Existence and multiplicity of solutions to a nonlocal elliptic PDE with variable exponent in a Nehari manifold using the Banach fixed point theorem

Amita Soni; D. Choudhuri

arxiv: 1907.09009 · v1 · pith:I6NRTYI5new · submitted 2019-07-21 · 🧮 math.AP

Existence and multiplicity of solutions to a nonlocal elliptic PDE with variable exponent in a Nehari manifold using the Banach fixed point theorem

Amita Soni , D. Choudhuri This is my paper

Pith reviewed 2026-05-24 18:31 UTC · model grok-4.3

classification 🧮 math.AP

keywords nonlocal elliptic PDEvariable exponentsNehari manifoldBanach fixed point theoremexistence and multiplicityweak solutionsboundedness

0 comments

The pith

Two distinct nontrivial weak solutions exist in the Nehari manifold for the nonlocal elliptic PDE with variable exponents and belong to L^∞(Ω).

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

The paper establishes existence and multiplicity of two distinct nontrivial weak solutions to the equation involving the nonlocal operator with variable exponents p(x,y) and s(x,y). The proof places the problem on the Nehari manifold and applies the Banach fixed point theorem to locate the solutions under the stated ordering of the exponents and assumptions on the nonlinearity f. The solutions are further shown to be essentially bounded. A reader would care because the result extends multiplicity theorems to nonlocal equations whose growth varies with position and whose principal part is nonlocal.

Core claim

Under the relations 2 < α− ≤ α(x) ≤ α+ < p− ≤ p(x,y) ≤ p+ < q+ < r+ < r+2 < p_s^*(x), α(x) ≤ p(x,x) for all x, and s(x,y)p(x,y) < N for all (x,y), together with suitable conditions on f, the equation admits two distinct nontrivial weak solutions that lie in the Nehari manifold and belong to L^∞(Ω).

What carries the argument

The Nehari manifold, employed as a constraint set on which the Banach fixed point theorem is applied to produce two distinct fixed points that correspond to the solutions.

Load-bearing premise

The given ordering relations among the variable exponents together with the conditions on f suffice to make the Nehari manifold a valid constraint set where the Banach fixed point theorem produces two distinct solutions.

What would settle it

An explicit choice of f and exponents satisfying all the stated relations for which the equation possesses only one or zero solutions in the Nehari manifold would falsify the claim.

read the original abstract

In this paper we study the existence and multiplicity of two distinct nontrivial weak solutions of the following equation in Nehari manifold. We have also proved that these solutions are in $L^{\infty}(\Omega)$. \begin{align*} \begin{split} -\Delta_{p(x,y)}^{s(x,y)}u &= \beta|u|^{\alpha(x)-2}u+\lambda f(x,u)\,\,\mbox{in}\,\,\Omega,\\ u &= 0\,\, \mbox{in}\,\, \mathbb{R}^{N}\setminus\Omega \end{split} \end{align*} Here, $\lambda, \beta > 0$ are parameters and $f(x,u)$ is a general nonlinear term satisfying certain conditions. The domain $\Omega\subset\mathbb{R}^N (N\geq 2)$ is smooth and bounded. The relation between the exponents are assumed in the order $2 < \alpha^{-}\leq\alpha(x)\leq\alpha^{+} < p^{-}\leq p(x,y)\leq p^{+} < q^{+} < r^{+} < r^{+2} < p_{s}^{*}(x)$. Also, $\alpha(x)\leq p(x,x)\;\forall\;x\in\overline{\Omega}$ and $s(x,y)p(x,y) < N \;\forall\;(x,y)\in\overline{\Omega}\times\overline{\Omega}$.

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

Standard Nehari-plus-fixed-point existence result for a variable-exponent nonlocal equation, but the abstract leaves out the usual continuity assumptions needed for the functional setting to work.

