Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-22T18:18:43.442Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic solutions for a fractional asymptotically linear problem

Published online by Cambridge University Press:  26 December 2018

Vincenzo Ambrosio  and
Giovanni Molica Bisci
Vincenzo Ambrosio
Affiliation:
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, via delle Scienze 206, 33100 Udine, Italy (vincenzo.ambrosio@uniud.it)
Giovanni Molica Bisci
Affiliation:
Dipartimento PAU, Università ‘Mediterranea’ di Reggio Calabria, Salita Melissari, Feo di Vito, 89100 Reggio Calabria, Italy (gmolica@unirc.it)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the pseudo-index theory developed by Bartolo, Benci and Fortunato [11] after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in the literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.

Keywords

Fractional Laplacianvariational methodsperiodic solutionsasymptotically linear problemnonresonant problemspseudo-genus35A1535S1535B10
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 593 - 615
Copyright
Copyright © Royal Society of Edinburgh 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Amann, H. and Zehnder, E.. Nontrivialssolutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa 7 (1980), 539–603.Google Scholar
2Ambrosio, V.. Periodic solutions for a pseudo-relativistic Schrödinger equation. Nonlinear Anal. TMA 120 (2015), 262–284.Google Scholar
3Ambrosio, V.. Periodic solutions for the non-local operator pseudo-relativistic ( − Δ + m 2)s − m 2s with m≥0. Topol. Methods Nonlinear Anal. 49 (2017), 75–104.Google Scholar
4Ambrosio, V.. Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator. J. Math. Phys. 57 (2016), 051502, 18 pp.Google Scholar
5Ambrosio, V.. Periodic solutions for a superlinear fractional problem without the Ambrosetti–Rabinowitz condition. Discrete Contin. Dyn. Syst. 37 (2017a), 2265–2284.Google Scholar
6Ambrosio, V. and Molica Bisci, G.. Periodic solutions for nonlocal fractional equations. Commun. Pure Appl. Anal. 16 (2017b), 331–334.Google Scholar
7Autuori, G. and Pucci, P.. Existence of entire solutions for a class of quasilinear elliptic equations. NoDEA Nonlinear Differ. Equ. Appl. 20 (2013a), 977–1009.Google Scholar
8Autuori, G. and Pucci, P.. Elliptic problems involving the fractional Laplacian in ℝN. J. Differ. Equ. 255 (2013b), 2340–2362.Google Scholar
9Bartolo, R. and Molica Bisci, G.. Asymptotically linear fractional p-Laplacian equations. Ann. Mat. Pura Appl. (4) 196 (2017), 427–442.Google Scholar
10Bartolo, R. and Molica Bisci, G.. A pseudo-index approach to fractional equations. Expo. Math. 33 (2015), 502–516.Google Scholar
11Bartolo, P., Benci, V. and Fortunato, D.. Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity. Nonlinear Anal. 7 (1983), 981–1012.Google Scholar
12Bartolo, R., Candela, A. M. and Salvatore, A.. p-Laplacian problems with nonlinearities interacting with the spectrum. Nonlinear Differ. Equ. Appl. 20 (2013), 1701–1721.Google Scholar
13Bartolo, R., Candela, A. M. and Salvatore, A.. Perturbed asymptotically linear problems. Ann. Mat. Pura Appl. 193 (2014), 89–101.Google Scholar
14Benci, V.. On the critical point theory for indefinite functionals in the presence of symmetries. Trans. Am. Math. Soc. 274 (1982), 533–572.Google Scholar
15Binlin, Z., Molica Bisci, G. and Servadei, R.. Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28 (2015), 2247–2264.Google Scholar
16Brändle, C., Colorado, E., de Pablo, A. and Sánchez, U.. A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 39–71.Google Scholar
17Cabré, X. and Tan, J.. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224 (2010), 2052–2093.Google Scholar
18Caffarelli, L. A. and Silvestre, L.. An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32 (2007), 1245–1260.Google Scholar
19Carmona, R., Masters, W. C. and Simon, B.. Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Func. Anal 91 (1990), 117–142.Google Scholar
20Di Nezza, E., Palatucci, G. and Valdinoci, E.. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573.Google Scholar
21Fiscella, A., Servadei, R. and Valdinoci, E.. A resonance problem for non-local elliptic operators. Z. Anal. Anwendungen 32 (2013), 411–431.Google Scholar
22Kuusi, T., Mingione, G. and Sire, Y.. Nonlocal equations with measure data. Comm. Math. Phys. 337 (2015a), 1317–1368.Google Scholar
23Kuusi, T., Mingione, G. and Sire, Y.. Nonlocal self-improving properties. Anal. PDE 8 (2015b), 57–114.Google Scholar
24Isernia, T.. Positive solution for nonhomogeneous sublinear fractional equations in ℝN, Complex. Var. Elliptic Equ. 63 (2018), 689–714.Google Scholar
25Lieb, E. H. and Loss, M.. Analysis (Providence, RI: American Mathematical Society, 2001).Google Scholar
26Molica Bisci, G., Rădulescu, V. and Servadei, R.. Variational Methods for Nonlocal Fractional Problems. With a Foreword by Jean Mawhin. In Encyclopedia of mathematics and its applications, vol. 162 (Cambridge: Cambridge University Press, 2016), ISBN 9781107111943.Google Scholar
27Molica Bisci, G., Repovš, D. and Servadei, R.. Nontrivial solutions of superlinear nonlocal problems. Forum Math 28 (2016), 1095–1110.Google Scholar
28Palatucci, G. and Pisante, A.. Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50 (2014), 799–829.Google Scholar
29Palatucci, G. and Pisante, A.. A global compactness type result for Palais–Smale sequences in fractional Sobolev spaces. Nonlinear Anal. 117 (2015), 1–7.Google Scholar
30Pucci, P. and Saldi, S.. Critical stationary Kirchhoff equations in ℝN involving nonlocal operators, Rev. Mat. Iberoam. 32 (2016), 1–22.Google Scholar
31Rabinowitz, P. H.. Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in Math., vol 65 (Providence: Amer. Math. Soc., 1984).Google Scholar
32Riesz, F. and Szökefalvi-Nagy, B.. Lecons d'analyse fonctionnelle (Budapest: Académie des Sciences de Hongrie, Akadémiai Kiadó, 1952), viii+449 pp.Google Scholar
33Ryznar, M.. Estimate of Green function for relativistic α-stable processes. Potential Anal. 17 (2002), 1–23.Google Scholar
34Secchi, S.. Ground state solutions for nonlinear fractional Schrödinger equations in ℝN. J. Math. Phys. 54 (2013a), 031501.Google Scholar
35Secchi, S.. Perturbation results for some nonlinear equations involving fractional operators. Differ. Equ. Appl. 5 (2013b), 221–236.Google Scholar
36Servadei, R. and Valdinoci, E.. Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389 (2012), 887–898.Google Scholar
37Servadei, R. and Valdinoci, E.. Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33 (2013), 2105–2137.Google Scholar
38Struwe, M.. Variational Methods. Applications to nonlinear partial differential equations and Hamiltonian systems, 4th edn, Ergeb. Math. Grenzgeb. (4), vol. 34 (Berlin: Springer-Verlag, 2008).Google Scholar
39Vázquez, J. L.. Nonlinear diffusion with fractional Laplacian operators, in Nonlinear partial differential equations. Abel Symp. 7 (2012), 271–298.Google Scholar
40Yosida, K.. Functional analysis, Die Grundlehren der Mathematischen Wissenschaften (New York, Berlin: Band 123 Academic Press, Inc., Springer-Verlag, 1965), xi+458 pp.Google Scholar