Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell Gauged O(3) Sigma model (2002.02685v3)

Abstract: 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, $$-\Delta u + A_0(\prod^k_{j=1}|x-p_j|^{2n_j} )^{-a} \frac{e^u}{(1+e^u)^{1+a}} = 4\pi\sum_{j=1}^k n_j\delta_{p_j} - 4\pi\sum^l_{j=1}m_j\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2.\qquad(E)$$ In this equation the ${\delta_{p_j}}{j=1}^k$ (resp. ${\delta{q_j}}{j=1}^l$ ) 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 ${u_\beta}{\beta\in(-2(N-M),\, -2]}$, where $u\beta$ is the unique non-topological solution such that $u_{\beta}=\beta\ln |x|+O(1)$ for $-2(N-M)<\beta<-2$ 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 $N,\, M,\, \beta$ and on the gravitational interaction associated to $a$.