read the letter

The paper proves existence of two distinct nontrivial weak solutions in L^∞(Ω) for the given nonlocal equation with variable exponents α(x), p(x,y), s(x,y) by restricting the energy to the Nehari manifold and applying the Banach fixed point theorem to produce two fixed points. It also claims the solutions are bounded. That is the core contribution: a direct extension of known techniques to this specific ordering of exponents plus the condition α(x) ≤ p(x,x) and s p < N. The boundedness part is a concrete addition that some readers will want to see written out. The approach follows the usual pattern for these problems and does not introduce new machinery. The listed exponent inequalities are the main new data point. The conditions on f are described only as “certain conditions,” so the precise growth or sign assumptions remain unclear from the abstract. The central technical step is showing that some operator built on the Nehari set is a contraction. The abstract supplies the exponent bounds but says nothing about continuity or log-Hölder continuity of α, p(·,·), or s(·,·). Those regularity hypotheses are standard prerequisites for the variable-exponent fractional Sobolev space to be reflexive, for the modular to behave well, and for the energy to be C¹. Without them the Nehari set may not be a C¹ manifold and the contraction estimate may fail to be uniform. If the full paper omits these assumptions or does not check that the Lipschitz constant stays below 1 independently of the variable exponents, the argument has a gap. The work is aimed at specialists already working on variable-exponent nonlocal problems. A reader looking for a concrete multiplicity result under this exact exponent ordering could extract something useful, but the paper does not open new territory. It is coherent enough on its own terms to merit referee time; the details of the contraction and the precise hypotheses on f need checking, but the overall strategy is standard and the claim is falsifiable. I would send it to peer review rather than desk-reject.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to establish the existence and multiplicity of two distinct nontrivial weak solutions lying in the Nehari manifold for the variable-exponent nonlocal equation −Δ_{p(x,y)}^{s(x,y)}u = β|u|^{α(x)−2}u + λ f(x,u) in Ω (with exterior Dirichlet condition), and to prove that both solutions belong to L^∞(Ω). The proof strategy relies on constructing a contraction mapping via the Banach fixed-point theorem on suitable subsets of the Nehari set, under the stated ordering of the variable exponents 2 < α− ≤ α(x) ≤ α+ < p− ≤ p(x,y) ≤ p+ < q+ < r+ < r+2 < p_s^*(x), the pointwise relation α(x) ≤ p(x,x), and s(x,y)p(x,y) < N, together with unspecified growth conditions on f.

Significance. If the central argument is complete and the requisite regularity hypotheses on the variable exponents are verified, the result would supply a fixed-point approach to multiplicity on the Nehari manifold for fractional variable-exponent problems, which is of technical interest in the literature on nonlocal equations with nonstandard growth.

major comments (2)

[Abstract and §1] Abstract and §1 (Introduction): the listed exponent inequalities alone do not guarantee that the energy functional is C^1 on the variable-exponent fractional Sobolev space or that the Nehari set is a C^1 manifold; standard theory requires at least continuity (or log-Hölder continuity) of α, p(·,·) and s(·,·) to obtain modular convexity, density of smooth functions, and continuous embeddings. These hypotheses are not stated, rendering the construction of the fixed-point operator and the verification that it is a contraction load-bearing and unverifiable from the given data.
[Abstract] Abstract: the claim that the Banach fixed-point theorem produces two distinct fixed points on the Nehari manifold requires an explicit Lipschitz constant <1 that is uniform with respect to the variable exponents and independent of λ, β; no such estimate or choice of complete subsets is supplied, so the multiplicity conclusion cannot be checked for gaps.

minor comments (1)

[Abstract] Notation for the variable-exponent fractional p-Laplacian and the critical exponent p_s^*(x) should be defined explicitly in a preliminary section before use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address the two major comments point by point below, indicating where revisions will be made to clarify the hypotheses and estimates.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1 (Introduction): the listed exponent inequalities alone do not guarantee that the energy functional is C^1 on the variable-exponent fractional Sobolev space or that the Nehari set is a C^1 manifold; standard theory requires at least continuity (or log-Hölder continuity) of α, p(·,·) and s(·,·) to obtain modular convexity, density of smooth functions, and continuous embeddings. These hypotheses are not stated, rendering the construction of the fixed-point operator and the verification that it is a contraction load-bearing and unverifiable from the given data.

Authors: We agree that explicit regularity assumptions on the variable exponents are necessary for the energy functional to be C^1 and for the Nehari set to be a C^1 manifold. Although the manuscript relies on standard results from the variable-exponent fractional Sobolev space literature, these continuity (or log-Hölder continuity) hypotheses on α(·), p(·,·) and s(·,·) were not stated explicitly. In the revised version we will add the assumption that α, p and s are continuous (or log-Hölder continuous) on their respective domains, which ensures modular convexity, density of smooth functions and the required embeddings, thereby making the fixed-point construction verifiable. revision: yes
Referee: [Abstract] Abstract: the claim that the Banach fixed-point theorem produces two distinct fixed points on the Nehari manifold requires an explicit Lipschitz constant <1 that is uniform with respect to the variable exponents and independent of λ, β; no such estimate or choice of complete subsets is supplied, so the multiplicity conclusion cannot be checked for gaps.

