Existence and decays of solutions for fractional Schrödinger equations with decaying potentials

We revisit the following fractional Schrödinger equation \begin{align}\label{1a} \varepsilon^{2s}(-Δ)^su +Vu=u^{p-1},,,,u>0,\ \ \ $\mathrm{i}\mathrm{n}$\ \R^N, \end{align} where $\epsilon >0$ is a small parameter, $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian, $s\in (0,1)$, $p\in (2,2_s^{\ast })$, $2_s^{\ast }=\frac{2N}{N-2s}$, $N>2s$, $V\in C({\mathbb{R}}^N,[0,+\mathrm{\infty }))$ is a potential. Under various decay assumptions on $V$, we introduce a uniform penalization argument combined with a comparison principle and iteration process to detect an explicit threshold value $p_{\ast }$, such that the above problem admits positive concentration solutions if $p\in (p_{\ast },2_s^{\ast })$, while it has no positive weak solutions for $p\in (2,p_{\ast })$ if $p_{\ast }>2$, where the threshold $p_{\ast }\in [2,2_s^{\ast })$ can be characterized explicitly by \begin{equation*}\label{qdj111} p_=\left{\begin{array}{l} 2+\frac {2s}{N-2s} \ \ \ \text { if } \lim\limits_{|x| \to \infty} (1+|x|^{2s})V(x)=0,\vspace{1mm} 2+\frac ω{N+2s-ω} \text { if } 0<\inf (1+|x|^ω)V(x)\le \sup (1+|x|^ω)V(x)< \infty \text { for some } ω\in [0, 2s],\vspace{1mm} 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text { if } \inf V(x)\log(e+|x|^2)>0. \end{array}\right. \end{equation} Moreover, corresponding to the various decay assumptions of $V(x)$, we obtain the decay properties of the solutions at infinity.