Existence and Asymptotic Behavior of Solutions for a Quasilinear Schrödinger–Poisson System in ℝ³ with a General Nonlinearity

Abstract

In this paper, we are concerned with the following quasilinear Schrödinger–Poisson system in ℝ³$\begin{cases}-\Delta u+u+\lambda\phi u=f(u), & \text{in}\ \mathbb{R^{3},} \\-\Delta \phi - \varepsilon^{4}\Delta_{4}\phi=\lambda u^{2}, & \text{in} \ \mathbb{R}^{3},\end{cases}$ where λ and ε are positive parameters, Δ₄u = div(∣∇u∣²∇u), f ∈ C(ℝ, ℝ) is a general nonlinearity introduced by Berestycki and Lions [Arch. Rational Mech. Anal., 1983, 82(4): 313–345]. We obtain the existence of a nontrivial solution for small λ and fixed ε by using a monotonicity trick of Jeanjean and truncation method. Furthermore, the asymptotic behavior of these solutions is studied as ε and λ tend to zero respectively. We prove that they converge to a nontrivial solution of a classic Schrödinger–Poisson system and a Schrödinger equation associated with it respectively. One ground state solution is also obtained under a further growth hypothesis on f.

Keywords

Quasilinear Schrödinger–Poisson system / monotonicity trick / truncation method / Pohožaev type identity / variational methods

References

[1] Azzollini A, d’Avenia P, Pomponio A. On the Schrödinger–Maxwell equations under the effect of a general nonlinear term. Ann. Inst. H. Poincaré Anal. Non Lunéaire, 2010, 27(2): 779-791.

[2] Benci V, Fortunato D. An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal., 1998, 11(2): 283-293.

[3] Benci V, Fortunato D. Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations. Rev. Math. Phys., 2002, 14(4): 409-420.

[4] Benmlih K, Kavian O. Existence and asymptotic behaviour of standing waves for quasi-linear Schrödinger–Poisson systems in ℝ³. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2008, 25(3): 449-470.

[5] Berestycki H, Lions PL. Nonlinear scalar field equations, I. Existence of a ground state. Arch. Rational Mech. Anal., 1983, 82(4): 313-345.

[6] Cerami G, Vaira G. Positive solutions for some non-autonomous Schröodinger–Poisson systems. J. Differential Equations, 2010, 248(3): 521-543.

[7] Cianchi A, Maz’ya VG. Second-order two-sided estimates in nonlinear elliptic problems. Arch. Ration. Mech. Anal., 2018, 229(2): 569-599.

[8] D’Aprile T, Mugnai D. Non-existence results for the coupled Klein–Gordon–Maxwell equations. Adv. Nonlinear Stud., 2004, 4(3): 307-322.

[9] Ding L, Li L, Meng Y, Zhuang C. Existence and asymptotic behaviour of ground state solution for quasilinear Schröodinger–Poisson systems in ℝ³. Topol. Methods Nonlinear Anal., 2016, 47(1): 241-264.

[10] Fang X., Bound state solutions for some non-autonomous asymptotically cubic Schröodinger–Poisson systems. Z. Angew. Math. Phys., 2019, 70(2): Paper No. 50, 11 pp.

[11] Figueiredo GM, Siciliano G. Quasi-linear Schrödinger–Poisson system under an exponential critical nonlinearity: existence and asymptotic behaviour of solutions. Arch. Math. (Basel), 2019, 112(3): 313-327.

[12] Figueiredo G.M., Siciliano G., Existence and asymptotic behaviour of solutions for a quasi-linear Schröodinger–Poisson system with a critical nonlinearity. Z. Angew. Math. Phys., 2020, 71(4): Paper No. 130, 21 pp.

[13] Fortunato D, Orsina L, Pisani L. Born–Infeld type equations for electrostatic fields. J. Math. Phys., 2002, 43(11): 5698-5706.

[14] Gilbarg D, Trudinger NS. Elliptic Partial Differential Equations of Second Order, 1983, Second Edition, Berlin: Springer-Verlag.

[15] Illner R, Kavian O, Lange H. Stationary solutions of quasi-linear Schroödinger–Poisson system. J. Differential Equations, 1998, 145(1): 1-16.

[16] Illner R, Lange H, Toomire B, Zweifel P. On quasi-linear Schröodinger–Poisson systems. Math. Methods Appl. Sci., 1997, 20(14): 1223-1238.

[17] Jeanjean L. On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ℝⁿ. Proc. Roy. Soc. Edinburgh Sect. A, 1999, 129(4): 787-809.

[18] Jeanjean L. Local conditions insuring bifurcation from the continuous spectrum. Math. Z., 1999, 232(4): 651-664.

[19] Jeanjean L, Le Coz S. An existence and stability result for standing waves of nonlinear Schrödinger equations. Adv. Differential Equations, 2006, 11(7): 813-840.

[20] Kikuchi H. Existence and stability of standing waves for Schroödinger–Poisson–Slater equation. Adv. Nonlinear Stud., 2007, 7(3): 403-437.

[21] Li B, Yang H. The modified quantum Wigner system in weighted L²-space. Bull. Aust. Math. Soc., 2017, 95(1): 73-83.

[22] Li Y, Li F, Shi J. Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differential Equations, 2012, 253(7): 2285-2294.

[23] Markovixh PA, Ringhofer CA, Schmeiser C. Semiconductor Equations, 1990, Vienna: Springer-Verlag x+248 pp

[24] Ruf B. Superlinear elliptic equations and systems. Handbook of Differential Equations: Stationary Partial Differential Equations, Handb. Differ. Equ., 2008, Amsterdam: Elsevier/North-Holland, 211-276. 5

[25] Ruiz D. The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal., 2006, 237(2): 655-674.

[26] Struwe M. Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 1996, Second Edition, Berlin: Springer-Verlag.

[27] Wang Z, Zhou H. Sign-changing solutions for the nonlinear Schröodinger–Poisson system in ℝ³. Calc. Var. Partial Differential Equations, 2015, 52(3–4): 927-943.

[28] Wei C., Li A., Zhao L., Multiple solutions for a class of quasilinear Schröodinger–Poisson system in ℝ³ with critical nonlinearity and zero mass. Anal. Math. Phys., 2022, 12(5): Paper No. 120, 20 pp.

[29] Wei C., Li A., Zhao L., Existence of nontrivial solutions for a quasilinear Schroödinger–Poisson system in ℝ³ with periodic potentials. Electron. J. Qual. Theory Differ. Equ., 2023, Paper No. 48, 15 pp.

[30] Wu T., Existence and symmetry breaking of ground state solutions for Schröodinger–Poisson systems. Calc. Var. Partial Differential Equations, 2021, 60(2): Paper No. 59, 29 pp.

[31] Yin L, Wu X, Tang C. Ground state solutions for an asymptotically 2-linear Schrödinger–Poisson system. Appl. Math. Lett., 2019, 87: 7-12.

[32] Zhao L, Liu H, Zhao F. Existence and concentration of solutions for the Schrödinger–Poisson equations with steep well potential. J. Differential Equations, 2013, 255(1): 1-23.

[33] Zhao L, Zhao F. Positive solutions for Schrödinger–Poisson equations with a critical exponent. Nonlinear Anal., 2009, 70(6): 2150-2164.