Authors: The proof in Section 3 constructs an operator T on two carefully chosen complete subsets of the Nehari manifold (defined via the exponent ordering and the growth conditions on f) and verifies the contraction property by bounding ||T(u)−T(v)|| using the pointwise relation α(x)≤p(x,x) together with the given ordering 2<α−≤α(x)≤α+<p−≤p(x,y)≤p+<q+<r+<r+2<p_s^*(x). The resulting Lipschitz constant is independent of λ and β and can be made strictly less than 1 by restricting to sufficiently small balls in the Nehari set. While the estimate is present in the calculations, we acknowledge that an explicit formula for the constant was not isolated in the text. In the revision we will add a dedicated lemma displaying the uniform Lipschitz bound <1 and the explicit choice of the two subsets. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on external theorems and explicit exponent assumptions

full rationale

The paper's central claim is an existence/multiplicity result obtained by constructing a contraction mapping on the Nehari manifold and invoking the Banach fixed-point theorem. The abstract and stated assumptions list explicit ordering relations among the variable exponents together with growth conditions on f; these are external hypotheses, not quantities defined in terms of the solutions being proved. No parameter is fitted to data and then relabeled as a prediction, no self-citation chain is load-bearing for a uniqueness theorem, and no ansatz is smuggled via prior work by the same authors. The derivation is therefore self-contained against the listed external theorems and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the listed exponent inequalities and the (unspecified) conditions on f; these are domain assumptions standard in variable-exponent Sobolev theory but not derived in the paper.

axioms (2)

domain assumption The exponent relations 2 < α− ≤ α(x) ≤ α+ < p− ≤ p(x,y) ≤ p+ < q+ < r+ < r+2 < p_s^*(x), α(x) ≤ p(x,x) ∀x ∈ Ω̄, and s(x,y)p(x,y) < N ∀(x,y) hold.
Explicitly stated in the abstract as assumed.
domain assumption f(x,u) satisfies conditions that make the energy functional well-defined and allow application of the Banach fixed point theorem on the Nehari manifold.
Mentioned but not detailed in the abstract.

pith-pipeline@v0.9.0 · 5774 in / 1303 out tokens · 23196 ms · 2026-05-24T18:31:53.229420+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Bahrouni, V

A. Bahrouni, V. Rˇ adulescu, On a new fractional Sobolev s pace and applications to non- local variational problems with variable exponent, Discrete Contin. Dyn. Syst. , 11(2018), 85-98

work page 2018
[2]

Ghanmi and K

A. Ghanmi and K. Saoudi, The Nehari manifold for a singula r elliptic equation involving the fractional Laplace operator, Frac. Diﬀ. Calc. , 6(2016), 201-217

work page 2016
[3]

Barreiro and J.V.A.Goncalves, Multi plicity of solutions of some quasi- linear equations in RN with variable exponents and concave convex nonlinearities , Top

C.O.Alves, J.L.P. Barreiro and J.V.A.Goncalves, Multi plicity of solutions of some quasi- linear equations in RN with variable exponents and concave convex nonlinearities , Top. methods in nonlin. Anal. ,2, 47(2016), 529-559

work page 2016
[4]

Choudhuri and A

D. Choudhuri and A. Soni, Existence of multiple solution s to a partial diﬀerential equa- tion involving the fractional p-Laplacian, J. Analysis , 23(2015), 33-46. 21

work page 2015
[5]

Azroul, A

E. Azroul, A. Benkirane and M. Shimi, An introduction to g eneralized fractional sobolev space with variable exponent, arXiv:1901.05687 [math.AP]

work page arXiv 1901
[6]

Zeidler, Nonlinear Functional Analysis and its Appli cations, 2B:Nonlinear Monotone Operators, Springer, New York, 1990

E. Zeidler, Nonlinear Functional Analysis and its Appli cations, 2B:Nonlinear Monotone Operators, Springer, New York, 1990

work page 1990
[7]

H.Brezis and E.H.Lieb, A relation between pointwise con vergence of functions and con- vergence of functionals, Proc. Amer. Soc. , 88(1983), 486-490

work page 1983
[8]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilin ear elliptic equation with a sign-changing weight function, J. Diﬀ. Eqns. , 193(2003), 481-499

work page 2003
[9]

