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

Abstract

The existence of singular limit solutions are investigated by establishing a new Liouville type theorem for nonlinear elliptic system by using the Pohozaev type identity and the nonlinear domain decomposition method.

Blow up solutions for 2-dimensional semilinear elliptic of liouville type with nonlinear gradient term

“…where λ > 0 and z 0 ∈ R 2 . The system of equations (5), under slightly more general conditions which include (7) as a special case, find their applications in the physics of charged particle beams. See [8,17,18].…”

“…A natural question that arises; can one build a solution u 1 respectively u 2 that concentrates in z 1 , z 2 respectively in z 2 , z 3 . A positive answer on this question was given in [5] where the authors studied the following problem…”

“…More specifically, u 1 blow up in z 1 , z 2 and u 2 blow up in z 2 , z 3 , where {z 1 , z 2 , z 3 } ⊂ Ω which will be chosen later. In the following Lemma, we give a necessary condition about the position of the points (z 1 , z 2 , z 3 ), using techniques inspired by the work of Suzuki [26] and Baraket et al [5].…”

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

“…They proved the existence of singular limit solutions which blow-up on different points as ρ tends to 0 using the nonlinear domain decomposition method. In dimension 2, the L-system has been interested by several authors [7,10,16,17,19]. Recently, the authors in [7] considered the following problem…”

“…In dimension 2, the L-system has been interested by several authors [7,10,16,17,19]. Recently, the authors in [7] considered the following problem…”