Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents

Yi  He; Gongbao  Li

doi:10.3934/dcds.2016.36.731

Article Contents

2016, Volume 36, Issue 2: 731-762. Doi: 10.3934/dcds.2016.36.731

This issue Previous Article Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term Next Article Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs

Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents

Yi He^1, and
Gongbao Li^1,

1.
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079

Received: March 2014

Published: May 2017

Abstract Related Papers Cited by

Abstract

We study the existence, concentration and multiplicity of weak solutions to the quasilinear Schrödinger equation with critical Sobolev growth \begin{equation*} \left\{ \begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u - {\varepsilon ^2}\Delta (u^2)u = W(x){u^{q - 1}} + {u^{2\cdot{2^*} - 1}} {\text{ in }}{\mathbb{R}^N},\\ u > 0{\text{ in }}{\mathbb{R}^N},\\ \end{gathered} \right. \end{equation*} where $\varepsilon $ is a small positive parameter, $N \ge 3$, ${2^ * } = \frac{{2N}} {{N - 2}}$, $4 < q < 2 \cdot {2^ * }$, $\min V > 0$ and $\inf W > 0$. Under proper assumptions, we obtain the existence and concentration phenomena of soliton solutions of the above problem. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple soliton solutions by employing the topology of the set where the potentials $V(x)$ attains its minimum and $W(x)$ attains its maximum.

Keywords:

Mathematics Subject Classification: Primary: 35J20, 35J60, 35J92.

Citation:

Related Papers

Cited by

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]	V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.doi: 10.1007/BF01234314.
[3]	V. Benci and G. Cerami, Existence of positive solutions of the equation $ - \Delta u + a(x)u = u^{(N + 2) / (N - 2)}$ in $\mathbbR^N$, J. Funct. Anal., 88 (1990), 90-117.doi: 10.1016/0022-1236(90)90120-A.
[4]	A. Borovskii and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP, 77 (1983), 562-573.
[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]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[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]	F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves, Phys. Rep., 189 (1990), 165-223.doi: 10.1016/0370-1573(90)90093-H.
[13]	H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.doi: 10.1002/cpa.3160360405.
[14]	T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460.doi: 10.1006/jfan.1993.1133.
[15]	K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993.doi: 10.1007/978-1-4612-0385-8.
[16]	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.
[17]	S. Cingolani, N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.
[18]	D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on Single-Peaked solutions of a semilinear problem, Ann. Inst. Henri Poincaré., 15 (1998), 73-111.doi: 10.1016/S0294-1449(99)80021-3.
[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]	Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82.doi: 10.1007/s00229-011-0530-1.
[21]	Y. Ding and F. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, 30 (2007), 231-249.doi: 10.1007/s00526-007-0091-z.
[22]	W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308.doi: 10.1007/BF00282336.
[23]	Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504, 27pp.doi: 10.1063/1.4774153.
[24]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.doi: 10.1016/0022-247X(74)90025-0.
[25]	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.
[26]	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.
[27]	R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys.B , 37 (1980), 83-87.doi: 10.1007/BF01325508.
[28]	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.
[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]	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.
[34]	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.
[35]	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.
[36]	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.
[37]	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.
[38]	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.
[39]	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.
[40]	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.
[41]	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.
[42]	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.
[43]	M. del Pino and P. L. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.doi: 10.1007/BF01189950.
[44]	A. Pomponio and S. Secchi, On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results, J. Differential Equations, 207 (2004), 229-266.doi: 10.1016/j.jde.2004.06.015.
[45]	P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.doi: 10.1512/iumj.1986.35.35036.
[46]	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.
[47]	G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A., 110 (1982), 41-80.doi: 10.1016/0378-4371(82)90104-2.
[48]	P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.doi: 10.1007/BF00946631.
[49]	B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.doi: 10.1103/PhysRevE.50.R687.
[50]	S. Takeno and S. Homma, Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitation, Progr. Theoret. Physics, 65 (1981), 172-189.
[51]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244.doi: 10.1007/BF02096642.
[52]	Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations, Nonlinear Differ. Equ. Appl., 19 (2012), 19-47.doi: 10.1007/s00030-011-0116-3.