Singular Limits for 2-Dimensional Elliptic Problem With Exponentially Dominated Nonlinearity and Singular Data

Abstract

We study existence of solutions with singular limits for a 2-dimensional semilinear elliptic problem with exponential dominated nonlinearity and a singular source term given by Dirac masses, imposing Dirichlet boundary condition. This paper extends previous results obtained in [3, 8]. We mainly use the method of domain decomposition.

Singular Limits for 2-Dimensional Elliptic Problem with Exponentially Dominated Nonlinearity and a Quadratic Convection Term

“…This paper extends previous results obtained in [1], [3], [4] and some references therein for related issues. …”

“…This type of equation has been studied by Bartolucci et al in [8] and by Chen and Lin in [10]. They obtained the existence, sharp estimates and construction of multiple bubbles to (4). That the construction of nontrivial branches of solutions of such semilinear elliptic equations with exponential nonlinearities is equivalent to prove the existence of a conformal change of metric for which the corresponding mean curvature surfaces in the Euclidean space is non constant function.…”

Singular Limit Solutions for a 4-dimensional Semilinear Elliptic System of Liouville Type

“…Here g is the Green's function defined as the solution of    −∆ z g(z, z ) = 8πδ z=z in Ω, g(z, z ) = 0 on ∂Ω and h is its smooth part defined by h(z, z ) = g(z, z ) + 4 ln |z − z |. Some generalizations can be found in [3,8,14]. In dimension 4, other authors were motivated by similar problems, we refer the reader to [2,4,11,12].…”

“…Theorem 1.4. Let Ω be a regular open subset of R 4 and x 1 , x 2 , x 3 ∈ Ω be given disjoint points. Suppose that (u ρ 1 , u ρ 2 ) is a one parameter family of solutions of (1.1), such that…”

Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case

“…Some generalizations can be found in [2,4,13,14]. The blow-up analysis for the equation (9) is well understood thanks to the work of Suzuki [24] and Nagasaki-Suzuki [23], which permits to localize the blow-up set of singular limit solutions (up to a subsequence) as critical point of a functional given by the Green's functions.…”