Partial Blow-up Phenomena in the $SU(3)$ Toda System on Riemann Surfaces

Abstract: This work studies the partial blow-up phenomena for the $SU(3)$ Toda system on compact Riemann surfaces with smooth boundary. We consider the following coupled Liouville system with Neumann boundary conditions: $$ -\Delta_g u_1 = 2\rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - \frac 1 {|\Sigma|g}\right) - \rho_2\left( \frac{V_2 e^{u_2}}{\int{\Sigma} V_2 e^{u_2} \, dv_g} - \frac{1}{|\Sigma|g}\right) \text{in} \,\mathring\Sigma$$ and $$ -\Delta_g u_2 = 2\rho_2\left( \frac{V_2 e^{u_2}}{\int{\Sigma} V_2 e^{u_2} \, dv_g} - \frac{1}{|\Sigma|g}\right) - \rho_1\left( \frac{V_1 e^{u_1}}{\int{\Sigma} V_1 e^{u_1} \, dv_g} - \frac{1}{|\Sigma|g}\right) \text{in} \,\mathring\Sigma$$ with boundary conditions $ \partial{\nu_g} u_1 = \partial_{\nu_g} u_2 = 0 \text{ on} \, \partial \Sigma,$ where $(\Sigma, g)$ is a compact Riemann surface with the interior $\mathring\Sigma$ and smooth boundary $\partial\Sigma$, $\rho_i$ is a non-negative parameter and $V_i$ is a smooth positive function for $i=1,2$. We construct a family of blow-up solutions via the Lyapunov-Schmidt reduction and variational methods, wherein one component remains uniformly bounded from above, while the other exhibits partial blow-ups at a prescribed number of points, both in the interior and on the boundary. This construction is based on the existence of a non-degeneracy solution of a so-called shadow system. Moreover, we establish the existence of partial blow-up solutions in three cases: (i) for any $\rho_2>0$ sufficiently small; (ii) for generic $V_1, V_2$ and any $\rho_2\in (0,2\pi)$; (iii) for generic $V_1, V_2$, the Euler characteristic $\chi(\Sigma)<1$ and any $\rho_2\in (2\pi,+\infty)\setminus 2\pi \mathbb{N}_+$.