Given a smooth bounded domain Ω ⊂ R3 , we consider the following nonlinear Schrödinger-Poisson type ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ −Δu + ϕu − |u|p−2u = ωu in λΩ, −Δϕ= u2 in λΩ, u > 0 in λΩ, u = ϕ= 0 on ∂(λΩ), ∫︁λΩ u2 dx = ρ2 in the expanding domain λΩ ⊂ R3,λ > 1 and p ∈ (2,3), in the unknowns (u,ϕ,ω). We show that, for arbitrary large values of the expanding parameter λ and arbitrary small values of the mass ρ > 0, the number of solutions is at least the Ljusternick-Schnirelmann category of λΩ. Moreover we show that as λ → +∞ the solutions found converge to a ground state of the problem in the whole space R3
Small normalised solutions for a Schrödinger-Poisson system in expanding domains: Multiplicity and asymptotic behaviour
Siciliano, Gaetano
2025-01-01
Abstract
Given a smooth bounded domain Ω ⊂ R3 , we consider the following nonlinear Schrödinger-Poisson type ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ −Δu + ϕu − |u|p−2u = ωu in λΩ, −Δϕ= u2 in λΩ, u > 0 in λΩ, u = ϕ= 0 on ∂(λΩ), ∫︁λΩ u2 dx = ρ2 in the expanding domain λΩ ⊂ R3,λ > 1 and p ∈ (2,3), in the unknowns (u,ϕ,ω). We show that, for arbitrary large values of the expanding parameter λ and arbitrary small values of the mass ρ > 0, the number of solutions is at least the Ljusternick-Schnirelmann category of λΩ. Moreover we show that as λ → +∞ the solutions found converge to a ground state of the problem in the whole space R3File in questo prodotto:
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