POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRÖDINGER EQUATIONS*

doi:10.1007/s10473-023-0308-z

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Qing Guo, Leiga Zhao. POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRÖDINGER EQUATIONS^*[J].Acta mathematica scientia,Series B, 2023, 43(3): 1116-1130.

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[1] Alves C O, Miyagaki O H. Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}$^N via penalization method. Calc Var Partial Differential Equations, 2016, 55: Art 47
[2] Ambrosio V.Mountain pass solutions for the fractional Berestycki-Lions problem. Adv Differential Equations, 2018, 23: 455-488
[3] Ambrosio V, d'Avenia P. Nonlinear fractional magnetic Schrüodinger equation: existence and multiplicity. J Differential Equations, 2018, 264: 3336-3368
[4] Amick C J, Toland J F.Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane. Acta Math, 1991, 167: 107-126
[5] Bahri A, Li Y Y.On the min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}$^N. Rev Mat Iberoamericana, 1990, 6: 1-15
[6] Benci V, Cerami G.Positive solutions of semilinear elliptic problems in exterior domains. Arch Rat Mech Anal, 1987, 99: 283-300
[7] Caffarelli L, Silvestre L.An extension problem related to the fractional Laplacian. Comm in Part Diff Equa, 2007, 32: 1245-1260
[8] Cao D M.Positive solutions and bifurcation from essential spectrum of semilinear elliptic equation on $\mathbb{R}$^N. Nonlinear Analysis TMA, 1990, 15: 1045-1052
[9] Cerami G, Passaseo D.The effect of concentrating potentials in some singularly perturbed problems. Calc Var Partial Differential Equations, 2003, 17: 257-281
[10] Cerami G, Vaira G.Positive solutions for some nonautonomous Schrüodinger-Poisson systems. J Diff Equations, 2010, 248: 521-543
[11] Chang X, Wang Z-Q.Ground state of scalar field equations involving fractional Laplacian with general nonlinearity. Nonlinearity, 2013, 26: 479-494
[12] Chen W, Li C, Li Y.A direct method of moving planes for the fractional Laplacian. Adv Math, 2017, 308: 404-437
[13] Cheng M.Bound state for the fractional Schrüodinger equation with unbounded potential. J Math Phys, 2012, 53: 043507
[14] Cho Y, Ozawa T.A note on the existence of a ground state solution to a fractional Schrüodinger equation. Kyushu J Math, 2013, 67: 227-236
[15] Dipierro S, Palatucci G, Valdinoci E.Existence and symmetry results for a Schrüodinger type problem involving the fractional Laplacian. Matematiche (Catania), 2013, 68: 201-216
[16] Évéquoz G, Fall M M.Positive solutions to some nonlinear fractional Schrüodinger equations via a min-max procedure. arXiv:1312.7068
[17] Fall M M, Valdinoci E.Uniqueness and nondegeneracy of positive solutions of (-△)^su + u = u^p in $\mathbb{R}$^N when s is close to 1. Comm Math Phys, 2014, 329: 383-404
[18] Fall M M, Jarohs S.Overdetermined problems with fractional Laplacian. ESAIM Control Optim Calc Var, 2015, 21: 924-938
[19] Felmer P, Quaas A, Tan J G.Positive solutions of nonlinear Schrüodinger equation with the fractional Laplacian. Proc Roy Soc Edinburgh Sect A, 2012, 142: 1237-1262
[20] Feng B.Ground states for the fractional Schrüodinger equation. Electron J Differ Eq, 2013, 127: 1-11
[21] Frank R L, Lenzmann E.Uniqueness of non-linear ground states for fractional Laplacians in R. Acta Mathematica, 2013, 210: 261-318
[22] Frank R L, Lenzmann E, Silvestre L.Uniqueness of radial solutions for the fractional Laplacian. Commun Pure Appl Math, 2016, 69: 1671-1726
[23] He X, Zou W. Existence and concentration result for the fractional Schrüodinger equations with critical nonlinearities. Calc Var Partial Differential Equations, 2016, 55: Art 91
[24] Kwong M K.Uniqueness of positive solutions of △u - u + u^p = 0 in $\mathbb{R}$^N. Arch Rational Mech Anal, 1989, 105: 243-266
[25] Jarohs S.Symmetry of solutions to nonlocal nonlinear boundary value problems in radial sets. Nonlinear Differential Equations Appl, 2016, 23(3): 1-22
[26] Jin T, Li Y, Xiong J.On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions. J Eur Math Soc, 2014, 16: 1111-1171
[27] Laskin N.Fractional quantum mechanics and Lévy path integrals. Phys Lett A, 2000, 268: 298-305
[28] Li Q, Teng K, Wu X.Ground states for fractional Schrüodinger equations with critical growth. J Math Phys, 2018, 59: 033504
[29] Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1: 109-145, 223-283
[30] Molica Bisci G, Rădulescu V, Servadei R.Variational Methods for Nonlocal Fractional Problems. Cambridge: Cambridge University Press, 2016
[31] Nezza Di E, Palatucci G, Valdinoci E.Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136 : 521-573
[32] Niu M, Tang Z.Least energy solutions for nonlinear Schrüodinger equation involving the fractional Laplacian and critical growth. Discrete Contin Dyn Syst Ser A, 2017, 37: 3963-3987
[33] Rabinowitz P H.On a class of nonlinear Schrüodinger equations. Z Angew Math Phys, 1992 43: 270-291
[34] Shang X, Zhang J.Ground states for fractional Schrüodinger equations with critical growth. Nonlinearity, 2014, 27: 187-207
[35] Secchi S.Ground state solutions for nonlinear fractional Schrüodinger equations in $\mathbb{R}$^N. J Math Phys, 2013, 54: 031501
[36] Secchi S.On fractional Schrüodinger equations in $\mathbb{R}$^N without the Ambrosetti-Rabinowitz condition. Topol Method Nonl Anal, 2016, 47: 19-41
[37] Struwe M. Variational Methods.Berlin: Springer-Verlag, 1996
[38] Willem M. Minimax Theorems. Boston: Birkhüauser, 1996
[39] Yan S, Yang J, Yu X.Equations involving fractional Laplacian operator: Compactness and application. J Funct Anal, 2015, 269: 47-79
[40] Yang P, Liu J, Tang C.Ground state solutions for non-local fractional Schrodinger equations. Electron J Differ Eq, 2015, 223: 1-16

POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRÖDINGER EQUATIONS^*

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