Weak limit and blowup of approximate solutions to H-systems

Let H be a continuous, real valued function on R3 having a finite limit H0 at infinity. Fixing a domain Ω in R2, we study the behaviour of a sequence (u^{n}) of approximate solutions to the H-system $\Delta u=2H(u)u_{x}\wedge u_{y}$ in Ω. Assuming that sup_{p\in R3}|(H(p)-H0)p| <1, we show that the weak limit u of the sequence (u^{n}) solves the H-system and u^{n} converges to u strongly in H1 apart from a countable set S made by isolated points. Moreover, if in addition H(p)=H0+o(1/|p|) as |p| tends to infinity, then in correspondence of each point of S we prove that the sequence (u^{n}) blows either an H-bubble or an H0-sphere. Here by H-bubble we mean a non constant solution of the H-system on R2 with finite Dirichlet integral, whereas by H0-sphere we mean an H0-bubble, which in fact, parametrizes a sphere of mean curvature H0.