论文标题
$ \ mathbb {r}^{4} $的两组分椭圆系统的站立波
Standing waves for two-component elliptic system with critical growth in $\mathbb{R}^{4}$: the attractive case
论文作者
论文摘要
在本文中，我们考虑了以下具有关键增长\ BEGIN {等式*}的两个两个组件椭圆系统 \ begin {case} -ΔU+（v_1（x）+λ） -ΔV+（v_2（x）+λ）v =μ_2v^{3}+βvu^{2}，\ \ x \ in \ athbb {r}^4， ％u \ geq 0，\ \ v \ geq 0 \ \ text {in} \ \ r^4。 \ end {cases} \ end {equation*}其中$ v_j（x）\ in l^{2}（\ mathbb {r}^4）$是非负势，而非线性系数$β，μ_j$，$ j $，$ j = 1,2 $，为正。在这里，我们还假设$λ> 0 $。通过变异方法与度理论相结合，我们证明了关于假设$β> \ max \ {μ_1，μ_2\} $的阳性解决方案的存在和多样性的结果。这些结果将半线性schrödinger方程的结果推广到Cerami和Passaseo（Siam J.Math。Anal。，28，867-885，（1997））到上述椭圆系统，同时扩展了存在的存在，同时扩展了Liu和liu的存在（Colc。var。var。partialiare diention方程，59：59：145：145：145：145，（20202020202020））。
In this paper, we consider the following two-component elliptic system with critical growth \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ)u=μ_1u^{3}+βuv^{2}, \ \ x\in \mathbb{R}^4, -Δv+(V_2(x)+λ)v=μ_2v^{3}+βvu^{2}, \ \ x\in \mathbb{R}^4 , % u\geq 0, \ \ v\geq 0 \ \text{in} \ \R^4. \end{cases} \end{equation*} where $V_j(x) \in L^{2}(\mathbb{R}^4)$ are nonnegative potentials and the nonlinear coefficients $β,μ_j$, $j=1,2$, are positive. Here we also assume $λ>0$. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of positive solutions under the hypothesis $β>\max\{μ_1,μ_2\}$. These results generalize the results for semilinear Schrödinger equation on half space by Cerami and Passaseo (SIAM J. Math. Anal., 28, 867-885, (1997)) to the above elliptic system, while extending the existence result from Liu and Liu (Calc. Var. Partial Differential Equations, 59:145, (2020)).
