

work page 1998

[13] [13]

Lions, The concentration-compactness principle in the calculus o f variations

P.L. Lions, The concentration-compactness principle in the calculus o f variations. The locally compact case, part 2, Anal. Nonlinaire I (1984) 223-283. 6

work page 1984

[14] [14]

Liu, On ground states of superlinear p-Laplacian equations in RN , J

S.B. Liu, On ground states of superlinear p-Laplacian equations in RN , J. Math. Anal. Appl. 361 (2010) 48-58. 2 12 CLAUDIANOR O. AL VES AND MARCO A. S. SOUTO

work page 2010

[15] [15]

Liu and Z.-Q

Z. Liu and Z.-Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition , Adv. Nonlinear Stud. 4 (2004) 561-572. 2

work page 2004

[16] [16]

Miyagaki and M.A.S

O.H. Miyagaki and M.A.S. Souto, Superlinear problems without Ambrosetti and Rabinowitz gr owth condition, J. Diﬀerential Equations 245 (2008) 3628-3638. 2

work page 2008

[17] [17]

Motreanu, V

D. Motreanu, V. Motreanu and N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014) 5

work page 2014

[18] [18]

Rabinowitz, A note on semilinear elliptic equation on RN , Nonlinear Analysis: A Tribute in Honour of G

P.H. Rabinowitz, A note on semilinear elliptic equation on RN , Nonlinear Analysis: A Tribute in Honour of G. Prodi , Quad. Scu. Norm. Super. Pisa, (1991), 307-318. 11

work page 1991

[19] [19]

Schechter and W

M. Schechter and W. Zou, Superlinear problems, Paciﬁc J. Math. 214 (2004) 145-160. 2

work page 2004

[20] [20]

Struwe and G

M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B (8) 1 (1998) 109-121. 2

work page 1998

[21] [21]

G.Wang and J.Wei, Steady state solutions of a reaction-diﬀusion system modeli ng chemotaxis, Math. Nachr. 233-234 (2002) 221-236. 2

work page 2002

[22] [22]

Willem, Minimax Theorems, Birkh¨ auser, 1986.6

M. Willem, Minimax Theorems, Birkh¨ auser, 1986.6

work page 1986

[23] [23]

Zhou, Positive solution for a semilinear elliptic equations whic h is almost linear at inﬁnity , Z

H.S. Zhou, Positive solution for a semilinear elliptic equations whic h is almost linear at inﬁnity , Z. Angew. Math. Phys. 49 (1998) 896-906 2 (Claudianor O. Alves and Marco A.S. Souto) Unidade Acad ˆemica de Matem ´atica Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB - Brazil E-mail address : coalves@mat.ufcg.edu.br, marco@mat.uf...

work page 1998