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http://dx.doi.org/10.4134/BKMS.b150923

MULTIPLE SOLUTIONS FOR A CLASS OF QUASILINEAR SCHRÖDINGER SYSTEM IN ℝN  

Chen, Caisheng (College of Science Hohai University)
Chen, Qiang (College of Science Hohai University)
Publication Information
Bulletin of the Korean Mathematical Society / v.53, no.6, 2016 , pp. 1753-1769 More about this Journal
Abstract
This paper is concerned with the quasilinear $Schr{\ddot{o}}dinger$ system $$(0.1)\;\{-{\Delta}u+a(x)u-{\Delta}(u^2)u=Fu(u,v)+h(x)\;x{\in}{\mathbb{R}}^N,\\-{\Delta}v+b(x)v-{\Delta}(v^2)v=Fv(u,v)+g(x)\;x{\in}{\mathbb{R}}^N,$$ where $N{\geq}3$. The potential functions $a(x),b(x){\in}L^{\infty}({\mathbb{R}}^N)$ are bounded in ${\mathbb{R}}^N$. By using mountain pass theorem and the Ekeland variational principle, we prove that there are at least two solutions to system (0.1).
Keywords
quasilinear $Schr{\ddot{o}}dinger$ system; mountain pass theorem; Ekeland's variational principle;
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1 C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrodinger equations with potential vanishing at infinity, J. Differential Equations 254 (2013), no. 4, 1977-1991.   DOI
2 A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.   DOI
3 H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486-490.   DOI
4 C. S. Chen, Multiple solutions for a class of quasilinear Schrodinger equations in ${\mathbb{R}}^N$, J. Math. Phys. 56 (2015), no. 7, 071507, 14 pp.
5 C. S. Chen, J. Huang, and L. Liu, Multiple solutions to the nonhomogeneous p-Kirchhoff elliptic equation with concave-convex nonlinearities, Appl. Math. Lett. 26 (2013), no. 7, 754-759.   DOI
6 M. Colin and L. Jeanjean, Solutions for a quasilinear Schrodinger equation: a dual approach, Nonlinear Anal. 56 (2004), no. 2, 213-226.   DOI
7 X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrodinger equation, J. Differential Equations 254 (2013), no. 4, 2015-2032.   DOI
8 Y. Guo and Z. Tang, Ground state solutions for quasilinear Schrodinger systems, J. Math. Anal. Appl. 389 (2012), no. 1, 322-339.   DOI
9 S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan 50 (1981), 3262-3267.   DOI
10 E.W. Laedke, K. H. Spatschek, and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys. 24 (1983), no. 12, 2764-2769.   DOI
11 J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrodinger equations I, Proc. Amer. Math. Soc. 131 (2003), no. 2, 441-448.   DOI
12 J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, Soliton solutions for quasilinear Schrodinger equation II, J. Differential Equations 187 (2003), no. 2, 473-493.   DOI
13 J. M. Bezerra do O, O. H. Miyagaki, and S. H. M. Soares, Soliton solutions for quasilinear Schrodinger equations: the critical exponential case, Nonlinear Anal. 67 (2007), no. 12, 3357-3372.   DOI
14 M. Poppenberg, K. Schmitt, and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrodinger equations, Calc. Var. Partial Differential Equations 14 (2002), no. 3, 329-344.   DOI
15 U. Severo and E. da Silva, On the existence of standing wave solutions for a class of quasilinear Schrodinger systems, J. Math. Anal. Appl. 412 (2014), no. 2, 763-775.   DOI
16 M. Struwe, Variational Methods, third ed., Springer-Verlag, New York, 2000.
17 M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.