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September  2015, 14(5): 1803-1816. doi: 10.3934/cpaa.2015.14.1803

Positive solution for quasilinear Schrödinger equations with a parameter

1. 

Business School of Hunan University, Changsha, Hunan 410082, China

Received  September 2014 Revised  February 2015 Published  June 2015

In this paper, we study the following quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2)^{(2-\alpha)/2}}=\mathrm{g}(x,u), \end{eqnarray} where $1 \le \alpha \le 2$, $N \ge 3$, $V\in C(R^N, R)$ and $\mathrm{g}\in C(R^N\times R, R)$. By using a change of variables, we get new equations, whose respective associated functionals are well defined in $H^1(R^N)$ and satisfy the geometric hypotheses of the mountain pass theorem. Using the special techniques, the existence of positive solutions is studied.
Citation: GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803
References:
[1]

J. M. Bezerra do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018.

[2]

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.

[3]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Analysis: Theorey, Methods $&$ Applications, 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.

[4]

Y. Cheng and J. Yang, Positive solution to a class of relativistic nonlinear Schrödinger equation, J. Math. Anal. Appl., 411 (2014), 665-674. doi: 10.1016/j.jmaa.2013.10.006.

[5]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, Journal of the physical Society of Japan, 50 (1981), 3262-3267.

[6]

P. L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact cases, part I and part II, Ann. Inst. H. Poincaré Anal. Non Linëaire, 1 (1984), 109-145, 223-283.

[7]

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

[8]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, Journal of Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.

[9]

E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, Journal of Mathematical Physics, 24 (1983), 2764-2769. doi: 10.1063/1.525675.

[10]

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

[11]

J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-892. doi: 10.1081/PDE-120037335.

[12]

A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc., 42 (1977), 1823-1835.

[13]

J. M. do Ó and U. Secero, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Cale. Var. Partial Differential Equations, 38 (2010), 275-315. doi: 10.1007/s00526-009-0286-6.

[14]

M. Porkolab and M. V. Goldman, Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.

[15]

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), 329-344. doi: 10.1007/s005260100105.

[16]

D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233. doi: 10.1088/0951-7715/23/5/011.

[17]

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

[18]

E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 722-744. doi: 10.1007/s00526-009-0299-1.

[19]

Xian Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632. doi: 10.1016/j.jde.2014.01.026.

[20]

M. B. Yang, Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities, Nonlinear Analysis, 75 (2012), 5362-5373. doi: 10.1016/j.na.2012.04.054.

[21]

J. Zhang, X. H. Tang and W. Zhang, Existence of infinitely many solutions for a quasilinear elliptic equation, Applied Mathematics Letters, 37 (2014), 131-135. doi: 10.1016/j.aml.2014.06.010.

[22]

J. Zhang, X. H. Tang and W. Zhang, Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, Journal of Mathematical Analysis and Applications, 420 (2014), 1762-1775. doi: 10.1016/j.jmaa.2014.06.055.

show all references

References:
[1]

J. M. Bezerra do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018.

[2]

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.

[3]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Analysis: Theorey, Methods $&$ Applications, 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.

[4]

Y. Cheng and J. Yang, Positive solution to a class of relativistic nonlinear Schrödinger equation, J. Math. Anal. Appl., 411 (2014), 665-674. doi: 10.1016/j.jmaa.2013.10.006.

[5]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, Journal of the physical Society of Japan, 50 (1981), 3262-3267.

[6]

P. L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact cases, part I and part II, Ann. Inst. H. Poincaré Anal. Non Linëaire, 1 (1984), 109-145, 223-283.

[7]

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

[8]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, Journal of Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.

[9]

E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, Journal of Mathematical Physics, 24 (1983), 2764-2769. doi: 10.1063/1.525675.

[10]

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

[11]

J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-892. doi: 10.1081/PDE-120037335.

[12]

A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc., 42 (1977), 1823-1835.

[13]

J. M. do Ó and U. Secero, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Cale. Var. Partial Differential Equations, 38 (2010), 275-315. doi: 10.1007/s00526-009-0286-6.

[14]

M. Porkolab and M. V. Goldman, Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.

[15]

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), 329-344. doi: 10.1007/s005260100105.

[16]

D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233. doi: 10.1088/0951-7715/23/5/011.

[17]

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

[18]

E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 722-744. doi: 10.1007/s00526-009-0299-1.

[19]

Xian Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632. doi: 10.1016/j.jde.2014.01.026.

[20]

M. B. Yang, Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities, Nonlinear Analysis, 75 (2012), 5362-5373. doi: 10.1016/j.na.2012.04.054.

[21]

J. Zhang, X. H. Tang and W. Zhang, Existence of infinitely many solutions for a quasilinear elliptic equation, Applied Mathematics Letters, 37 (2014), 131-135. doi: 10.1016/j.aml.2014.06.010.

[22]

J. Zhang, X. H. Tang and W. Zhang, Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, Journal of Mathematical Analysis and Applications, 420 (2014), 1762-1775. doi: 10.1016/j.jmaa.2014.06.055.

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