论文标题
来自重力麦克斯韦（3）Sigma模型的半线性椭圆方程的解决方案的定性特性
Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell Gauged O(3) Sigma model
论文作者
论文摘要
本文专用于以下半线性方程的研究，该方程的测量数据起源于重力麦克斯韦（Maxwell）$ o（3）$ sigma模型，$$-ΔU + a_0（\ prod^k_ = 1} | x-p_j | x-p_j |^|^{2n_j} {2n_j} {2n_j}）^{ - a} \ frac {e^u} {（1+e^u）^{1+a}} =4π\ sum_ {j = 1}^kn_jΔ__{p_j}-4π\ sum^l_ { \ Mathbb {r}^2。 $ \ {p_j \} _ {j = 1}^k $，（resp。$ \ {q_j \} _ {j = 1}^l $），$ n_j $和$ m_j $是正插图，而$ a $ a $是一个非听觉的实际数字。我们设置$ n = \ sum^k_ {j = 1} n_j $和$ m = \ sum^l_ {j = 1} m_j $。 在以前的作品\ cite {c，y2}中，已经建立了$（e）$的解决方案的某些定性属性。我们在本文中的目的是研究$ a> 0 $的更一般情况。这种情况的其他困难来自以下事实：非线性不再是单调，并且数据签署了措施。结果，我们不能再通过单调性方法直接构建解决方案，并结合了超溶液和亚树种技术。取而代之的是，我们开发了一种新的和独立的方法，使我们能够强调引力在测量的$ O（3）$ sigma模型中所扮演的角色。如果没有引力术语，即如果$ a = 0 $，问题$（e）$具有层的解决方案结构$ \ {u_β\ \} _ {β\ in（-2（n-m），\，-2]} $ $ -2（n-m）<β<-2 $和$ u _ { - 2} = - 2 \ ln | x | -2 | -2 \ ln \ ln \ ln | x |+o（1）分别在无穷大，相反，当$ a> 0 $时。即，$ u $倾向于在无穷大的$ - \ infty $中，而不是II型的非访问解决方案，这些解决方案倾向于$ \ infty $ infinity $ \ infty $。
This article is devoted to the study of the following semilinear equation with measure data which originates in the gravitational Maxwell gauged $O(3)$ sigma model, $$-Δu + A_0(\prod^k_{j=1}|x-p_j|^{2n_j} )^{-a} \frac{e^u}{(1+e^u)^{1+a}} = 4π\sum_{j=1}^k n_jδ_{p_j} - 4π\sum^l_{j=1}m_jδ_{q_j} \quad{\rm in}\;\; \mathbb{R}^2.\qquad(E)$$ In this equation the $\{δ_{p_j}\}_{j=1}^k$ (resp. $\{δ_{q_j}\}_{j=1}^l$ ) are Dirac masses concentrated at the points $\{p_j\}_{j=1}^k$, (resp. $\{q_j\}_{j=1}^l$), $n_j$ and $m_j$ are positive integers, and $a$ is a nonnegative real number. We set $ N=\sum^k_{j=1}n_j $ and $M= \sum^l_{j=1}m_j$. In previous works \cite{C,Y2}, some qualitative properties of solutions of $(E)$ with $a=0$ have been established. Our aim in this article is to study the more general case where $a>0$. The additional difficulties of this case come from the fact that the nonlinearity is no longer monotone and the data are signed measures. As a consequence we cannot anymore construct directly the solutions by the monotonicity method combined with the supersolutions and subsolutions technique. Instead we develop a new and self-contained approach which enables us to emphasize the role played by the gravitation in the gauged $O(3)$ sigma model. Without the gravitational term, i.e. if $a=0$, problem $(E)$ has a layer's structure of solutions $\{u_β\}_{β\in(-2(N-M),\, -2]}$, where $u_β$ is the unique non-topological solution such that $u_β=β\ln |x|+O(1)$ for $-2(N-M)<β<-2$ and $u_{-2}=-2\ln |x|-2\ln\ln |x|+O(1)$ at infinity respectively. On the contrary, when $a>0$, the set of solutions to problem $(E)$ has a much richer structure: besides the topological solutions, there exists a sequence of non-topological solutions in type I, i.e. such that $u $ tends to $-\infty$ at infinity, and of non-topological solutions of type II, which tend to $\infty$ at infinity. The existence of these types of solutions depends on the values of the parameters $N,\, M,\, β$ and on the gravitational interaction associated to $a$.
