Normalized Solutions for an Anisotropic Nonlinear Schrödinger Equation with Potentials

In this paper, we study the existence of normalized solutions for anisotropic nonlinear Schrödinger equation with potentials which are bounded and converge to positive constants at infinity. In the mass subcritical case, we show the energy functional is bounded below on the L2-sphere and prove the existence of a global minimizer. In the critical case, we establish a similar result under a condition on the mass. For the supercritical case, we introduce a Pohozaev-Nehari manifold and prove the existence of a positive-energy solution via the minimax methods. The compactness is recovered through detailed analysis involving the associated limit problem and strict monotonicity conditions on the potentials. To the best of our knowledge, this is the first study on the existence of normalized solutions for anisotropic nonlinear Schrödinger equation, and our approach provides a unified variational framework for handling anisotropic fractional operators with competing nonlinearities.