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: $$-{\Delta }_g{u}_1=2{\rho }_1\left(\frac{V_1{e}^{u_1}}{{\int }_{\Sigma }{V}_1{e}^{u_1}d{v}_g}-\frac{1}{|\Sigma {|}_g}\right)-{\rho }_2\left(\frac{V_2{e}^{u_2}}{{\int }_{\Sigma }{V}_2{e}^{u_2}d{v}_g}-\frac{1}{|\Sigma {|}_g}\right)\text{in}\mathring{\Sigma }$$ and $$-{\Delta }_g{u}_2=2{\rho }_2\left(\frac{V_2{e}^{u_2}}{{\int }_{\Sigma }{V}_2{e}^{u_2}d{v}_g}-\frac{1}{|\Sigma {|}_g}\right)-{\rho }_1\left(\frac{V_1{e}^{u_1}}{{\int }_{\Sigma }{V}_1{e}^{u_1}d{v}_g}-\frac{1}{|\Sigma {|}_g}\right)\text{in}\mathring{\Sigma }$$ 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 ${\rho }_2\in (2\pi ,+\infty )\setminus 2\pi {\mathbb{N}}_{+}$.