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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

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

[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.  Google Scholar

[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.  Google Scholar

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