# Article

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Keywords:
existence results; genus theory; fractional $p$-Kirchhoff problem
Summary:
We use the genus theory to prove the existence and multiplicity of solutions for the fractional $p$-Kirchhoff problem $$\begin {cases} \displaystyle -\biggl [M \biggl (\int _{Q}\dfrac {\vert u(x)-u(y)\vert ^{p}}{\vert x-y \vert ^{N+ps}} {\rm d}x {\rm d}y\biggr )\biggr ]^{p-1} (-\Delta )_{p}^{s}u=\lambda h(x,u) \quad \text {in}\ \Omega , \\ u=0 \quad \text {on}\ \mathbb {R}^N \setminus \Omega , \end {cases}$$ where $\Omega$ is an open bounded smooth domain of $\mathbb {R}^N$, $p>1$, $N>ps$ with $s\in (0,1)$ fixed, $Q = \mathbb {R}^{2N}\setminus (C\Omega \times C\Omega )$, $\lambda > 0$ is a numerical parameter, $M$ and $h$ are continuous functions.
References:
[1] Autuori, G., Colasuonno, F., Pucci, P.: On the existence of stationary solutions for higher order $p$-Kirchhoff problems. Commun. Contemp. Math. 16 (2014), Article ID 1450002, 43 pages. DOI 10.1142/S0219199714500023 | MR 3253900 | Zbl 1325.35129
[2] Caffarelli, L.: Nonlocal equations, drifts and games. Nonlinear Partial Differential Equations. Abel Symposia, vol. {\it 7} H. Holden et al. Springer, Heidelberg (2012), 37-52. DOI 10.1007/978-3-642-25361-4_3 | MR 3289358 | Zbl 1266.35060
[3] Castro, A.: Metodos variacionales y analisis functional no linear. X Colóquio Colombiano de Matematicas. Monograph published by the Colombian Math. Society, Paipa (1980), Spain.
[4] Chen, J., Cheng, B., Tang, X.: New existence of multiple solutions for nonhomogeneous Schrödinger-Kirchhoff problems involving the fractional $p$-Laplacian with sign-changing potential. Rev. Real Acad. Cien. Exact., Fís. Nat., Serie A. Mat. (2016), 1-24. DOI 10.1007/s13398-016-0372-5 | MR 3742996
[5] Chen, W., Deng, S.: Existence of solutions for a Kirchhoff type problem involving the fractional $p$-Laplace operator. Electron. J. Qual. Theory Differ. Equ. 2015 (2015), Article ID 87, 8 pages. DOI 10.14232/ejqtde.2015.1.87 | MR 3434217 | Zbl 1349.35088
[6] Cheng, K., Gao, Q.: Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in $\mathbb{R^{N}}$. Avaible at https://arxiv.org/abs/1701.03862v1
[7] Clarke, D. C.: A variant of the Lusternik-Schnirelman theory. Math. J., Indiana Univ. 22 (1972), 65-74. DOI 10.1512/iumj.1972.22.22008 | MR 0296777 | Zbl 0228.58006
[8] Colasuonno, F., Pucci, P.: Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 5962-5974. DOI 10.1016/j.na.2011.05.073 | MR 2833367 | Zbl 1232.35052
[9] Corrêa, F. J. S. A., Figueiredo, G. M.: On an elliptic equation of $p$-Kirchhoff-type via variational methods. Bull. Aust. Math. Soc. 74 (2006), 263-277. DOI 10.1017/S000497270003570X | MR 2260494 | Zbl 1108.45005
[10] Corrêa, F. J. S. A., Figueiredo, G. M.: On a $p$-Kirchhoff equation via Krasnoselskii's genus. Appl. Math. Lett. 22 (2009), 819-822. DOI 10.1016/j.aml.2008.06.042 | MR 2523587 | Zbl 1171.35371
[11] Nezza, E. Di, Palatucci, G., Valdinoci, E.: Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521-573. DOI 10.1016/j.bulsci.2011.12.004 | MR 2944369 | Zbl 1252.46023
[12] Dreher, M.: The Kirchhoff equation for the $p$-Laplacian. Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), 217-238. MR 2272915 | Zbl 1178.35006
[13] Goyal, S., Sreenadh, K.: Nehari manifold for non-local elliptic operator with concave-convex nonlinearities and sign-changing weight functions. Proc. Indian Acad. Sci., Math. Sci. 125 (2015), 545-558. DOI 10.1007/s12044-015-0244-5 | MR 3432207 | Zbl 1332.35375
[14] Kavian, O.: Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques. Mathématiques et Applications. Springer, Paris (1993). MR 1276944 | Zbl 0797.58005
[15] Krasnoselsk'ii, M. A.: Topological Methods in the Theory of Nonlinear Integral Equations. International Series of Monographs on Pure and Applied Mathematics 45. Pergamon Press, Oxford; MacMillan, New York (1964). MR 0159197 | Zbl 0111.30303
[16] Bisci, G. Molica, Radulescu, V. D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and its Applications 162. Cambridge University Press, Cambridge (2016). DOI 10.1017/CBO9781316282397 | MR 3445279 | Zbl 1356.49003
[17] Ourraoui, A.: On a $p$-Kirchhoff problem involving a critical nonlinearity. C. R. Math., Acad. Sci. Paris 352 (2014), 295-298. DOI 10.1016/j.crma.2014.01.015 | MR 3186916 | Zbl 1298.35096
[18] Peral, I.: Multiplicity of solutions for the $p$-Laplacian. Second School of Nonlinear Functional Analysis and Applications to Differential Equations, ICTP, Trieste (1997).
[19] Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics 65. AMS, Providence (1984). DOI 10.1090/cbms/065 | MR 0845785 | Zbl 0609.58002
[20] Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389 (2012), 887-898. DOI 10.1016/j.jmaa.2011.12.032 | MR 2879266 | Zbl 1234.35291
[21] Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367 (2015), 67-102. DOI 10.1090/S0002-9947-2014-05884-4 | MR 3271254 | Zbl 1323.35202
[22] Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60 (2007), 67-112. DOI 10.1002/cpa.20153 | MR 2270163 | Zbl 1141.49035
[23] Wang, L., Zhang, B.: Infinitely many solutions for Schrodinger-Kirchhoff type equations involving the fractional $p$-Laplacian and critical exponent. Electron. J. Differ. Equ. 2016 (2016), Paper No. 339, 18 pages. MR 3604784 | Zbl 1353.35307
[24] Zhang, L., Chen, Y.: Infinitely many solutions for sublinear indefinite nonlocal elliptic equations perturbed from symmetry. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 151 (2017), 126-144. DOI 10.1016/j.na.2016.12.001 | MR 3596674 | Zbl 06675023

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