Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term

Abstract

Given Ω bounded open regular set of ℝ 2 and x 1 , x 2 , ..., x m Ω, we give a sufficient condition for the problemto have a positive weak solution in Ω with u = 0 on ∂Ω, which is singular at each x i as the parameters r, l > 0 tend to 0 and where f(u) is dominated exponential nonlinearities functions. 2000 Mathematics Subject Classification: 35J60; 53C21; 58J05.

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

“…in which case (1) can be written as 2 w =Qe 4w , the solutions of which give rise to the conformal metric g w = e 2w g eucl whose Q-curvature is given byQ. There is by now extensive literature about this problem, and we refer to [10] and [15] for references and recent developments.…”

“…In particular, if we take λ = O(ε 2/3 ), then condition (A λ ) is satisfied. Under assumption (A λ ), we can treat equation (2) as a perturbation of the following equation:…”

“…We are interested in studying equations of type 2 w + Q λ (w) -24 e w + 2 4(1-γ ) ε 8(1-γ ) ((1 + ε 2 )τ ) 4(1-γ ) e γ w = 0 (28) inB R ε,λ . Given ϕ ∈ C 4,α (S 3 ) and ψ ∈ C 2,α (S 3 ).…”

“…Similarly, given some boundary dataφ := (φ i ) ∈ (C 4,α (S 3 )) m andψ = (ψ i ) ∈ (C 2,α (S 3 )) m satisfying (27) and (40), some parametersη := (η i ) ∈ R m satisfying (41), we can use the result of Proposition 5 to find a solution u ext of (provided ε ∈ (0, ε κ )) 2 u + Q λ (u)ρ 4 e u + e γ u = 0, in B r ε,λ x j , ∀1 ≤ j ≤ m.…”