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 $(S,g)$, $$ \Delta_g u+\lambda \Biggl(ue^{u^2}-{1\over |S|} \int_S ue^{u^2} dv_g\Biggl)=0,\quad\text{in $S$},\qquad \int_S u\,dv_g=0, $$ where $\lambda>0$ is a small parameter, $|S|$ is the area of $S$, $\Delta_g$ is the Laplace-Beltrami operator and $dv_g$ is the area element. Given any integer $k\geq 1$, under general conditions on $S$ we find a bubbling solution $u_\lambda$ which blows up at exactly $k$ points in $S$, as $\lambda \to0$. 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.