• 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
  • DCDS Home
  • This Issue
  • Next Article
    Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs
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:
[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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves,, Phys. Rep., 189 (1990), 165. 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. 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. doi: 10.1006/jfan.1993.1133.

[15]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems,, Birkhäuser, (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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324. 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. 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. 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. 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. 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. doi: 10.1512/iumj.2005.54.2502.

[30]

S. Kurihura, Large-amplitude quasi-solitons in superfluids films,, J. Phys. Soc. Jpn., 50 (1981), 3262. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[45]

P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. 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. 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. 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. doi: 10.1007/BF00946631.

[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.

[50]

S. Takeno and S. Homma, Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitation,, Progr. Theoret. Physics, 65 (1981), 172.

[51]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Commun. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

F. Bass and N. N. Nasanov, Nonlinear electromagnetic spin waves,, Phys. Rep., 189 (1990), 165. 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. 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. doi: 10.1006/jfan.1993.1133.

[15]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems,, Birkhäuser, (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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324. 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. 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. 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. 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. 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. doi: 10.1512/iumj.2005.54.2502.

[30]

S. Kurihura, Large-amplitude quasi-solitons in superfluids films,, J. Phys. Soc. Jpn., 50 (1981), 3262. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[45]

P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. 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. 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. 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. doi: 10.1007/BF00946631.

[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.

[50]

S. Takeno and S. Homma, Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitation,, Progr. Theoret. Physics, 65 (1981), 172.

[51]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Commun. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642.

[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.

[1]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

[2]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107

[3]

Daniele Cassani, João Marcos do Ó, Abbas Moameni. Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations. Communications on Pure & Applied Analysis, 2010, 9 (2) : 281-306. doi: 10.3934/cpaa.2010.9.281

[4]

Minbo Yang, Yanheng Ding. Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 429-449. doi: 10.3934/cpaa.2013.12.429

[5]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99

[6]

Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099

[7]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[8]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004

[9]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[10]

Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273

[11]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054

[12]

Marcos L. M. Carvalho, José Valdo A. Goncalves, Claudiney Goulart, Olímpio H. Miyagaki. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 83-106. doi: 10.3934/cpaa.2019006

[13]

César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535

[14]

Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025

[15]

Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417

[16]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487

[17]

Teresa D'Aprile. Some existence and concentration results for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2002, 1 (4) : 457-474. doi: 10.3934/cpaa.2002.1.457

[18]

Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254

[19]

Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393

[20]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]