Gabriella Tarantello, Università di Roma "Tor Vergata"¶
We call this problem $(P)_k$:
$$ -\Delta u = \frac{4}{k^2} e^u (1 - e^u) - 4 \pi \sum_{j=1}^N \delta_{p_j} $$Theorem (G. Tarantello, Birkhauser PNDE 2008): There exists a critical value of the Chern-Simons parameter $k_c = k_c (N, |M|) > 0$ such that $(P)_k$ admits a soluiton $\iff k \in (0, k_c]$. Moreover, if $0<k<k_c \implies (P)_k $ admits at least two solutions $u_{\pm, k}$
We call this problem $(1)_\rho$
$$-\Delta w = \rho ( \frac{e^w}{\int_M e^w} - \frac{1}{|M|} ) - (4 \pi \sum_{j=1}^{N_j} \alpha_j \delta_{p_j} ) - \frac{4 \pi N}{|M|} $$$$ \int_M w = 0$$Here there is no multiplicity among the $p_j$. Rather the multiplicity is accounted for by the $\alpha_j$. $$ N = \sum_{j=1}^{N_j} \alpha_j, ~\alpha_j > -1. $$
The set of solutions of $(1)_\rho$ is compact provided that $\rho \in [0,L]\backslash \Gamma$ where $\Gamma$ is a discrete set. In particular, if $\alpha_j \in \mathbb{N} \bigcup \{ 0 \} \implies \Gamma = 8 \pi \mathbb{N}.$
"Leray-Schauder degree" of the regular part of the solutions is independent of $\rho \in (8 \pi (k-1), 8\pi k)$, ~k \in \mathbb{N}$. The degree depends *only* on $k$ here.
Malchiodi-Djiali
Interesting analytical features connected to $\chi(M)$, the Euler characteristic!
Interesting dichotomy in the concentration-compactness analysis. When the $\alpha_j$ are non-integer, there are interesting excesses that accumulate into the blowup concentrations.
blow-up analysis up to suitable scaling, bubbles are formed which are described as solutions of a Liouville problem:
$$ -\Delta u = e^u - 4 \pi \alpha \delta_0, ~{\mbox{in}}~ \mathbb{R}^2$$$$ \int_{\mathbb{R}^2} e^u < \infty$$where $\alpha =0$ if $z_k \notin \{ p_1, \dots, p_N \}$ OR $\alpha = \alpha_i$ if $z_k = p_j$.
We have a whole systems extension of all this! Not all of what I said persists but some of it extends. The work is "much more delicate".
Systerm versions of these results toward the applications to non-abelian vortices.
Coupling matrix (2x2 case). $G = U(1) \times SU(N)$. In the case $N=2$ and $k =3$, you get the Toda System. Most of the results described in these lectures (in the "regular" case $\alpha_j = 0$) extend to the Toda System. ... connections to Algebraic Geometry here. When you are away from the Today System, we don't really know what's going on.
Q: Time varying extensions?
$$ \frac{d}{dt} u = $$Look at the mean field...
$$ \int_M \frac{1}{2} |\nabla u |^2 - \rho \ln \frac{1}{|M|} \int h_0 e^u = E(u). $$Now, study the gradient flow of this problem.
Q: Well-posedness for the Chern-Simons system?
A: Chae et. al. have some classical local existence for wave generlizations for given CAuchy data.