A Mountain-Pass Algorithm for Nonlocal Problems with Super-quadratic Nonlinearities

Gabriela Jaramillo; Joshua M. Siktar; Loic Cappanera

arxiv: 2605.23106 · v1 · pith:BZFM4LTQnew · submitted 2026-05-21 · 🧮 math.AP

A Mountain-Pass Algorithm for Nonlocal Problems with Super-quadratic Nonlinearities

Loic Cappanera , Gabriela Jaramillo , Joshua M. Siktar This is my paper

Pith reviewed 2026-05-25 05:07 UTC · model grok-4.3

classification 🧮 math.AP

keywords nonlocal equationsmountain pass theoremsuper-quadratic nonlinearityexistence of solutionsDirichlet boundary conditionsNeumann boundary conditionsconvolution kernels

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The pith

The Mountain Pass Theorem establishes existence of nontrivial solutions to the nonlocal equation -Lu = f(x,u) with super-quadratic f for both Dirichlet and Neumann boundary conditions.

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

The paper proves that nontrivial solutions exist for the equation -Lu = f(x,u) where L arises from radially symmetric L1 convolution kernels with finite second moments and f is super-quadratic. It applies the classical Mountain Pass Theorem to locate critical points of the associated energy functional. The result covers both homogeneous nonlocal Dirichlet and nonlocal Neumann boundary conditions. A sympathetic reader would care because the mild kernel assumptions permit kernels with algebraic decay or sign changes that appear in biological and physical models, and the work adds numerical simulations that adapt gradient descent to the energy landscape for odd-degree nonlinearities.

Core claim

Using the strong nonlinearities present in the equation, we prove the existence of nontrivial solutions using the classical Mountain Pass Theorem, a central result in minimax theory that equates solutions of our equation to critical points of a corresponding energy functional. This existence result holds with both homogeneous nonlocal Dirichlet and nonlocal Neumann boundary conditions.

What carries the argument

The Mountain Pass Theorem applied to the energy functional associated with the nonlocal problem.

If this is right

Nontrivial solutions exist for the nonlocal problem under the stated kernel and nonlinearity assumptions.
The existence holds simultaneously for homogeneous nonlocal Dirichlet and nonlocal Neumann boundary conditions.
Numerical approximations are feasible by running gradient descent on the energy functional for nonlinearities whose maximal degree in u is odd.
The result applies to kernels with algebraic decay and to sign-changing kernels.

Where Pith is reading between the lines

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

The same variational argument could be tested on nonlocal operators that are not pure convolutions but still satisfy the finite-second-moment condition.
The numerical scheme might be used to track how solution amplitude changes when the kernel support or decay rate is varied.
Specific models in population dynamics could be checked by substituting kernels already validated in the literature into the existence theorem.

Load-bearing premise

The nonlinearity f must be super-quadratic so the energy functional possesses the mountain-pass geometry needed for the theorem to apply.

What would settle it

An explicit radially symmetric L1 kernel with finite second moment together with a super-quadratic f for which the energy functional fails to satisfy the Palais-Smale condition or the mountain-pass geometry would falsify the existence claim.

read the original abstract

In this paper we consider a nonlinear equation $-\mathcal{L} u(x) = f(x, u(x))$ with a super-quadratic nonlinearity, $f$, and a nonlocal operator, $\mathcal{L}$, generated by a special class of radially symmetric $L^1$ convolution kernels with finite second moments. The assumptions on this operator are mild and allow for a variety of kernels used in biological and physical applications, including kernels with algebraic decay and sign changing kernels. Using the strong nonlinearities present in the equation, we prove the existence of nontrivial solutions using the classical Mountain Pass Theorem, a central result in minimax theory that equates solutions of our equation to critical points of a corresponding energy functional. This existence result holds with both homogeneous nonlocal Dirichlet and nonlocal Neumann boundary conditions. We supplement these theoretical results with numerical simulations for various nonlinearities with odd maximal degree in the unknown $u$. The numerical scheme exploits the resulting energy landscape which allows one to adapt a gradient descent algorithm.

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 gives a standard mountain-pass existence result for nonlocal equations under a broad but specific class of kernels, with some added numerics.

