New concentration phenomena for SU(3)SU(3) Toda system

In this paper, we consider the following SU(3)SU(3) Toda system:equation(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|)h1(x)=h2(x)=h(|x|) is a positive smooth function, and B1(0)B1(0) is a unit ball in R2R2. In [29], the authors construct a family of solutions (u1,ρ,u2,ρ)(u1,ρ,u2,ρ) of (0.1) for h≡1h≡1, such that the solutions blow up at the origin with mass (8π,4π)(8π,4π), after some scaling with the limiting profile −Δwi=|x|αi−2ewi−Δwi=|x|αi−2ewi in R2R2, ∫R2|x|αi−2ewi<∞∫R2|x|αi−2ewi<∞, where αi=23−iαi=23−i for i=1,2i=1,2. In that paper, the authors ask whether there exist blow-up solutions of mass (8π,4π)(8π,4π), with the limiting profileΔw+2ew=0in R2,∫R2ew<∞ such that u1u1 is the sum of two bubbles and u2u2 has one bubble. In this paper, we prove that such solution exists.