Abstract
We consider the following Liouville-type equation with exponential Neumann boundary condition:{−Δu˜=ε2K(x)e2u˜,x∈D,∂u˜∂n+1=εκ(x)eu˜,x∈∂D, where D⊂R2 is the unit disk, ε2K(x)>0 and εκ(x)>0 stand for the prescribed Gaussian curvature and geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if κ(x)+K(x)+κ(x)2 (x∈∂D) has a local extremum point, which is a new result for exponential Neumann boundary problems.
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