# American Institute of Mathematical Sciences

November  2015, 14(6): 2487-2508. doi: 10.3934/cpaa.2015.14.2487

## 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

Received  June 2015 Revised  June 2015 Published  September 2015

This paper is concerned with a type of quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-p\Delta(|u|^{2p})|u|^{2p-2}u=\lambda|u|^{q-2}u+|u|^{2p2^{*}-2}u, \end{eqnarray} where $\lambda>0, N\ge3, 4p < q < 2p2^*, 2^*=\frac{2N}{N-2}, 1< p < +\infty$. For any given $k \ge 0$, by using a change of variables and Nehari minimization, we obtain a sign-changing minimizer with $k$ nodes.
Citation: Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
##### 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.
 [1] Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004 [2] Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125 [3] Monica Lazzo, Paul G. Schmidt. Nodal properties of radial solutions for a class of polyharmonic equations. Conference Publications, 2007, 2007 (Special) : 634-643. doi: 10.3934/proc.2007.2007.634 [4] Jianqing Chen, Qian Zhang. Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term. Communications on Pure and Applied Analysis, 2022, 21 (2) : 669-686. doi: 10.3934/cpaa.2021193 [5] Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure and Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99 [6] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [7] Xiang-Dong Fang. Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1603-1615. doi: 10.3934/cpaa.2017077 [8] Edcarlos D. Silva, Jefferson S. Silva. Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5441-5470. doi: 10.3934/dcds.2020234 [9] João Marcos do Ó, Abbas Moameni. Solutions for singular quasilinear Schrödinger equations with one parameter. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1011-1023. doi: 10.3934/cpaa.2010.9.1011 [10] Guofa Li, Yisheng Huang. Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021214 [11] Haiyan Wang. Positive radial solutions for quasilinear equations in the annulus. Conference Publications, 2005, 2005 (Special) : 878-885. doi: 10.3934/proc.2005.2005.878 [12] Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921 [13] Dumitru Motreanu, Viorica V. Motreanu, Abdelkrim Moussaoui. Location of Nodal solutions for quasilinear elliptic equations with gradient dependence. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 293-307. doi: 10.3934/dcdss.2018016 [14] Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597 [15] Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885 [16] Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273 [17] Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025 [18] Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure and Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004 [19] Yi He, Gongbao Li. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 731-762. doi: 10.3934/dcds.2016.36.731 [20] Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054

2020 Impact Factor: 1.916