Existence and uniqueness of ground state solutions for the planar Schr\"odinger-Newton equation on the disc

TL;DR

This paper proves the existence, symmetry, and uniqueness of positive ground state solutions for the planar Schr"odinger-Newton equation on a disc, and analyzes their behavior as the disc radius grows large.

Contribution

It establishes the first rigorous proof of existence, symmetry, and uniqueness of ground states for this equation on a disc, including their asymptotic behavior.

Findings

Existence of positive ground state solutions on the disc.

Radial symmetry of solutions proven.

Unique ground state solutions converge to trivial as radius increases.

Abstract

This paper is concerned with the existence and qualitative properties of positive ground state solutions for the planar Schr\"odinger-Newton equation on the disc. First, we prove the existence and radial symmetry of all the positive ground state solutions by employing the symmetric decreasing rearrangement and Talenti's inequality. Next, we develop Newton's theorem and then use the contraction mapping principle to establish the uniqueness of the positive ground state solution for the Schr\"odinger-Newton equation on the disc in the two dimensional case. Finally, we show that the unique positive ground state solution converges to the trivial solution as the radius $R$ tending to infinity, which is totally different from the higher dimensional case in \cite{Guo-Wang-Yi}.