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Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source
Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion
1. | Department of Mathematics, Huazhong Normal University, Wuhan 430079 |
2. | Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA |
3. | Department of Mathematics, Huazhong Normal University, Wuhan, 430079, China |
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. |
[2] |
S. Bae, H. O. Choi and D. H. Pahk, Existence of nodal solutions of nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect., A 137 (2007), 1135-1155.
doi: 10.1017/S0308210505000727. |
[3] |
T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\R^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[4] |
T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrodinger equations with steep potential well and indefinite potential, Discrete Contin. Dyn. Syst., 33, (2012), 7-26, |
[5] |
G. Bianchi, J. Chabrowski and A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal. TMA, 25 (1995), 41-59.
doi: 10.1016/0362-546X(94)E0070-W. |
[6] |
João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio 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. |
[7] |
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. |
[8] |
H. Brezis and E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[9] |
D. Cao and X. Zhu, On the existence and nodal character of semilinear elliptic equations, Acta. Math. Sci., 8 (1988), 345-359. |
[10] |
X. L. Chen and 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. |
[11] |
G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Func. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[12] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[13] |
M. Conti, L. Merizzi and S. Terracini, Radial solutions of superlinear equations on $\R^N$. I. A global variational approach, Arch. Ration. Mech. Anal., 153 (2000), 291-316.
doi: 10.1007/s002050050015. |
[14] |
A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[15] |
Y. Deng, The existence and nodal character of solutions in $\R ^N$ for semilinear elliptic equations involving critical Sobolev exponents, Acta. Math. Sci., 9 (1989), 385-402. |
[16] |
Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $\R^N$, Commun. Math. Sci., 9 (2011), 859-878.
doi: 10.4310/CMS.2011.v9.n3.a9. |
[17] |
Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504.
doi: 10.1063/1.4774153. |
[18] |
Y. Deng and W. Shuai, Positive solutions for quasilinear Schrodinger equations with critical growth and potential vanishing at infinity, Commun. Pure Appl. Anal., 13 (2014), 2273-2287.
doi: 10.3934/cpaa.2014.13.2273. |
[19] |
P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrodinger equation, Commun. Pure Appl. Anal., 13 (2014), 2395-2406.
doi: 10.3934/cpaa.2014.13.2395. |
[20] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1998. |
[21] |
S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. |
[22] |
Xiang-Qing Liu, Jia-Quan Liu and Zhi Qiang Wang, Quasilinear elliptic equations with critical growth via pertubation method, Journal Differential Equations, 254 (2013), 102-124
doi: 10.1016/j.jde.2012.09.006. |
[23] |
E. Laedke, K. 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. |
[24] |
J. Liu and Z. 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. Liu and Z. Wang, Symmetric solutions to a modified nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 121-133.
doi: 10.1088/0951-7715/21/1/007. |
[26] |
J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[27] |
J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[28] |
X. Q. Liu and J. Q. Liu, Quasilinear elliptic equations via perturbation meathod, Proc. Amer. Math. Soc. 141 (2013), 253-263.
doi: 10.1090/S0002-9939-2012-11293-6. |
[29] |
C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7. |
[30] |
A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\R^N$, J. Differential Equations, 229 (2006), 570-587.
doi: 10.1016/j.jde.2006.07.001. |
[31] |
A. Moameni, Soliton solutions for quasilinear Schrodinger equations involving supercritical exponent in $\R^N$, Commun. Pure Appl. Anal., 7 (2007), 89-105.
doi: 10.3934/cpaa.2008.7.89. |
[32] |
Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175. |
[33] |
M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[34] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. |
[35] |
Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrodinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[36] |
Marco A. S. Souto and Sergio H. M. Soares, Ground state solutions for quasilinear stationary Schrodinger equations with critical growth, Commun. Pure Appl. Anal., 12 (2012), 99-116.
doi: 10.3934/cpaa.2013.12.99. |
[37] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. |
[38] |
T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations, 27 (2006), 421-437.
