Abstract
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.
Highlights
Introduction and main resultIn this paper, we consider the following logarithmic Schrödinger–Poisson system with singular term, in Ω, −∆φ = u2, in Ω, (1.1)u = φ = 0, on ∂Ω, where Ω ⊂ R3 is a smooth bounded domain with boundary ∂Ω, 0 < γ < 1, p ∈ (4, 6) and λ > 0 is a real parameter.L
The existence and multiplicity of positive solutions for this equation are considered under some suitable condition by the critical point theory for non-smooth functional and supper-and sub-solutions method
We find that most of Schrödinger–Poisson system contain only power terms and not the logarithmic terms |t|p−2t log |t|
Summary
We consider the following logarithmic Schrödinger–Poisson system with singular term U > 0, u = φ = 0, in Ω, in Ω, in Ω, on ∂Ω, where Ω ⊂ R3 is a smooth bounded domain with boundary ∂Ω, η = ±1, γ ∈ (0, 1) is a constant, μ > 0 is a parameter and he proved the existence and uniqueness result for η = 1 and multiplicity of solutions for η = −1 and μ > 0 small enough by using Nehari manifold. The existence and multiplicity of positive solutions for this equation are considered under some suitable condition by the critical point theory for non-smooth functional and supper-and sub-solutions method.
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