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Uniqueness of the ground state solutions of quasilinear Schrödinger equations. (English) Zbl 1232.35067
Summary: We are concerned with the uniqueness result of positive solutions for a class of quasilinear elliptic equations arising from plasma physics. We convert a quasilinear elliptic equation into a semilinear one and show the unique existence of a positive radial solution for the original equation under the suitable conditions on the power of nonlinearity and quasilinearity. We also investigate the non-degeneracy of a positive radial solution for a converted semilinear elliptic equation.
35J62Quasilinear elliptic equations
35A02Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness
82D10Plasmas (statistical mechanics)
35J61Semilinear elliptic equations
35Q41Time-dependent Schrödinger equations, Dirac equations
[1]Brizhik, L.; Eremko, A.; Piette, B.; Zakrzewski, W. J.: Electron self-trapping in a discrete two-dimensional lattice, Physica D 159, 71-90 (2001) · Zbl 0983.81533 · doi:10.1016/S0167-2789(01)00332-3
[2]Kurihara, S.: Large-amplitude quasi-solitons in superfluid films, J. phys. Soc. Japan 50, 3262-3267 (1981)
[3]Adachi, S.; Watanabe, T.: G-invariant positive solutions for a quasilinear Schrödinger equation, Adv. diff. Eqns. 16, 289-324 (2011) · Zbl 1223.35162
[4]Colin, M.; Jeanjean, L.: Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear anal. TMA. 56, 213-226 (2004) · Zbl 1035.35038 · doi:10.1016/j.na.2003.09.008
[5]Liu, J. -Q.; Wang, Z. -Q.: Soliton solutions for quasilinear Schrödinger equations, Proc. amer. Math. soc. 131, 441-448 (2003) · Zbl 1229.35269 · doi:10.1090/S0002-9939-02-06783-7
[6]Liu, J. -Q.; Wang, Y. -Q.; Wang, Z. -Q.: Soliton solutions for quasilinear Schrödinger equations, II, J. diff. Eqns. 187, 473-493 (2003) · Zbl 1229.35268 · doi:10.1016/S0022-0396(02)00064-5
[7]Poppenberg, M.; Schmitt, K.; Wang, Z. -Q.: On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. var. PDE 14, 329-344 (2002) · Zbl 1052.35060 · doi:10.1007/s005260100105
[8]Liu, J. -Q.; Wang, Y. -Q.; Wang, Z. -Q.: Solutions for quasi-linear Schrödinger equations via the Nehari method, Comm. PDE 29, 879-901 (2004) · Zbl 1140.35399 · doi:10.1081/PDE-120037335
[9]Colin, M.; Jeanjean, L.; Squassina, M.: Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity 23, 1353-1385 (2010) · Zbl 1192.35163 · doi:10.1088/0951-7715/23/6/006
[10]Cortázar, C.; Elgueta, M.; Felmer, P.: Uniqueness of positive solutions of Δu+f(u)=0 in RN, N3, Arch. rat. Mech. anal. 142, 127-141 (1998) · Zbl 0912.35059 · doi:10.1007/s002050050086
[11]Kwong, M.; Zhang, L.: Uniqueness of the positive solution of Δu+f(u)=0 in an annulus, Diff. int. Eqns. 4, 583-599 (1991) · Zbl 0724.34023
[12]Mcleod, K.; Serrin, J.: Uniqueness of positive radial solutions of Δu+f(u)=0 in RN, Arch. rat. Mech. anal. 99, 115-145 (1987) · Zbl 0667.35023 · doi:10.1007/BF00275874
[13]Serrin, J.; Tang, M.: Uniqueness of ground states for quasilinear elliptic equations, Indiana univ. Math. J. 49, 897-923 (2000) · Zbl 0979.35049 · doi:10.1512/iumj.2000.49.1893
[14]Berestycki, H.; Gallouët, T.; Kavian, O.: Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. sci. Paris sér. I math. 297, 307-310 (1983) · Zbl 0544.35042
[15]Berestycki, H.; Lions, P. L.: Nonlinear scalar fields equations, I. Existence of a ground state, Arch. rational mech. Anal. 82, 313-345 (1983) · Zbl 0533.35029
[16]Hirata, J.; Ikoma, N.; Tanaka, K.: Nonlinear scalar field equations in RN: mountain pass and symmetric mountain pass approaches, Top. methods in nonlinear anal. 35, 253-276 (2010) · Zbl 1203.35106
[17]Byeon, J.; Jeanjean, L.; Maris, M.: Symmetry and monotonicity of least energy solutions, Calc. var. PDE 36, 481-492 (2009) · Zbl 1226.35041 · doi:10.1007/s00526-009-0238-1
[18]Gidas, B.; Ni, W. M.; Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in RN, Adv. math. Studies A 7, 369-402 (1981) · Zbl 0469.35052
[19]Bates, P.; Shi, J.: Existence and instability of spike layer solutions to singular perturbation problems, J. funct. Anal. 196, 429-482 (2002) · Zbl 1010.47036 · doi:10.1016/S0022-1236(02)00013-7