Abstract:In this paper, we consider the existence of positive solution to the Schrödinger-Poisson system with singular term
where Ω⊂R3 is a smooth bounded domain with boundary ∂Ω, λ is a real parameter and γ∈(0,1), p∈(2, 6). Firstly, perturbation technique is used to solve the problem that the functional with singular term cannot be differentiable at zero. Secondly, by applying the Ekeland variational principal and mountain pass lemma, the perturbation functional corresponding to this problem has a local minimum and critical point of the mountain path type. Finally, the existence of two positive solutions is obtained by estimating the sequence with a consistent lower bound and taking the limit of perturbation.
Benci V, Fortunato D F. Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations[J]. Reviews in Mathematical Physics, 2002, 14(4):409-420.
[2]
Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations[J]. Topol Methods Nonlinear Anal, 1998(11):283-293.
[3]
Zhang Q. Existence, uniqueness and multiplicity of positive solutions for Schrödinger-Poisson system with singularity[J]. Journal of Mathematical Analysis and Applications, 2016(437):160-180.
[4]
Lei C Y, Liao J F. Multiple positive solutions for Schrödinger-Poisson system involving singularity and critical exponent[J]. Mathematical Methods in the Applied Sciences, 2019(42):2417-2430.
[5]
Fan H N. Multiple positive solutions for Schrodinger-Poisson systems involving concave-convex nonlinearities[J]. Electronic Journal of Differential Equations, 2019(86):1-19.
[6]
Wang L L. Multiple positive solutions for a kind of singular Schrödinger-Poisson system[J]. Applicable Analysis, 2018, 99(2):270-284.
[7]
Sun Y J, Wu S P, Long Y M. Combined effects of singular and superlinear nonlinearities in some singular boundary value problems[J]. Journal of Differential Equations, 2001, 176(2):511-531.
[8]
Zhao L G, Zhao F K. Positive solutions for Schrödinger-Poisson equations with a critical exponent[J]. Nonlinear Analysis-Theory Methods & Applications, 2009, 70(6):2150-2164.
[9]
Zhang J F, Lei C Y, Guo L T. Positive solutions for a nonlocal Schrödinger-Newton system involving critical nonlinearity[J]. Computers & Mathematics with Applications, 2018, 76(8):1966-1974.
[10]
高歌, 闫宝强. 一类奇异边值问题的正解[J]. 应用泛函分析学报, 2016(1):50-59.
[11]
Jiang Y, Zhou H. Schrödinger-Poisson system with singular potential[J]. Journal of Mathematical Analysis and Applications, 2014, 417(1):411-438.
[12]
Azzollini A, D'Avenia P, Luisi V. Generalized Schrödinger-Poisson type systems[J]. Communications on Pure and Applied Analysis, 2010(2):283-293.