Abstract
By using the standard scaling arguments, we show that the infimum of the following minimization problem:Iρ2=inf{(1/2)∫ℝ3|∇u|2dx+(1/4)∬ℝ3(|u(x)|2|u(y)|2/|x-y|)dx dy− (1/p)∫ℝ3|u|pdx:u∈Bρ}can be achieved forp∈(2,3)andρ>0small, whereBρ:={u∈H1(ℝ3):∥u∥2=ρ}. Moreover, the properties ofIρ2/ρ2and the associated Lagrange multiplierλρare also given ifp∈(2,8/3].
Highlights
In this paper, we consider the nonlinear Schrodinger-Poisson type equation:−Δu + (|x|−1 ∗ |u|2) u − |u|p−2u = λu, in R3, (1)where λ ∈ R is a parameter, p ∈ (2, 6), and ∗ denotes the convolution
In the range p ∈ (2, 3), Bellazzini and Siciliano showed in [18] that (8) holds for ρ > 0 small, where they developed a new abstract theorem which guarantees the following condition (MD) for s > 0 small: (MD) The function s → Is2 /s2 is monotone decreasing. We remark that their abstract theorem heavily relies on the behavior of Iρ2 near zero; that is, to use the abstract theorem, one has to verify some extra conditions, such as ρ → Iρ2 is continuous; lim ρ→0
We introduce a new subset Bρ ∩ P of Bρ, we consider the minimization problem (7) constrained on Bρ ∩ P instead of Bρ, and we use the standard scaling arguments to prove that (8) holds for ρ > 0 small
Summary
We consider the nonlinear Schrodinger-Poisson type equation:. where λ ∈ R is a parameter, p ∈ (2, 6), and ∗ denotes the convolution. In the case that the frequency λ is a fixed and assigned parameter, the critical points of the following functional defined in H1(R3; R):. (MD) The function s → Is2 /s2 is monotone decreasing We remark that their abstract theorem heavily relies on the behavior of Iρ2 near zero; that is, to use the abstract theorem, one has to verify some extra conditions, such as ρ → Iρ2 is continuous; lim ρ→0.
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