Existence and multiplicity of nontrivial solutions for Schr??dinger-Poisson systems on bounded domains

摘要

In this paper, we concern with the following Schr??dinger-Poisson system: $$ extstyleegin{cases} -Delta u+phi u = f(x,u) , & xinOmega, -Deltaphi=u^{2}, & xinOmega, u=phi=0, & x inpartialOmega, end{cases} $$ where ?? is a smooth bounded domain in (mathbb{R}^{3}). Under more appropriate assumptions on??f, we obtain new results on the existence of nontrivial solutions and infinitely many solutions by using the mountain pass theorem and the symmetric mountain pass theorem, respectively. We extend and improve some recent results in the literature.KeywordsSchr??dinger-Poisson system??variational methods??mountain pass theorem??symmetric mountain pass theorem??MSC35J20??35J60??1 Introduction and preliminariesConsider the the following Schr??dinger-Poisson system: $$ extstyleegin{cases} -Delta u+phi u = f(x,u) , & xinOmega, -Deltaphi=u^{2}, & x inOmega, u=phi=0, & x inpartialOmega, end{cases} $$ (1.1) where ?? is a smooth bounded domain in (mathbb{R}^{3}), and (fin C(Omegaimesmathbb{R},mathbb{R})).System (1.1) is related to the stationary analogue of the nonlinear parabolic Schr??dinger-Poisson system $$ extstyleegin{cases} -irac{partialpsi}{partial t}=-Deltapsi+phi(x)psi- ert psi ert ^{p-2}psi & ext{in } Omega, -Deltaphi= ert psi ert ^{2} & ext{in }Omega, psi=phi=0 & ext{on } partialOmega. end{cases} $$ (1.2) The first equation in (1.2) is called the Schr??dinger equation, which describes quantum particles interacting with the electromagnetic field generated by the motion. An interesting class of Schr??dinger equations is the case where the potential (phi(x)) is determined by the charge of the wave function itself, that is, when the second equation in (1.2) (Poisson equation) holds. For more details as regards the physical relevance of the Schr??dinger-Poisson system, we refer to [1, 2, 3, 4].Recently, Schr??dinger-Poisson systems on unbounded domains or on the whole space (mathbb{R}^{N}) have attracted a lot of attention. Many solvability conditions on the nonlinearity have been given to obtain the existence and multiplicity of solutions for Schr??dinger-Poisson systems in (mathbb{R}^{N}), we refer the readers to [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and references therein.