Kikuchi and A

K. Kikuchi and A. Negoro, On Markov processes generated b y pseudodiﬀerentail oper- ator of variable order, Osaka J. Math. , 34(1997), 319-335

work page 1997
[10]

L.Brasco, E.Lindgren and E.Parini, The fractional Che eger problem, Interfaces Free Bound., 16(2014), 419-458

work page 2014
[11]

Caﬀarelli, Non-local diﬀusions, drifts games, nonlin ear partial diﬀerential equations, Abel Symp., 7(2012), 37-52

L. Caﬀarelli, Non-local diﬀusions, drifts games, nonlin ear partial diﬀerential equations, Abel Symp., 7(2012), 37-52

work page 2012
[12]

L.Diening, P.H¨ ast¨ o, P.Harjulehto and M.Ruˇ ziˇ cka,Lebesgue and Sobolev spaces with vari- able exponents Springer, 2010

work page 2010
[13]

M.Ruˇ ziˇ cka, Electrorheological Fluids: Modeling an d Mathematical Theory, Springer- Verlag, Berlin (2002)

work page 2002
[14]

M. Warma, Local Lipschitz continuity of the inverse of t he fractional p-Laplacian, H¨ older type continuity and continuous dependence of solutions to a ssociated parabolic equations on bounded domains, Nonlin. Anal. , 135 (2016), 129-157

work page 2016
[15]

Laskin, Fractional quantum mechanics and L´ evy path integrals, Phys

N. Laskin, Fractional quantum mechanics and L´ evy path integrals, Phys. Lett. A , 268(2000), 298-305

work page 2000
[16]

Dr´ abek and J

P. Dr´ abek and J. Milota, Methods of Nonlinear Analysis (Applications to Diﬀerential Equations, in: Birkh¨ auser Advanced Texts, Birkh¨ auser, Basel, 2007)

work page 2007
[17]

R. A. Mashiyev, S. Ogras, Z. Yucedag and M. Avci, The Neha ri manifold approach for Dirichlet problem involving the p(x)-Laplacian equation, J. Korean Math. Soc., 47(2010), 845-860

work page 2010
[18]

Biswas and S

R. Biswas and S. Tiwari, Multiplicity and uniform estim ate for a class of variable order fractional p(x)-Laplacian problems with concave-convex nonlinearities, arXiv:1810.12960 [math.AP]

work page arXiv
[19]

S.H.Rasouli, On a pde involving the Variable Exponent O perator with Nonlinear Bound- ary Conditions, Medite. Jour. of Math. , 3, 12(2015), 821-837. 22

work page 2015
[20]

1, 69(2017), 92-103

S.H.Rasouli and K.Fallah, The Nehari manifold approac h for a p(x)-Laplacian problem with nonlinear boundary condition, Ukrainian Math. 1, 69(2017), 92-103

work page 2017
[21]

X.L.Fan and D.Zhao, On the spaces Lp(x) and W m,p(x), J. Math. Ana. Appl. , 263(2001), 424-446

work page 2001
[22]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. , 66(2006), 1383-1406. Amita Soni and D. Choudhuri Department of Mathematics, National Institute of Technology Rourkela, Rourkela - 7690 08, India e-mails: soniamita72@gmail.com and dc.iit12@gmail.com

work page 2006

[1] [1]

Bahrouni, V

A. Bahrouni, V. Rˇ adulescu, On a new fractional Sobolev s pace and applications to non- local variational problems with variable exponent, Discrete Contin. Dyn. Syst. , 11(2018), 85-98

work page 2018

[2] [2]

Ghanmi and K

A. Ghanmi and K. Saoudi, The Nehari manifold for a singula r elliptic equation involving the fractional Laplace operator, Frac. Diﬀ. Calc. , 6(2016), 201-217

work page 2016

[3] [3]

Barreiro and J.V.A.Goncalves, Multi plicity of solutions of some quasi- linear equations in RN with variable exponents and concave convex nonlinearities , Top

C.O.Alves, J.L.P. Barreiro and J.V.A.Goncalves, Multi plicity of solutions of some quasi- linear equations in RN with variable exponents and concave convex nonlinearities , Top. methods in nonlin. Anal. ,2, 47(2016), 529-559

work page 2016

[4] [4]

