Existence and multiplicity results for nonlinear Schrödinger-Poisson equation with general potential

Abstract

This paper is concerned with the Schrödinger-Poisson equation

-\mathrm{\Delta }u+V(x)u+\phi (x)u=f(x,u),x\in {ℝ}^{\mathrm{3}},

-\mathrm{\Delta }\phi =u^{\mathrm{2}},\underset{\left|x\right|\to +\infty }{\lim }\phi (x)=\mathrm{0.}

Under certain hypotheses on V and a general spectral assumption, the existence and multiplicity of solutions are obtained via variational methods.

Keywords

Schrdinger-Poisson equation / Morse index / variational method

References

[1] Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J Math Anal Appl, 2008, 345: 90–108

[2] Bartsch T, Pankov A, Wang Z Q. Nonlinear Schrödinger equations with steep potential well. Commun Contemp Math, 2001, 3: 549–569

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

[4] Chen S, Wang C. Existence of multiple nontrivial solutions for a Schrödinger-Poisson system. J Math Anal Appl, 2014, 411: 787–793

[5] Conley C, Zehnder E. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm Pure Appl Math, 1984, 37: 207–253

[6] D'Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134: 893–906

[7] Dong Y. Index theory, nontrivial solutions and asymptotically linear second order Hamiltonian systems. J Differential Equations, 2005, 214: 233–255

[8] Dong Y. Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations. Calc Var Partial Differential Equations, 2010, 38: 75–109

[9] Ekeland I. Convexity Methods in Hamiltonian Mechanics. Berlin: Springer, 1990

[10] Jiang Y, Zhou H. Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251: 582–608

[11] Liu Z L, Guo S. On ground state solutions for the Schrödinger-Poisson equations with critical growth. J Math Anal Appl, 2014, 412: 435–448

[12] Liu Z L, Heerden F A, Wang Z Q. Nodal type bounded states of Schrödinger equations via invariant set and minimax methods. J Differential Equations, 2005, 214: 358–390

[13] Liu Z L, Su J, Weth T. Compactness results for Schrödinger equations with asymptotically linear terms. J Differential Equations, 2006, 231: 201–212

[14] Liu Z L, Wang Z Q, Zhang J. Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system. Ann Mat Pura Appl, 2016, 195: 775–794

[15] Long Y. Index Theory for Symplectic Paths with Applications. Progr Math, Vol 207. Basel: Birkhauser, 2002

[16] Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg Conf Ser Math, No 65. Providence: Amer Math Soc, 1986

[17] Reed M, Simon B. Methods of Modern Mathematical Physics IV: Analysis of Operators. New York: Academic Press, 1978

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

[19] Shan Y. Morse index and multiple solutions for the asymptotically linear Schrödinger type equation. Nonlinear Anal, 2013, 89: 170–178

[20] Simon B. Schrödinger semigroups. Bull Amer Math Soc, 1982, 7: 447–526

[21] Stuart C A, Zhou H S. Global branch of solutions for nonlinear Schrödinger equations with deepening potential well. Proc Lond Math Soc, 2006, 92: 655–681

[22] Sun J. Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations. J Math Anal Appl, 2012, 390: 514–522

[23] Sun J, Chen H, Nieto J J. On ground state solutions for some non-autonomous Schrödinger-Poisson systems. J Differential Equations, 2012, 252: 3365–3380

[24] Wang Z, Zhou H S. Positive solutions for nonlinear Schrödinger equations with deepening potential well. J Eur Math Soc (JEMS), 2009, 11: 545–573

[25] Zhang Q, Xu B. Multiple solutions for Schrödinger-Poisson systems with indefinite potential and combined nonlinearity. J Math Anal Appl, 2017, 455: 1668–1687

[26] Zhao L, Zhao F. On the existence of solutions for the Schrödinger-Poisson equations. J Math Anal Appl, 2008, 346: 155–169

[27] Zou W. Sign-changing saddle point. J Funct Anal, 2005, 219: 433–468

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