×

Existence and multiplicity of solutions for a fractional \(p\)-Laplacian problem of Kirchhoff type via Krasnoselskii’s genus. (English) Zbl 1463.35493

Summary: We use the genus theory to prove the existence and multiplicity of solutions for the fractional \(p\)-Kirchhoff problem \[ \begin{cases}\displaystyle -\left[ M\left( \int_Q\frac{| u(x)-u(y)|^p}{| x-y|^{N+ps}}dx\, dy\right)\right]^{p-1}(-\Delta)_p^su=\lambda h(x,u)\quad\text{in}\;\Omega,\\ u=0\quad\text{on}\;\mathbb{R}^N\setminus\Omega,\end{cases} \] 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.

MSC:

35R11 Fractional partial differential equations
35R09 Integro-partial differential equations
35A15 Variational methods applied to PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
PDFBibTeX XMLCite
Full Text: DOI

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 · Zbl 1325.35129
[2] Caffarelli, L., Nonlocal equations, drifts and games, Nonlinear Partial Differential Equations. Abel Symposia, vol. 7 H. Holden et al. Springer, Heidelberg (2012), 37-52 · 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 · Zbl 1380.35085
[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 · 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 · 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 · 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 · 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 · 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 · Zbl 1252.46023
[12] Dreher, M., The Kirchhoff equation for the \(p\)-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), 217-238 · 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 · Zbl 1332.35375
[14] Kavian, O., Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques, Mathématiques et Applications. Springer, Paris (1993) · 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) · 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) · 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 · 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) · Zbl 0609.58002
[20] Servadei, R.; Valdinoci, E., Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887-898 · Zbl 1234.35291
[21] Servadei, R.; Valdinoci, E., The Brezis-Nirenberg result for the fractional Laplacian, Trans. Am. Math. Soc. 367 (2015), 67-102 · 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 · 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 · 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 · Zbl 1359.35036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.