POSITIVE SOLUTIONS FOR A FRACTIONAL MAGNETIC SCHRÖDINGER EQUATIONS WITH SINGULAR NONLINEARITY AND STEEP POTENTIAL

The paper deals with the following magnetic Schrödinger equation with singular nonlinearity and steep potential

$\left\{ \begin{array}{l} ( - \Delta )_A^su + {V_\lambda }(x)u = \mu f(x){u^{ - \gamma }} + g(x){u^{p - 1}},{\rm{in}}\;\;{{\mathbb{R}}^N},\\ u > 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{in}}\;\;{{\mathbb{R}}^N}, \end{array} \right.$

where $ (-\Delta)_A^{s} $ is the fractional magnetic Laplacian operator with $ 0<s<1 $, and $ 0<\gamma<1 $, $ 2<p<2_s^{*} $ $ \left(2_s^{*} = \frac{2N}{N-2s}\ \ \mathrm{for} \ \ N>2s \right) $, the potential $ V_{\lambda}(x) = \lambda V^{+}(x)-V^{-}(x) $ with $ V^{\pm} = \max\{\pm V, 0\} $, $ \lambda, \mu>0 $ are parameters, $ f\in L^{\frac{p}{p+\gamma-1}}( \mathbb{R}^N) $ is a positive weight, while $ g\in L^{\infty}( \mathbb{R}^N) $ is a sign-changing function. By applying the Nehari manifold and fibering map, we obtain the existence of at least two positive solutions, where some new estimates will be established. Recent some results from the literature are extended.