The authors state that the following problem \[ (*):\;\varepsilon^{2s}(-\Delta)^su+V(x)u=f(u)\text{ in }\mathbb R^n, \] has a positive solution \(u_\varepsilon\in H^s(\mathbb R^n)\) for all \(\varepsilon\in (0,\varepsilon_0)\), where \(\varepsilon_0>0\), and that if \(x_\varepsilon\) is an associated global maximum point, then \(\displaystyle \lim_{\varepsilon\to0}V(x_\varepsilon)=V_0=\inf_{\xi\in \Lambda}V(\xi)<\min_{\zeta\in\partial \Lambda}V(\zeta)\) where \(\Lambda\) is a bounded set in \(\mathbb R^n\) and \(\partial \Lambda\) is its boundary. Apropos of \(H^s(\mathbb R^n),(-\Delta)^s\), they are, respectively, the fractional Sobolev space and the fractional Laplacian operator for \(s\in (0,1)\). Furthermore, the authors assume that \(\displaystyle \liminf\limits_{|\xi|\to\infty}V(\xi)>\inf_{\xi\in\mathbb R^n}V(\xi)=V_1>0\). With regard to \(f\), the authors suppose that it satisfies meaningful conditions (Theorem 1.1). The novelty of the results is that the authors consider \(V\) and \(f\) belonging to a particular large class of functions. The authors take recourse in using Caffarelli and Silvester’s method, through which looking for a solution of \((*)\) reduces to seeking a solution of \[ (**):\;-\operatorname{div}(y^{1-2s}\nabla w)=0\text{ in }\mathbb R_+^{n+1}\text{ and } \]
\[ -k_s\frac{\partial w}{\partial \nu}=-V(\varepsilon x)w+g(\varepsilon x,\omega)\text{ on }\mathbb R^n\times\{0\}. \] Here, \(k_s\) is a parameter relying on \(s\) given in terms of the gamma function and \(\displaystyle \frac{\partial w}{\partial \nu}:=\lim_{y\to0^+}y^{1-2s}\frac{\partial w}{\partial y}(x,y)=-\frac{1}{k_s}(-\Delta)^su(x)\), and \(g(x,t)=\mathcal{X}_\Lambda(x)f(t)+(1-\mathcal{X}_\Lambda)\widetilde{f}(t)\) for \((x,t)\in \mathbb R^n\times \mathbb R\), where \(\widetilde{f}(t)=f(t)\text{ if }t\leq a\text{ and }\widetilde{f}(t)=\displaystyle\frac{V_1}{k}t\text{ if }t\geq a>0\) such that \(k,a\) are fixed numbers, and \(\mathcal{X}_\Lambda\) stands for the indicator function on \(\Lambda\). Among the tools used by the authors, they show that \(E_\varepsilon(\cdot)\), the energy functional associated to \((**)\) satisfies the Palais-Small compactness condition and has a nonnegative critical point on \(X^{1,s}\), the completion of the set of smooth functions with compact support on the closure of \(\mathbb R^{n+1}\), with respect to a suitable norm (Lemma 2.2 and Theorem 2.1).