Weak limit and blowup of approximate solutions to H-systems

Let be a continuous function such that H(p)→H0∈R as |p|→+∞. Fixing a domain Ω in R2 we study the behaviour of a sequence (un) of approximate solutions to the H-system Δu=2H(u)ux∧uy in Ω. Assuming that supp∈R3|(H(p)−H0)p|<1, we show that the weak limit of the sequence (un) solves the H-system and un→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|→+∞, then in correspondence of each point of S we prove that the sequence (un) blows either an H-bubble or an H0-sphere.