In this paper, we are interested in the existence and concentration of normalized solutions for the following logarithmic Schrödinger–Bopp–Podolsky type system involving the 𝑝-Laplacian in ℝ3: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −𝜀𝑝Δ𝑝𝑢+𝑍(𝑥)|𝑢|𝑝−2𝑢−𝜅𝜙𝑢=𝜆|𝑢|𝑝−2𝑢+|𝑢|𝑝−2𝑢log|𝑢|𝑝 in ℝ3, −𝜀2Δ𝜙+𝑎2𝜀4Δ2𝜙=4𝜋𝑢2 in ℝ3, ∫ℝ3|𝑢|𝑝𝑑𝑥=𝑑𝑝𝜀3, where Δ𝑝 ⋅ =div(|∇⋅|𝑝−2∇ ⋅) denotes the usual 𝑝-Laplacian operator, 𝑍 is a given external potential, 𝜅 >0 a constant, 𝑎 >0 is the Bopp–Podolsky constant and 𝜀 >0 is a small parameter. The unknowns are 𝑢,𝜙 :ℝ3 →ℝ and the Lagrange multiplier 𝜆 ∈ℝ. If 𝑝 ∈[2, 12 5 ), we obtain, via the variational method, that the number of positive solutions depends on the profile of 𝑍 and the solutions concentrate around the global minimum points of 𝑍 in the semiclassical limit as 𝜀 →0+.
Existence and concentration of normalized solutions for logarithmic Schrödinger–Bopp–Podolsky type system
Gaetano Siciliano
;
2025-01-01
Abstract
In this paper, we are interested in the existence and concentration of normalized solutions for the following logarithmic Schrödinger–Bopp–Podolsky type system involving the 𝑝-Laplacian in ℝ3: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −𝜀𝑝Δ𝑝𝑢+𝑍(𝑥)|𝑢|𝑝−2𝑢−𝜅𝜙𝑢=𝜆|𝑢|𝑝−2𝑢+|𝑢|𝑝−2𝑢log|𝑢|𝑝 in ℝ3, −𝜀2Δ𝜙+𝑎2𝜀4Δ2𝜙=4𝜋𝑢2 in ℝ3, ∫ℝ3|𝑢|𝑝𝑑𝑥=𝑑𝑝𝜀3, where Δ𝑝 ⋅ =div(|∇⋅|𝑝−2∇ ⋅) denotes the usual 𝑝-Laplacian operator, 𝑍 is a given external potential, 𝜅 >0 a constant, 𝑎 >0 is the Bopp–Podolsky constant and 𝜀 >0 is a small parameter. The unknowns are 𝑢,𝜙 :ℝ3 →ℝ and the Lagrange multiplier 𝜆 ∈ℝ. If 𝑝 ∈[2, 12 5 ), we obtain, via the variational method, that the number of positive solutions depends on the profile of 𝑍 and the solutions concentrate around the global minimum points of 𝑍 in the semiclassical limit as 𝜀 →0+.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
