Singularly perturbed fractional Schr\"{o}dinger equation involving a general critical nonlinearity

Abstract

In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schr\"{o}dinger problem \begin{align*} \varepsilon^{2s}(-\Delta)^su+V(x)u=f(u) \ \ \ \mbox{in} \ \ \ \mathbb{R}^N, \end{align*} where $N>2s$ and the nonlinearity $f$ has critical growth. By using the variational approach, we construct a localized bound-state solution concentrating around an isolated component of the positive minimum point of $V$ as $\varepsilon\rightarrow 0$. Our result improves the study made in X. He and W. Zou ({\it Calc. Var. Partial Differential Equations}. 55-91(2016)), in the sense that, in the present paper, the {\it Ambrosetti-Rabinowitz} condition and {\it monotonicity} condition on $f(t)/t$ are not required.