Multiple of solutions for nonlocal elliptic equations with critical exponent driven by the Fractional \(p\)-Laplacian of order \(s\). (English) Zbl 1474.35656

Summary: In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional \(p\)-Laplacian of order \(s\). We show the above result when \(\lambda > 0\) is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.


35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35B33 Critical exponents in context of PDEs
Full Text: DOI


[1] Caffarelli, L. A.; Salsa, S.; Silvestre, L., Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Inventiones Mathematicae, 171, 2, 425-461 (2008) · Zbl 1148.35097
[2] Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60, 1, 67-112 (2007) · Zbl 1141.49035
[3] Caffarelli, L.; Roquejoffre, J.-M.; Savin, O., Nonlocal minimal surfaces, Communications on Pure and Applied Mathematics, 63, 9, 1111-1144 (2010) · Zbl 1248.53009
[4] Sire, Y.; Valdinoci, E., Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, Journal of Functional Analysis, 256, 6, 1842-1864 (2009) · Zbl 1163.35019
[5] Wei, Y. H.; Su, X. F., Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calculus of Variations and Partial Differential Equations, 52, 1-2, 95-124 (2015) · Zbl 1317.35285
[6] Guan, Q.-Y.; Ma, Z.-M., Boundary problems for fractional Laplacians, Stochastics and Dynamics, 5, 3, 385-424 (2005) · Zbl 1077.60036
[7] Laskin, N., Fractional quantum mechanics and Lévy path integrals, Physics Letters A, 268, 4-6, 298-305 (2000) · Zbl 0948.81595
[8] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136, 5, 521-573 (2012) · Zbl 1252.46023
[9] Xiping, Z., Nontrivial solution of quasilinear elliptic equations involving critical Sobolev exponent, Science in China Series A-Mathematics, Physics, Astronomy & Technological Science, 31, 10, 1166-1181 (1988) · Zbl 0677.35039
[10] Li, G. B., The Existence of Nontrivial Solution of Quasilinear Elliptic Pde of Variational Type (1987), Hubei, China: Wuhan University, Hubei, China
[11] Garcia Azorero, J.; Peral Alonso, I., Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Transactions of the American Mathematical Society, 323, 2, 877-895 (1991) · Zbl 0729.35051
[12] Gongbao, L.; Guo, Z., Multiple solutions for the p&q-Laplacian problem with critical exponent, Acta Mathematica Scientia, 29, 4, 903-918 (2009) · Zbl 1212.35125
[13] Figueiredo, G. M., Existence and multiplicity of solutions for a class of p & q elliptic problems with critical exponent, Mathematische Nachrichten, 286, 11-12, 1129-1141 (2013) · Zbl 1278.35094
[14] Brasco, L.; Squassina, M., Optimal solvability for a nonlocal problem at critical growth, Journal of Differential Equations, 264, 3, 2242-2269 (2018) · Zbl 1386.35428
[15] Perera, K.; Squassina, M.; Yang, Y., Bifurcation and multiplicity results for critical fractional p-Laplacian problems, Mathematische Nachrichten, 289, 2-3, 332-342 (2016) · Zbl 1336.35050
[16] Mawhin, J.; Molica Bisci, G., A Brezis-Nirenberg type result for a nonlocal fractional operator, Journal Of The London Mathematical Society-Second Series, 95, 1, 73-93 (2017) · Zbl 1398.35276
[17] Servadei, R.; Valdinoci, E., The Brezis-Nirenberg result for the fractional Laplacian, Transactions of the American Mathematical Society, 367, 1, 67-102 (2015) · Zbl 1323.35202
[18] Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society, 88, 3, 486-490 (1983) · Zbl 0526.46037
[19] Kavian, O., Introduction à la théorie des points critiques et applications aux problèmes elliptiques (1993), Paris, France: Springer-Verlag, Paris, France · Zbl 0797.58005
[20] Struwe, M., Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (1990), Berlin, Germany: Springer, Berlin, Germany · Zbl 0746.49010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.