Qualitative Properties of Positive Solutions of a Mixed Order Nonlinear Schrodinger Equation

In this paper, we deal with the mixed local/nonlocal Schrödinger equation −u |x|→∆>limu+ 0+∞in(u−(x∆)R) n =su0+u = upin Rn, where n 2, s ∈ (0, 1), and p ∈ 1, nn−+22 . 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 −∆ + (−∆)s. In this context, we provide self-contained proofs of these results based on Fourier analysis techniques. © 2025 American Institute of Mathematical Sciences. All rights reserved.