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Ground state solutions for quasilinear stationary Schrödinger equations with critical growth

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  • 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.
    Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35J10.


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  • [1]

    S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differential Equations, 16 (2011), 289-324.


    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), 537-554.doi: 10.1007/s00526-011-0422-y.


    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), 307-310.


    H.Berestycki and P. L. Lions, Nonlinear scalar field equations, I - existence of a ground state, Arch. Rat. Mech. Analysis, 82 (1983), 313-346.doi: 10.1007/BF00250555.


    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), 71-90.doi: 10.1016/S0167-2789(01)00332-3.


    D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent $\mathbbR^2$, Comm. Part. Diff. Equations, 17 (1992), 407-435.doi: 10.1080/03605309208820848.


    S. Coleman, V. Glazer and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys, 58 (1978), 211-221.doi: 10.1007/BF01609421.


    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.


    M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.doi: 10.1016/j.na.2003.09.008.


    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), 275-315.doi: 10.1007/s00526-009-0286-6.


    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), 3357-3372.doi: 10.1016/j.na.2006.10.018.


    L. Jeanjean, L. and K. Tanaka, A Remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Mathematical Society, 131 (2003), 2399-2408.doi: 10.1090/S0002-9939-02-06821-1.


    P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201.doi: 10.4171/RMI/6.


    J. Liu, Y. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473-493.doi: 10.1016/S0022-0396(02)00064-5.


    J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448.


    J. Liu, Y. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.doi: 10.1081/PDE-120037335.


    J. MoserA sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092. doi: 10.1512/iumj.1971.20.20101.


    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), 329-344.doi: 10.1007/s005260100105.


    E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.doi: 10.1007/s00526-009-0299-1.


    N. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.

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