论文标题
与混合耦合的椭圆系统的符号溶液的能量估计
Energy estimates for seminodal solutions to an elliptic system with mixed couplings
论文作者
论文摘要
我们研究半线性椭圆方程的系统 $ - Δu_i+ u_i = \ sum_ {j = 1}^\ ellβ_{ij} | u_j |^p |^p | u_i | u_i |^{p-2} u_i，\ qquad u_i \ in h^1（\ mathb {\ mathb {r}^n），\ qquad e $ n o $ 4 $，$ 1 <p <\ frac {n} {n-2} $，矩阵$（β_{ij}）$是对称的，并且承认块分解，使每个块中的条目为正或零，所有其他条目都是负的。 我们在$（β_{ij}）$上提供简单的条件，该条件可以保证存在完全非平凡的解决方案，即所有其组成部分的解决方案。 我们建立了完全非平凡的解决方案，该系统具有正面和非放置签名组件的规定组合，并且当系统最多有两个块时，我们会为它们的能量提供上限。 我们将具有积极和非放置标志改变组件的解决方案得出到奇异扰动的椭圆方程系统$$ - \ varepsilon^2ΔU_I+ u_i = \ sum_ = \ sum_ {J = 1} u_i \在h^1_0（b_1（0））中，\ qquad i = 1，\ ldots，\ ell，$$在单位球中表现出两种不同种类的渐近行为：解决方案的组成部分将其成分为$ \ varepsilon \ varepsilon \ to varepsilon \ to couppledents，其组成部分仍然保留了所有的限制。
We study the system of semilinear elliptic equations $$-Δu_i+ u_i = \sum_{j=1}^\ell β_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1(\mathbb{R}^N),\qquad i=1,\ldots,\ell,$$ where $N\geq 4$, $1<p<\frac{N}{N-2}$, and the matrix $(β_{ij})$ is symmetric and admits a block decomposition such that the entries within each block are positive or zero and all other entries are negative. We provide simple conditions on $(β_{ij})$, which guarantee the existence of fully nontrivial solutions, i.e., solutions all of whose components are nontrivial. We establish existence of fully nontrivial solutions to the system having a prescribed combination of positive and nonradial sign-changing components, and we give an upper bound for their energy when the system has at most two blocks. We derive the existence of solutions with positive and nonradial sign-changing components to the system of singularly perturbed elliptic equations $$-\varepsilon^2Δu_i+ u_i = \sum_{j=1}^\ell β_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1_0(B_1(0)),\qquad i=1,\ldots,\ell,$$ in the unit ball, exhibiting two different kinds of asymptotic behavior: solutions whose components decouple as $\varepsilon\to 0$, and solutions whose components remain coupled all the way up to their limit.