read the letter

The main takeaway is that this paper establishes existence of nontrivial solutions via the mountain pass theorem for nonlocal problems with super-quadratic nonlinearities, where the operator is generated by radially symmetric L1 kernels that have finite second moments. This covers both homogeneous nonlocal Dirichlet and Neumann boundary conditions, and they back it up with some numerical examples using gradient descent on the energy functional. What stands out as new is the kernel class itself. The assumptions are mild enough to include kernels with algebraic decay and sign changes, which appear in biological and physical models, yet still guarantee the quadratic part gives a Hilbert space and the compact embeddings for the Palais-Smale condition. The paper does a good job laying out how the super-quadratic growth gives the mountain pass geometry and how the kernel properties make the functional well-defined. The numerical part uses the energy landscape to run a gradient descent, which is a reasonable way to find the critical points they prove exist. The soft spots are minor. The argument follows the classical mountain pass theorem closely once the space is set up, so there is no new technique in the variational part. The numerics are supplementary and do not include error analysis or comparisons to other methods, which is fine for a theory paper but limits how much they add. If the kernel class turns out to require more work to verify the conditions than the abstract suggests, that could be a point for referees to check. This paper is for people already working on variational methods for nonlocal equations. Someone looking for existence results in a setting that matches certain application kernels would find it useful. It deserves a serious referee because the result is cleanly stated and the setting is relevant. I would recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript considers the nonlocal equation −ℒu(x)=f(x,u(x)) with super-quadratic nonlinearity f and nonlocal operator ℒ generated by radially symmetric L¹ convolution kernels with finite second moments. It proves existence of nontrivial solutions via the classical Mountain Pass Theorem applied to the associated energy functional, for both homogeneous nonlocal Dirichlet and Neumann boundary conditions. Numerical simulations are included for nonlinearities with odd maximal degree, using a gradient-descent scheme adapted to the energy landscape.

Significance. If the claims hold, the work extends the range of nonlocal operators to which the Mountain Pass Theorem applies under mild kernel hypotheses that cover algebraic decay and sign-changing kernels arising in applications. The finite-second-moment condition is used to secure the Hilbert structure and compact embeddings needed for the Palais-Smale condition. The numerical component supplies concrete illustrations of the theoretical existence result.

minor comments (3)

[Title and Abstract] The title refers to a 'Mountain-Pass Algorithm,' yet the abstract and introduction emphasize the existence theorem; a short paragraph clarifying how the numerical gradient-descent scheme constitutes the algorithmic contribution would improve alignment.
[Numerical simulations] In the numerical section, the discretization of the nonlocal operator ℒ (e.g., quadrature rule for the convolution kernel) is not specified; adding one or two sentences on the implementation would aid reproducibility.
[Introduction] The statement that the kernel assumptions 'allow for a variety of kernels' would benefit from an explicit example of a sign-changing kernel satisfying the finite-second-moment condition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper applies the classical Mountain Pass Theorem directly to the energy functional I associated with the nonlocal operator L generated by the stated class of kernels. The superquadratic growth of f supplies the geometry (I(0)=0, existence of e with I(e)<0, positive inf on small sphere), while finite-second-moment and radial L1 assumptions ensure the quadratic term defines a Hilbert space and yields compact embeddings for the Palais-Smale condition. Both Dirichlet and Neumann cases are handled by the same variational argument. No equation or step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the result is an instance of a standard theorem on a new but explicitly axiomatized operator class. Numerical simulations are presented separately and do not enter the existence proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The existence proof rests on domain assumptions about the kernel class and the growth of f; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The nonlocal operator L is generated by radially symmetric L1 convolution kernels with finite second moments.
This is the structural hypothesis that permits the Mountain Pass geometry to hold for the energy functional.
domain assumption The nonlinearity f is super-quadratic.
Required to ensure the energy functional satisfies the geometric conditions of the Mountain Pass Theorem.

pith-pipeline@v0.9.0 · 5705 in / 1380 out tokens · 37953 ms · 2026-05-25T05:07:58.455750+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages · 1 internal anchor

[1]

A mountain pass algorithm for quasilinear boundary value problem withp-Laplacian.Math- ematics and Computers in Simulation, 189:291–304, 2021

Michaela Bailová and Ji ˇrí Bouchala. A mountain pass algorithm for quasilinear boundary value problem withp-Laplacian.Math- ematics and Computers in Simulation, 189:291–304, 2021

work page 2021
[2]

A new approach to solving a quasilinear boundary value problem withp-Laplacian using optimization.Applications of Mathematics, 68(4):425–439, 2023

Michaela Bailová and Ji ˇrí Bouchala. A new approach to solving a quasilinear boundary value problem withp-Laplacian using optimization.Applications of Mathematics, 68(4):425–439, 2023. A MOUNTAIN-PASS ALGORITHM FOR NONLOCAL PROBLEMS WITH SUPER-QUADRATIC NONLINEARITIES 33

work page 2023
[3]

On some nonlocal evolution equations arising in materials science.Nonlinear dynamics and evolution equations, 48:13–52, 2006

Peter W Bates. On some nonlocal evolution equations arising in materials science.Nonlinear dynamics and evolution equations, 48:13–52, 2006

work page 2006
[4]

Complexity and coexistence in a simple spatial model for arid savanna ecosystems.Theoretical Ecology, 6(2):131–141, 2013

Mara Baudena and Max Rietkerk. Complexity and coexistence in a simple spatial model for arid savanna ecosystems.Theoretical Ecology, 6(2):131–141, 2013

work page 2013
[5]

