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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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] S. Kurihara, Large-amplitude quasi-solitons in superfluid films, Journal of the physical Society of Japan, 50 (1981), 3262-3267. Google Scholar [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.  Google Scholar [7] A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Letters, 27 (1978), 517-520. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc., 42 (1977), 1823-1835. Google Scholar [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.  Google Scholar [14] M. Porkolab and M. V. Goldman, Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] S. Kurihara, Large-amplitude quasi-solitons in superfluid films, Journal of the physical Society of Japan, 50 (1981), 3262-3267. Google Scholar [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.  Google Scholar [7] A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Letters, 27 (1978), 517-520. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc., 42 (1977), 1823-1835. Google Scholar [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.  Google Scholar [14] M. Porkolab and M. V. Goldman, Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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