A priori estimates for an elliptic equation with exponential nonlinearity

Abstract

We establish compactness and existence results for the following class of nonlinear elliptic problems: −Δu=K(x) e^u in Ω, with ∫_ΩK(x) e^u=λ, where Ω is a smooth two‐dimensional bounded domain with nontrivial topology and K(x)= $\prod_{j=1}^{m}$ |x−x_j|^2α_j e^σ(x) is a nonnegative function which admits a zero at the point x_j∈Ω with multiplicity α_j∈(0,1), for j=1,…,m. We consider the case where λ∈(8π,min _j=1,…,m4π(3−α_j))\{8π(1+α_j)}_j=1,…,m and extend the work of Struwe and Tarantello [34] and Ding et al. [12].