Large Energy Bubble Solutions for Supercritical Fractional Schrödinger Equation with Double Potentials

Abstract

We consider the following supercritical fractional Schrödinger equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u + V(y) u=Q(y)u^{2_s^*-1+\varepsilon }, \;u>0, &{}\hbox { in } {\mathbb {R}}^{N},\\ u \in D^s( {\mathbb {R}}^{N}), \end{array}\right. } \end{aligned}$$(*)

where \(2_s^*=\frac{2N}{N-2s},\; N> 4s\), \(0< s < 1\), \((y',y'') \in {\mathbb {R}}^{2} \times {\mathbb {R}}^{N-2}\), \(V(y) = V(|y'|,y'')\) and \(Q(y) = Q(|y'|,y'') \not \equiv 0\) are two bounded non-negative functions. Under some suitable assumptions on the potentials V and Q, we will use the finite-dimensional reduction argument and some local Pohozaev type identities to prove that for \(\varepsilon > 0\) small enough, the problem \((*)\) has a large number of bubble solutions whose functional energy is in the order \(\varepsilon ^{-\frac{N-4s}{(N-2s)^2}}.\)