Mountain pass solutions for nonlocal equations.Ann

G Molica Bisci and V Radulescu. Mountain pass solutions for nonlocal equations.Ann. Acad. Sci. Fenn. Math, 39(2):579–592, 2014

work page 2014
[6]

Three weak solutions for nonlocal fractional equations.Advanced Nonlinear Studies, 14(3):619–629, 2014

Giovanni Molica Bisci and Bruno Antonio Pansera. Three weak solutions for nonlocal fractional equations.Advanced Nonlinear Studies, 14(3):619–629, 2014

work page 2014
[7]

Cambridge University Press, 2016

Giovanni Molica Bisci, Vicen¸ tiu D R ˘adulescu, and Raffaella Servadei.Variational methods for nonlocal fractional problems, volume 162. Cambridge University Press, 2016

work page 2016
[8]

Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data

Dorota Bors. Application of mountain pass theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data.arXiv preprint arXiv:1408.0618, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014
[9]

Thin front limit of an integro-differential Fisher–KPP equation with fat-tailed kernels.SIAM Journal on Mathematical Analysis, 50(3):3365–3394, 2018

Emeric Bouin, Jimmy Garnier, Christopher Henderson, and Florian Patout. Thin front limit of an integro-differential Fisher–KPP equation with fat-tailed kernels.SIAM Journal on Mathematical Analysis, 50(3):3365–3394, 2018

work page 2018
[10]

Théorie et applications.Analyse Fonctionnelle, 1983

Haim Brezis. Théorie et applications.Analyse Fonctionnelle, 1983

work page 1983
[11]

Springer, 2011

Haim Brezis.Functional analysis, Sobolev spaces and partial differential equations, volume 2. Springer, 2011

work page 2011
[12]

Alfonso Bueno-Orovio, David Kay, Vicente Grau, Blanca Rodriguez, and Kevin Burrage. Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization.Journal of The Royal Society Interface, 11(97):20140352, 08 2014

work page 2014
[13]

Bullock, Laura Mallada González, Riin Tamme, Lars Götzenberger, Steven M

James M. Bullock, Laura Mallada González, Riin Tamme, Lars Götzenberger, Steven M. White, Meelis Pärtel, and Danny A. P. Hooftman. A synthesis of empirical plant dispersal kernels.Journal of Ecology, 105(1):6–19, 2017

work page 2017
[14]

Non-isothermal nonlocal phase-field models with a double-obstacle potential.arXiv preprint arXiv:2310.07861, 2023

Olena Burkovska. Non-isothermal nonlocal phase-field models with a double-obstacle potential.arXiv preprint arXiv:2310.07861, 2023

work page arXiv 2023
[15]

On a nonlocal Cahn–Hilliard model permitting sharp interfaces.Mathematical Models and Methods in Applied Sciences, 31(09):1749–1786, 2021

Olena Burkovska and Max Gunzburger. On a nonlocal Cahn–Hilliard model permitting sharp interfaces.Mathematical Models and Methods in Applied Sciences, 31(09):1749–1786, 2021

work page 2021
[16]

Analysis and simulations of a nonlocal Gray-Scott model.arXiv preprint arXiv:2212.10648, 2022

Loic Cappanera, Gabriela Jaramillo, and Cory Ward. Analysis and simulations of a nonlocal Gray-Scott model.arXiv preprint arXiv:2212.10648, 2022

work page arXiv 2022
[17]

Combined effects in nonlinear elliptic equations involving fractional operators.Journal of Pseudo-Differential Operators and Applications, 14(3):35, 2023

Rym Chammem, Abdeljabbar Ghanmi, and Mahfoudh Mechergui. Combined effects in nonlinear elliptic equations involving fractional operators.Journal of Pseudo-Differential Operators and Applications, 14(3):35, 2023

work page 2023
[18]

A mountain pass method for the numerical solution of semilinear wave equations.Nu- merische Mathematik, 64(1):487–509, 1993

YS Choi, PJ McKenna, and M Romano. A mountain pass method for the numerical solution of semilinear wave equations.Nu- merische Mathematik, 64(1):487–509, 1993

work page 1993
[19]

A mountain pass method for the numerical solution of semilinear elliptic problems

Yung Sze Choi and P Joseph McKenna. A mountain pass method for the numerical solution of semilinear elliptic problems. Nonlinear Analysis: Theory, Methods & Applications, 20(4):417–437, 1993

work page 1993
[20]

Evolutionary stability of ideal free nonlocal dispersal.Journal of Biological Dynamics, 6(2):395–405, 2012

Salomé Martínez Chris Cosner, Juan Dávila. Evolutionary stability of ideal free nonlocal dispersal.Journal of Biological Dynamics, 6(2):395–405, 2012

work page 2012
[21]

A variant of the Lusternik-Schnirelman theory.Indiana University Mathematics Journal, 22(1):65–74, 1972

