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December  2016, 6(4): 551-593. doi: 10.3934/mcrf.2016016

Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity

1. 

School of Mathematics and Statistics, South-Central University For Nationalities, Wuhan, 430074

2. 

Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079

Received  January 2016 Revised  March 2016 Published  October 2016

We are concerned with a class of singularly perturbed quasilinear Schrödinger equations of the following form: \[ - {\varepsilon ^2}\Delta u - {\varepsilon ^2}\Delta ({u^2})u + V(x)u = h(u),{\text{ }}u > 0{\text{ in }}{\mathbb{R}^N}, \] where $\varepsilon $ is a small positive parameter, $N \ge 3$ and the nonlinearity $h$ is of critical growth. We construct a family of positive solutions ${u_\varepsilon } \in {H^1}({\mathbb{R}^N})$ of the above problem which concentrates around local minima of $V$ as $\varepsilon \to 0$ under certain assumptions on $h$. Our result especially solves the above problem in the case where $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(2 < q \le 4,{\text{ }}\lambda > 0)$ and completes the study made in some recent works in the sense that, in those papers only the case where $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(4 < q < 2 \cdot {2^ * },{\text{ }}\lambda > 0)$ was considered. Moreover, our main results extend also the arguments used in Byeon and Jeanjean [14], which deal with Schrödinger equations with subcritical nonlinearities, to the quasilinear Schrödinger equations with critical nonlinearities.
Citation: Yi He, Gongbao Li. Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity. Mathematical Control and Related Fields, 2016, 6 (4) : 551-593. doi: 10.3934/mcrf.2016016
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show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

J. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984.

[3]

F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves, Phys. Rep., 189 (1990), 165-223. doi: 10.1016/0370-1573(90)90093-H.

[4]

V. Benci and G. Cerami, Existence of positive solutions of the equation $ - \Delta u + a(x)u = u^{\frac {N + 2} {N - 2}}$ in $\mathbbR^N$, J. Funct. Anal., 88 (1990), 90-117. doi: 10.1016/0022-1236(90)90120-A.

[5]

H. Berestycki, T. Gallouët and O. Kavian, Equations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math., 297 (1983), 307-310.

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[7]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.

[8]

J. M. Bezerra do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. doi: 10.1016/j.jde.2009.11.030.

[9]

A. Borovskii and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP, 77 (1983), 562-573.

[10]

A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schröndinger equation, Commun. Math. Phys., 189 (1997), 73-105. doi: 10.1007/s002200050191.

[11]

H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids, B5 (1993), 3539-3550.

[12]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.

[13]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[14]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal., 185 (2007), 185-200. doi: 10.1007/s00205-006-0019-3.

[15]

J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differential Equations, 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8.

[16]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.

[17]

S. Cingolani and N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.

[18]

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.

[19]

X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett., 70 (1993), 2082-2085.

[20]

G. M. Figueiredo, N. Ikoma and J. R. Santos Junior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8.

[21]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.

[22]

C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Commun. Partial Differential Equations, 21 (1996), 787-820. doi: 10.1080/03605309608821208.

[23]

J. Garcia Azorero and I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a non-symmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895. doi: 10.1090/S0002-9947-1991-1083144-2.

[24]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[25]

J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbbR^N$: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276.

[26]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys., 37 (1980), 83-87. doi: 10.1007/BF01325508.

[27]

Y. He and G. Li, The existence and concentration of weak solutions to a class of $p$-Laplacian type problems in unbounded domains, Sci. China Math., 57 (2014), 1927-1952. doi: 10.1007/s11425-014-4830-2.

[28]

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

[29]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbbR^N$, Indiana Univ. Math. J., 54 (2005), 443-464. doi: 10.1512/iumj.2005.54.2502.

[30]

S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn., 50 (1981), 3262-3267. doi: 10.1143/JPSJ.50.3262.

[31]

A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons in superfluid films, Phys. Rep., 194 (1990), 117-238.

[32]

G. Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27-36. doi: 10.5186/aasfm.1990.1521.

[33]

G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on $\mathbbR^N$, Commun. Partial Differential Equations, 14 (1989), 1291-1314. doi: 10.1080/03605308908820654.

[34]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 1 (1984), 223-283.

[35]

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

[36]

X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263. doi: 10.1090/S0002-9939-2012-11293-6.

[37]

X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124. doi: 10.1016/j.jde.2012.09.006.

[38]

E. Laedke and K. Spatschek, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769. doi: 10.1063/1.525675.

[39]

J. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7.

[40]

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

[41]

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.

[42]

J. M. do Ó and 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.

[43]

V. G. Makhankov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.

[44]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical growth in $\mathbbR^N$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.

[45]

O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbbR^N$ with critical growth, Nonlinear Anal., 29 (1997), 773-781. doi: 10.1016/S0362-546X(96)00087-9.

[46]

E. S. Noussair, C. A. Swanson and J. F. Yang, Quasilinear elliptic problems with critical exponents, Nonlinear Anal., 20 (1993), 285-301. doi: 10.1016/0362-546X(93)90164-N.

[47]

W. M. Ni and I. Takagi, On the shape of least energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. doi: 10.1002/cpa.3160440705.

[48]

W. M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.

[49]

W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768. doi: 10.1002/cpa.3160480704.

[50]

Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class $( V )_a$, Commun. Partial Differential Equations, 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.

[51]

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