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February  2016, 36(2): 731-762. doi: 10.3934/dcds.2016.36.731

## Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents

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

Received  March 2014 Published  August 2015

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.
Citation: Yi He, Gongbao Li. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 731-762. doi: 10.3934/dcds.2016.36.731
##### References:
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Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schröndinger equation,, Commun. Math. Phys., 189 (1997), 73. doi: 10.1007/s002200050191. Google Scholar [7] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.1090/S0002-9939-1983-0699419-3. Google Scholar [8] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [9] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250556. Google Scholar [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. doi: 10.1016/j.jde.2009.11.030. Google Scholar [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. Google Scholar [12] F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves,, Phys. Rep., 189 (1990), 165. doi: 10.1016/0370-1573(90)90093-H. Google Scholar [13] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar [14] T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation,, J. Funct. Anal., 117 (1993), 447. doi: 10.1006/jfan.1993.1133. Google Scholar [15] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems,, Birkhäuser, (1993). doi: 10.1007/978-1-4612-0385-8. Google Scholar [16] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar [17] S. Cingolani, N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,, Topol. Methods Nonlinear Anal., 10 (1997), 1. Google Scholar [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. doi: 10.1016/S0294-1449(99)80021-3. Google Scholar [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. Google Scholar [20] Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities,, Manuscripta Math., 140 (2013), 51. doi: 10.1007/s00229-011-0530-1. Google Scholar [21] Y. Ding and F. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity,, Calc. Var. Partial Differential Equations, 30 (2007), 231. doi: 10.1007/s00526-007-0091-z. Google Scholar [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. doi: 10.1007/BF00282336. Google Scholar [23] Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4774153. Google Scholar [24] I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324. doi: 10.1016/0022-247X(74)90025-0. Google Scholar [25] A. Floer and A.Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar [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. doi: 10.1090/S0002-9947-1991-1083144-2. Google Scholar [27] R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, Z. Phys.B , 37 (1980), 83. doi: 10.1007/BF01325508. Google Scholar [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. doi: 10.1007/s11425-014-4830-2. Google Scholar [29] L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbbR^N$,, Indiana Univ. Math. J., 54 (2005), 443. doi: 10.1512/iumj.2005.54.2502. Google Scholar [30] S. Kurihura, Large-amplitude quasi-solitons in superfluids films,, J. Phys. Soc. Jpn., 50 (1981), 3262. doi: 10.1143/JPSJ.50.3262. Google Scholar [31] A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons in superfluid films,, Phys. Rep., 194 (1990), 117. Google Scholar [32] G. Li, Some properties of weak solutions of nonlinear scalar field equations,, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27. doi: 10.5186/aasfm.1990.1521. Google Scholar [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. Google Scholar [34] X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method,, J. Differential Equations, 254 (2013), 102. doi: 10.1016/j.jde.2012.09.006. Google Scholar [35] E. Laedke and K. Spatschek, Evolution theorem for a class of perturbed envelope soliton solutions,, J. Math. Phys., 24 (1983), 2764. doi: 10.1063/1.525675. Google Scholar [36] J. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I,, Proc. Amer. Math. Soc., 131 (2003), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar [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. doi: 10.1081/PDE-120037335. Google Scholar [38] J. Liu, Y. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar [39] O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbbR^N$ with critical growth,, Nonlinear Anal., 29 (1997), 773. doi: 10.1016/S0362-546X(96)00087-9. Google Scholar [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. doi: 10.1007/s00526-009-0286-6. Google Scholar [41] V. G. Makhankov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory,, Phys. Rep., 104 (1984), 1. doi: 10.1016/0370-1573(84)90106-6. Google Scholar [42] E. S. Noussair, C. A. Swanson and J. F. Yang, Quasilinear elliptic problems with critical exponents,, Nonlinear Anal., 20 (1993), 285. doi: 10.1016/0362-546X(93)90164-N. Google Scholar [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. doi: 10.1007/BF01189950. Google Scholar [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. doi: 10.1016/j.jde.2004.06.015. Google Scholar [45] P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036. Google Scholar [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. doi: 10.1007/s005260100105. Google Scholar [47] G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain,, Phys. A., 110 (1982), 41. doi: 10.1016/0378-4371(82)90104-2. Google Scholar [48] P. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [49] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, Phys. Rev. E, 50 (1994), 687. doi: 10.1103/PhysRevE.50.R687. Google Scholar [50] S. Takeno and S. Homma, Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitation,, Progr. Theoret. Physics, 65 (1981), 172. Google Scholar [51] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Commun. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar [52] Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations,, Nonlinear Differ. Equ. Appl., 19 (2012), 19. doi: 10.1007/s00030-011-0116-3. Google Scholar

