August  2017, 37(8): 4213-4230. doi: 10.3934/dcds.2017179

Positive ground state solutions for a quasilinear elliptic equation with critical exponent

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

* Corresponding author: Yinbin Deng

Received  October 2016 Revised  March 2017 Published  April 2017

In this paper, we study the following quasilinear elliptic equation with critical Sobolev exponent:
$ -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=|u|^{2^*-2}u+|u|^{p-2}u, \quad x\in {{\mathbb{R}}^{N}}, $
which models the self-channeling of a high-power ultra short laser in matter, where N ≥ 3; 2 < p < 2* = $\frac{{2N}}{{N -2}}$ and V (x) is a given positive potential. Combining the change of variables and an abstract result developed by Jeanjean in [14], we obtain the existence of positive ground state solutions for the given problem.
Citation: Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
References:
[1]

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

[2]

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.  Google Scholar

[3]

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.  Google Scholar

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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 B, 5 (1993), 3539-3550. doi: 10.1063/1.860828.  Google Scholar

[5]

H. Brezis and E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[6]

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

[7]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

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

[9]

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

[10]

Y. DengS. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262.  doi: 10.1016/j.jde.2015.09.021.  Google Scholar

[11]

Y. Deng, S. Peng and S. Yan, Solitary wave solutions to a quasilinear Schrödinger equation modeling the self-channeling of a high-power ultrashort laser in matter, submitted. Google Scholar

[12]

M. F. FurtadoL. A. Maia and E. S. Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud., 8 (2008), 353-373.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

[13]

J. P. García Azorero and Alonso I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

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L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${{\mathbb{R}}.{N}}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

[15]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on ${{\mathbb{R}}^{N}}$, Indiana Univ. Math. J., 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.  Google Scholar

[16]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.   Google Scholar

[17]

E. LaedkeK. 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

[18]

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

[19]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1987), 109-145.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[20]

J. LiuY. Wang and Z. 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

[21]

J. LiuY. Wang and Z. 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

[22]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I, I, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[23]

X. LiuJ. Liu and Z. 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.  Google Scholar

[24]

X. LiuJ. Liu and Z. 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

[25]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in ${{\mathbb{R}}.{N}}$, J. Differential Equations, 229 (2006), 570-587.  doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[26]

M. PoppenbergK. Schmitt and Z. 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

[27]

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.  Google Scholar

[28]

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

[29]

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

[30]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[31]

J. YangY. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (2013), 071502, 19pp.  doi: 10.1063/1.4811394.  Google Scholar

[32]

J. Zhang and W. Zou, A Berestycki-Lions theorem revisited, Commun. Contemp. Math., 14 (2012), 1250033, 14 pp.  doi: 10.1142/S0219199712500332.  Google Scholar

[33]

X. Zhu and D. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci., 9 (1989), 307-328.   Google Scholar

show all references

References:
[1]

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

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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 B, 5 (1993), 3539-3550. doi: 10.1063/1.860828.  Google Scholar

[5]

H. Brezis and E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[6]

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

[7]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[8]

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

[9]

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

[10]

Y. DengS. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262.  doi: 10.1016/j.jde.2015.09.021.  Google Scholar

[11]

Y. Deng, S. Peng and S. Yan, Solitary wave solutions to a quasilinear Schrödinger equation modeling the self-channeling of a high-power ultrashort laser in matter, submitted. Google Scholar

[12]

M. F. FurtadoL. A. Maia and E. S. Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud., 8 (2008), 353-373.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

[13]

J. P. García Azorero and Alonso I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

[14]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${{\mathbb{R}}.{N}}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

[15]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on ${{\mathbb{R}}^{N}}$, Indiana Univ. Math. J., 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.  Google Scholar

[16]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.   Google Scholar

[17]

E. LaedkeK. 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

[18]

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

[19]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1987), 109-145.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[20]

J. LiuY. Wang and Z. 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

[21]

J. LiuY. Wang and Z. 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

[22]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I, I, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[23]

X. LiuJ. Liu and Z. 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.  Google Scholar

[24]

X. LiuJ. Liu and Z. 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

[25]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in ${{\mathbb{R}}.{N}}$, J. Differential Equations, 229 (2006), 570-587.  doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[26]

M. PoppenbergK. Schmitt and Z. 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

[27]

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.  Google Scholar

[28]

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

[29]

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

[30]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[31]

J. YangY. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (2013), 071502, 19pp.  doi: 10.1063/1.4811394.  Google Scholar

[32]

J. Zhang and W. Zou, A Berestycki-Lions theorem revisited, Commun. Contemp. Math., 14 (2012), 1250033, 14 pp.  doi: 10.1142/S0219199712500332.  Google Scholar

[33]

X. Zhu and D. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci., 9 (1989), 307-328.   Google Scholar

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