Multiple positive solutions for Schrödinger‐Poisson system involving singularity and critical exponent

2019 ◽  
Vol 42 (7) ◽  
pp. 2417-2430 ◽  
Author(s):  
Chun‐Yu Lei ◽  
Jia‐Feng Liao
Keyword(s):  
Critical Exponent ◽  
Positive Solutions ◽  
Multiple Positive Solutions ◽  
Poisson System

Existence of multiple positive solutions for a semilinear equation with critical exponent

1992 ◽  
Vol 122 (1-2) ◽  
pp. 161-175 ◽  
Author(s):  
Deng Yinbing
Keyword(s):  
Critical Exponent ◽  
Unique Solution ◽  
Positive Solutions ◽  
Positive Constant ◽  
Multiple Positive Solutions ◽  
Semilinear Equation ◽  
Image Position

SynopsisIn this paper we discuss the problemWe show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).

scholarly journals Multiple positive solutions for a logarithmic Schrödinger–Poisson system with singular nonelinearity

2021 ◽  
pp. 1-15
Author(s):  
Linyan Peng ◽  
Hongmin Suo ◽  
Deke Wu ◽  
Hongxi Feng ◽  
Chunyu Lei
Keyword(s):  
Variational Method ◽  
Critical Point ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Multiple Positive Solutions ◽  
Poisson System ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this article, we devote ourselves to investigate the following logarithmic Schrödinger–Poisson systems with singular nonlinearity { − Δ u + ϕ u = | u | p−2 u log ⁡ | u | + λ u γ , i n   Ω , − Δ ϕ = u 2 , i n   Ω , u = ϕ = 0 , o n   ∂ Ω , where Ω is a smooth bounded domain with boundary 0 < γ < 1 , p ∈ ( 4 , 6 ) and λ > 0 is a real parameter. By using the critical point theory for nonsmooth functional and variational method, the existence and multiplicity of positive solutions are established.

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