Classical solutions to a Hartree type system. Issue 12 (27th December 2021) - Le, Phuong
Abstract: This paper is concerned with positive classical solutions to the nonlocal system of Hartree type − Δ u = 1 | x | n − α ∗ v p v p − 1 in R n, − Δ v = 1 | x | n − β ∗ u q u q − 1 in R n, \begin{equation*} \def\eqcellsep{&}\begin{array}{rcl} \hspace*{85pt}-\Delta u &=& \displaystyle {\left(\frac{1}{|x|^{n-\alpha }} * v^p\right)} v^{p-1} \quad \text{ in }\mathbb {R}^n, \hspace*{-85pt}\vspace*{6pt}\\ \hspace*{85pt} -\Delta v &=& \displaystyle {\left(\frac{1}{|x|^{n-\beta }} * u^q\right)} u^{q-1} \quad \text{ in }\mathbb {R}^n, \hspace*{-85pt}\end{array} \end{equation*} where n ≥ 3 $n\ge 3$ and 0 < α, β < n $0<\alpha, \beta <n$ . We prove that the system has no positive solution if 1 < p ≤ n + α n − 2 $1<p\le \frac{n+\alpha }{n-2}$, 1 < q ≤ n + β n − 2 $1<q\le \frac{n+\beta }{n-2}$ and ( p, q ) ≠ n + α n − 2, n + β n − 2 $(p, q) \ne \left(\frac{n+\alpha }{n-2}, \frac{n+\beta }{n-2}\right)$ . We also classify all positive solutions to the system in the critical case ( p, q ) = n + α n − 2, n + β n − 2 $(p, q)=\left(\frac{n+\alpha }{n-2}, \frac{n+\beta }{n-2}\right)$ . The main tool we use is the method of moving spheres in integral forms. Our results can be seen as the Hartree type counterpart of the classical work "D. G. de Figueiredo and P. L. Felmer, A Liouville‐type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994),  387–397".