论文标题
与Hardy-Littlewood-Sobolev关键指数的耦合Hartree系统的高能量阳性解决方案
High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents
论文作者
论文摘要
我们研究了耦合的hartree系统$$ \ left \ {\ oken {array} {ll}-ΔU+ v_1（x） ＆\ mbox {in} \ \ \ \ \ \ \ \ \ \}^n，\\ [1mm]-ΔV + v_2（x）v =α_2\ big（| x |^{ - 4} \ ast v^{2} \ big） ＆\ \ mbox {in} \ \ \ \ m} {r}^n，\ end {array} \ right。 $ n $ n \ geq 5 $，$β> \ max \ {α_1，α_2\} \ geq \ min \ {α_1，α_2\}> 0 $和$ v_1，\ in l^{n/2}（n/2}（n/2}） l _ {\ text {loc}}}^{\ infty}（\ mathbb {r}^n）$是非负势。从强大的小木 - 贝布莱夫不等式的意义上讲，该系统至关重要。对于具有$ v_1 = v_2 = 0 $的系统，我们采用整体形式的移动球体参数来对正面解决方案进行分类，并证明正面解决方案的独特性到翻译和扩张，这具有独立的兴趣。然后，使用独特性属性，我们建立了全局紧凑度引理的非本地版本，并证明了系统的高能量阳性解决方案，假设$ | v_1 | _ {l^{l^{n/2}（\ Mathbb {r}^n）}很小。
We study the coupled Hartree system $$ \left\{\begin{array}{ll} -Δu+ V_1(x)u =α_1\big(|x|^{-4}\ast u^{2}\big)u+β\big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -Δv+ V_2(x)v =α_2\big(|x|^{-4}\ast v^{2}\big)v +β\big(|x|^{-4}\ast u^{2}\big)v &\mbox{in}\ \mathbb{R}^N, \end{array}\right. $$ where $N\geq 5$, $β>\max\{α_1,α_2\}\geq\min\{α_1,α_2\}>0$, and $V_1,\,V_2\in L^{N/2}(\mathbb{R}^N)\cap L_{\text{loc}}^{\infty}(\mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|_{L^{N/2}(\mathbb{R}^N)}+|V_2|_{L^{N/2}(\mathbb{R}^N)}>0$ is suitably small.
