Qualitative properties of positive solutions of a mixed order nonlinear Schrödinger equation

In this paper, we deal with the mixed local/nonlocal Schrödinger equation$ \begin{align} \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{align} $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 also make use 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.