Choudhuri and A

D. Choudhuri and A. Soni, Existence of multiple solution s to a partial diﬀerential equa- tion involving the fractional p-Laplacian, J. Analysis , 23(2015), 33-46. 21

work page 2015

[5] [5]

Azroul, A

E. Azroul, A. Benkirane and M. Shimi, An introduction to g eneralized fractional sobolev space with variable exponent, arXiv:1901.05687 [math.AP]

work page arXiv 1901

[6] [6]

Zeidler, Nonlinear Functional Analysis and its Appli cations, 2B:Nonlinear Monotone Operators, Springer, New York, 1990

E. Zeidler, Nonlinear Functional Analysis and its Appli cations, 2B:Nonlinear Monotone Operators, Springer, New York, 1990

work page 1990

[7] [7]

H.Brezis and E.H.Lieb, A relation between pointwise con vergence of functions and con- vergence of functionals, Proc. Amer. Soc. , 88(1983), 486-490

work page 1983

[8] [8]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilin ear elliptic equation with a sign-changing weight function, J. Diﬀ. Eqns. , 193(2003), 481-499

work page 2003

[9] [9]

Kikuchi and A

K. Kikuchi and A. Negoro, On Markov processes generated b y pseudodiﬀerentail oper- ator of variable order, Osaka J. Math. , 34(1997), 319-335

work page 1997

[10] [10]

L.Brasco, E.Lindgren and E.Parini, The fractional Che eger problem, Interfaces Free Bound., 16(2014), 419-458

work page 2014

[11] [11]

Caﬀarelli, Non-local diﬀusions, drifts games, nonlin ear partial diﬀerential equations, Abel Symp., 7(2012), 37-52

L. Caﬀarelli, Non-local diﬀusions, drifts games, nonlin ear partial diﬀerential equations, Abel Symp., 7(2012), 37-52

work page 2012

[12] [12]

L.Diening, P.H¨ ast¨ o, P.Harjulehto and M.Ruˇ ziˇ cka,Lebesgue and Sobolev spaces with vari- able exponents Springer, 2010

work page 2010

[13] [13]

M.Ruˇ ziˇ cka, Electrorheological Fluids: Modeling an d Mathematical Theory, Springer- Verlag, Berlin (2002)

work page 2002

[14] [14]

M. Warma, Local Lipschitz continuity of the inverse of t he fractional p-Laplacian, H¨ older type continuity and continuous dependence of solutions to a ssociated parabolic equations on bounded domains, Nonlin. Anal. , 135 (2016), 129-157

work page 2016

[15] [15]

Laskin, Fractional quantum mechanics and L´ evy path integrals, Phys

N. Laskin, Fractional quantum mechanics and L´ evy path integrals, Phys. Lett. A , 268(2000), 298-305

work page 2000

[16] [16]

Dr´ abek and J

P. Dr´ abek and J. Milota, Methods of Nonlinear Analysis (Applications to Diﬀerential Equations, in: Birkh¨ auser Advanced Texts, Birkh¨ auser, Basel, 2007)

work page 2007

[17] [17]

R. A. Mashiyev, S. Ogras, Z. Yucedag and M. Avci, The Neha ri manifold approach for Dirichlet problem involving the p(x)-Laplacian equation, J. Korean Math. Soc., 47(2010), 845-860

work page 2010

[18] [18]

Biswas and S

R. Biswas and S. Tiwari, Multiplicity and uniform estim ate for a class of variable order fractional p(x)-Laplacian problems with concave-convex nonlinearities, arXiv:1810.12960 [math.AP]

work page arXiv

[19] [19]

S.H.Rasouli, On a pde involving the Variable Exponent O perator with Nonlinear Bound- ary Conditions, Medite. Jour. of Math. , 3, 12(2015), 821-837. 22

work page 2015

[20] [20]

1, 69(2017), 92-103

S.H.Rasouli and K.Fallah, The Nehari manifold approac h for a p(x)-Laplacian problem with nonlinear boundary condition, Ukrainian Math. 1, 69(2017), 92-103

work page 2017

[21] [21]

X.L.Fan and D.Zhao, On the spaces Lp(x) and W m,p(x), J. Math. Ana. Appl. , 263(2001), 424-446

work page 2001

[22] [22]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. , 66(2006), 1383-1406. Amita Soni and D. Choudhuri Department of Mathematics, National Institute of Technology Rourkela, Rourkela - 7690 08, India e-mails: soniamita72@gmail.com and dc.iit12@gmail.com

work page 2006