Abstract
This paper is dedicated to studying the following fractional Kirchhoff-type equation
where \(a, b>0\), either \(N=2\) and \(\alpha \in (1/2,1)\) or \(N=3\) and \(\alpha \in (3/4,1)\) holds, \(V\in \mathcal {C}(\mathbb {R}^{N}, [0,\infty ))\) and \(f\in \mathcal {C}(\mathbb {R}^N\times \mathbb {R}, \mathbb {R})\). By combining the constraint variational method with some new inequalities, we prove that the above problem possesses a radial sign-changing solution \(u_b\) for \(b\ge 0\) without the usual Nehari-type monotonicity condition on f, and its energy is strictly larger than twice that of the ground state radial solutions of Nehari-type. Moreover, we establish the convergence property of \(u_b\) as \(b\searrow 0\). In particular, our results unify both asymptotically cubic and super-cubic cases, which improve and complement the existing ones in the literature.
Article PDF
Similar content being viewed by others
References
Alves, C.O., Nòbrega, A.B.: Nodal ground state solution to a biharmonic equation via dual method. J. Differ. Equ. 260, 5174–5201 (2016)
Ambrosio, V., Isernia, T.: A multiplicity result for a fractional Kirchhoff equation in \({\mathbb{R}}^N\) with a general nonlinearity. Commun. Contemp. Math. 1, 1750054 (2017). https://doi.org/10.1142/S0219199717500547
Applebaum, D.: Lévy processes, from probability fo finance and quantum groups. Notices Am. Math. Soc. 51, 1336–1347 (2004)
Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996)
Bartsch, T., Liu, Z., Weth, T.: Sign changing solutions of superlinear Schrödinger equations. Commun. Partial Differ. Equ. 29, 25–42 (2004)
Bartsch, T., Weth, T.: Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 259–281 (2005)
Bernstein, S.: Sur une class d’équationsfonctionnelles aux dérivéespartielles. Bull. Acad. Sci. URSS Sér. Math. (Izv. Akad. Nauk SSSR) 4, 17–26 (1940)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation. Adv. Differ. Equ. 6, 701–730 (2001)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Chang, X., Wang, Z.Q.: Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian. J. Differ. Equ. 256, 2965–2992 (2014)
Chang, K., Gao, Q.: Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in \({\mathbb{R}}^N\), arXiv:1701.03862 (2017)
Chen, S.T., Tang, X.H.: Ground state solutions for asymptotically periodic Kirchhoff-type equations with asymptotically cubic or super-cubic nonlinearities. Mediterr. J. Math. 14, 209 (2017)
Chen, S.T., Tang, X.H.: Improved results for Klein–Gordon–Maxwell systems with general nonlinearity. Discrete Contin. Dyn. Syst. A 38, 2333–2348 (2018)
Chen, S.T., Tang, X.H.: Infinitely many solutions and least energy solutions for Klein–Gordon–Maxwell systems with general superlinear nonlinearity. Comput. Math. Appl. 75, 3358–3366 (2018)
Cheng, B.T., Tang, X.H.: Ground state sign-changing solutions for asymptotically 3-linear Kirchhoff-type problems. Complex Var. Elliptic 62, 1093–1116 (2017)
Chipot, M., Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30, 4619–4627 (1997)
Deng, Y., Peng, S., Shuai, W.: Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in \({\mathbb{R}}^3\). J. Funct. Anal. 269, 3500–3527 (2015)
Deng, Y., Shuai, W.: Sign-changing solutions for non-local elliptic equations involving the fractional Laplacian. Adv. Differ. Equ. 23, 109–134 (2018)
Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Le Matematiche 68, 201–216 (2013)
Felmer, P., Quaas, A., Tan, J.G.: Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142, 1237–1262 (2012)
Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)
Jin, H., Liu, W.: Fractional Kirchhoff equation with a general critical nonlinearity. Appl. Math. Lett. 74, 140–146 (2017)
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Laskin, N.: Fractional schrödinger equation. Phys. Rev. E 66, 056108 (2002)
Lions, P.L.: Symètrie et compacitè dans les espaces de Sobolev. J. Funct. Anal. 49, 315–334 (1982)
Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos. Inst. Mat., Univ. Fed. Rio de Janeiro, 1997, in: North-Holland Math. Stud., vol. 30. North-Holland, Amsterdam, pp. 284–346 (1978)
Liu, Z.S., Squassina, M., Zhang, J.J.: Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension. Nonlinear Differ. Equ. Appl. 24, 50 (2017)
Luo, H.X.: Sign-changing solutions for non-local elliptic equations. Electron. J. Differ. Equ. 2017, 1–15 (2017)
Mao, A.M., Zhang, Z.T.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009)
Mao, A.M., Luan, S.X.: Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems. J. Math. Anal. Appl. 383, 239–243 (2011)
Miranda, C.: Un’osservazione su un teorema di Brouwer. Boll. Unione Mat. Ital. 3, 5–7 (1940)
Molc̆anov, S.A., Ostrovskii, E.: Symmetric stable processes as traces of degenerate diffusion processes. Teor. Verojatnost. i Primenen. 14, 127–130 (1969)
Noussair, E.S., Wei, J.: On the effect of the domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem. Indiana Univ. Math. J. 46, 1321–1332 (1997)
Nyamoradi, N.: Existence of three solutions for Kirchhoff nonlocal operators of elliptic type. Math. Commun. 18, 489–502 (2013)
Shuai, W.: Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J. Differ. Equ. 259, 1256–1274 (2015)
Tang, X.H., Cheng, B.T.: Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J. Differ. Equ. 261(4), 2384–2402 (2016)
Tang, X.H., Chen, S.T.: Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equ. 56, 110 (2017)
Tang, X.H., Chen, S.T.: Ground state solutions of Nehari–Pohozaev type for Schrödinger–Poisson problems with general potentials. Discrete Contin. Dyn. Syst. A 37, 4973–5002 (2017)
Tang, X.H., Lin, X.Y., Yu, J.S.: Nontrivial solutions for Schrödinger equation with local super-quadratic conditions. J. Dyn. Differ. Equ. 1, 1–15 (2018). https://doi.org/10.1007/s10884-018-9662-2
Teng, K., Wang, K., Wang, R.: A sign-changing solution for nonlinear problems involving the fractional Laplacian. Electron. J. Differ. Equ. 109, 1–12 (2015)
Wang, Z., Zhou, H.S.: Radial sign-changing solution for fractional Schrödinger equation. Discrete Contin. Dyn. Syst. 36, 499–508 (2016)
Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
Xiang, M., Zhang, B., Guo, X.: Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem. Nonlinear Anal. 120, 299–313 (2015)
Zhang, Z.T., Perera, K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317(2), 456–463 (2006)
Zhang, W., Tang, X.H., Mi, H.: On fractional Schrödinger equation with periodic and asymptotically periodic conditions. Comput. Math. Appl. 74, 1321–1332 (2017)
Zou, W.M.: Sign-Changing Critical Point Theory. Springer, New York (2008)
Acknowledgements
This work was supported by the NNSF (11701487, 11626202), Hunan Provincial Natural Science Foundation of China (2016JJ6137), Scientific Research Fund of Hunan Provincial Education Department (15B223). The authors thank the anonymous referees for their valuable suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, S., Tang, X. & Liao, F. Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions. Nonlinear Differ. Equ. Appl. 25, 40 (2018). https://doi.org/10.1007/s00030-018-0531-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-018-0531-9