论文标题
与Hardy-Littlewood-Sobolev关键指数的耦合的Hartree系统：高能量阳性解决方案的存在和多样性
A coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent: existence and multiplicity of high energy positive solutions
论文作者
论文摘要
本文与Hardy-Littlewood-Sobolev关键指数有关耦合的Hartree系统 \ begin {equation*} \ begin {case} -ΔU+（v_1（x）+λ_1）u =μ_1（| x |^{ - 4}*u^{2}） -ΔV+（v_2（x）+λ_2）v =μ_2（| x |^{ - 4}*v^{2}）v+β（| x | x |^{ - 4}*u^{2}）v，\ \ \＆x \ \ end {cases} \ end {equation*}其中$ n \ geq 5 $，$λ_1$，$λ_2\ geq 0 $ with $λ_1+λ_2\ neq 0 $，$ v_1（x）（x），v_ {2}（2}（x）（x）（x）（x）（x）（x）\ in l^{\ frac $ nonny nony nonnoy $μ_1$，$μ_2$，$β$是正常数。这种系统源自Bose-Einstein凝结理论和非线性光学的数学模型。通过变异方法与度理论相结合，我们证明了关于假设$β> \ max \ {μ_1，μ_2\} $的高能量正溶液的存在和多样性的结果
This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ_1)u=μ_1(|x|^{-4}*u^{2})u+β(|x|^{-4}*v^{2})u, \ \ &x\in R^N, -Δv+(V_2(x)+λ_2)v=μ_2(|x|^{-4}*v^{2})v+β(|x|^{-4}*u^{2})v, \ \ &x\in R^N, \end{cases} \end{equation*} where $N\geq 5$, $λ_1$, $λ_2\geq 0$ with $λ_1+λ_2\neq 0$, $V_1(x), V_{2}(x)\in L^{\frac{N}{2}}(R^N)$ are nonnegative functions and $μ_1$, $μ_2$, $β$ are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of high energy positive solutions under the hypothesis $β>\max\{μ_1,μ_2\}$
