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

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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

Abstract
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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 and 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), 289-324.
[2]	S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833. doi: 10.1016/j.na.2011.09.015.
[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), 255-279.
[4]	A. Ambrosetti and Z. Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbb{R}^{N}$, Disc. Cont. Dyn. Syst., 9 (2003), 55-68. doi: 10.3934/dcds.2003.9.55.
[5]	P. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264. doi: 10.1016/S0022-1236(02)00013-7.
[6]	H. Berestycki and P. L. Lions, Nonlinear scalar fields equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[7]	J. Byeon, L. Jeanjean and M. Mariş, Symmetry and monotonicity of least energy solutions, Calc. Var. PDE, 36 (2009), 481-492. doi: 10.1007/s00526-009-0238-1.
[8]	L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D, 159 (2001), 71-90.
[9]	M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.
[10]	M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. doi: 10.1088/0951-7715/23/6/006.
[11]	J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbb{R}^{N}$: mountain pass and symmetric mountain pass approaches, Top. Methods in Nonlinear Anal., 35 (2010), 253-276.
[12]	L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^{N}$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.
[13]	S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
[14]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rat. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[15]	J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Diff. Eqns., 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.
[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), 879-901. doi: 10.1081/PDE-120037335.
[17]	W. M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problems, Duke Math. J., 70 (1992), 247-281. doi: 10.1215/S0012-7094-93-07004-4.
[18]	T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem, II, J. Diff. Eqns., 158 (1999), 94-151. doi: 10.1016/S0022-0396(99)80020-5.
[19]	M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. PDE, 14 (2002), 329-344. doi: 10.1007/s005260100105.
[20]	J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893.

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References:

[1]	S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Diff. Eqns., 16 (2011), 289-324.
[2]	S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833. doi: 10.1016/j.na.2011.09.015.
[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), 255-279.
[4]	A. Ambrosetti and Z. Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbb{R}^{N}$, Disc. Cont. Dyn. Syst., 9 (2003), 55-68. doi: 10.3934/dcds.2003.9.55.
[5]	P. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264. doi: 10.1016/S0022-1236(02)00013-7.
[6]	H. Berestycki and P. L. Lions, Nonlinear scalar fields equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[7]	J. Byeon, L. Jeanjean and M. Mariş, Symmetry and monotonicity of least energy solutions, Calc. Var. PDE, 36 (2009), 481-492. doi: 10.1007/s00526-009-0238-1.
[8]	L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D, 159 (2001), 71-90.
[9]	M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.
[10]	M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. doi: 10.1088/0951-7715/23/6/006.
[11]	J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbb{R}^{N}$: mountain pass and symmetric mountain pass approaches, Top. Methods in Nonlinear Anal., 35 (2010), 253-276.
[12]	L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^{N}$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.
[13]	S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
[14]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rat. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[15]	J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Diff. Eqns., 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.
[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), 879-901. doi: 10.1081/PDE-120037335.
[17]	W. M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problems, Duke Math. J., 70 (1992), 247-281. doi: 10.1215/S0012-7094-93-07004-4.
[18]	T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem, II, J. Diff. Eqns., 158 (1999), 94-151. doi: 10.1016/S0022-0396(99)80020-5.
[19]	M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. PDE, 14 (2002), 329-344. doi: 10.1007/s005260100105.
[20]	J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893.

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