# 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 & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
##### References:
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Wang, Soliton solutions for generalized quasilinear Schrodinger equations,, \emph{Nonlinear Anal. TMA., 80 (2013), 194. doi: 10.1016/j.na.2012.10.005. Google Scholar [36] Marco A. S. Souto and Sergio H. M. Soares, Ground state solutions for quasilinear stationary Schrodinger equations with critical growth,, \emph{Commun. Pure Appl. Anal.}, 12 (2012), 99. doi: 10.3934/cpaa.2013.12.99. Google Scholar [37] W. A. Strauss, Existence of solitary waves in higher dimensions,, \emph{Comm. Math. Phys., 55 (1977), 149. Google Scholar [38] T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations,, \emph{Calc. Var. Partial Differential Equations, 27 (2006), 421. doi: 10.1007/s00526-006-0015-3. Google Scholar

show all references

##### References:
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Google Scholar [6] João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations, 248 (2010), 722. doi: 10.1016/j.jde.2009.11.030. Google Scholar [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,, \emph{Phys. Fluids B, 5 (1993), 3539. Google Scholar [8] H. Brezis and E. Lieb, A relation between pointwise convergence of function and convergence of functional,, \emph{Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.2307/2044999. Google Scholar [9] D. Cao and X. Zhu, On the existence and nodal character of semilinear elliptic equations,, \emph{Acta. Math. Sci., 8 (1988), 345. Google Scholar [10] X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,, \emph{Phys. Rev. Lett., 70 (1993), 2082. Google Scholar [11] G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents,, \emph{J. Func. Anal., 69 (1986), 289. doi: 10.1016/0022-1236(86)90094-7. Google Scholar [12] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach,, \emph{Nonlinear Anal. TMA., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar [13] M. Conti, L. Merizzi and S. Terracini, Radial solutions of superlinear equations on $\R^N$. I. A global variational approach,, \emph{Arch. Ration. Mech. Anal., 153 (2000), 291. doi: 10.1007/s002050050015. Google Scholar [14] A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation,, \emph{Comm. Math. Phys., 189 (1997), 73. doi: 10.1007/s002200050191. Google Scholar [15] Y. Deng, The existence and nodal character of solutions in $\R ^N$ for semilinear elliptic equations involving critical Sobolev exponents,, \emph{Acta. Math. Sci., 9 (1989), 385. Google Scholar [16] Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $\R^N$,, \emph{Commun. Math. Sci., 9 (2011), 859. doi: 10.4310/CMS.2011.v9.n3.a9. Google Scholar [17] Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, \emph{Journal of Mathematical Physics, 54 (2013). doi: 10.1063/1.4774153. Google Scholar [18] Y. Deng and W. Shuai, Positive solutions for quasilinear Schrodinger equations with critical growth and potential vanishing at infinity,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 2273. doi: 10.3934/cpaa.2014.13.2273. Google Scholar [19] P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrodinger equation,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 2395. doi: 10.3934/cpaa.2014.13.2395. Google Scholar [20] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, {Springer-Verlag, (1998). Google Scholar [21] S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan, 50 (1981), 3262. Google Scholar [22] Xiang-Qing Liu, Jia-Quan Liu and Zhi Qiang Wang, Quasilinear elliptic equations with critical growth via pertubation method,, \emph{Journal Differential Equations, 254 (2013), 102. doi: 10.1016/j.jde.2012.09.006. Google Scholar [23] E. Laedke, K. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions,, \emph{J. Math. Phys., 24 (1983), 2764. doi: 10.1063/1.525675. Google Scholar [24] J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I.,, \emph{Proc. Amer. Math. Soc., 131 (2003), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar [25] J. Liu and Z. Wang, Symmetric solutions to a modified nonlinear Schrödinger equation,, \emph{Nonlinearity, 21 (2008), 121. doi: 10.1088/0951-7715/21/1/007. Google Scholar [26] J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II,, \emph{J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar [27] J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, \emph{Comm. Partial Differential Equations, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar [28] X. Q. Liu and J. Q. Liu, Quasilinear elliptic equations via perturbation meathod,, \emph{Proc. Amer. Math. Soc.} \textbf{141} (2013), 141 (2013), 253. doi: 10.1090/S0002-9939-2012-11293-6. Google Scholar [29] C. Miranda, Un'osservazione su un teorema di Brouwer,, \emph{Boll. Un. Mat. Ital., 3 (1940), 5. Google Scholar [30] A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\R^N$,, \emph{J. Differential Equations, 229 (2006), 570. doi: 10.1016/j.jde.2006.07.001. Google Scholar [31] A. Moameni, Soliton solutions for quasilinear Schrodinger equations involving supercritical exponent in $\R^N$,, \emph{Commun. Pure Appl. Anal., 7 (2007), 89. doi: 10.3934/cpaa.2008.7.89. Google Scholar [32] Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations,, \emph{Acta Math., 105 (1961), 141. Google Scholar [33] M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, \emph{Calc. Var. Partial Differential Equations, 14 (2002), 329. doi: 10.1007/s005260100105. Google Scholar [34] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E, 50 (1994), 687. Google Scholar [35] Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrodinger equations,, \emph{Nonlinear Anal. TMA., 80 (2013), 194. doi: 10.1016/j.na.2012.10.005. Google Scholar [36] Marco A. S. Souto and Sergio H. M. Soares, Ground state solutions for quasilinear stationary Schrodinger equations with critical growth,, \emph{Commun. Pure Appl. Anal.}, 12 (2012), 99. doi: 10.3934/cpaa.2013.12.99. Google Scholar [37] W. A. Strauss, Existence of solitary waves in higher dimensions,, \emph{Comm. Math. Phys., 55 (1977), 149. Google Scholar [38] T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations,, \emph{Calc. Var. Partial Differential Equations, 27 (2006), 421. doi: 10.1007/s00526-006-0015-3. Google Scholar
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