Open Access

Normalized homoclinic solutions of discrete nonlocal double phase problems

Mingqi Xiang

https://orcid.org/0000-0002-0712-7149

College of Science, Civil Aviation, University of China, Tianjin 300300, P. R. China

E-mail Address: xiangmingqi_hit@163.com

Search for more papers by this author

Yunfeng Ma

College of Science, Civil Aviation, University of China, Tianjin 300300, P. R. China

E-mail Address: Martayr1921@163.com

Search for more papers by this author

, and

Miaomiao Yang

School of Mathematics and Statistics, Qilu University of Technology (Shandong, Academy of Sciences), Jinan 250353, P. R. China

E-mail Address: 3d07@163.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S1664360724500036Cited by:8 (Source: Crossref)

Abstract

The aim of this paper is to discuss the existence of normalized solutions to the following nonlocal double phase problems driving by the discrete fractional Laplacian:

where

$α, β \in (0, 1)$ ,

$ω : ℤ \to (0, \infty)$ ,

$1 < p \leq q < \infty$ ,

$λ, μ \in ℝ$ ,

$h \in ℓ^{\frac{q}{q - r}} (ℤ)$ if

$1 < r < q$ ,

$h \in ℓ^{\infty} (ℤ)$ if

$r > q$ , and

${(- Δ_{𝔻})}_{κ}^{s}$ (

$s = α$ or

$β$ ,

$κ = p$ or

$q$ ) is the discrete fractional

$κ$ -Laplacian. By variational methods, we discuss the existence of non-negative normalized homoclinic solutions under the conditions that the nonlinear term satisfies sublinear growth or superlinear growth conditions. In particular, we establish the compactness of the associated energy functional of the problem without weights. Our paper is the first time to deal with the existence of normalized solutions for discrete double phase problems.

Communicated by Vicentiu Radulescu

Keywords:

