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September  2020, 40(9): 5441-5470. doi: 10.3934/dcds.2020234

Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure

Universidade Federal de Goiás, 74001-970, Goiás-GO, Brazil

* Corresponding author: Edcarlos D. Silva

Received  November 2019 Revised  April 2020 Published  June 2020

Fund Project: The first author is partially supported by CNPq grant 429955/2018-9

It is established multiplicity of solutions for critical quasilinear Schrödinger equations defined in the whole space using a linking structure. The main difficulty comes from the lack of compactness of Sobolev embedding into Lebesgue spaces. Moreover, the potential is bounded from below and above by positive constants. In order to overcome these difficulties we employ Lions Concentration Compactness Principle together with some fine estimates for the energy functional restoring some kind of compactness.

Citation: Edcarlos D. Silva, Jefferson S. Silva. Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5441-5470. doi: 10.3934/dcds.2020234
References:
[1]

C. O. AlvesY. Wang and Y. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differential Equations, 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973) 349–381. doi: 10.1016/0022-1236(73)90051-7.

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${\bf{R}}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.

[4]

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Mathematics and its Applications (Soviet Series), 66, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4.

[5]

A. de BouardN. Hayashi and J.-C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.

[6]

L. Brull and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 278-288. 

[7]

X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma, Phys. Rev. Lett., 70 (1993), 2082-2085.  doi: 10.1103/PhysRevLett.70.2082.

[8]

S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.  doi: 10.12775/TMNA.1997.019.

[9]

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.

[10]

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.

[11]

Y. DengS. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147.  doi: 10.1016/j.jde.2014.09.006.

[12]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, in Differential Equations, Lecture Notes in Math., 957, Springer, Berlin-New York, 1982, 34–47. doi: 10.1007/BFb0066233.

[13]

M. F. FurtadoE. D. Silva and M. L. Silva, Quasilinear Schrödinger equations with asymptotically linear nonlinearities, Adv. Nonlinear Stud., 14 (2014), 671-686.  doi: 10.1515/ans-2014-0309.

[14]

M. F. FurtadoE. D. Silva and M. L. Silva, Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin, Z. Angew. Math. Phys., 66 (2015), 277-291.  doi: 10.1007/s00033-014-0406-9.

[15]

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.

[16]

A. M. KosevichB. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.  doi: 10.1016/0370-1573(90)90130-T.

[17]

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

[18]

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

[19]

L. D. Landau and E. M. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, Addison-Wesley Series in Advanced Physics, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958.

[20]

M. Lazzo, Existence and multiplicity results for a class of nonlinear elliptic problems in $\mathbb{R}^N$, Dicrete Contin. Dyn. Syst., (2003), 526–535. doi: 10.3934/proc.2003.2003.526.

[21]

Q. Li and X. Wu, Existence, multiplicity, and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth, J. Math. Phys., 58 (2017), 30pp. doi: 10.1063/1.4982035.

[22]

H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.  doi: 10.1016/j.na.2009.01.171.

[23]

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

[24]

A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520. 

[25]

X. LiuJ. Liu and Z.-Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669.  doi: 10.1007/s00526-012-0497-0.

[26]

X. LiuJ. Liu and Z.-Q. Wang, Ground states for quasilinear elliptic equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669.  doi: 10.1007/s00526-012-0497-0.

[27]

J. Q. 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.

[28]

J.-Q. LiuY.-Q. 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.

[29]

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

[30]

S. Liu and J. Zhou, Standing waves for quasilinear Schrödinger equations with indefinite potentials, J. Differential Equations, 265 (2018), 3970-3987.  doi: 10.1016/j.jde.2018.05.024.

[31]

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

[32]

A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc. Jpn., 42 (1977), 1824-1835.  doi: 10.1143/JPSJ.42.1824.

[33]

J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.  doi: 10.3934/cpaa.2009.8.621.

[34]

J. M. do ÓO. 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.

[35]

M. PoppenbergK. 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.

[36]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.

[37]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[38]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal., 80 (2013), 194-201.  doi: 10.1016/j.na.2012.10.005.

[39]

Y. Shen and Y. Wang, A class of generalized quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 15 (2016), 853-870.  doi: 10.3934/cpaa.2016.15.853.

[40]

E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal., 16 (1991), 455-477.  doi: 10.1016/0362-546X(91)90070-H.

[41]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.

[42]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.

[43]

E. D. Silva and J. S. Silva, Quasilinear Schrödinger equations with nonlinearities interacting with high eigenvalues, J. Math. Phys., 60 (2019), 24pp. doi: 10.1063/1.5091810.

