A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

Abstract

In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularity. Precisely, let (Σ,D) be such a surface with divisor $D=\Sigma_{i=1}^m\beta_{i}p_{i}$, where β_i > −1 and p_i ∈ Σ for i = 1, …, m, and g be a metric representing D. Denote b₀ = 4π(1 + min_1⩽i⩽mβ_i). Suppose ψ : Σ → ℝ is a continuous function with ∫_Σψdv_g ≠ 0 and define

$$\lambda _1^{**} (\sum ,g) = \mathop {\inf }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi udv_g = 0,\smallint _\sum u^2 dv_g = 1} \int_\sum {\left| {\nabla _g u} \right|^2 dv_g .}$$

Then for any$\alpha\in[0,\lambda_1^{**}(\Sigma, g))$, we have

$$\mathop {\sup }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi u = 0,\smallint _\sum \left| {\nabla _g u} \right|^2 dv_g - \alpha \smallint _\sum u^2 dv_g \leqslant 1} \int_\sum {e^{b_0 u^2 } dv_g < + \infty .}$$

When b > b₀, the integrals $\int_\sum {e^{bu^2 } dv_g }$ are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.