David C Clark and David Gilbarg. A variant of the Lusternik-Schnirelman theory.Indiana University Mathematics Journal, 22(1):65–74, 1972

work page 1972
[22]

Multiple solutions for thep-Laplace operator with critical growth

Pablo L De Nápoli, Julián Fernández Bonder, and Analía Silva. Multiple solutions for thep-Laplace operator with critical growth. Nonlinear Analysis: Theory, Methods & Applications, 71(12):6283–6289, 2009

work page 2009
[23]

Variational methods with application to one-dimensional boundary value problems and numerical evaluations

Marco Degiovanni. Variational methods with application to one-dimensional boundary value problems and numerical evaluations. InApplied Mathematical Problems in Geophysics: Cetraro, Italy 2019, pages 85–123. Springer, 2022

work page 2019
[24]

Springer, 2017

Serena Dipierro, María Medina, Enrico Valdinoci, et al.Fractional elliptic problems with critical growth in the whole of Rn. Springer, 2017

work page 2017
[25]

Chapter 5 - the phase field method for geometric moving interfaces and their numerical approx- imations

Qiang Du and Xiaobing Feng. Chapter 5 - the phase field method for geometric moving interfaces and their numerical approx- imations. In Andrea Bonito and Ricardo H. Nochetto, editors,Geometric Partial Differential Equations - Part I, volume 21 of Handbook of Numerical Analysis, pages 425–508. Elsevier, 2020

work page 2020
[26]

Sherratt

Lukas Eigentler and Jonathan A. Sherratt. Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal.Journal of Mathematical Biology, 77(3):739–763, 2018

work page 2018
[27]

Existence of pulses in excitable media with nonlocal coupling.Advances in Mathematics, 270:400– 456, 2015

Gregory Faye and Arnd Scheel. Existence of pulses in excitable media with nonlocal coupling.Advances in Mathematics, 270:400– 456, 2015. 34 LOIC CAPPANERA, GABRIELA JARAMILLO, JOSHUA M. SIKTAR

work page 2015
[28]

Infinitely many positive solutions for a nonlocal problem.Applied Mathematics Letters, 84:49–55, 2018

Guangze Gu, Wei Zhang, and Fukun Zhao. Infinitely many positive solutions for a nonlocal problem.Applied Mathematics Letters, 84:49–55, 2018

work page 2018
[29]

Traveling waves in a nonlocal dispersal predator-prey model.Discrete & Continu- ous Dynamical Systems-Series S, 14(9), 2021

Yu-Xia Hao, Wan-Tong Li, and Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model.Discrete & Continu- ous Dynamical Systems-Series S, 14(9), 2021

work page 2021
[30]

Bard Ermentrout

Stewart Heitmann and G. Bard Ermentrout. Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with mexican hat connectivity.Biological Cybernetics, 109(3):333–347, 2015

work page 2015
[31]

Hutson, S

V . Hutson, S. Martinez, K. Mischaikow, and G. T. Vickers. The evolution of dispersal.Journal of Mathematical Biology, 47(6):483– 517, 2003

work page 2003
[32]

The effect of impurities on striped phases.Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 149(1):131–168, 2019

Gabriela Jaramillo, Arnd Scheel, and Qiliang Wu. The effect of impurities on striped phases.Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 149(1):131–168, 2019

work page 2019
[33]

A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations

Ryuji Kajikiya. A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. Journal of Functional Analysis, 225(2):352–370, 2005

work page 2005
[34]

Mexican hats and pinwheels in visual cortex.Proceedings of the National Academy of Sciences, 100(5):2848–2853, 2003

Kukjin Kang, Michael Shelley, and Haim Sompolinsky. Mexican hats and pinwheels in visual cortex.Proceedings of the National Academy of Sciences, 100(5):2848–2853, 2003

work page 2003
[35]

Existence and symmetry result for fractionalp-Laplacian inR n.Communications on Pure and Applied Analysis, 16(1):99–114, 2016

César E Torres Ledesma. Existence and symmetry result for fractionalp-Laplacian inR n.Communications on Pure and Applied Analysis, 16(1):99–114, 2016

work page 2016
[36]

Frithjof Lutscher, Elizaveta Pachepsky, and Mark A. Lewis. The effect of dispersal patterns on stream populations.SIAM Review, 47(4):749–772, 2005

work page 2005
[37]

Variational methods for nonpositive mixed local–nonlocal operators.Frac- tional Calculus and Applied Analysis, 26(3):943–961, 2023

Alberto Maione, Dimitri Mugnai, and Eugenio Vecchi. Variational methods for nonpositive mixed local–nonlocal operators.Frac- tional Calculus and Applied Analysis, 26(3):943–961, 2023

work page 2023
[38]

Analysis of a scalar nonlocal peridynamic model with a sign changing kernel.Discrete & Con- tinuous Dynamical Systems-B, 18(5):1415, 2013

