A double mean field equation related to a curvature prescription problem

Abstract

We study a double mean field-type PDE related to a prescribed curvature problem on compacts surfaces with boundary:Here ρ and ρ are real parameters and K, h are smooth positive functions on Σ and ∂Σ respectively.We provide a general blow-up analysis, then a Moser-Trudinger inequality, which gives energy-minimizing solutions for some range of parameters. Finally, we provide existence of min-max solutions for a wider range of parameters, which is dense in the plane if Σ is not simply connected.2010 Mathematics Subject Classification. 35J20, 58J32.

Existence of critical points for a mean field functional on a compact Riemann surface with boundary

“…The above equation is related to the problem of prescribing geodesic curvature on surfaces with nonempty boundary via conformal deformation of the metric. Some results on solvability of (1.3) are in [3,4,18,24,38], where some crucial ideas were introduced by using variational techniques considering different potentials. From the variational structure of (1.3), the energy functional is…”

“…Let us recall some topological notions which have been described in detail in [3,15,25]. For k ∈ N + , the kth set of formal barycenters of ∂Σ is defined by…”