AMSC: 49M25, 35A15, 35R11, 35K05

References

1. R. P. Agarwal, K. Perera and D. O’Regan , Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (2004) 69–73. Crossref, Google Scholar
2. C. O. Alves , On the existence of multiple normalized solutions to a class of elliptic problems in whole $ℝ^{N}$ , Z. Angew. Math. Phys. 73 (2022) 97. Crossref, Google Scholar
3. V. Ambrosio and V. D. Rǎdulescu , Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl. 142 (2020) 101–145. Crossref, Google Scholar
4. D. Applebaum , Lévy processes — From probability to finance quantum groups, Notices Amer. Math. Soc. 51 (2004) 1336–1347. Google Scholar
5. G. Autuori and P. Pucci , Elliptic problems involving the fractional Laplacian in $ℝ^{N}$ , J. Differential Equations 255 (2013) 2340–2362. Crossref, Google Scholar
6. H. Berestycki and P. L. Lions , Nonlinear scalar field equation II, existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983) 247–375. Google Scholar
7. D. Bonheure, F. Colasuonno and J. Földes , On the Born-Infeld equation for electrostatic fields with a superposition of point charges, Ann. Mat. Pura Appl. 198(3) (2019) 749–772. Crossref, Google Scholar
8. A. Cabada, A. Iannizzotto and S. Tersian , Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl. 356 (2009) 418–428. Crossref, Google Scholar
9. L. Caffarelli , Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia 7 (2012) 37–52. Crossref, Google Scholar
10. L. Caffarelli and L. Silvestre , An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) 1245–1260. Crossref, Google Scholar
11. J. Cheng, P. Chen and L. Zhang , Homoclinic solutions for a differential inclusion system involving the $p (t)$ -Laplacian, Adv. Nonlinear Anal. 12(1) (2023) 20220272-1–26. Google Scholar
12. O. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona , Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math. 330 (2018) 688–738. Crossref, Google Scholar
13. G. Devillanova and G. Carlo Marano , A free fractional viscous oscillator as a forced standard damped vibration, Fract. Calc. Appl. Anal. 19(2) (2016) 319–356. Crossref, Google Scholar
14. J. Diblik and E. Korobko , Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type, Adv. Nonlinear Anal. 12(1) (2023) 20230105-1–23. Google Scholar
15. N. Dimitrov and S. Tersian , Existence of homoclinic solutions for nonlinear fourth order $p$ -Laplacian difference equation, Discrete Contin. Dyn. Syst. Ser. B 25 (2020) 555–567. Google Scholar
16. E. Di Nezza, G. Palatucci and E. Valdinoci , Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012) 521–573. Crossref, Google Scholar
17. M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler , Banach Space Theory (Springer, 2011). Crossref, Google Scholar
18. A. Fiscella and E. Valdinoci , A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014) 156–170. Crossref, Google Scholar
19. D. Goel, D. Kumar and K. Sreenadh , Regularity and multiplicity results for fractional (p, q)-Laplacian equations, Commun. Contemp. Math. 22(8) (2020) 1950065. Link, Google Scholar
20. A. Iannizzotto and V. Rădulescu , Positive homoclinic solutions for the discrete $p$ -Laplacian with a coercive weight function, Differ. Integral Equ. 27 (2014) 35–44. Google Scholar
21. A. Iannizzotto and S. A. Tersian , Multiple homoclinic solutions for the discrete $p$ –Laplacian via critical point theory, J. Math. Anal. Appl. 403 (2013) 173–182. Crossref, Google Scholar
22. M. Izydorek, J. Janczewska and J. Mawhin , Homoclinics for singular strong force Lagrangian systems, Adv. Nonlinear Anal. 9 (2020) 644–653. Crossref, Google Scholar
23. L. Jeanjean , Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (1997) 1633–1659. Crossref, Google Scholar
24. C. Ju and B. Zhang , On fractional discrete $p$ -Laplacian equations via Clark’s theorem, Appl. Math. Comput. 434 (2022) 127443. Crossref, Google Scholar
25. X. Ju, H. Die and M. Xiang , The Nehari manifold method for discrete fractional $p$ -Laplacian equations, Adv. Difference Equations 559 (2020) 2020. Google Scholar
26. N. Laskin , Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000) 298–305. Crossref, Google Scholar
27. S. Lenardi and N. S. Papageorgiou , Positive solutions for a class of singular $(p, q)$ -equations, Adv. Nonlinear Anal. 12 (2023) 20220300. Crossref, Google Scholar
28. Q. Li, J. Nie and W. Zhang , Existence and asymptotics of normalized ground states for a Sobolev critical Kirchhoff equation, J. Geom. Anal. 33(4) (2023) 126-1–22. Crossref, Google Scholar
29. Q. Li and W. Zou , The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the $L^{2}$ -subcritical and $L^{2}$ -supercritical cases, Adv. Nonlinear Anal. 11 (2022) 1531–1551. Crossref, Google Scholar
30. H. Luo and Z. Zhang , Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differential Equations 59 (2020) 143. Crossref, Google Scholar
31. S. E. Manouni, G. Marino and P. Winkert , Existence results for double phase problems depending on Robin and Steklov eigenvalues for the $p$ -Laplacian, Adv. Nonlinear Anal. 11 (2022) 304–320. Crossref, Google Scholar
32. P. Marcellini and K. Miller , Elliptic versus parabolic regularization for the equation of prescribed mean curvature, J. Differential Equations 137(1) (1997) 1–53. Crossref, Google Scholar
33. M. Mihăilescu, V. Rădulescu and S. Tersian , Homoclinic solutions of difference equations with variable exponents, Topol. Methods Nonlinear Anal. 38 (2011) 277–289. Google Scholar
34. G. Molica Bisci, V. Rădulescu and R. Servadei , Variational Methods for Nonlocal Fractional Problems (Cambridge University Press, Cambridge, 2015). Google Scholar
35. B. Noris, H. Tavares and G. Verzini , Normalized solutions for nonlinear Schrödinger systems on bounded domains, Nonlinearity 32(3) (2019) 1044–1072. Crossref, Google Scholar
36. N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš , On a class of parametric $(p, 2)$ -equations, Appl. Math. Optim. 75(2) (2017) 193–228. Crossref, Google Scholar
37. N. S. Papageorgiou, J. Zhang and W. Zhang , Solutions with sign information for noncoercive double phase equations, J. Geom. Anal. 34 (2024) 14. Crossref, Google Scholar
38. R. Servadei and E. Valdinoci , Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012) 887–898. Crossref, Google Scholar
39. N. Soave , Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279(6) (2020) 108610. Crossref, Google Scholar
40. E. Valdinoci , From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. Se MA 49 (2009) 33–44. Google Scholar
41. C. Wang and J. Sun , Normalized solutions for the $p$ -Laplacian equation with a trapping potential, Adv. Nonlinear Anal. 12 (2023) 20220291. Crossref, Google Scholar
42. M. Willem , Minimax Theorem (Birkhäuser, Boston, 1996). Crossref, Google Scholar
43. M. Xiang and Y. Ma , Existence and stability of normalized solutions for nonlocal double phase problems, J. Geom. Anal. 34 (2024) 46. Crossref, Google Scholar
44. M. Xiang, V. D. Rǎdulescu and B. Zhang , Existence results for singular fractional $p$ -Kirchhoff problems, Acta Math. Sci. 42 (2022) 1209–1224. Crossref, Google Scholar
45. M. Xiang and B. Zhang , Homoclinic solutions for fractional discrete Laplacian equations, Nonlinear Anal. 198 (2020) 111886. Crossref, Google Scholar
46. B. Zhen and J. Yu , Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency, Adv. Nonlinear Anal. 11 (2022) 212–224. Crossref, Google Scholar
47. M. Zhen, B. Zhang and V. D. Rǎdulescu , Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case, Discrete Contin. Dyn. Syst. 41 (2021) 2653–2676. Crossref, Google Scholar

Vol. 14, No. 02

Metrics

Downloaded 252 times

History

Received 9 January 2024

Revised 31 March 2024

Accepted 3 April 2024

Published: 27 April 2024

Information

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Keywords

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Normalized homoclinic solutions of discrete nonlocal double phase problems

Abstract

Recommended