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

Previous Article
Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems
CPAA Home
This Issue
Next Article
The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction

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
Related Papers

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

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

[1]	Yinbin Deng, Qi Gao. Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 367-380. doi: 10.3934/dcds.2009.24.367
[2]	Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251
[3]	Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179
[4]	Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483
[5]	Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391
[6]	Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027
[7]	Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027
[8]	Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[9]	Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[10]	Long Wei. Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 925-946. doi: 10.3934/cpaa.2008.7.925
[11]	Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053
[12]	Haitao Yang. On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 187-198. doi: 10.3934/cpaa.2005.4.197
[13]	Fang-Fang Liao, Chun-Lei Tang. Four positive solutions of a quasilinear elliptic equation in $ R^N$. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2577-2600. doi: 10.3934/cpaa.2013.12.2577
[14]	Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[15]	Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361
[16]	Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[17]	Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735
[18]	Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861
[19]	Yan Zhang. Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5183-5199. doi: 10.3934/dcds.2016025
[20]	Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47

2017 Impact Factor: 0.884

American Institute of Mathematical Sciences