New concentration phenomena for SU(3) Toda system

In this paper, we consider the following SU(3) Toda system:(0.1){δu1+2ρ2h1eu1-ρ2h2eu2=0in B1(0),δu2+2ρ2h2eu2-ρ2h1eu1=0in B1(0),u1=u2=0on ∂B1(0), where ρ is a positive parameter which will go to zero, h₁(x)=h₂(x)=h(|x|) is a positive smooth function, and B₁(0) is a unit ball in R2. In [29], the authors construct a family of solutions (u_1,ρ, u_2,ρ) of (0.1) for h≡1, such that the solutions blow up at the origin with mass (8π, 4π), after some scaling with the limiting profile -δwi=|x|αi-2ewi in R2, ∫R2|x|αi-2ewi<∞, where α_i=2^3-i for i=1, 2. In that paper, the authors ask whether there exist blow-up solutions of mass (8π, 4π), with the limiting profileδw+2ew=0in R2,∫R2ew<∞ such that u₁ is the sum of two bubbles and u₂ has one bubble. In this paper, we prove that such solution exists.