# American Institute of Mathematical Sciences

January  2013, 12(1): 99-116. doi: 10.3934/cpaa.2013.12.99

## Ground state solutions for quasilinear stationary Schrödinger equations with critical growth

 1 Universidade Federal de Campina Grande, Departamento de Matemática e Estatística, 58429-000 Campina Grande, PB, Brazil 2 Departmento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 13560-970, São Carlos, SP

Received  August 2010 Revised  July 2012 Published  September 2012

We establish the existence of ground state solution for quasilinear Schrödinger equations involving critical growth. The method used here is minimizing the gradient integral norm in a manifold defined by integrals involving the primitive of the nonlinearity function.
Citation: Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99
##### References:
 [1] S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation,, Adv. Differential Equations, 16 (2011), 289. Google Scholar [2] C. O. Alves, M. S. Montenegro and M. A. S. Souto, Existence of a ground state solution for a nonlinear scalar field equation with critical growth,, Calc. Var. Partial Differential Equations, 43 (2012), 537. doi: 10.1007/s00526-011-0422-y. Google Scholar [3] H. Berestycki and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan,, C. R. Acad. Sci. Paris Ser. I Math, 297 (1984), 307. Google Scholar [4] H.Berestycki and P. L. Lions, Nonlinear scalar field equations, I - existence of a ground state,, Arch. Rat. Mech. Analysis, 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [5] L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice,, Physica D: Nonlinear Phenomena, 159 (2001), 71. doi: 10.1016/S0167-2789(01)00332-3. Google Scholar [6] D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent $\mathbbR^2$,, Comm. Part. Diff. Equations, 17 (1992), 407. doi: 10.1080/03605309208820848. Google Scholar [7] S. Coleman, V. Glazer and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations,, Comm. Math. Phys, 58 (1978), 211. doi: 10.1007/BF01609421. Google Scholar [8] 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 [9] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar [10] J. M. do Ó, U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two,, Calc. Var. Partial differential Equations, 38 (2010), 275. doi: 10.1007/s00526-009-0286-6. Google Scholar [11] J. M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case,, Nonlinear Anal., 67 (2007), 3357. doi: 10.1016/j.na.2006.10.018. Google Scholar [12] L. Jeanjean, L. and K. Tanaka, A Remark on least energy solutions in $\mathbbR^N$,, Proc. Amer. Mathematical Society, 131 (2003), 2399. doi: 10.1090/S0002-9939-02-06821-1. Google Scholar [13] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I,, Rev. Mat. Iberoamericana, 1 (1985), 145. doi: 10.4171/RMI/6. Google Scholar [14] J. Liu, Y. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar [15] J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I,, Proc. Amer. Math. Soc., 131 (2003), 441. Google Scholar [16] J. Liu, Y. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, Comm. Partial Differential Equations, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar [17] J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077. doi: 10.1512/iumj.1971.20.20101. Google Scholar [18] M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, Calc. Var. Partial differential Equations, 14 (2002), 329. doi: 10.1007/s005260100105. Google Scholar [19] E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, Calc. Var. Partial Differential Equations, 39 (2010), 1. doi: 10.1007/s00526-009-0299-1. Google Scholar [20] N. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar

show all references

##### References:
 [1] S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation,, Adv. Differential Equations, 16 (2011), 289. Google Scholar [2] C. O. Alves, M. S. Montenegro and M. A. S. Souto, Existence of a ground state solution for a nonlinear scalar field equation with critical growth,, Calc. Var. Partial Differential Equations, 43 (2012), 537. doi: 10.1007/s00526-011-0422-y. Google Scholar [3] H. Berestycki and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan,, C. R. Acad. Sci. Paris Ser. I Math, 297 (1984), 307. Google Scholar [4] H.Berestycki and P. L. Lions, Nonlinear scalar field equations, I - existence of a ground state,, Arch. Rat. Mech. Analysis, 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [5] L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice,, Physica D: Nonlinear Phenomena, 159 (2001), 71. doi: 10.1016/S0167-2789(01)00332-3. Google Scholar [6] D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent $\mathbbR^2$,, Comm. Part. Diff. Equations, 17 (1992), 407. doi: 10.1080/03605309208820848. Google Scholar [7] S. Coleman, V. Glazer and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations,, Comm. Math. Phys, 58 (1978), 211. doi: 10.1007/BF01609421. Google Scholar [8] 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 [9] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar [10] J. M. do Ó, U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two,, Calc. Var. Partial differential Equations, 38 (2010), 275. doi: 10.1007/s00526-009-0286-6. Google Scholar [11] J. M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case,, Nonlinear Anal., 67 (2007), 3357. doi: 10.1016/j.na.2006.10.018. Google Scholar [12] L. Jeanjean, L. and K. Tanaka, A Remark on least energy solutions in $\mathbbR^N$,, Proc. Amer. Mathematical Society, 131 (2003), 2399. doi: 10.1090/S0002-9939-02-06821-1. Google Scholar [13] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I,, Rev. Mat. Iberoamericana, 1 (1985), 145. doi: 10.4171/RMI/6. Google Scholar [14] J. Liu, Y. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar [15] J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I,, Proc. Amer. Math. Soc., 131 (2003), 441. Google Scholar [16] J. Liu, Y. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, Comm. Partial Differential Equations, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar [17] J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077. doi: 10.1512/iumj.1971.20.20101. Google Scholar [18] M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, Calc. Var. Partial differential Equations, 14 (2002), 329. doi: 10.1007/s005260100105. Google Scholar [19] E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, Calc. Var. Partial Differential Equations, 39 (2010), 1. doi: 10.1007/s00526-009-0299-1. Google Scholar [20] N. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar
 [1] Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 [2] Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345 [3] Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745 [4] Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345 [5] Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003 [6] Scipio Cuccagna, Masaya Maeda. On weak interaction between a ground state and a trapping potential. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3343-3376. doi: 10.3934/dcds.2015.35.3343 [7] Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048 [8] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813 [9] Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195 [10] Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943 [11] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71 [12] 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 [13] Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991 [14] Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure & Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030 [15] Dengfeng Lü. Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1781-1795. doi: 10.3934/cpaa.2016014 [16] Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417 [17] Jian Zhang, Wen Zhang. Existence and decay property of ground state solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110 [18] Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120 [19] Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131 [20] Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139

2018 Impact Factor: 0.925