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

We revisit the following fractional Schrödinger equation 0.1 \(\begin{aligned} \varepsilon ^{2s}(-\Delta )^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \textrm{in}\ {\mathbb {R}}^N, \end{aligned}\) $$\begin{array}{c}\hfill {\epsilon }^{2s}{(-\mathrm{\Delta })}^{s}u+Vu=u^{p-1},u>0,\text{in}{\mathbb{R}}^N,\end{array}$$ where \(\varepsilon >0\) $\epsilon >0$ is a small parameter, \((-\Delta )^s\) ${(-\mathrm{\Delta })}^s$ denotes the fractional Laplacian, \(s\in (0,1)\) $s\in (0,1)$ , \(p\in (2, 2_s^*)\) $p\in (2,2_s^{\ast })$ , \(2_s^*=\frac{2N}{N-2s}\) $2_s^{\ast }=\frac{2N}{N-2s}$ , \(N>2s\) $N>2s$ , \(V\in C\big ({\mathbb {R}}^N, [0, +\infty )\big )\) $V\in C({\mathbb{R}}^N,[0,+\infty ))$ is a general potential. Under various assumptions on V(x) at infinity, including V(x) decaying with various rate at infinity, we introduce a unified penalization argument and give a complete result on the existence and nonexistence of positive solutions. More precisely, we combine a comparison principle with iteration process to detect an explicit threshold value \(p_*\) $p_{\ast }$ , such that the above problem admits positive concentration solutions if \(p\in (p_*, \,2_s^*)\) $p\in (p_{\ast },2_s^{\ast })$ , while it has no positive weak solutions for \(p\in (2,\,p_*)\) $p\in (2,p_{\ast })$ if \(p_*>2\) $p_{\ast }>2$ , where the threshold \(p_*\in [2, 2^*_s)\) $p_{\ast }\in [2,2_s^{\ast })$ can be characterized explicitly by

\(\begin{aligned} p_*=\left\{ \begin{array}{ll} 2+\frac{2s}{N-2s} &{}\quad \text{ if } \lim \limits _{|x| \rightarrow \infty } (1+|x|^{2s})V(x)=0,\\ 2+\frac{\omega }{N+2s-\omega } &{}\quad \text{ if } 0\!<\!\inf (1\!+\!|x|^\omega )V(x)\!\le \! \sup (1\!+\!|x|^\omega )V(x)\!<\! \infty \text{ for } \text{ some } \omega \!\in \! [0, 2s],\\ 2&{}\quad \text{ if } \inf V(x)\log (e+|x|^2)>0. \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill p_{\ast }=\left\{\begin{array}{cc}2+\frac{2s}{N-2s}\hfill & \text{if}\underset{|x|\to \infty }{lim}{(1+|x|}^{2s})V(x)=0,\hfill \\ 2+\frac{\omega }{N+2s-\omega }\hfill & {\text{if}0<inf(1+|x|}^{\omega }{)V(x)\le sup(1+|x|}^{\omega })V(x)<\infty \text{for}\text{some}\omega \in [0,2s],\hfill \\ 2\hfill & \text{if}infV(x)log(e+|x{|}^2)>0.\hfill \end{array}\right)\end{array}$$ Moreover, corresponding to the various decay assumptions of V(x), we obtain the decay properties of the solutions at infinity. Our results reveal some new phenomena on the existence and decays of the solutions to this type of problems.