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Global solutions of shock reflection problem for the pressure gradient system
Existence results for quasilinear Schrödinger equations with a general nonlinearity
1. | College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001, China |
2. | Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China |
3. | School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China |
$ \begin{equation*} \label{eq0.1}-\Delta u+V(x)u- \Delta(u^2)u = h(u)\ \ \mbox{in}\ {\mathbb{R}}^N,\tag{A} \end{equation*} $ |
$ N\geq 3 $ |
$ V: {\mathbb{R}}^N\to{\mathbb{R}} $ |
$ h: {\mathbb{R}}\to{\mathbb{R}} $ |
$ V $ |
$ h $ |
References:
[1] |
S. Adachi, M. Shibata and T. Watanabe,
Blow-up phenomena and asymptotic profiles of ground states of quasilinear elliptic equations with $H^1$-supercritical nonlinearities, J. Differ. Equ., 256 (2014), 1492-1514.
doi: 10.1016/j.jde.2013.11.004. |
[2] |
S. Adachi and T. Watanabe,
$G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differ. Equ., 16 (2011), 289-324.
|
[3] |
C. O. Alves, Y. J. Wang and Y. T. Shen,
Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.
doi: 10.1016/j.jde.2015.02.030. |
[4] |
D. Arcoya, L. Boccardo and L. Orsina,
Critical points for functionals with quasilinear singular Euler-Lagrange equations, Calc. Var. Partial Differ. Equ., 47 (2013), 159-180.
doi: 10.1007/s00526-012-0514-3. |
[5] |
A. Azzollini and A. Pomponio,
On the Schrödinger equation in ${\mathbb{R}}^N$ under the effect of a general nonlinear term, Indiana Univ. Math. J., 58 (2009), 1361-1378.
doi: 10.1512/iumj.2009.58.3576. |
[6] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, Ⅰ, Existence of a ground state, Arch. Ration. Meth. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[7] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[8] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. Theory Methods Appl., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[9] |
Y. B. Deng, S. J. Peng and S. S. Yan,
Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260 (2016), 1228-1262.
doi: 10.1016/j.jde.2015.09.021. |
[10] |
J. M. do Ó and U. Severo,
Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differ. Equ., 38 (2010), 275-315.
doi: 10.1007/s00526-009-0286-6. |
[11] |
X. D. Fang and A. Szulkin,
Multiple solutions for a quasilinear Schrödinger equation, J. Differ. Equ., 254 (2013), 2015-2032.
doi: 10.1016/j.jde.2012.11.017. |
[12] |
E. Gloss,
Existence and concentration of positive solutions for a quasilinear equation in ${\mathbb{R}}^N$, J. Math. Anal. Appl., 371 (2010), 465-484.
doi: 10.1016/j.jmaa.2010.05.033. |
[13] |
Y. X. Guo and Z. W. Tang,
Multi-bump bound state solutions for the quasilinear Schrödinger equation with critical frequency, Pac. J. Math., 270 (2014), 49-77.
doi: 10.2140/pjm.2014.270.49. |
[14] |
X. M. He, A. X. Qian and W. M. Zou,
Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.
doi: 10.1088/0951-7715/26/12/3137. |
[15] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${\mathbb{R}}^N$, Proc. R. Soc. Edinb. Sect. A Math., 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[16] |
L. Jeanjean,
Local conditions insuring bifurcation from the continuous spectrum, Math. Z., 232 (1999), 651-664.
doi: 10.1007/PL00004774. |
[17] |
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. |
[18] |
Y. T. Jing, Z. L. Liu and Z. Q. Wang, Multiple solutions of a parameter-dependent quasilinear elliptic equation, Calc. Var. Partial Differ. Equ., 55 (2016), 150.
doi: 10.1007/s00526-016-1067-7. |
[19] |
S. Kurihara,
Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jpn., 50 (1981), 3262-3267.
doi: 10.1143/JPSJ.50.3801. |
[20] |
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. |
[21] |
J. Q. Liu, X. Q. Liu and Z. Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun. Partial Differ. Equ., 39 (2014), 2216-2239.
doi: 10.1080/03605302.2014.942738. |
[22] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differ. Equ., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[23] |
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. |
[24] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[25] |
J. Q. Liu and Z. Q. Wang,
Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differ. Equ., 257 (2014), 2874-2899.
doi: 10.1016/j.jde.2014.06.002. |
[26] |
J. Q. Liu, Z. Q. Wang and Y. X. Guo,
Multibump solutions for quasilinear elliptic equations, J. Funct. Anal., 262 (2012), 4040-4102.
doi: 10.1016/j.jfa.2012.02.009. |
[27] |
X. Q. Liu, J. Q. Liu and Z. Q. Wang,
Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.
