# American Institue of Mathematical Sciences

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

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, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. [2] H. Berestycki, P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. [3] João M. Bezerra do ó, Olímpio. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. [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. [5] H. Brezis, E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490. [6] X. L. Chen, 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. [7] M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226. [8] A. De Bouard, N. Hayashi, J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105. [9] Y. Deng, S. Peng, S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147. [10] Y. Deng, S. Peng, S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262. [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. [12] M. F. Furtado, L. A. Maia, E. S. Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud., 8 (2008), 353-373. [13] J. P. García Azorero, Alonso I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. [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. [15] L. Jeanjean, K. Tanaka, A positive solution for a nonlinear Schrödinger equation on ${{\mathbb{R}}^{N}}$, Indiana Univ. Math. J., 54 (2005), 443-464. [16] S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. [17] E. Laedke, K. Spatschek, L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769. [18] H. F. Lins, E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905. [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. [20] J. Liu, Y. Wang, Z. Wang, Soliton solutions for quasilinear Schrödinger equations. Ⅱ, J. Differential Equations, 187 (2003), 473-493. [21] J. Liu, Y. Wang, Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. [22] J. Liu, Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I, I, Proc. Amer. Math. Soc., 131 (2003), 441-448. [23] X. Liu, J. Liu, Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124. [24] X. Liu, J. Liu, Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669. [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. [26] M. Poppenberg, K. Schmitt, Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344. [27] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. [28] Y. Shen, Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201. [29] E. A. B. Silva, G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. [30] M. Willem, Minimax Theorems, Birkh¨auser, Boston, (1996). [31] J. Yang, Y. Wang, A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (), 071502. [32] J. Zhang, W. Zou, A Berestycki-Lions theorem revisited, Commun. Contemp. Math., 14 (), 1250033-14 pp. [33] X. Zhu, D. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci., 9 (), 307-328.

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##### References:
 [1] A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. [2] H. Berestycki, P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. [3] João M. Bezerra do ó, Olímpio. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. [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. [5] H. Brezis, E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490. [6] X. L. Chen, 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. [7] M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226. [8] A. De Bouard, N. Hayashi, J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105. [9] Y. Deng, S. Peng, S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147. [10] Y. Deng, S. Peng, S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262. [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. [12] M. F. Furtado, L. A. Maia, E. S. Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud., 8 (2008), 353-373. [13] J. P. García Azorero, Alonso I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. [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. [15] L. Jeanjean, K. Tanaka, A positive solution for a nonlinear Schrödinger equation on ${{\mathbb{R}}^{N}}$, Indiana Univ. Math. J., 54 (2005), 443-464. [16] S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. [17] E. Laedke, K. Spatschek, L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769. [18] H. F. Lins, E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905. [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. [20] J. Liu, Y. Wang, Z. Wang, Soliton solutions for quasilinear Schrödinger equations. Ⅱ, J. Differential Equations, 187 (2003), 473-493. [21] J. Liu, Y. Wang, Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. [22] J. Liu, Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I, I, Proc. Amer. Math. Soc., 131 (2003), 441-448. [23] X. Liu, J. Liu, Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124. [24] X. Liu, J. Liu, Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669. [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. [26] M. Poppenberg, K. Schmitt, Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344. [27] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. [28] Y. Shen, Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201. [29] E. A. B. Silva, G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. [30] M. Willem, Minimax Theorems, Birkh¨auser, Boston, (1996). [31] J. Yang, Y. Wang, A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (), 071502. [32] J. Zhang, W. Zou, A Berestycki-Lions theorem revisited, Commun. Contemp. Math., 14 (), 1250033-14 pp. [33] X. Zhu, D. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci., 9 (), 307-328.
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