Standing waves of fractional Schrödinger equations with critical nonlinearities and decaying potentials

In this paper, we study the following nonlinear fractional Schrödinger equation$ \begin{align} \varepsilon^{2s}(-\Delta)^s u+V(x) u = f(u), \ \ x\in \mathbb R^N, \end{align} ~~~~~(1)$where $ s\in(0, 1), N>2s, \varepsilon>0 $ is a small parameter, $ (-\Delta)^s $ is the fractional Laplacian operator, $ V\in C( \mathbb R^N, [0, +\infty)) $ is a general potential which may vanish at infinity and $ f $ is a general nonlinear function including both critical and subcritical cases, which may not satisfy the Ambrosetti-Rabinowitz condition and the monotonicity condition on $ \frac {f(t)}{t} $. Under some weaker assumptions on $ f $ and $ V $, we establish the existence of positive solutions $ u_\varepsilon $ for problem (1) concentrating at a local minimum of $ V $ as $ \varepsilon\rightarrow0 $. Furthermore, the decay estimates for the positive solutions $ u_\varepsilon $ depending on the decay rates of the potential $ V $ are obtained. The proofs of our main results are much more complicated than those in [3,15,21,32] due to the double effects of decaying potentials and nonlinearities without the $ (AR) $ condition and the monotonicity condition on $ \frac {f(t)}{t} $. Indeed, we establish a unified approach to deal with equation (1) in which the nonlinearity $ f $ satisfies Berestycki-Lions type conditions and the potential $ V $ is non-vanishing or vanishing at infinity.