Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-08T06:16:43.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Least Energy Nodal Solutions for a Defocusing Schrödinger Equation with Supercritical Exponent

Part of: Elliptic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  16 August 2018

Minbo Yang ,
Carlos Alberto Santos  and
Jiazheng Zhou
Minbo Yang
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People's Republic of China (mbyang@zjnu.edu.cn)
Carlos Alberto Santos*
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília DF, Brazil (csantos@unb.br; jiazzheng@gmail.com)
Jiazheng Zhou
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília DF, Brazil (csantos@unb.br; jiazzheng@gmail.com)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the existence of least energy nodal solution for the defocusing quasilinear Schrödinger equation

$$-\Delta u - u \Delta u^2 + V(x)u = a(x)[g(u) + \lambda \vert u \vert ^{p-2}u] \hbox{in} {\open R}^N,$$
where λ≥0 is a real parameter, V(x) is a non-vanishing function, a(x) can be a vanishing positive function at infinity, the nonlinearity g(u) is of subcritical growth, the exponent p≥22*, and N≥3. The proof is based on a dual argument on Nehari manifold by employing a deformation argument and an $L</italic>^{\infty}({\open R}^{N})$-estimative.

Keywords

quasilinear Schrödinger equationleast energy nodal solutionssupercritical growthMiranda's Theorem

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 1 - 23
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bartsch, T. and Weth, T., A note on additional properties of sign changing solutions to superlinear elliptic equations, Topolog. Meth. Nonliner. Anal. 22 (2003), 1–14.Google Scholar
2Bartsch, T., Weth, T. and Willem, M., Partial symmetry of least energy nodal solution to some variational problems, J. Anal. Math. 1 (2005), 1–18.Google Scholar
3Castro, A., Cossio, J. and Neuberger, J. M., A sign-changing solution for a superlinear Dirichlet problem, Rocky Mt. J. Math. 27(4) (1997), 1041–1053.Google Scholar
4Chabrowski, J., On nodal radial solutions of an elliptic problem involving critical Sobolev exponent, Commentat. Math. Univ. Carolinae 37 (1996), 1–16.Google Scholar
5Colin, M. and Jeanjean, L., Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal. 56 (2004), 213–226.Google Scholar
6Deng, Y., Li, Y. and Yan, X., Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion, Commun. Pure Appl. Anal. 14(6) (2015), 2487–2508.Google Scholar
7Deng, Y., Peng, S. and Wang, J., Infinitely many sign-changing solutions for quasilinear Schrödinger equations in ${\open R} N$, Commun. Math. Sci. 9(3) (2011), 859–878.Google Scholar
8Deng, Y., Peng, S. and Wang, J., Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys. 54(1) (2013), 011504.Google Scholar
9Deng, Y., Peng, S. and Wang, J., Nodal soliton solutions for generalized quasilinear Schrödinger equations, J. Math. Phys. 55 (2014), 051501.Google Scholar
10Deng, Y. and Wei, S., Existence and concentration behavior of sign-changing solutions for quasilinear Schrödinger equations, Sci. China Math. 59(6) (2016), 1095–1112.Google Scholar
11Kurihura, S., Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan 50 (1981), 3262–3267.Google Scholar
12Liu, J., Liu, X. and Wang, Z., Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Diff. Equ. 39(12) (2014), 2216–2239.Google Scholar
13Liu, X., Liu, J. and Wang, Z., Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013), 253–263.Google Scholar
14Liu, X., Liu, J. and Wang, Z., Quasilinear elliptic equations with critical growth via perturbation method, J. Diff. Equ. 254(1) (2013), 102–124.Google Scholar
15Liu, J. and Wang, Z. Q., Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc. 131 (2002), 441–448.Google Scholar
16Liu, J., Wang, Y. and Wang, Z., Soliton solutions for quasilinear Schrödinger equations II, J. Diff. Equ. 187 (2003), 473–493.Google Scholar
17Liu, J., Wang, Y. and Wang, Z. Q., Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Diff. Equ. 29 (2004), 879–901.Google Scholar
18Li, F., Zhu, X. and Liang, Z., Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation, J. Math. Anal. Appl. 443(1) (2016), 11–38.Google Scholar
19Maia, L. A., Miyagaki, O. H. and Soares, S. H. M., A sign-changing solution for an asymptotically linear Schrödinger equation, Proc. Edinb. Math. Soc. 58(2) (2015), 697–716.Google Scholar
20Poppenberg, M., Schmitt, K. and Wang, Z. Q., On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 329–344.Google Scholar
21Silva, E. B. and Vieira, G. F., Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations 39 (2010), 1–33.Google Scholar
22Struwe, M., Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math. 39 (1982), 233–240.Google Scholar
23Willem, M., Minimax theorems, Volume 24 (Birkhäuser, 1996).Google Scholar
24Yang, M., Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities, Nonlinear Anal. 75 13 (2012), 5362–5373.Google Scholar