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

TL;DR

This paper investigates the existence and concentration of solutions for a singularly perturbed fractional Schrödinger equation with critical nonlinearity, using variational methods without requiring the Ambrosetti-Rabinowitz condition.

Contribution

It extends previous results by removing the need for the Ambrosetti-Rabinowitz and monotonicity conditions on the nonlinearity.

Findings

Existence of localized bound-state solutions concentrating near minimum points of V.

Solutions are constructed without the Ambrosetti-Rabinowitz condition.

Improves upon prior work by relaxing nonlinear growth conditions.

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.