Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential

TL;DR

This paper proves the uniqueness of positive solutions for a class of fractional nonlinear elliptic equations with harmonic potential, addressing an open question in the mathematical analysis of such equations.

Contribution

It establishes the first rigorous proof of uniqueness for positive solutions to fractional elliptic equations with harmonic potential, expanding understanding in this area.

Findings

Proves uniqueness of positive solutions under specified conditions.

Addresses and resolves an open problem from previous research.

Provides a mathematical framework for fractional elliptic equations with harmonic potential.

Abstract

In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, \end{align*} where $n \geq 1$, $0<s<1$, $\omega>-\lambda_{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, and $\lambda_{1,s}>0$ is the lowest eigenvalue of the operator $(-\Delta)^s + |x|^2$. This solves an open question raised in \cite{SS} concerning the uniqueness of solutions to the equation.