Construction of singular limits for four-dimensional elliptic problems with exponentially dominated nonlinearity

Construction of singular limits for a strongly perturbed four-dimensional Navier problem with exponentially dominated nonlinearity and nonlinear terms

“…Recently, in [4], the authors constructed non-minimal solutions of problem (9) with singular limit as the parameter ρ tends to 0. Their results, which give the converse of case (3) given in the last theorem, can be stated as follows.…”

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

“…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]. Wei in [20], has studied the behavior of solutions to the following nonlinear eigenvalue problem for the biharmonic operator ∆ 2 in R 4 .…”

“…are given (the values of τ − l and τ + l , for l = 1, 2, 3 will be fixed later). First, we consider some set of boundary data ϕ i := (ϕ i 1 , ϕ i 2 ) ∈ (C 4,α (S 3 )) 2 and ψ i := (ψ i 1 , ψ i 2 ) ∈ (C 2,α (S 3 )) 2 . Let ε ∈ (0, ε κ ) and according to the result of Propositions 3.10, 3.11 and 3.13, we can find, u int := (u int,1 , u int,2 ) a solution of (3.7) in B rε ( x 1 ) ∪ B rε ( x 2 ) ∪ B rε ( x 3 ), which can be decomposed as…”

“…Here τ i > 0 is a constant which will be fixed later. Given ϕ i := (ϕ i 1 , ϕ i 2 ) ∈ (C 4,α (S 3 )) 2 and ψ i := (ψ i 1 , ψ i 2 ) ∈ (C 2,α (S 3 )) 2 such that (ϕ i 1 , ψ i 1 ) and (ϕ i 2 , ψ i 2 ) satisfy (3.5). We denote by u = u ε=1,τ i =1 .…”

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

“…We are interested in solutions which "concentrate" as the parameters ε and λ tend to 0 (in a sense that will be determined later). The aim of this paper is to study the influence of the non-linear gradient term |∇u| 2 . It turns out that the presence of this term can have significant influence on the existence of a solution, as well as on its asymptotic behavior.…”