Tadele Mengesha and Qiang Du. Analysis of a scalar nonlocal peridynamic model with a sign changing kernel.Discrete & Con- tinuous Dynamical Systems-B, 18(5):1415, 2013

work page 2013
[39]

Pinto and G

David J. Pinto and G. Bard Ermentrout. Spatially structured activity in synaptically coupled neuronal networks: I. traveling fronts and pulses.SIAM Journal on Applied Mathematics, 62(1):206–225, 2001

work page 2001
[40]

The mountain pass theorem: Theme and variations.Differential Equations, Lecture Notes in Mathematics, 957:237–271, 1982

Paul Rabinowitz. The mountain pass theorem: Theme and variations.Differential Equations, Lecture Notes in Mathematics, 957:237–271, 1982

work page 1982
[41]

Number 65

Paul H Rabinowitz et al.Minimax methods in critical point theory with applications to differential equations. Number 65. American Mathematical Soc., 1986

work page 1986
[42]

Mountain pass solutions for non-local elliptic operators.Journal of Mathematical Analysis and Applications, 389(2):887–898, 2012

Raffaella Servadei and Enrico Valdinoci. Mountain pass solutions for non-local elliptic operators.Journal of Mathematical Analysis and Applications, 389(2):887–898, 2012

work page 2012
[43]

Variational methods for non-local operators of elliptic type.Discrete & Continuous Dynamical Systems, 33(5):2105, 2013

Raffaella Servadei and Enrico Valdinoci. Variational methods for non-local operators of elliptic type.Discrete & Continuous Dynamical Systems, 33(5):2105, 2013

work page 2013
[44]

Sherratt

Jonathan A. Sherratt. Invasion generates periodic traveling waves (wavetrains) in predator-prey models with nonlocal dispersal. SIAM Journal on Applied Mathematics, 76(1):293–313, 2016

work page 2016
[45]

Joshua M. Siktar. Existence of solutions for fractional optimal control problems with superlinear-subcritical controls.arXiv preprint arXiv:2408.09586, 2024

work page arXiv 2024
[46]

Springer, 2000

Michael Struwe and M Struwe.Variational methods, volume 991. Springer, 2000

work page 2000
[47]

Spatial organization of vegetation arising from non- local excitation with local inhibition in tropical rainforests.Physica D: Nonlinear Phenomena, 238(13):1061–1067, 2009

Sally Thompson, Gabriel Katul, John Terborgh, and Patricia Alvarez-Loayza. Spatial organization of vegetation arising from non- local excitation with local inhibition in tropical rainforests.Physica D: Nonlinear Phenomena, 238(13):1061–1067, 2009

work page 2009
[48]

Propagation dynamics in periodic predator-prey systems with nonlocal dispersal

Shi-Liang Wu, Liyan Pang, and Shigui Ruan. Propagation dynamics in periodic predator-prey systems with nonlocal dispersal. Journal de Mathámatiques Pures et Appliqués, 170:57–95, 2023

work page 2023
[49]

Traveling waves in a nonlocal dispersal SIR epidemic model.Nonlinear Analysis: Real World Applications, 23:129–147, 2015

Fei-Ying Yang, Wan-Tong Li, and Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal SIR epidemic model.Nonlinear Analysis: Real World Applications, 23:129–147, 2015

work page 2015
[50]

Traveling waves in a nonlocal dispersal Kermack-McKendrick epi- demic model.Discrete and Continuous Dynamical Systems - B, 18(7):1969–1993, 2013

Fei-Ying Yang, Yan Li, Wan-Tong Li, and Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epi- demic model.Discrete and Continuous Dynamical Systems - B, 18(7):1969–1993, 2013

work page 1969
[51]

Spatial and temporal dynamics of a nonlocal viral infection model.SIAM Journal on Applied Mathematics, 78(4):1954–1980, 2018

Guangyu Zhao and Shigui Ruan. Spatial and temporal dynamics of a nonlocal viral infection model.SIAM Journal on Applied Mathematics, 78(4):1954–1980, 2018

work page 1954

[1] [1]

A mountain pass algorithm for quasilinear boundary value problem withp-Laplacian.Math- ematics and Computers in Simulation, 189:291–304, 2021

Michaela Bailová and Ji ˇrí Bouchala. A mountain pass algorithm for quasilinear boundary value problem withp-Laplacian.Math- ematics and Computers in Simulation, 189:291–304, 2021

work page 2021

[2] [2]

A new approach to solving a quasilinear boundary value problem withp-Laplacian using optimization.Applications of Mathematics, 68(4):425–439, 2023

Michaela Bailová and Ji ˇrí Bouchala. A new approach to solving a quasilinear boundary value problem withp-Laplacian using optimization.Applications of Mathematics, 68(4):425–439, 2023. A MOUNTAIN-PASS ALGORITHM FOR NONLOCAL PROBLEMS WITH SUPER-QUADRATIC NONLINEARITIES 33

work page 2023

[3] [3]

