July  2011, 10(4): 1037-1054. doi: 10.3934/cpaa.2011.10.1037

Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071E01103, China

2. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084e/pjp, China

3. 

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received  April 2010 Revised  September 2010 Published  April 2011

By using a change of variable, the quasilinear Schrödinger equation is reduced to semilinear elliptic equation. Then, Mountain Pass theorem without $(PS)_c$ condition in a suitable Orlicz space is employed to prove the existence of positive standing wave solutions for a class of quasilinear Schrödinger equations involving critical Sobolev-Hardy exponents.
Citation: Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
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show all references

References:
[1]

Nonlinear Anal., 56 (2004), 781-791. doi: 10.1016/j.na.2003.06.003.  Google Scholar

[2]

Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

[3]

Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018.  Google Scholar

[4]

Comm. Pure and Applied Anal., 8 (2009), 621-644. doi: 10.3934/cpaa.2009.8.621.  Google Scholar

[5]

J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[6]

Trans. Amer. Math. Soc., 352 (2000), 5703-5743. doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar

[7]

Z. Phys. B, 37 (1980), 83-87. doi: 10.1007/BF01325508.  Google Scholar

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Nonlinear Anal., 66 (2007), 241–252. doi: 10.1016/j.na.2005.11.028.  Google Scholar

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Phys. Rep., 194 (1990), 117-238. doi: 10.1016/0370-1573(90)90130-T.  Google Scholar

[10]

J. Phys. Soc. Japan, 50 (1981), 3262-3267. doi: 10.1143/jpsj.50.3262.  Google Scholar

[11]

J. Math. Phys., 24 (1983), 2764-2769. doi: 10.1063/1.525675.  Google Scholar

[12]

JETP Lett., 27 (1978), 517-520. doi: 0021-3640778/2710-0517.  Google Scholar

[13]

Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.  Google Scholar

[14]

Proc. Amer. Math. Soc., 131 (2002), 441-448. doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[15]

J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/s0022-0396(02)0064-5.  Google Scholar

[16]

Phys. Rep, 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[17]

Springer-Verlag, 1989.  Google Scholar

[18]

Nonlinear Anal., 29 (1997), 773-781. doi: 10.1016/s0362-546x(96)00087-9.  Google Scholar

[19]

J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[20]

Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009.  Google Scholar

[21]

Calc. Var. Partial Differential Equations 14 (2002), 329-344. doi: 10.1007/s005260100105.  Google Scholar

[22]

Physica A, 110 (1982), 41-80. doi: 10.1016/0378-4371(82)90104-2.  Google Scholar

[23]

Progr. Theoret. Phys., 65 (1981), 172-189. doi: 10.1143/PTP.65.172.  Google Scholar

[24]

Z angew Math. Phy., 43 (1992), 272-291. doi: 10.1007/BF00946631.  Google Scholar

[25]

Nonlinear Anal., 71 (2009), 6157-6169. doi: 10.1016/j.na.2009.06.006.  Google Scholar

[26]

Applied Mathematics and Computation, 216 (2010), 849-856. doi: 10.1016/j.amc.2010.01.091.  Google Scholar

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