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

Abstract

We study an elliptic equation related to the Moser-Trudinger inequality on a compact Riemann surface $(S,g)$, $${\Delta }_{g}u+\lambda (u{e}^{u^2}-\frac{1}{|S|}{\int }_{S}u{e}^{u^2}d{v}_g)=0,\text{in S},{\int }_{S}ud{v}_g=0,$$ where $\lambda >0$ is a small parameter, $|S|$ is the area of $S$, ${\Delta }_g$ is the Laplace-Beltrami operator and $d{v}_g$ is the area element. Given any integer $k\ge 1$, under general conditions on $S$ we find a bubbling solution $u_{\lambda }$ which blows up at exactly $k$ points in $S$, as $\lambda \to 0$. 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.

41 pages