On some nonlocal evolution equations arising in materials science.Nonlinear dynamics and evolution equations, 48:13–52, 2006

Peter W Bates. On some nonlocal evolution equations arising in materials science.Nonlinear dynamics and evolution equations, 48:13–52, 2006

work page 2006

[4] [4]

Complexity and coexistence in a simple spatial model for arid savanna ecosystems.Theoretical Ecology, 6(2):131–141, 2013

Mara Baudena and Max Rietkerk. Complexity and coexistence in a simple spatial model for arid savanna ecosystems.Theoretical Ecology, 6(2):131–141, 2013

work page 2013

[5] [5]

Mountain pass solutions for nonlocal equations.Ann

G Molica Bisci and V Radulescu. Mountain pass solutions for nonlocal equations.Ann. Acad. Sci. Fenn. Math, 39(2):579–592, 2014

work page 2014

[6] [6]

Three weak solutions for nonlocal fractional equations.Advanced Nonlinear Studies, 14(3):619–629, 2014

Giovanni Molica Bisci and Bruno Antonio Pansera. Three weak solutions for nonlocal fractional equations.Advanced Nonlinear Studies, 14(3):619–629, 2014

work page 2014

[7] [7]

Cambridge University Press, 2016

Giovanni Molica Bisci, Vicen¸ tiu D R ˘adulescu, and Raffaella Servadei.Variational methods for nonlocal fractional problems, volume 162. Cambridge University Press, 2016

work page 2016

[8] [8]

Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data

Dorota Bors. Application of mountain pass theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data.arXiv preprint arXiv:1408.0618, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[9] [9]

Thin front limit of an integro-differential Fisher–KPP equation with fat-tailed kernels.SIAM Journal on Mathematical Analysis, 50(3):3365–3394, 2018

Emeric Bouin, Jimmy Garnier, Christopher Henderson, and Florian Patout. Thin front limit of an integro-differential Fisher–KPP equation with fat-tailed kernels.SIAM Journal on Mathematical Analysis, 50(3):3365–3394, 2018

work page 2018

[10] [10]

Théorie et applications.Analyse Fonctionnelle, 1983

Haim Brezis. Théorie et applications.Analyse Fonctionnelle, 1983

work page 1983

[11] [11]

Springer, 2011

Haim Brezis.Functional analysis, Sobolev spaces and partial differential equations, volume 2. Springer, 2011

work page 2011

[12] [12]

Alfonso Bueno-Orovio, David Kay, Vicente Grau, Blanca Rodriguez, and Kevin Burrage. Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization.Journal of The Royal Society Interface, 11(97):20140352, 08 2014

work page 2014

[13] [13]

Bullock, Laura Mallada González, Riin Tamme, Lars Götzenberger, Steven M

James M. Bullock, Laura Mallada González, Riin Tamme, Lars Götzenberger, Steven M. White, Meelis Pärtel, and Danny A. P. Hooftman. A synthesis of empirical plant dispersal kernels.Journal of Ecology, 105(1):6–19, 2017

work page 2017

[14] [14]

Non-isothermal nonlocal phase-field models with a double-obstacle potential.arXiv preprint arXiv:2310.07861, 2023

Olena Burkovska. Non-isothermal nonlocal phase-field models with a double-obstacle potential.arXiv preprint arXiv:2310.07861, 2023

work page arXiv 2023

[15] [15]

On a nonlocal Cahn–Hilliard model permitting sharp interfaces.Mathematical Models and Methods in Applied Sciences, 31(09):1749–1786, 2021

Olena Burkovska and Max Gunzburger. On a nonlocal Cahn–Hilliard model permitting sharp interfaces.Mathematical Models and Methods in Applied Sciences, 31(09):1749–1786, 2021

work page 2021

[16] [16]

Analysis and simulations of a nonlocal Gray-Scott model.arXiv preprint arXiv:2212.10648, 2022

Loic Cappanera, Gabriela Jaramillo, and Cory Ward. Analysis and simulations of a nonlocal Gray-Scott model.arXiv preprint arXiv:2212.10648, 2022

work page arXiv 2022

[17] [17]

Combined effects in nonlinear elliptic equations involving fractional operators.Journal of Pseudo-Differential Operators and Applications, 14(3):35, 2023

Rym Chammem, Abdeljabbar Ghanmi, and Mahfoudh Mechergui. Combined effects in nonlinear elliptic equations involving fractional operators.Journal of Pseudo-Differential Operators and Applications, 14(3):35, 2023

work page 2023

[18] [18]

A mountain pass method for the numerical solution of semilinear wave equations.Nu- merische Mathematik, 64(1):487–509, 1993

YS Choi, PJ McKenna, and M Romano. A mountain pass method for the numerical solution of semilinear wave equations.Nu- merische Mathematik, 64(1):487–509, 1993

work page 1993

[19] [19]

A mountain pass method for the numerical solution of semilinear elliptic problems

