Qualitative properties of positive solutions of a mixed order nonlinear Schr\"{o}dinger equation

TL;DR

This paper investigates positive solutions of a mixed local/nonlocal Schrödinger equation, establishing their existence, decay behavior, and symmetry properties using Fourier analysis and kernel properties.

Contribution

It proves the existence of positive solutions for the mixed Schrödinger equation and analyzes their decay and symmetry, introducing new regularity results and kernel-based methods.

Findings

Existence of positive solutions established.

Solutions exhibit power-type decay at infinity.

Solutions are radially symmetric under certain conditions.

Abstract

In this paper, we deal with the following mixed local/nonlocal Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{ll} - \Delta u + (-\Delta)^s u+u = u^p \quad \hbox{in $\mathbb{R}^n$,} u>0 \quad \hbox{in $\mathbb{R}^n$,} \lim\limits_{|x|\to+\infty}u(x)=0, \end{array} \right. \end{equation*} where $n\geqslant2$, $s\in (0,1)$ and $p\in\left(1,\frac{n+2}{n-2}\right)$. The existence of positive solutions for the above problem is proved, relying on some new regularity results. In addition, we study the power-type decay and the radial symmetry properties of such solutions. The methods make use also of some basic properties of the heat kernel and the Bessel kernel associated with the operator $- \Delta + (-\Delta)^s$: in this context, we provide self-contained proofs of these results based on Fourier analysis techniques.