doi: 10.1007/s00526-006-0015-3. |
show all references
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. |
[2] |
S. Bae, H. O. Choi and D. H. Pahk, Existence of nodal solutions of nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect., A 137 (2007), 1135-1155.
doi: 10.1017/S0308210505000727. |
[3] |
T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\R^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[4] |
T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrodinger equations with steep potential well and indefinite potential, Discrete Contin. Dyn. Syst., 33, (2012), 7-26, |
[5] |
G. Bianchi, J. Chabrowski and A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal. TMA, 25 (1995), 41-59.
doi: 10.1016/0362-546X(94)E0070-W. |
[6] |
João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio 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. |
[7] |
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. |
[8] |
H. Brezis and E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[9] |
D. Cao and X. Zhu, On the existence and nodal character of semilinear elliptic equations, Acta. Math. Sci., 8 (1988), 345-359. |
[10] |
X. L. Chen and 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. |
[11] |
G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Func. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[12] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[13] |
M. Conti, L. Merizzi and S. Terracini, Radial solutions of superlinear equations on $\R^N$. I. A global variational approach, Arch. Ration. Mech. Anal., 153 (2000), 291-316.
doi: 10.1007/s002050050015. |
[14] |
A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[15] |
Y. Deng, The existence and nodal character of solutions in $\R ^N$ for semilinear elliptic equations involving critical Sobolev exponents, Acta. Math. Sci., 9 (1989), 385-402. |
[16] |
Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $\R^N$, Commun. Math. Sci., 9 (2011), 859-878.
doi: 10.4310/CMS.2011.v9.n3.a9. |
[17] |
Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504.
doi: 10.1063/1.4774153. |
[18] |
Y. Deng and W. Shuai, Positive solutions for quasilinear Schrodinger equations with critical growth and potential vanishing at infinity, Commun. Pure Appl. Anal., 13 (2014), 2273-2287.
doi: 10.3934/cpaa.2014.13.2273. |
[19] |
P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrodinger equation, Commun. Pure Appl. Anal., 13 (2014), 2395-2406.
doi: 10.3934/cpaa.2014.13.2395. |
[20] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1998. |
[21] |
S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. |
[22] |
Xiang-Qing Liu, Jia-Quan Liu and Zhi Qiang Wang, Quasilinear elliptic equations with critical growth via pertubation method, Journal Differential Equations, 254 (2013), 102-124
doi: 10.1016/j.jde.2012.09.006. |
[23] |
E. Laedke, K. 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. |
[24] |
J. Liu and Z. 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. Liu and Z. Wang, Symmetric solutions to a modified nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 121-133.
doi: 10.1088/0951-7715/21/1/007. |
[26] |
J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[27] |
J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[28] |
X. Q. Liu and J. Q. Liu, Quasilinear elliptic equations via perturbation meathod, Proc. Amer. Math. Soc. 141 (2013), 253-263.
doi: 10.1090/S0002-9939-2012-11293-6. |
[29] |
C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7. |
[30] |
A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\R^N$, J. Differential Equations, 229 (2006), 570-587.
doi: 10.1016/j.jde.2006.07.001. |
[31] |
A. Moameni, Soliton solutions for quasilinear Schrodinger equations involving supercritical exponent in $\R^N$, Commun. Pure Appl. Anal., 7 (2007), 89-105.
doi: 10.3934/cpaa.2008.7.89. |
[32] |
Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175. |
[33] |
M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[34] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. |
[35] |
Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrodinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[36] |
Marco A. S. Souto and Sergio H. M. Soares, Ground state solutions for quasilinear stationary Schrodinger equations with critical growth, Commun. Pure Appl. Anal., 12 (2012), 99-116.
doi: 10.3934/cpaa.2013.12.99. |
[37] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. |
[38] |
T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations, 27 (2006), 421-437.
doi: 10.1007/s00526-006-0015-3. |
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