A class of generalized quasilinear Schrödinger equations

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May 2016, 15(3): 853-870. doi: 10.3934/cpaa.2016.15.853

A class of generalized quasilinear Schrödinger equations

Yaotian Shen ^1, and Youjun Wang ^2,

1.	Department of Mathematics, South China University of Technology, Guangzhou 510640
2.	School of Mathematics, South China University of Technology, Guangzhou 510640, China

Received June 2015 Revised November 2015 Published February 2015

Abstract
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We establish the existence of nontrivial solutions for the following quasilinear Schrödinger equation with critical Sobolev exponent: \begin{eqnarray} -\Delta u+V(x) u-\Delta [l(u^2)]l'(u^2)u= \lambda u^{\alpha2^*-1}+h(u),\ \ x\in \mathbb{R}^N, \end{eqnarray} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is a given potential and $l,h$ are real functions, $\lambda\geq 0$, $\alpha>1$, $2^*=2N/(N-2)$, $N\geq 3$. Our results cover two physical models $l(s)=s^{\frac{\alpha}{2}}$ and $l(s) = (1+s)^{\frac{\alpha}{2}}$ with $\alpha\geq 3/2$.

Keywords: Quasilinear Schrödinger equations, generalized., critical exponents.

Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35Q5.

Citation: Yaotian Shen, Youjun Wang. A class of generalized quasilinear Schrödinger equations. Communications on Pure and Applied Analysis, 2016, 15 (3) : 853-870. doi: 10.3934/cpaa.2016.15.853

References:

[1]	S. Adachia and T. Watanable, G-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Diff. Eqns., 16 (2011), 289-324.
[2]	S. Adachia and T. Watanabeb, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonl. Anal. TMA., 75 (2012), 819-833.
[3]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
[4]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Arch. Rational Mech. Anal., 82 (1983), 313-346. doi: 10.1007/BF00250555.
[5]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[6]	L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497. doi: 10.1088/0951-7715/16/4/317.
[7]	M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonl. Anal. TMA., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.
[8]	A. De Bouard, N. 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.
[9]	J. M. do Ó, O. Miyagaki and H. Olimpio, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. doi: 10.1016/j.jde.2009.11.030.
[10]	B. Hartmann and W. Zakzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev., 68 (2003), 1-9.
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[12]	L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
[13]	A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.
[14]	S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan., 50 (1981), 3262-3267.
[15]	E. W. Laedke, K. 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.
[16]	A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520.
[17]	J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448.
[18]	J. Q. Liu, Y. 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.
[19]	J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari Method, Commun. Partial Differ. Equ., 29 (2004), 879-901. doi: 10.1081/PDE-120037335.
[20]	V. G. Makhankov 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.
[21]	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.
[22]	G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Physica A, 110 (1982), 41-80. doi: 10.1016/0378-4371(82)90104-2.
[23]	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.
[24]	Y. T. Shen and X. K. Guo, The positive solutions of degenerate variational problem and degenerate elliptic equations, Chin. J. Cont. Mathematics, 14 (1993), 157-166.
[25]	Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonl. Anal. TMA., 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005.
[26]	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.
[27]	Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comp., 216 (2010), 849-856. doi: 10.1016/j.amc.2010.01.091.
[28]	Y. J. Wang and W. M, Zou, Bound states to critical quasilinear Schrödinger equations, Nonl. Diff. Equa. Appl., 19 (2012), 19-47.
[29]	J. Yang, Y. J. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys, 54 (2013), 071502.

show all references

References:

[1]	S. Adachia and T. Watanable, G-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Diff. Eqns., 16 (2011), 289-324.
[2]	S. Adachia and T. Watanabeb, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonl. Anal. TMA., 75 (2012), 819-833.
[3]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
[4]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Arch. Rational Mech. Anal., 82 (1983), 313-346. doi: 10.1007/BF00250555.
[5]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[6]	L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497. doi: 10.1088/0951-7715/16/4/317.
[7]	M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonl. Anal. TMA., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.
[8]	A. De Bouard, N. 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.
[9]	J. M. do Ó, O. Miyagaki and H. Olimpio, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. doi: 10.1016/j.jde.2009.11.030.
[10]	B. Hartmann and W. Zakzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev., 68 (2003), 1-9.
[11]	R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys., 37 (1980), 83-87. doi: 10.1007/BF01325508.
[12]	L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
[13]	A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.
[14]	S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan., 50 (1981), 3262-3267.
[15]	E. W. Laedke, K. 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.
[16]	A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520.
[17]	J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448.
[18]	J. Q. Liu, Y. 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.
[19]	J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari Method, Commun. Partial Differ. Equ., 29 (2004), 879-901. doi: 10.1081/PDE-120037335.
[20]	V. G. Makhankov 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.
[21]	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.
[22]	G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Physica A, 110 (1982), 41-80. doi: 10.1016/0378-4371(82)90104-2.
[23]	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.
[24]	Y. T. Shen and X. K. Guo, The positive solutions of degenerate variational problem and degenerate elliptic equations, Chin. J. Cont. Mathematics, 14 (1993), 157-166.
[25]	Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonl. Anal. TMA., 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005.
[26]	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.
[27]	Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comp., 216 (2010), 849-856. doi: 10.1016/j.amc.2010.01.091.
[28]	Y. J. Wang and W. M, Zou, Bound states to critical quasilinear Schrödinger equations, Nonl. Diff. Equa. Appl., 19 (2012), 19-47.
[29]	J. Yang, Y. J. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys, 54 (2013), 071502.

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