We study the constrained minimizing problem for the energy functional related to the nonlinear Choquard equation $$\begin{aligned} I(a) = \inf \left\{ E(\phi ) \ : \ \phi \in H^1\big ({\mathbb {R}}^N\big ), \Vert \phi \Vert ^2_{L^2} =a \right\} , \end{aligned}$$ I ( a ) = inf E ( ϕ ) : ϕ ∈ H 1 ( R N ) , ‖ ϕ ‖ L 2 2 = a , where $$N\ge 1$$ N ≥ 1 , $$a>0$$ a > 0 , $$\begin{aligned} E(\phi ) := \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla \phi (x)|^2 dx + \frac{1}{2} \int _{{\mathbb {R}}^N} V(x) |\phi (x)|^2 dx -\frac{1}{2p} \int _{{\mathbb {R}}^N} (I_\alpha * |\phi |^p)(x) |\phi (x)|^p dx \end{aligned}$$ E ( ϕ ) : = 1 2 ∫ R N | ∇ ϕ ( x ) | 2 d x + 1 2 ∫ R N V ( x ) | ϕ ( x ) | 2 d x - 1 2 p ∫ R N ( I α ∗ | ϕ | p ) ( x ) | ϕ ( x ) | p d x is the energy functional with $$0<\alpha 1$$ q > 1 if $$N=2$$ N = 2 and $$q=\frac{N}{2}$$ q = N 2 if $$N\ge 3$$ N ≥ 3 and (A2) for any $$c>0$$ c > 0 , $$|\{x \in {\mathbb {R}}^N \ : \ |V(x)| >c \}| <\infty $$ | { x ∈ R N : | V ( x ) | > c } | < ∞ . We first give a complete classification of existence and non-existence of minimizers for the problem. In the mass-critical case $$p=\frac{N+\alpha +2}{N}$$ p = N + α + 2 N , under an appropriate assumption of the external potential, we give a detailed description of the blow-up behavior of minimizers as the mass tends to a critical value.

中文翻译：

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具有奇异势的非线性 Choquard 方程的规定质量极小值的爆破行为

我们研究与非线性 Choquard 方程相关的能量函数的约束最小化问题 $$\begin{aligned} I(a) = \inf \left\{ E(\phi ) \ : \ \phi \in H^1\ big ({\mathbb {R}}^N\big ), \Vert \phi \Vert ^2_{L^2} =a \right\} , \end{aligned}$$ I ( a ) = inf E ( ϕ ) : ϕ ∈ H 1 ( RN ) , ‖ ϕ ‖ L 2 2 = a , 其中 $$N\ge 1$$ N ≥ 1 , $$a>0$$ a > 0 , $$\begin{aligned } E(\phi ) := \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla \phi (x)|^2 dx + \frac{1}{2 } \int _{{\mathbb {R}}^N} V(x) |\phi (x)|^2 dx -\frac{1}{2p} \int _{{\mathbb {R}}^ N} (I_\alpha * |\phi |^p)(x) |\phi (x)|^p dx \end{aligned}$$ E ( ϕ ) : = 1 2 ∫ RN | ∇ ϕ ( x ) | 2 dx + 1 2 ∫ RNV ( x ) | φ ( x ) | 2 dx - 1 2 p ∫ RN ( I α ∗ | ϕ | p ) ( x ) | φ ( x ) | pdx 是 $$0<\alpha 的能量泛函1$$ q > 1 如果 $$N=2$$ N = 2 并且 $$q=\frac{N}{2}$$ q = N 2 如果 $$N\ge 3$$ N ≥ 3 并且 ( A2) 对于任何 $$c>0$$ c > 0 , $$|\{x \in {\mathbb {R}}^N \ : \ |V(x)| >c \}| <\infty $$ | { x ∈ RN : | V ( x ) | > c } | < ∞ 。我们首先给出问题的极小值存在和不存在的完整分类。在质量临界情况下 $$p=\frac{N+\alpha +2}{N}$$ p = N + α + 2 N ，在适当的外势假设下，我们给出了打击的详细描述当质量趋于临界值时，最小化器的向上行为。