Bubbling solutions for Moser-Trudinger type equations on compact Riemann surfaces

We study an elliptic equation related to the Moser–Trudinger inequality on a compact Riemann surface (S,g), Δ _gu+λ(ue ^{u 2} −[Formula presented]∫Sue ^{u 2} dv _g)=0,in S,∫Sudv _g=0, where λ>0 is a small parameter, |S| is the area of S, Δ _g is the Laplace–Beltrami operator and dv _g is the area element. Given any integer k≥1, under general conditions on S we find a bubbling solution u _λ which blows up at exactly k points in S, as λ→0. When S is a flat two-torus in rectangular form, we find that either seven or nine families of such solutions do exist for k=2. In particular, in any square flat two-torus actually nine families of bubbling solutions with two bubbling points do exist. If S is a Riemann surface with non-constant Robin's function then at least two bubbling solutions with k=1 exists.