Semiclassical states for singularly perturbed Schrödinger-Poisson systems with a general Berestycki-Lions or critical nonlinearity
Keywords
35J60, 35J20, 35Q40Abstract
This paper is concerned with two classes of singularly perturbed Schrödinger-Poisson systems of the form \begin{equation*} \begin{cases} -\varepsilon^2\triangle u+u+ \phi u=f(u), & x\in {\mathbb{R}}^{3},\\ -\triangle \phi=u^2, & x\in \mathbb{R}^3, \end{cases} \end{equation*} and \begin{equation*} \begin{cases} -\varepsilon^2\triangle u+V(x)u+ \phi u=g(x,u)+K(x)u^5, & x\in {\mathbb{R}}^{3},\\ -\triangle \phi=u^2,& x\in \mathbb{R}^3, \end{cases} \end{equation*} where $\ep> 0$ is a small parameter. We prove that: (1) the first system admits a concentrating bounded state for small $\ep$, where $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies Berestycki-Lions assumptions which are almost necessary; (2) there exists a constant $\ep_0> 0$ determined by $V,K$ and $g$ such that for any $\ep\in (0,\ep_0]$ the second system has a nontrivial solution, where $V,K\in \mathcal{C}(\mathbb{R}^3,\mathbb{R})$, $V(x)\ge 0$, $K(x)> 0$, $g\in \mathcal{C}(\mathbb{R}^3\times \mathbb{R},\mathbb{R})$ is an indefinite function. Our results improve and complement the previous ones in the literature.References
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
A. Azzollini, P. d’Avenia and A. Pomponio, On the Schrödinger–Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Nonlineaire, vol. 27, Elsevier Masson, 2010.
A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger–Maxwell equations, J. Math. Anal. Appl. 345 (2008), 90–108.
T. Bartsch, Z.Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, Stationary partial differential equations, Vol. II 2 (2005), 1–55.
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), 283–293.
V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations, Rev. Math. Phys. 14 (1998), 283–293.
R. Benguria, H. Brezis and E.H. Lieb, The Thomas–Fermi–von Weizsäcker theory of atoms and molecules, Comm. Math. Phys. 79 (1981), 167–180.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–345.
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal. 185 (2007), 185–200.
S.T. Chen, A. Fiscella, P. Pucci and X.H. Tang, Semiclassical ground state solutions for critical Schrödinger–Poisson systems with lower perturbations, J. Differential Equations (2019), DOI: 10.1016/j.jde.2019.09.041.
S.T. Chen and X.H. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations (2019), DOI: 10.1016/j.jde.2019.08.036.
S.T. Chen and X.H. Tang, Improved results for Klein–Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst. 38 (2018), 2333–2348.
S.T. Chen and X.H. Tang, Berestycki–Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal. 9 (2020), 496–515.
T. D’Aprile and D. Mugnai, Non-existence results for the coupled Klein–Gordon–Maxwell equations, Adv. Nonlinear Stud. 4 (2004), 307–322.
T. D’Aprile and D. Mugnai, Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 893–906.
T. D’Aprile and J.C. Wei, On Bound States Concentrating on Spheres for the Maxwell–Schrödinger Equation, SIAM J. Math. Anal. 37 (2005), 321–342.
X.M. He and W.M. Zou, Existence and concentration of ground states for Schrödinger–Poisson equations with critical growth, J. Math. Phys. 53 (2012), 023702.
W.N. Huang and X.H. Tang, Semiclassical solutions for the nonlinear Schrödinger–Maxwell equations with critical growth, Taiwan. J. Math. 18 (2014), 1203–1217.
L. Jeanjean, 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 129 (1999), 787–809.
L. Jeanjean and S.L. Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations 11 (2006), 813–840.
L. Jeanjean and K. Tanka, A positive solution for a nonlinear Schrödinger equation on RN , Indiana Univ. Math. J. 54 (2005), 443–464.
W. Jeong and J. Seok, On perturbation of a functional with the mountain pass geometry, Calc. Var. Partial Differential Equations 49 (2014), 649–668.
E.H. Lieb, Thomas–Fermi and related theories and molecules, Rev. Modern Phys. 53 (1981), 603–641.
E.H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev inequality and related inequalities, Ann. of Math. 118 (1983), 349–374.
E.H. Lieb and M. Loss, Analysis, 2nd ed, American Mathematical Society, Providence, Rhode Island, 2001.
X.Y. Lin and X.H. Tang, Semiclassical solutions of perturbed p-Laplacian equations with critical nonlinearity, J. Math. Anal. Appl. 413 (2014), 438–449.
P.L. Lions, Solutions of Hartree–Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1984), 33–97.
Z.S. Liu, S.J. Guo, Y.Q. Fan, Multiple semiclassical states for coupled Schrödinger–Poisson equations with critical exponential growth, J. Math. Phys. 56 (2015), 041505.
P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor equations (1990).
D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), 655–674.
B. Simon, Schrödinger semigroups, Bull. Am. Math. Soc. 7 (1982), 447–526.
J.J. Sun and S.W. Ma, Ground state solutions for some Schrödinger–Poisson systems with periodic potentials, J. Differential Equations 260 (2016), 2119–2149.
X.H. Tang and S.T. Chen, Ground state solutions of Nehari–Pohoz̆aev type for Schrödinger–Poisson problems with general potentials, Disc. Contin. Dyn. Syst. 37 (2017), 4973–5002.
X.H. Tang and S.T. Chen, Ground state solutions of Nehari–Pohoz̆aev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations 56 (2017), 110.
X.H. Tang, X. Lin and J. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dynam. Differential Equations (2018), 1–15, DOI: https://doi.org/10.1007/s10884-018-9662-2.
X.H. Tang and X.Y. Lin, Existence of ground state solutions of Nehari–Pankov type to Schrödinger systems, Sci. China Math. 62 (2019), DOI: https://doi.org/10.1007/s11425017-9332-3.
M.B. Yang, Z.F. Shen and Y.H. Ding, Multiple semiclassical solutions for the nonlinear Maxwell–Schrödinger system, Nonlinear Anal. 71 (2009), 730–739.
J. Zhang, On the Schrödinger–Poisson equations with a general nonlinearity in the critical growth, Nonlinear Anal. 75 (2012), 6391–6401.
J.J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger–Poisson system with critical growth, J. Math. Phys. 55 (2014), 031507.
L.G. Zhao and F.K. Zhao, Positive solutions for Schrödinger–Poisson equations with a critical exponent, Nonlinear Anal. 70 (2009), 2150–2164.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0