Multiple normalized solutions for a quasi-linear Schrödinger equation via dual approach
DOI:
https://doi.org/10.12775/TMNA.2022.052Keywords
Quasi-linear Schrödinger equations, normalized solutions, dual method, the minimax principleAbstract
In this paper, we construct multiple normalized solutions of the following from quasi-linear Schrödinger equation: -\Delta u-\Delta(|u|^{2})u-\mu u=|u|^{p-2}u, \quad\text{in } \mathbb{R}^N, subject to a mass-subcritical constraint. In order to overcome non-smoothness of the associated variational formulation we make use of the dual approach. The constructed solutions possess energies being clustered at $0$ level which makes it difficult to use existing methods for non-smooth variational problems such as the variational perturbation approach.References
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. Differential Equations 256 (2014), 1492–1514.
S. Adachi and T. Watanabe, Uniqueness of ground state solutions for quasilinear Schrödinger equations, Nonlinear Anal. 75 (2012), 819–833.
A. Ambrosetti and Z.-Q. Wang, Positive solutions to a class of quasilinear elliptic equations on R, Discrete Contin. Dyn. Syst. 9 (2003), 55–68.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations I, existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–346.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations II, existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), 347–375.
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, New York, 2003.
T. Cazenave and P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549–561.
K.C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer–Verlag, Berlin, 2005.
J. Chen, Y. Li and Z.-Q. Wang, Stability of standing waves for a class of quasilinear Schrödinger equations, European J. Appl. Math. 23 (2012), 611–633.
M. Colin and L. Jeanjean, Solutions for quasilinear Schrödinger equation: a dual approach, Nonlinear Anal. 56 (2004), 213–226.
M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasilinear Schrödinger equations, Nonlinearity 23 (2010), 1353–1385.
X.D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations 254 (2013), 2015–2032.
J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in RN : mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), 253–276.
L. Jeanjean and S.-S. Lu, Nonradial normalized solutions for nonlinear scalar field equations, Nonlinearity 32 (2019), 4942–4966.
L. Jeanjean and T. Luo, Sharp non-existence results of prescribed L2 -norm solutions for some class of Schrödinger–Poisson and quasi-linear equations, Z. Angew. Math. Phys. 64 (2013), 937–954.
L. Jeanjean, T. Luo and Z.-Q. Wang, Multiple normalized solutions for quasilnear Schrödinger equations, J. Differential Equations 259 (2015), 3894–3928.
P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. H. Poincaré 1 (1984), 109–145.
P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part 2, Ann. Inst. H. Poincaré 1 (1984), 223–83.
J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc. 131 (2003), 441–448.
J. Liu and Z.-Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equations 257 (2014), 2874–2899.
J. Liu, Y. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations 187 (2003), 473–493.
J. Liu, Y. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), 879–901.
X.Q. Liu, J.Q. Liu and Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013), 253–263.
X.Q. Liu, J.Q and Liu, Z.-Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations 254 (2013), 102–124.
M. Poppenburg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 329–344.
P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC.
H. Ye and Y. Yu, The existence of normalized solutions for L2 -critical quasilinear Schrödinger equations, J. Math. Anal. Appl. 497 (2021), 124839.
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