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. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [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. doi: 10.1007/BF01234314. Google Scholar [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. doi: 10.1016/0022-1236(90)90120-A. Google Scholar [4] A. Borovskii and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter,, JETP, 77 (1983), 562. Google Scholar [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. Google Scholar [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. doi: 10.1007/s002200050191. Google Scholar [7] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.1090/S0002-9939-1983-0699419-3. Google Scholar [8] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [9] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250556. Google Scholar [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. doi: 10.1016/j.jde.2009.11.030. Google Scholar [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. Google Scholar [12] F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves,, Phys. Rep., 189 (1990), 165. doi: 10.1016/0370-1573(90)90093-H. Google Scholar [13] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar [14] T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation,, J. Funct. Anal., 117 (1993), 447. doi: 10.1006/jfan.1993.1133. Google Scholar [15] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems,, Birkhäuser, (1993). doi: 10.1007/978-1-4612-0385-8. Google Scholar [16] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar [17] S. Cingolani, N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,, Topol. Methods Nonlinear Anal., 10 (1997), 1. Google Scholar [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. doi: 10.1016/S0294-1449(99)80021-3. Google Scholar [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. Google Scholar [20] Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities,, Manuscripta Math., 140 (2013), 51. doi: 10.1007/s00229-011-0530-1. Google Scholar [21] Y. Ding and F. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity,, Calc. Var. Partial Differential Equations, 30 (2007), 231. doi: 10.1007/s00526-007-0091-z. Google Scholar [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. doi: 10.1007/BF00282336. Google Scholar [23] Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4774153. Google Scholar [24] I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324. doi: 10.1016/0022-247X(74)90025-0. Google Scholar [25] A. Floer and A.Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar [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. doi: 10.1090/S0002-9947-1991-1083144-2. Google Scholar [27] R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, Z. Phys.B , 37 (1980), 83. doi: 10.1007/BF01325508. Google Scholar [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. doi: 10.1007/s11425-014-4830-2. Google Scholar [29] L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbbR^N$,, Indiana Univ. Math. J., 54 (2005), 443. doi: 10.1512/iumj.2005.54.2502. Google Scholar [30] S. Kurihura, Large-amplitude quasi-solitons in superfluids films,, J. Phys. Soc. Jpn., 50 (1981), 3262. doi: 10.1143/JPSJ.50.3262. Google Scholar [31] A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons in superfluid films,, Phys. Rep., 194 (1990), 117. Google Scholar [32] G. Li, Some properties of weak solutions of nonlinear scalar field equations,, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27. doi: 10.5186/aasfm.1990.1521. Google Scholar [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. Google Scholar [34] X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method,, J. Differential Equations, 254 (2013), 102. doi: 10.1016/j.jde.2012.09.006. Google Scholar [35] E. Laedke and K. Spatschek, Evolution theorem for a class of perturbed envelope soliton solutions,, J. Math. Phys., 24 (1983), 2764. doi: 10.1063/1.525675. Google Scholar [36] J. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I,, Proc. Amer. Math. Soc., 131 (2003), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar [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. doi: 10.1081/PDE-120037335. Google Scholar [38] J. Liu, Y. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar [39] O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbbR^N$ with critical growth,, Nonlinear Anal., 29 (1997), 773. doi: 10.1016/S0362-546X(96)00087-9. Google Scholar [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. doi: 10.1007/s00526-009-0286-6. Google Scholar [41] V. G. Makhankov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory,, Phys. Rep., 104 (1984), 1. doi: 10.1016/0370-1573(84)90106-6. Google Scholar [42] E. S. Noussair, C. A. Swanson and J. F. Yang, Quasilinear elliptic problems with critical exponents,, Nonlinear Anal., 20 (1993), 285. doi: 10.1016/0362-546X(93)90164-N. Google Scholar [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. doi: 10.1007/BF01189950. Google Scholar [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. doi: 10.1016/j.jde.2004.06.015. Google Scholar [45] P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036. Google Scholar [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. doi: 10.1007/s005260100105. Google Scholar [47] G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain,, Phys. A., 110 (1982), 41. doi: 10.1016/0378-4371(82)90104-2. Google Scholar [48] P. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [49] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, Phys. Rev. E, 50 (1994), 687. doi: 10.1103/PhysRevE.50.R687. Google Scholar [50] S. Takeno and S. Homma, Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitation,, Progr. Theoret. Physics, 65 (1981), 172. Google Scholar [51] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Commun. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar [52] Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations,, Nonlinear Differ. Equ. Appl., 19 (2012), 19. doi: 10.1007/s00030-011-0116-3. Google Scholar
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