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

We consider the following supercritical fractional Schrödinger equation: * \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u + V(y) u=Q(y)u^{2_s^*-1+\varepsilon }, \;u>0, &{}\hbox { in } {\mathbb {R}}^{N},\\ u \in D^s( {\mathbb {R}}^{N}), \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(-\mathrm{\Delta })}^{s}u+V(y)u=Q(y)u^{2_s^{\ast }-1+\epsilon },u>0,\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ u\in D^s({\mathbb{R}}^N),\hfill \end{array}\right)\end{array}$$ where \(2_s^*=\frac{2N}{N-2s},\; N> 4s\) $2_s^{\ast }=\frac{2N}{N-2s},N>4s$ , \(0< s < 1\) $0<s<1$ , \((y',y'') \in {\mathbb {R}}^{2} \times {\mathbb {R}}^{N-2}\) $(y^{\prime },y^{\prime \prime })\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$ , \(V(y) = V(|y'|,y'')\) $V(y)=V(|y^{\prime }|,y^{\prime \prime })$ and \(Q(y) = Q(|y'|,y'') \not \equiv 0\) $Q(y)=Q(|y^{\prime }|,y^{\prime \prime })\not\equiv 0$ 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 \(\varepsilon > 0\) $\epsilon >0$ small enough, the problem \((*)\) $(\ast )$ has a large number of bubble solutions whose functional energy is in the order \(\varepsilon ^{-\frac{N-4s}{(N-2s)^2}}.\) ${\epsilon }^{-\frac{N-4s}{{(N-2s)}^2}}.$