Existence results for mean field equations

Let $\mathrm{\Omega }$ be an annulus. We prove that the mean field equation $-\mathrm{\Delta }\psi =\frac{e\text{sp}-\beta \psi }{\int \text{sb}\mathrm{\Omega }e\text{sp}-\beta \psi }$ admits a solution with zero boundary for $\beta \in (-16\pi ,-8\pi )$. This is a supercritical case for the Moser-Trudinger inequality.