Yung Sze Choi and P Joseph McKenna. A mountain pass method for the numerical solution of semilinear elliptic problems. Nonlinear Analysis: Theory, Methods & Applications, 20(4):417–437, 1993

work page 1993

[20] [20]

Evolutionary stability of ideal free nonlocal dispersal.Journal of Biological Dynamics, 6(2):395–405, 2012

Salomé Martínez Chris Cosner, Juan Dávila. Evolutionary stability of ideal free nonlocal dispersal.Journal of Biological Dynamics, 6(2):395–405, 2012

work page 2012

[21] [21]

A variant of the Lusternik-Schnirelman theory.Indiana University Mathematics Journal, 22(1):65–74, 1972

David C Clark and David Gilbarg. A variant of the Lusternik-Schnirelman theory.Indiana University Mathematics Journal, 22(1):65–74, 1972

work page 1972

[22] [22]

Multiple solutions for thep-Laplace operator with critical growth

Pablo L De Nápoli, Julián Fernández Bonder, and Analía Silva. Multiple solutions for thep-Laplace operator with critical growth. Nonlinear Analysis: Theory, Methods & Applications, 71(12):6283–6289, 2009

work page 2009

[23] [23]

Variational methods with application to one-dimensional boundary value problems and numerical evaluations

Marco Degiovanni. Variational methods with application to one-dimensional boundary value problems and numerical evaluations. InApplied Mathematical Problems in Geophysics: Cetraro, Italy 2019, pages 85–123. Springer, 2022

work page 2019

[24] [24]

Springer, 2017

Serena Dipierro, María Medina, Enrico Valdinoci, et al.Fractional elliptic problems with critical growth in the whole of Rn. Springer, 2017

work page 2017

[25] [25]

Chapter 5 - the phase field method for geometric moving interfaces and their numerical approx- imations

Qiang Du and Xiaobing Feng. Chapter 5 - the phase field method for geometric moving interfaces and their numerical approx- imations. In Andrea Bonito and Ricardo H. Nochetto, editors,Geometric Partial Differential Equations - Part I, volume 21 of Handbook of Numerical Analysis, pages 425–508. Elsevier, 2020

work page 2020

[26] [26]

Sherratt

Lukas Eigentler and Jonathan A. Sherratt. Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal.Journal of Mathematical Biology, 77(3):739–763, 2018

work page 2018

[27] [27]

Existence of pulses in excitable media with nonlocal coupling.Advances in Mathematics, 270:400– 456, 2015

Gregory Faye and Arnd Scheel. Existence of pulses in excitable media with nonlocal coupling.Advances in Mathematics, 270:400– 456, 2015. 34 LOIC CAPPANERA, GABRIELA JARAMILLO, JOSHUA M. SIKTAR

work page 2015

[28] [28]

Infinitely many positive solutions for a nonlocal problem.Applied Mathematics Letters, 84:49–55, 2018

Guangze Gu, Wei Zhang, and Fukun Zhao. Infinitely many positive solutions for a nonlocal problem.Applied Mathematics Letters, 84:49–55, 2018

work page 2018

[29] [29]

Traveling waves in a nonlocal dispersal predator-prey model.Discrete & Continu- ous Dynamical Systems-Series S, 14(9), 2021

Yu-Xia Hao, Wan-Tong Li, and Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model.Discrete & Continu- ous Dynamical Systems-Series S, 14(9), 2021

work page 2021

[30] [30]

Bard Ermentrout

Stewart Heitmann and G. Bard Ermentrout. Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with mexican hat connectivity.Biological Cybernetics, 109(3):333–347, 2015

work page 2015

[31] [31]

Hutson, S

V . Hutson, S. Martinez, K. Mischaikow, and G. T. Vickers. The evolution of dispersal.Journal of Mathematical Biology, 47(6):483– 517, 2003

work page 2003

[32] [32]

The effect of impurities on striped phases.Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 149(1):131–168, 2019

Gabriela Jaramillo, Arnd Scheel, and Qiliang Wu. The effect of impurities on striped phases.Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 149(1):131–168, 2019

work page 2019

[33] [33]

A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations

Ryuji Kajikiya. A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. Journal of Functional Analysis, 225(2):352–370, 2005

work page 2005

[34] [34]

Mexican hats and pinwheels in visual cortex.Proceedings of the National Academy of Sciences, 100(5):2848–2853, 2003

Kukjin Kang, Michael Shelley, and Haim Sompolinsky. Mexican hats and pinwheels in visual cortex.Proceedings of the National Academy of Sciences, 100(5):2848–2853, 2003

work page 2003

[35] [35]

Existence and symmetry result for fractionalp-Laplacian inR n.Communications on Pure and Applied Analysis, 16(1):99–114, 2016

César E Torres Ledesma. Existence and symmetry result for fractionalp-Laplacian inR n.Communications on Pure and Applied Analysis, 16(1):99–114, 2016

work page 2016

[36] [36]

