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

TL;DR

This paper constructs bubbling solutions for a Moser-Trudinger type elliptic equation on compact Riemann surfaces, showing existence of solutions with multiple blow-up points under various geometric conditions.

Contribution

It introduces new bubbling solutions for the equation on Riemann surfaces, including explicit solutions on flat tori and existence results based on Robin's function.

Findings

Existence of solutions with exactly k blow-up points as the parameter tends to zero.

Multiple families of bubbling solutions on flat tori, including nine for two points in a square torus.

At least two solutions with one blow-up point exist when the surface has non-constant Robin's function.

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.