带有次临界或临界增长的分数阶Schr?dinger-Poisson方程组非平凡解的存在性
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摘要
研究了一类带有次临界或临界增长的分数阶Schr?dinger-Poisson方程组,应用Nehari流形方法得到了非平凡解的存在性.
In this paper, we are concerned with a class of fractional Schrodinger-Poisson systems. We obtain a nontrivial solution via the Nehari manifold method.
In this paper, we are concerned with a class of fractional Schrodinger-Poisson systems. We obtain a nontrivial solution via the Nehari manifold method.
引文
[1] Ambrosetti, A., On Schr(o|¨)dinger-Poisson systems, Milan J. Math., 2005, 76:257-274.
[2] Azzollini, A. and Pomponio, A., Ground state solutions for the nonlinear Schr(o|¨)dinger-Maxwell equations, J.Math. Anal. Appl., 2005, 345:90-108.
[3] Barrios, B., Colorado, E. and Pablo, A.D., On some critical problems for the fractional Laplacian operator,J. Differential Equations, 2011, 252(11):6133-6162.
[4] Benci, V. and Fortunato, D., An eigenvalue problem for the Schr(o|¨)dinger-maxwell equations, Topol. Methods Nonlinear Anal., 1998, 11:283-293.
[5] Brezis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals,Proc. Amer. Math. Soc., 1983, 88(3):486-486.
[6] Brown, K.J. and Zhang, Y.P., The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 2003, 193(2):481-499.
[7] Gaetano, S., Multiple positive solutions for a Schr(o|¨)dinger-Poisson-Slater system, J. Math. Anal.Appl., 2010,365:288-299.
[8] Giovany, M.F. and Gaetano, S., Positive solutions for the fractional Laplacian in the almost critical case in a bounded domain, Nonlinear Anal. Real World Appl., 2017, 36:89-100.
[9] He, X.M., Multiplicity and concentration of positive solutions for the Schrodinger-Poisson equations, Z.Angew. Math. Phys., 2011, 62:869-889.
[10] Jiang, Y.S. and Zhou, H.S., Schrodinger-Poisson system with steep potential well, J. Differential Equations,2011, 251:582-608.
[11] Li, K.X., Existence of nontrivial solutions for nonlinear fractional Schr(o|¨)dinger-Poisson equations, Applied Mathematics Letters, 2017, 72:1-9.
[12] Liu, W.M., Existence of multi-bump solutions for the fractional Schr(o|¨)dinger-Poisson system, J. Math. Phys.,2016, 57(9):779-791.
[13] Liu, Z.S. and Zhang, J.J., Multiplicity and concentration of positive solutions for the fractional Schr(o|¨)dingerPoisson systems with critical growth,ESAIM Control Optim.Calc. Var., 2017, 23(4):1515-1542.
[14] Luo, H.X. and Tang, X.H., Infinitely many radial solutions for the fractional Schr(o|¨)dinger-Poisson systems,J. Nonlinear Sci. Appl., 2016, 9(6):3808-3821.
[15] Marco, G. and Anna, M.M., Low energy solutions for the limit of Schr(o|¨)dinger-Maxwell semiclassical systems,Progr. Nonlinear Differential Equations Appl., 2014, 85:287-300.
[16] Nezza, E.D., Palatucci, G. and Valdinoci, E., Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci.Math., 2012, 136:521-573.
[17] Servadei, R. and Valdinoci, E., The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math.Soc., 2015, 367:67-102.
[18] Teng, K.M., Existence of ground state solutions for the nonlinear fractional Schrodinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 2016, 261:3061-3106.
[19] Wang, Z.P. and Zhou, H.S., Positive solution for a nonlinear stationary Schrodinger-Poisson system in R~3,Discrete Contin. Dyn. Syst., 2007, 18:809-816.
[20] Wei, Z.L., Existence of infinitely many solutions for the fractional Schrodinger-Maxwell equations, 2015,arXiv:1508.03008vl.
[21] Yu, Y.Y., Zhao, F.K. and Zhao, L.G., The concentration behavior of ground state solutions for a fractional Schrodinger-Poisson system, Calc. Var. Partial Differential Equations, 2017, 56(4):116, 25 pp.
