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Multiplicity results for a fractional Kirchhoff equation involving sign-changing weight function. (English) Zbl 1361.35189

Summary: In this paper, we prove the existence and multiplicity of solutions for a fractional Kirchhoff equation involving a sign-changing weight function which generalizes the corresponding result of T.-F. Wu [Rocky Mt. J. Math. 39, No. 3, 995–1011 (2009; Zbl 1179.35129)]. Our main results are based on the method of a Nehari manifold.

MSC:

35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
47G20 Integro-differential operators

Citations:

Zbl 1179.35129
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Full Text: DOI

References:

[1] Wu, TF: Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mt. J. Math. 39, 995-1011 (2009) · Zbl 1179.35129 · doi:10.1216/RMJ-2009-39-3-995
[2] Autuori, G, Fiscella, A, Pucci, P: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699-714 (2015) · Zbl 1323.35015 · doi:10.1016/j.na.2015.06.014
[3] Chen, CY, Kuo, YC, Wu, TF: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876-1908 (2011) · Zbl 1214.35077 · doi:10.1016/j.jde.2010.11.017
[4] Fiscella, A, Valdinoci, E: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156-170 (2014) · Zbl 1283.35156 · doi:10.1016/j.na.2013.08.011
[5] Pucci, P, Saldi, S: Critical stationary Kirchhoff equations in RN \(\mathbb{R}^N\) involving nonlocal operators. Rev. Mat. Iberoam. 32, 1-22 (2016) · Zbl 1405.35045 · doi:10.4171/RMI/879
[6] Pucci, P, Xiang, M, Zhang, B: Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN \(\mathbb{R}^N\). Calc. Var. Partial Differ. Equ. 54(3), 2785-2806 (2015) · Zbl 1329.35338 · doi:10.1007/s00526-015-0883-5
[7] Mishra, PK, Sreenadh, K: Existence and multiplicity results for fractional p-Kirchhoff equation with sign changing nonlinearities. Adv. Pure Appl. Math. (2015). doi:10.1515/apam-2015-0018 · Zbl 1381.35229 · doi:10.1515/apam-2015-0018
[8] Di Nezza, E, Palatucci, G, Valdinoci, E: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521-573 (2012) · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004
[9] Drabek, P, Pohozaev, SI: Positive solutions for the p-Laplacian: application of the fibering method. Proc. R. Soc. Edinb. A 127, 703-726 (1997) · Zbl 0880.35045 · doi:10.1017/S0308210500023787
[10] Ni, WM, Takagi, I: On the shape of least energy solution to a Neumann problem. Commun. Pure Appl. Math. 44, 819-851 (1991) · Zbl 0754.35042 · doi:10.1002/cpa.3160440705
[11] Ekeland, I: On the variational principle. J. Math. Anal. Appl. 17, 324-353 (1974) · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0
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