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January  2014, 13(1): 97-118. doi: 10.3934/cpaa.2014.13.97

Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities

Shinji Adachi 1, , Masataka Shibata 2, and  Tatsuya Watanabe 3,

1. 

Department of Mathematical and Systems Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu, 432-8561, Japan
2. 

Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551
3. 

Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto-City, 603-8555, Japan

Received  April 2012 Revised  May 2013 Published  July 2013

We study the asymptotic behavior of the ground state for a class of quasilinear Schrödinger equations with general nonlinearities. By the variational argument and dual approach, we show the asymptotic non-degeneracy and uniqueness of the ground state.
Keywords: Quasilinear elliptic equation, asymptotic behavior..
Mathematics Subject Classification: Primary: 35J62, 35A15, 35B4.
Citation: Shinji Adachi, Masataka Shibata, Tatsuya Watanabe. Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 97-118. doi: 10.3934/cpaa.2014.13.97
References:
[1]

S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation,, Adv. Diff. Eqns., 16 (2011), 289.   Google Scholar

[2]

S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations,, Nonlinear Anal., 75 (2012), 819.  doi: 10.1016/j.na.2011.09.015.  Google Scholar

[3]

S. Adachi and T. Watanabe, Asymptotic properties of ground states of quasilinear Schrödinger equations with $H^1$-subcritical exponent,, Adv. Nonlinear Stud., 12 (2012), 255.   Google Scholar

[4]

A. Ambrosetti and Z. Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\R$,, Disc. Cont. Dyn. Syst., 9 (2003), 55.  doi: 10.3934/dcds.2003.9.55.  Google Scholar

[5]

P. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems,, J. Funct. Anal., 196 (2002), 211.  doi: 10.1016/S0022-1236(02)00013-7.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar fields equations, I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[7]

J. Byeon, L. Jeanjean and M. Mariş, Symmetry and monotonicity of least energy solutions,, Calc. Var. PDE, 36 (2009), 481.  doi: 10.1007/s00526-009-0238-1.  Google Scholar

[8]

L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice,, Physica D, 159 (2001), 71.   Google Scholar

[9]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach,, Nonlinear Anal. TMA., 56 (2004), 213.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[10]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations,, Nonlinearity, 23 (2010), 1353.  doi: 10.1088/0951-7715/23/6/006.  Google Scholar

[11]

J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\RN$: mountain pass and symmetric mountain pass approaches,, Top. Methods in Nonlinear Anal., 35 (2010), 253.   Google Scholar

[12]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\RN$,, Proc. Amer. Math. Soc., 131 (2003), 2399.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[13]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, J. Phys. Soc. Japan, 50 (1981), 3262.   Google Scholar

[14]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, Arch. Rat. Mech. Anal., 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar

[15]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Diff. Eqns., (2003), 473.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[16]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Solutions for quasi-linear Schrödinger equations via the Nehari method,, Comm. PDE, 29 (2004), 879.  doi: 10.1081/PDE-120037335.  Google Scholar

[17]

W. M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problems,, Duke Math. J., 70 (1992), 247.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[18]

T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem, II,, J. Diff. Eqns., 158 (1999), 94.  doi: 10.1016/S0022-0396(99)80020-5.  Google Scholar

[19]

M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, Calc. Var. PDE, 14 (2002), 329.  doi: 10.1007/s005260100105.  Google Scholar

[20]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations,, Indiana Univ. Math. J., 49 (2000), 897.  doi: 10.1512/iumj.2000.49.1893.  Google Scholar

show all references

References:
[1]

S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation,, Adv. Diff. Eqns., 16 (2011), 289.   Google Scholar

[2]

S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations,, Nonlinear Anal., 75 (2012), 819.  doi: 10.1016/j.na.2011.09.015.  Google Scholar

[3]

S. Adachi and T. Watanabe, Asymptotic properties of ground states of quasilinear Schrödinger equations with $H^1$-subcritical exponent,, Adv. Nonlinear Stud., 12 (2012), 255.   Google Scholar

[4]

A. Ambrosetti and Z. Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\R$,, Disc. Cont. Dyn. Syst., 9 (2003), 55.  doi: 10.3934/dcds.2003.9.55.  Google Scholar

[5]

P. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems,, J. Funct. Anal., 196 (2002), 211.  doi: 10.1016/S0022-1236(02)00013-7.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar fields equations, I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[7]

J. Byeon, L. Jeanjean and M. Mariş, Symmetry and monotonicity of least energy solutions,, Calc. Var. PDE, 36 (2009), 481.  doi: 10.1007/s00526-009-0238-1.  Google Scholar

[8]

L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice,, Physica D, 159 (2001), 71.   Google Scholar

[9]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach,, Nonlinear Anal. TMA., 56 (2004), 213.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[10]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations,, Nonlinearity, 23 (2010), 1353.  doi: 10.1088/0951-7715/23/6/006.  Google Scholar

[11]

J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\RN$: mountain pass and symmetric mountain pass approaches,, Top. Methods in Nonlinear Anal., 35 (2010), 253.   Google Scholar

[12]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\RN$,, Proc. Amer. Math. Soc., 131 (2003), 2399.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[13]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, J. Phys. Soc. Japan, 50 (1981), 3262.   Google Scholar

[14]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, Arch. Rat. Mech. Anal., 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar

[15]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Diff. Eqns., (2003), 473.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[16]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Solutions for quasi-linear Schrödinger equations via the Nehari method,, Comm. PDE, 29 (2004), 879.  doi: 10.1081/PDE-120037335.  Google Scholar

[17]

W. M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problems,, Duke Math. J., 70 (1992), 247.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[18]

T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem, II,, J. Diff. Eqns., 158 (1999), 94.  doi: 10.1016/S0022-0396(99)80020-5.  Google Scholar

[19]

M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, Calc. Var. PDE, 14 (2002), 329.  doi: 10.1007/s005260100105.  Google Scholar

[20]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations,, Indiana Univ. Math. J., 49 (2000), 897.  doi: 10.1512/iumj.2000.49.1893.  Google Scholar

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