Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system

Abstract

In this paper, we consider the Dirichlet problem for an elliptic system on a ball in R 2 . By investigating the properties for the corresponding linearized equations of solutions, and adopting the Pohozaev identity and Implicit Function Theorem, we show the uniqueness and the structure of solutions.

On the solutions to a Liouville-type system involving singularity

“…Proof We refer the reader to [8,9] for the proof of this lemma. In fact, from (3.8), it is easy to see that ψ 1 (r ) decreases strictly and hence φ 1 (r ) increases strictly near r = 0.…”

“…In physics, just to name but a few, it represents the electric potential induced by charge carriers in the theory of electrolytes [24], and the Newtonian potential of a cluster of self-gravitating mass distribution [1,4,25,26]. Moreover, it is also induced by a mean field equation which comes from the spherical Onsager vortex theory, bridging the gap between statistical mechanics of classical vortices and the random surface problem [7,21], and is considered to deal with topics closely related to the abelian model in the Chern-Simons theories [9,10,15,27].…”

On the uniqueness and structure of solutions to a coupled elliptic system

“…While such techniques have been widely used in proving the uniqueness of solutions of scalar equations (see [6,20,21,29,30]), the generalization to systems of equations is not straightforward due to the coupling. Here we use some ideas appearing in our earlier works [7,8].…”

“…By virtue of the method of moving planes, the Dirichlet problem of (1.1) in B R (0) with (u, v) = (0, 0) on ∂ B R (0) can be reduced to a corresponding system of ordinary differential equations with u(R) = v(R) = 0, see for example [2,5,8,14,22,36]. While the proof is standard, for the reader's convenience, we will present it in detail in Appendix A.…”