论文标题
具有关键和亚临界非线性的半线性椭圆系统的归一化基态
Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities
论文作者
论文摘要
在本文中，我们研究了以下系统最少能量的归一化解决方案：$$ \ begin {case}-ΔU+λ_1U=μ_1| u |^{p-2} u+βr_1| U | U |^{r_1-2} |^{r_1-2} | -ΔV+λ_2V=μ_2| v |^{q-2} v+βr_2| u |^{r_1} | v |^{r_2-2} v \ quad＆quad＆\ quad＆\ hbox {in} r^n} u^2 = a_1^2 \ quad \ hbox {and} \; \ int _ {\ mathbb r^n} v^2 = a_2^2，\ end {cases} $ there $ p，q，q，q，r_1+r_2 $可以很关键。为此，我们研究了Pohozaev歧管的几何形状和相关的最小化问题。根据$ A_1，A_2 $和$β$的某些假设，我们获得了上述系统的正归一化基态解决方案。我们解决了该领域的一些未解决的开放问题。
In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -Δu+λ_1u=μ_1 |u|^{p-2}u+βr_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -Δv+λ_2v=μ_2 |v|^{q-2}v+βr_2|u|^{r_1}|v|^{r_2-2}v\quad&\hbox{in}\;\mathbb R^N,\\ \int_{\mathbb R^N}u^2=a_1^2\quad\hbox{and}\;\int_{\mathbb R^N}v^2=a_2^2, \end{cases}$$ where $p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on $a_1,a_2$ and $β$, we obtain the existence of the positive normalized ground state solution to the above system. We have solved some unsolved open problems in this area.
