The numbers of positive solutions by the Lusternik-Schnirelmann category for a quasilinear elliptic system critical with Hardy terms. (English) Zbl 1475.35148

Khiddi, Mustapha

Abstr. Appl. Anal. 2019, Article ID 4829861, 9 p. (2019).

Summary: In this paper, we study the quasilinear elliptic system with Sobolev critical exponent involving both concave-convex and Hardy terms in bounded domains. By employing the technique introduced by V. Benci and G. Cerami [Arch. Ration. Mech. Anal. 114, No. 1, 79–93 (1991; Zbl 0727.35055)], we obtain at least \(\operatorname{cat}(\Omega) + 1\) distinct positive solutions.

MSC:

35J57	Boundary value problems for second-order elliptic systems
35B09	Positive solutions to PDEs
35B33	Critical exponents in context of PDEs
35J62	Quasilinear elliptic equations
35J92	Quasilinear elliptic equations with \(p\)-Laplacian

Citations:

Zbl 0727.35055

References:

[1]	Chu, C.-M.; Tang, C.-L., Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, 71, 11, 5118-5130 (2009) · Zbl 1176.35064 · doi:10.1016/j.na.2009.03.080
[2]	Ambrosetti, A.; Brézis, H.; Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems, Journal of Functional Analysis, 122, 2, 519-543 (1994) · Zbl 0805.35028 · doi:10.1006/jfan.1994.1078
[3]	Benmouloud, S.; Echarghaoui, R.; Sbaï, S. M., Multiplicity of positive solutions for a critical quasilinear elliptic system with concave and convex nonlinearities, Journal of Mathematical Analysis and Applications, 396, 1, 375-385 (2012) · Zbl 1255.35119 · doi:10.1016/j.jmaa.2012.05.078
[4]	Hsu, T.-S., Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities, Nonlinear Analysis: Theory, Methods & Applications, 71, 7-8, 2688-2698 (2009) · Zbl 1167.35356 · doi:10.1016/j.na.2009.01.110
[5]	Wu, T.-F., On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, Journal of Mathematical Analysis and Applications, 318, 1, 253-270 (2006) · Zbl 1153.35036 · doi:10.1016/j.jmaa.2005.05.057
[6]	Wang, L.; Wei, Q.; Kang, D., Existence and multiplicity of positive solutions to elliptic systems involving critical exponents, Journal of Mathematical Analysis and Applications, 383, 2, 541-552 (2011) · Zbl 1225.35092 · doi:10.1016/j.jmaa.2011.05.053
[7]	Chabrowski, J., On multiple solutions for the nonhomogeneous p-laplacian with a critical sobolev exponent, Differential and Integral Equations, 8, 4, 705-716 (1995) · Zbl 0814.35033
[8]	Han, P., Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Analysis: Theory, Methods & Applications, 61, 5, 735-758 (2005) · Zbl 1210.35021 · doi:10.1016/j.na.2005.01.030
[9]	Tarantello, G., On nonhomogeneous elliptic equations involving critical Sobolev exponent, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 9, 281-304 (2016), Elsevier · Zbl 0785.35046
[10]	Willem, M., Minimax theorems, progress in nonlinear differential equations and applications, 24 (1996) · Zbl 0856.49001
[11]	Struwe, M.; Struwe, M., Variational methods (1990), Springer · Zbl 0746.49010
[12]	Echarghaoui, R.; Khiddi, M.; Sbai, S. M., Multiple positive solutions for a Choquard equation involving both concave-convex and Hardy-Littlewood-Sobolev critical exponent, Differential Equations & Applications, 9, 4, 505-520 (2017) · Zbl 1401.35116 · doi:10.7153/dea-2017-09-34