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: $$ -Δ_g u_1 = 2ρ_1\left( \frac{V_1 e^{u_1}}{\int_Σ V_1 e^{u_1} \, dv_g} - \frac 1 {|Σ|_g}\right) - ρ_2\left( \frac{V_2 e^{u_2}}{\int_Σ V_2 e^{u_2} \, dv_g} - \frac{1}{|Σ|_g}\right) \text{in} \,\mathringΣ$$ and $$ -Δ_g u_2 = 2ρ_2\left( \frac{V_2 e^{u_2}}{\int_Σ V_2 e^{u_2} \, dv_g} - \frac{1}{|Σ|_g}\right) - ρ_1\left( \frac{V_1 e^{u_1}}{\int_Σ V_1 e^{u_1} \, dv_g} - \frac{1}{|Σ|_g}\right) \text{in} \,\mathringΣ$$ with boundary conditions $ \partial_{ν_g} u_1 = \partial_{ν_g} u_2 = 0 \text{ on} \, \partial Σ,$ where $(Σ, g)$ is a compact Riemann surface with the interior $\mathringΣ$ and smooth boundary $\partialΣ$, $ρ_i$ 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 $ρ_2>0$ sufficiently small; (ii) for generic $V_1, V_2$ and any $ρ_2\in (0,2π)$; (iii) for generic $V_1, V_2$, the Euler characteristic $χ(Σ)<1$ and any $ρ_2\in (2π,+\infty)\setminus 2π\mathbb{N}_+$.