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Doubly nonlocal system with Hardy–Littlewood–Sobolev critical nonlinearity
J. Giacomoni, , K. Sreenadh
Published in
Volume: 467
Issue: 1
Pages: 638 - 672
This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy–Littlewood–Sobolev inequality {(−Δ)su=λ|u|q−2u+(∫Ω[Formula presented]dy)|u|2μ ⁎−2uinΩ(−Δ)sv=δ|v|q−2v+(∫Ω[Formula presented]dy)|v|2μ ⁎−2vinΩu=v=0inRn∖Ω where Ω is a smooth bounded domain in Rn, n>2s, s∈(0,1), (−Δ)s is the well known fractional Laplacian, μ∈(0,n), 2μ ⁎=[Formula presented] is the upper critical exponent in the Hardy–Littlewood–Sobolev inequality, 1<q<2 and λ,δ>0 are real parameters. We study the fibering maps corresponding to the functional associated with (Pλ,δ) and show that minimization over suitable subsets of Nehari manifold renders the existence of at least two non trivial solutions of (Pλ,δ) for suitable range of λ and δ. © 2018 Elsevier Inc.}, author_keywords={Choquard equation; Fractional Laplacian; Hardy–Littlewood–Sobolev critical exponent; Nonlocal operator}, funding_details={Indo-French Centre for Applied MathematicsIndo-French Centre for Applied Mathematics, IFCAM, UMI CNRS 3494}, funding_text_1={The authors were funded by IFCAM (Indo-French Centre for Applied Mathematics) UMI CNRS 3494 under the project “Singular phenomena in reaction diffusion equations and in conservation laws”.}, references={Alves, C.O., Figueiredo, M.G., Yang, M., Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity (2016) Adv. Nonlinear Anal., 5 (4), pp. 331-345; Alves, C.O., Gao, F., Squassina, M., Yang, M., Singularly perturbed critical Choquard equations (2017) J. Differential Equations, 263, pp. 3943-3988; Applebaum, D., Lévy processes—from probability to finance and quantum groups (2004) Notices Amer. Math. Soc., 51 (11), pp. 1336-1347; Bonanno, C., D'Avenia, P., Ghimenti, M., Squassina, M., Soliton dynamics for the generalized Choquard equations (2014) J. Math. Anal. Appl., 417, pp. 180-199; Buffoni, B., Jeanjean, L., Stuart, C.A., Existence of a nontrivial solution to a strongly indefinite semilinear equation (1993) Proc. Amer. Math. Soc., 119 (1), pp. 179-186; Chen, Y.-H., Liu, C., Ground state solutions for non-autonomous fractional Choquard equations (2016) Nonlinearity, 29, pp. 1827-1842; Chen, W., Squassina, M., Critical nonlocal systems with concave-convex powers (2016) Adv. Nonlinear Stud., 16 (4), pp. 821-842; Choi, W., On strongly indefinite systems involving the fractional Laplacian (2015) Nonlinear Anal., 120, pp. 127-153; D'Avenia, P., Siciliano, G., Squassina, M., On fractional Choquard equations (2015) Math. Models Methods Appl. Sci., 25 (8), pp. 1447-1476; D'Avenia, P., Siciliano, G., Squassina, M., Existence results for a doubly nonlocal equation (2015) São Paulo J. Math. Sci., 9 (2), pp. 311-324; Fan, H., Multiple positive solutions for a fractional elliptic system with critical nonlinearities (2016) Bound. Value Probl., 2016, p. 196; Faria, L.F.O., Miyagaki, O.H., Pereira, F.R., Squassina, M., Zhang, C., The Brezis–Nirenberg problem for nonlocal systems (2015) Adv. Nonlinear Anal., 5 (1), pp. 85-103; Gao, F., Yang, M., On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents (2017) J. Math. Anal. Appl., 448 (2), pp. 1006-1041; Gao, F., Yang, M., On the Brezis–Nirenberg type critical problem for nonlinear Choquard equation (2018) Sci. China Math., 61 (7), pp. 1219-1242; Ghimenti, M., Van Schaftingen, J., Nodal solutions for the Choquard equation (2016) J. Funct. Anal., 271, pp. 107-135; Ghimenti, M., Moroz, V., Schaftingen, J.V., Least action nodal solutions for the quadratic Choquard equation (2017) Proc. Amer. Math. Soc., 145 (2), pp. 737-747; Giacomoni, J., Mishra, P.K., Sreenadh, K., Critical growth fractional elliptic systems with exponential nonlinearity (2016) Nonlinear Anal., 136, pp. 117-135; Giacomoni, J., Mukherjee, T., Sreenadh, K., Positive solutions of fractional elliptic equation with critical and singular nonlinearity (2017) Adv. Nonlinear Anal., 6 (3), pp. 327-354; Guo, Z., Luo, S., Zou, W., On critical systems involving frcational Laplacian (2017) J. Math. Anal. Appl., 446 (1), pp. 681-706; He, X., Squassina, M., Zou, W., The Nehari manifold for fractional systems involving critical nonlinearities (2016) Commun. Pure Appl. Anal., 15 (4), pp. 1285-1308; Lieb, E., Loss, M., Analysis (2001) Graduate Studies in Mathematics, , AMS Providence, Rhode Island; Lü, D., Xu, G., On nonlinear fractional Schrödinger equations with Hartree-type nonlinearity (2018) Appl. Anal., 97 (2), pp. 255-273; Molica Bisci, G., Radulescu, V.D., Servadei, R., Variational Methods for Nonlocal Fractional Problems (2016) Encyclopedia of Mathematics and Its Applications, , Cambridge University Press Cambridge; Moroz, V., Schaftingen, J.V., A guide to the Choquard equation (2017) J. Fixed Point Theory Appl., 19 (1), pp. 773-813; Mukherjee, T., Sreenadh, K., Critical growth fractional elliptic problem with singular nonlinearities (2016) Electron. J. Differential Equations, 54, pp. 1-23; Ros-Oton, X., Nonlocal equations in bounded domains: a survey (2016) Publ. Mat., 60, pp. 3-26; Ros-Oton, X., Serra, J., The extremal solution for the fractional Laplacian (2014) Calc. Var., 50, pp. 723-750; Servadei, R., Valdinoci, E., The Brezis–Nirenberg result for the fractional Laplacian (2015) Trans. Amer. Math. Soc., 367 (1), pp. 67-102; Shen, Z., Gao, F., Yang, M., Multiple solutions for nonhomogeneous Choquard equation involving Hardy–Littlewood–Sobolev critical exponent (2017) Z. Angew. Math. Phys.; Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator (2007) Comm. Pure Appl. Math., 60, pp. 67-112; Tarantello, G., On nonhomogeneous elliptic equations involving critical Sobolev exponent (1992) Ann. Inst. H. Poincaré Anal. Non Linéaire, 9, pp. 281-304; Wang, K., Wei, J., On the uniqueness of solutions of a nonlocal elliptic system (2016) Math. Ann., 365 (1-2), pp. 105-153; Xiang, M., Zhang, B., Rădulescu, V.D., Existence of solutions for perturbed fractional p-Laplacian equations (2016) J. Differential Equations, 260 (2), pp. 1392-1413; Zhang, X., Zhang, B., Xiang, M., Ground states for fractional Schrödinger equations involving a critical nonlinearity (2016) Adv. Nonlinear Anal., 5 (3), pp. 293-314}, correspondence_address1={Giacomoni, J.; Université de Pau et des Pays de l'Adour, avenue de l'université, France; email: jacques.giacomoni@univ-pau.fr}, publisher={Academic Press Inc.}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},
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