[22] Yu, Y.Y., Zhao, F.K. and Zhao, L.G., The existence and multiplicity of solutions of a fractional Schr(o|¨)dingerPoisson system with critical growth, Sci. China Math., 2018, 61(6):1039-1062.
[23] Zhang, J.J., do O, J.M. and Squassina, M., Fractional Schr(o|¨)dinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 2016, 16(1):15-30.
[2] Azzollini, A. and Pomponio, A., Ground state solutions for the nonlinear Schr(o|¨)dinger-Maxwell equations, J.Math. Anal. Appl., 2005, 345:90-108.
[3] Barrios, B., Colorado, E. and Pablo, A.D., On some critical problems for the fractional Laplacian operator,J. Differential Equations, 2011, 252(11):6133-6162.
[4] Benci, V. and Fortunato, D., An eigenvalue problem for the Schr(o|¨)dinger-maxwell equations, Topol. Methods Nonlinear Anal., 1998, 11:283-293.
[5] Brezis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals,Proc. Amer. Math. Soc., 1983, 88(3):486-486.
[6] Brown, K.J. and Zhang, Y.P., The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 2003, 193(2):481-499.
[7] Gaetano, S., Multiple positive solutions for a Schr(o|¨)dinger-Poisson-Slater system, J. Math. Anal.Appl., 2010,365:288-299.
[8] Giovany, M.F. and Gaetano, S., Positive solutions for the fractional Laplacian in the almost critical case in a bounded domain, Nonlinear Anal. Real World Appl., 2017, 36:89-100.
[9] He, X.M., Multiplicity and concentration of positive solutions for the Schrodinger-Poisson equations, Z.Angew. Math. Phys., 2011, 62:869-889.
[10] Jiang, Y.S. and Zhou, H.S., Schrodinger-Poisson system with steep potential well, J. Differential Equations,2011, 251:582-608.
[11] Li, K.X., Existence of nontrivial solutions for nonlinear fractional Schr(o|¨)dinger-Poisson equations, Applied Mathematics Letters, 2017, 72:1-9.
[12] Liu, W.M., Existence of multi-bump solutions for the fractional Schr(o|¨)dinger-Poisson system, J. Math. Phys.,2016, 57(9):779-791.
[13] Liu, Z.S. and Zhang, J.J., Multiplicity and concentration of positive solutions for the fractional Schr(o|¨)dingerPoisson systems with critical growth,ESAIM Control Optim.Calc. Var., 2017, 23(4):1515-1542.
[14] Luo, H.X. and Tang, X.H., Infinitely many radial solutions for the fractional Schr(o|¨)dinger-Poisson systems,J. Nonlinear Sci. Appl., 2016, 9(6):3808-3821.
[15] Marco, G. and Anna, M.M., Low energy solutions for the limit of Schr(o|¨)dinger-Maxwell semiclassical systems,Progr. Nonlinear Differential Equations Appl., 2014, 85:287-300.
[16] Nezza, E.D., Palatucci, G. and Valdinoci, E., Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci.Math., 2012, 136:521-573.
[17] Servadei, R. and Valdinoci, E., The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math.Soc., 2015, 367:67-102.
[18] Teng, K.M., Existence of ground state solutions for the nonlinear fractional Schrodinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 2016, 261:3061-3106.
[19] Wang, Z.P. and Zhou, H.S., Positive solution for a nonlinear stationary Schrodinger-Poisson system in R~3,Discrete Contin. Dyn. Syst., 2007, 18:809-816.
[20] Wei, Z.L., Existence of infinitely many solutions for the fractional Schrodinger-Maxwell equations, 2015,arXiv:1508.03008vl.
[21] Yu, Y.Y., Zhao, F.K. and Zhao, L.G., The concentration behavior of ground state solutions for a fractional Schrodinger-Poisson system, Calc. Var. Partial Differential Equations, 2017, 56(4):116, 25 pp.
[22] Yu, Y.Y., Zhao, F.K. and Zhao, L.G., The existence and multiplicity of solutions of a fractional Schr(o|¨)dingerPoisson system with critical growth, Sci. China Math., 2018, 61(6):1039-1062.
[23] Zhang, J.J., do O, J.M. and Squassina, M., Fractional Schr(o|¨)dinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 2016, 16(1):15-30.