doi: 10.1090/S0002-9939-2012-11293-6. |
[28] |
V. G. Makhankov and V. K. Fedyanin,
Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. Rev. Sec. Phys. Lett., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[29] |
M. Poppenberg, K. Schmitt and Z. Q. Wang,
On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[30] |
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. |
[31] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
show all references
References:
[1] |
S. Adachi, M. Shibata and T. Watanabe,
Blow-up phenomena and asymptotic profiles of ground states of quasilinear elliptic equations with $H^1$-supercritical nonlinearities, J. Differ. Equ., 256 (2014), 1492-1514.
doi: 10.1016/j.jde.2013.11.004. |
[2] |
S. Adachi and T. Watanabe,
$G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differ. Equ., 16 (2011), 289-324.
|
[3] |
C. O. Alves, Y. J. Wang and Y. T. Shen,
Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.
doi: 10.1016/j.jde.2015.02.030. |
[4] |
D. Arcoya, L. Boccardo and L. Orsina,
Critical points for functionals with quasilinear singular Euler-Lagrange equations, Calc. Var. Partial Differ. Equ., 47 (2013), 159-180.
doi: 10.1007/s00526-012-0514-3. |
[5] |
A. Azzollini and A. Pomponio,
On the Schrödinger equation in ${\mathbb{R}}^N$ under the effect of a general nonlinear term, Indiana Univ. Math. J., 58 (2009), 1361-1378.
doi: 10.1512/iumj.2009.58.3576. |
[6] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, Ⅰ, Existence of a ground state, Arch. Ration. Meth. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[7] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[8] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. Theory Methods Appl., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[9] |
Y. B. Deng, S. J. Peng and S. S. Yan,
Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260 (2016), 1228-1262.
doi: 10.1016/j.jde.2015.09.021. |
[10] |
J. M. do Ó and U. Severo,
Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differ. Equ., 38 (2010), 275-315.
doi: 10.1007/s00526-009-0286-6. |
[11] |
X. D. Fang and A. Szulkin,
Multiple solutions for a quasilinear Schrödinger equation, J. Differ. Equ., 254 (2013), 2015-2032.
doi: 10.1016/j.jde.2012.11.017. |
[12] |
E. Gloss,
Existence and concentration of positive solutions for a quasilinear equation in ${\mathbb{R}}^N$, J. Math. Anal. Appl., 371 (2010), 465-484.
doi: 10.1016/j.jmaa.2010.05.033. |
[13] |
Y. X. Guo and Z. W. Tang,
Multi-bump bound state solutions for the quasilinear Schrödinger equation with critical frequency, Pac. J. Math., 270 (2014), 49-77.
doi: 10.2140/pjm.2014.270.49. |
[14] |
X. M. He, A. X. Qian and W. M. Zou,
Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.
doi: 10.1088/0951-7715/26/12/3137. |
[15] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${\mathbb{R}}^N$, Proc. R. Soc. Edinb. Sect. A Math., 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[16] |
L. Jeanjean,
Local conditions insuring bifurcation from the continuous spectrum, Math. Z., 232 (1999), 651-664.
doi: 10.1007/PL00004774. |
[17] |
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. |
[18] |
Y. T. Jing, Z. L. Liu and Z. Q. Wang, Multiple solutions of a parameter-dependent quasilinear elliptic equation, Calc. Var. Partial Differ. Equ., 55 (2016), 150.
doi: 10.1007/s00526-016-1067-7. |
[19] |
S. Kurihara,
Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jpn., 50 (1981), 3262-3267.
doi: 10.1143/JPSJ.50.3801. |
[20] |
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. |
[21] |
J. Q. Liu, X. Q. Liu and Z. Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun. Partial Differ. Equ., 39 (2014), 2216-2239.
doi: 10.1080/03605302.2014.942738. |
[22] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differ. Equ., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[23] |
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. |
[24] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[25] |
J. Q. Liu and Z. Q. Wang,
Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differ. Equ., 257 (2014), 2874-2899.
doi: 10.1016/j.jde.2014.06.002. |
[26] |
J. Q. Liu, Z. Q. Wang and Y. X. Guo,
Multibump solutions for quasilinear elliptic equations, J. Funct. Anal., 262 (2012), 4040-4102.
doi: 10.1016/j.jfa.2012.02.009. |
[27] |
X. Q. Liu, J. Q. Liu and Z. Q. Wang,
Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.
doi: 10.1090/S0002-9939-2012-11293-6. |
[28] |
V. G. Makhankov and V. K. Fedyanin,
Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. Rev. Sec. Phys. Lett., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[29] |
M. Poppenberg, K. Schmitt and Z. Q. Wang,
On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[30] |
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. |
[31] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
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