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Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$
Multiplicity of solutions for elliptic systems via local Mountain Pass method
| 1. | Universidade Federal da Campina Grande, Departamento de Matemática, 58109-970, Campina Grande - PB, Brazil |
| 2. | Universidade Federal do Pará, Departamento de Matemática, 66075-100, Belém - PA, Brazil |
| 3. | Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília - DF |
$-\varepsilon^{2} \Delta u +W(x)u=Q_{u}(u,v)$ in $\mathbb{R}^N,$
$-\varepsilon^{2} \Delta v +V(x)v=Q_{v}(u,v)$ in $\mathbb{R}^N, $
$u,v \in H^{1}(\mathbb{R}^N),u(x),v(x)>0$ for each $x \in \mathbb{R}^N, $
where $\varepsilon>0$, $W$ and $V$ are positive potentials and $Q$ is a homogeneous function with subcritical growth. We relate the number of solutions with the topology of the set where $W$ and $V$ attain their minimum values. In the proof we apply Ljusternik-Schnirelmann theory.
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