[44]

E. D. Silva and J. S. Silva, Existence of solution for quasilinear Schrödinger equations using a linking structure, preprint.

[45]

M. A. S. Souto and S. H. M. Soares, Ground state solutions for quasilinear stationary Schrödinger equations with critical growth, Commun. Pure Appl. Anal., 12 (2013), 99-116.  doi: 10.3934/cpaa.2013.12.99.

[46]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

show all references

References:
[1]

C. O. AlvesY. Wang and Y. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differential Equations, 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973) 349–381. doi: 10.1016/0022-1236(73)90051-7.

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${\bf{R}}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.

[4]

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Mathematics and its Applications (Soviet Series), 66, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4.

[5]

A. de BouardN. Hayashi and J.-C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.

[6]

L. Brull and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 278-288. 

[7]

X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma, Phys. Rev. Lett., 70 (1993), 2082-2085.  doi: 10.1103/PhysRevLett.70.2082.

[8]

S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.  doi: 10.12775/TMNA.1997.019.

[9]

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.

[10]

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.

[11]

Y. DengS. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147.  doi: 10.1016/j.jde.2014.09.006.

[12]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, in Differential Equations, Lecture Notes in Math., 957, Springer, Berlin-New York, 1982, 34–47. doi: 10.1007/BFb0066233.

[13]

M. F. FurtadoE. D. Silva and M. L. Silva, Quasilinear Schrödinger equations with asymptotically linear nonlinearities, Adv. Nonlinear Stud., 14 (2014), 671-686.  doi: 10.1515/ans-2014-0309.

[14]

M. F. FurtadoE. D. Silva and M. L. Silva, Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin, Z. Angew. Math. Phys., 66 (2015), 277-291.  doi: 10.1007/s00033-014-0406-9.

[15]

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.

[16]

A. M. KosevichB. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.  doi: 10.1016/0370-1573(90)90130-T.

[17]

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

[18]

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

[19]

L. D. Landau and E. M. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, Addison-Wesley Series in Advanced Physics, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958.

[20]

M. Lazzo, Existence and multiplicity results for a class of nonlinear elliptic problems in $\mathbb{R}^N$, Dicrete Contin. Dyn. Syst., (2003), 526–535. doi: 10.3934/proc.2003.2003.526.

[21]

Q. Li and X. Wu, Existence, multiplicity, and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth, J. Math. Phys., 58 (2017), 30pp. doi: 10.1063/1.4982035.

[22]

H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.  doi: 10.1016/j.na.2009.01.171.

[23]

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

[24]

A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520. 

[25]

X. LiuJ. Liu and Z.-Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669.  doi: 10.1007/s00526-012-0497-0.

[26]

X. LiuJ. Liu and Z.-Q. Wang, Ground states for quasilinear elliptic equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669.  doi: 10.1007/s00526-012-0497-0.

[27]

J. Q. 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.

[28]

J.-Q. LiuY.-Q. 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.

[29]

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

[30]

S. Liu and J. Zhou, Standing waves for quasilinear Schrödinger equations with indefinite potentials, J. Differential Equations, 265 (2018), 3970-3987.  doi: 10.1016/j.jde.2018.05.024.

[31]

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

[32]

A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc. Jpn., 42 (1977), 1824-1835.  doi: 10.1143/JPSJ.42.1824.

[33]

J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.  doi: 10.3934/cpaa.2009.8.621.

[34]

J. M. do ÓO. 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.

[35]

M. PoppenbergK. 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.

[36]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.

[37]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[38]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal., 80 (2013), 194-201.  doi: 10.1016/j.na.2012.10.005.

[39]

Y. Shen and Y. Wang, A class of generalized quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 15 (2016), 853-870.  doi: 10.3934/cpaa.2016.15.853.

[40]

E. A. B. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal., 16 (1991), 455-477.  doi: 10.1016/0362-546X(91)90070-H.

[41]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.

[42]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.

[43]

E. D. Silva and J. S. Silva, Quasilinear Schrödinger equations with nonlinearities interacting with high eigenvalues, J. Math. Phys., 60 (2019), 24pp. doi: 10.1063/1.5091810.

[44]

E. D. Silva and J. S. Silva, Existence of solution for quasilinear Schrödinger equations using a linking structure, preprint.

[45]

M. A. S. Souto and S. H. M. Soares, Ground state solutions for quasilinear stationary Schrödinger equations with critical growth, Commun. Pure Appl. Anal., 12 (2013), 99-116.  doi: 10.3934/cpaa.2013.12.99.

[46]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

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