论文标题
无限弱耦合的合作椭圆系统的解决方案
Solutions to indefinite weakly coupled cooperative elliptic systems
论文作者
论文摘要
我们研究椭圆系统\ begin {等式*} \ begin {case}-ΔU_1-κ__1U_1=μ_1| U_1 | u_1 |^{p -2} u_1 +λ +λα| U_1 | u_1 |^{α-2{α -2} | μ_2| u_2 |^{p-2} u_2 +λβ| u_1 |^α| u_2 | u_2 |^{β-2} u_2，\\ u_1，\\ u_2 \ in d^{1,2} _0 _0（ω），\ end eend {cases} \ end {eque} $ \ mathbb {r}^n $，$ n \ geq 3 $，$κ__1，κ_2\ in \ mathbb {r} $，$μ_1，μ_2，μ_2，λ> 0 $，$α，$α，β> 1 $，和$α+β= p \ le 2^p \ le 2^*：n} 对于（2,2^*）$，我们建立了基态的存在，并以$λ$足够大的$λ$进行了大量的完全非平凡的解决方案。 如果$ p = 2^*$和$κ__1，κ_2> 0 $，我们确定$λ$的地面状态，如果$ n \ ge5 $或$ n = 4 $，并且都不是$ -BEN = 4 $，n = 4 $ and n n = 4 $，则既不是$-Δ$ in $ω$。
We study the elliptic system \begin{equation*} \begin{cases} -Δu_1 - κ_1u_1 = μ_1|u_1|^{p-2}u_1 + λα|u_1|^{α-2}|u_2|^βu_1, \\ -Δu_2 - κ_2u_2 = μ_2|u_2|^{p-2}u_2 + λβ|u_1|^α|u_2|^{β-2}u_2, \\ u_1,u_2\in D^{1,2}_0(Ω), \end{cases} \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N\geq 3$, $κ_1,κ_2\in\mathbb{R}$, $μ_1,μ_2,λ>0$, $α,β>1$, and $α+ β= p\le 2^*:=\frac{2N}{N-2}$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $λ$ sufficiently large. If $p=2^*$ and $κ_1,κ_2>0$ we establish the existence of a ground state for $λ$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $κ_1$ nor $κ_2$ are Dirichlet eigenvalues of $-Δ$ in $Ω$.
