Abstract
In this article, we consider the following Kirchhoff equation involving fractional Laplacian
where \(a,\ b>0\) are constants, \(\frac{3}{4}<s<1,\) \([u]_s\) is the so-called Gagliardo (semi)norm of u, \(4<p<2^*_{s}=\frac{6}{3-2s}\) and \(\Omega \subset {\mathbb {R}}^3\) is an exterior domain with smooth boundary \(\partial \Omega \ne \emptyset .\) By establishing a global compactness lemma of the fractional Kirchhoff equation in exterior domains, we verify the compactness of Palais–Smale sequences corresponding to above problem at higher energy level interval. Then combining some crucial estimates and barycentric function, we determine the existence of positive bound state solutions provided that \({\mathbb {R}}^3\backslash \Omega \) is contained in a small ball. In addition, we point out that the main result can be extended to fractional Sobolev critical case with small parameter.
Similar content being viewed by others
References
Akahori T, Murata M (2022) Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions. NoDEA Nonlinear Differ Equ Appl 29:54
Akahori T, Ibrahim S, Ikoma N et al (2019) Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies. Calc Var Partial Differ Equ 58:32
Alves CO, Freitas L (2017) Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Milan J Math 85:309–330
Alves CO, Bisci GM, Ledesma CT (2020) Existence of solutions for a class of fractional elliptic problems on exterior domains. J Differ Equ 268:7183–7219
Ambrosio V, Isernia T (2018) A multiplicity result for a fractional Kirchhoff equation in \({\mathbb{R} }^N\) with a general nonlinearity. Commun Contemp Math 20:17
Applebaum D (2004) Lévy processes-from probability to finance and quantum groups. Not Am Math Soc 51:1336–1347
Arosio A, Panizzi S (1996) On the wellposedness of the Kirchhoff string. Trans Amer Math Soc 348:305–330
Autuori G, Fiscella A, Pucci P (2015) Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal 125:699–714
Benci V, Cerami G (1987) Positive solutions of some nonlinear elliptic problems in exterior domains. Arch Rational Mech Anal 99:283–300
Chang K (2005) Methods in Nonlinear Analysis. Springer, Berlin
Chen P, Liu X (2021) Positive solutions for Kirchhoff equation in exterior domains. J Math Phys 62:18
Chen S, Tang X, Liao F (2018) Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions. NoDEA Nonlinear Differ Equ Appl 25:23
Correia JN, Figueiredo GM (2020) Existence of positive solution for a fractional elliptic equation in exterior domain. J Differ Equ 268:1946–1973
Correia JN, Oliveira CP (2022) Existence of a positive solution for a class of fractional elliptic problems in exterior domains involving critical growth. J Math Anal Appl 506:34
de Souza M, Severo UB, Luiz do Rêgo T (2022) On solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity. Commun Contemp Math 24:38
Di Nezza E, Palatucci G, Valdinoci E (2012) Hitchhiker’s guide to the fractional sobolev spaces. Bull Sci Math 136:521–573
Fiscella A, Valdinoci E (2014) A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal 94:156–170
Guo Z (2015) Ground states for Kirchhoff equations without compact condition. J Differ Equ 259:2884–2902
He X, Zou W (2009) Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal 70:1407–1414
He X, Zou W (2012) Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({\mathbb{R} }^3\). J Differ Equ 252:1813–1834
Hofer H (1982) Variational and topological methods in partially ordered Hilbert spaces. Math Ann 261:493–514
Jia L, Li X, Ma S (2023) Existence of positive solutions for Kirchhoff-type problem in exterior domains. Proc Edinb Math Soc 66:182–217
Kirchhoff G (1883) Mechanik. Teuber, Leipzig
Laskin N (2000) Fractional quantum mechanics and Lévy path integrals. Phys Lett A 268:298–305
Li G, Ye H (2014) Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb{R} }^{3}\). J Differ Equ 257:566–600
Li G, Luo P, Peng S et al (2020) A singularly perturbed Kirchhoff problem revisited. J Differ Equ 268:541–589
Molica Bisci G, Rădulescu VD, Servadei R (2016) Variational methods for nonlocal fractional problems, Vol. 162, Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge University Press
Nezza E, Palatucci G, Valdinoci E (2012) Hitchhiker’s guide to the fractional Sobolev spaces. Bull Sci Math 136:521–573
Rabinowitz P (1974) Variational methods for nonlinear eigenvalue problems. In: Prodi G (ed) Eigenvalues of Nonlinear Problems. CIME, pp 141–195
Wang Y, Yuan R, Zhang Z (2023) Positive solutions for Kirchhoff equation in exterior domains with small Sobolev critical perturbation. Complex Var Elliptic Equ. https://doi.org/10.1080/17476933.2023.2209730
Willem M (1996) Minimax theorems. Birkhäuser, Boston
Wu K, Gu G (2022) Existence of positive solutions for fractional Kirchhoff equation. Z Angew Math Phys 73:13
Xiang M, Zhang B, Zhang X (2016) A nonhomogeneous fractional \(p\)-Kirchhoff type problem involving critical exponent in \({\mathbb{R} }^N\). Adv Nonlinear Stud 17:611–640
Xiang M, Rădulescu VD, Zhang B (2019) Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc Var Partial Differ Equ 58:27
Yang Z, Zhai H, Zhao F (2023) On the fractional Kirchhoff equation with critical Sobolev exponent. Appl Math Lett 141:8
Yu S, Tang C, Zhang Z (2023) Normalized solutions of mass subcritical fractional Schrödinger equations in exterior domains. J Geom Anal 33:30
Acknowledgements
The authors would like to thank anonymous referees for the careful reading of this paper and for several useful comments.
Funding
Fumei Ye acknowledges the support through the special subsidy from Chongqing human resources and Social Security Bureau, and the Natural Science Foundation of Chongqing, China (No. CSTB2023NSCQ-BHX0226). Chun-Lei Tang acknowledges the support through the National Natural Science Foundation of China (No. 12371120).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors do not have conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Project supported by the National Natural Science Foundation of China (No.12371120) and Chongqing Postdoctoral Science Foundation Project (No. CSTB2023NSCQ-BHX0226).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ye, F., Yu, S. & Tang, CL. Positive solutions for the fractional Kirchhoff type problem in exterior domains. Comp. Appl. Math. 43, 191 (2024). https://doi.org/10.1007/s40314-024-02719-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-024-02719-1