Concentration solutions to the singularly prescribed Gaussian and geodesic curvatures problem

[摘要] We consider the following Liouville-type equation with exponential Neumann boundary condition: {-Delta(u) over tilde = epsilon K-2(x)e(2 (u) over tilde,) x is an element of D, partial derivative(u) over tilde + 1 = epsilon kappa(x)e (x) over tilde , x is an element of partial derivative D, where D subset of R-2 is the unit disk, epsilon K-2(x) > 0 and epsilon kappa (x) > 0 stand for the prescribed Gaussian curvature and geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if kappa(x) + congruent to K(x) + Kappa (x)2 (x is an element of partial derivative D) has a local extremum point, which is a new result for exponential Neumann boundary problems. (C) 2021 Elsevier Inc. All rights reserved. MSC: 35J20; 35R01

[发布日期] 2021-11-15

[关键词] Concentration;Existence;Exponential Neumann boundary condition