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

Abstract

In this paper, we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear Schrödinger equation. We establish the existence and uniqueness of solutions to the Dirichlet problem on the ball. In addition, the nonexistence of the ground state solutions under certain conditions on the nonlinearities and the complete structure of different types of solutions to the shooting problem are proved.

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.…”

Existence, uniqueness and stability of positive solutions to sublinear elliptic systems

“…Several authors have taken that approach for the existence of the solutions [30,33,34] and much success has been achieved for Lane-Emden systems. A more general approach, using shooting method and linearized equations for the radial case, has been taken by the present authors [6,7]. Korman [21] obtained a uniqueness and exact multiplicity result for the one-dimensional case.…”

“…Korman [21] obtained a uniqueness and exact multiplicity result for the one-dimensional case. A more general approach, using shooting method and linearized equations for the radial case, has been taken by the present authors [6,7].…”