Frithjof Lutscher, Elizaveta Pachepsky, and Mark A. Lewis. The effect of dispersal patterns on stream populations.SIAM Review, 47(4):749–772, 2005

work page 2005

[37] [37]

Variational methods for nonpositive mixed local–nonlocal operators.Frac- tional Calculus and Applied Analysis, 26(3):943–961, 2023

Alberto Maione, Dimitri Mugnai, and Eugenio Vecchi. Variational methods for nonpositive mixed local–nonlocal operators.Frac- tional Calculus and Applied Analysis, 26(3):943–961, 2023

work page 2023

[38] [38]

Analysis of a scalar nonlocal peridynamic model with a sign changing kernel.Discrete & Con- tinuous Dynamical Systems-B, 18(5):1415, 2013

Tadele Mengesha and Qiang Du. Analysis of a scalar nonlocal peridynamic model with a sign changing kernel.Discrete & Con- tinuous Dynamical Systems-B, 18(5):1415, 2013

work page 2013

[39] [39]

Pinto and G

David J. Pinto and G. Bard Ermentrout. Spatially structured activity in synaptically coupled neuronal networks: I. traveling fronts and pulses.SIAM Journal on Applied Mathematics, 62(1):206–225, 2001

work page 2001

[40] [40]

The mountain pass theorem: Theme and variations.Differential Equations, Lecture Notes in Mathematics, 957:237–271, 1982

Paul Rabinowitz. The mountain pass theorem: Theme and variations.Differential Equations, Lecture Notes in Mathematics, 957:237–271, 1982

work page 1982

[41] [41]

Number 65

Paul H Rabinowitz et al.Minimax methods in critical point theory with applications to differential equations. Number 65. American Mathematical Soc., 1986

work page 1986

[42] [42]

Mountain pass solutions for non-local elliptic operators.Journal of Mathematical Analysis and Applications, 389(2):887–898, 2012

Raffaella Servadei and Enrico Valdinoci. Mountain pass solutions for non-local elliptic operators.Journal of Mathematical Analysis and Applications, 389(2):887–898, 2012

work page 2012

[43] [43]

Variational methods for non-local operators of elliptic type.Discrete & Continuous Dynamical Systems, 33(5):2105, 2013

Raffaella Servadei and Enrico Valdinoci. Variational methods for non-local operators of elliptic type.Discrete & Continuous Dynamical Systems, 33(5):2105, 2013

work page 2013

[44] [44]

Sherratt

Jonathan A. Sherratt. Invasion generates periodic traveling waves (wavetrains) in predator-prey models with nonlocal dispersal. SIAM Journal on Applied Mathematics, 76(1):293–313, 2016

work page 2016

[45] [45]

Joshua M. Siktar. Existence of solutions for fractional optimal control problems with superlinear-subcritical controls.arXiv preprint arXiv:2408.09586, 2024

work page arXiv 2024

[46] [46]

Springer, 2000

Michael Struwe and M Struwe.Variational methods, volume 991. Springer, 2000

work page 2000

[47] [47]

Spatial organization of vegetation arising from non- local excitation with local inhibition in tropical rainforests.Physica D: Nonlinear Phenomena, 238(13):1061–1067, 2009

Sally Thompson, Gabriel Katul, John Terborgh, and Patricia Alvarez-Loayza. Spatial organization of vegetation arising from non- local excitation with local inhibition in tropical rainforests.Physica D: Nonlinear Phenomena, 238(13):1061–1067, 2009

work page 2009

[48] [48]

Propagation dynamics in periodic predator-prey systems with nonlocal dispersal

Shi-Liang Wu, Liyan Pang, and Shigui Ruan. Propagation dynamics in periodic predator-prey systems with nonlocal dispersal. Journal de Mathámatiques Pures et Appliqués, 170:57–95, 2023

work page 2023

[49] [49]

Traveling waves in a nonlocal dispersal SIR epidemic model.Nonlinear Analysis: Real World Applications, 23:129–147, 2015

Fei-Ying Yang, Wan-Tong Li, and Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal SIR epidemic model.Nonlinear Analysis: Real World Applications, 23:129–147, 2015

work page 2015

[50] [50]

Traveling waves in a nonlocal dispersal Kermack-McKendrick epi- demic model.Discrete and Continuous Dynamical Systems - B, 18(7):1969–1993, 2013

Fei-Ying Yang, Yan Li, Wan-Tong Li, and Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epi- demic model.Discrete and Continuous Dynamical Systems - B, 18(7):1969–1993, 2013

work page 1969

[51] [51]

Spatial and temporal dynamics of a nonlocal viral infection model.SIAM Journal on Applied Mathematics, 78(4):1954–1980, 2018

Guangyu Zhao and Shigui Ruan. Spatial and temporal dynamics of a nonlocal viral infection model.SIAM Journal on Applied Mathematics, 78(4):1954–1980, 2018

work page 1954