Construction of infinitely many solutions for fractional Schrödinger equation with double potentials

We consider the following fractional Schrödinger equation involving critical exponent: \(\begin{aligned} (-\Delta )^su+V(y)u=Q(y)u^{2_s^*-1}, \;u>0, \; \hbox { in } \mathbb {R}^{N},\; u \in D^s(\mathbb {R}^N), \end{aligned}\) $$\begin{array}{c}\hfill {(-\mathrm{\Delta })}^{s}u+V(y)u=Q(y)u^{2_s^{\ast }-1},u>0,\text{in}{\mathbb{R}}^N,u\in D^s({\mathbb{R}}^N),\end{array}$$ where \(2_s^*=\frac{2N}{N-2s}\) $2_s^{\ast }=\frac{2N}{N-2s}$ , \((y',y'') \in \mathbb {R}^{2} \times \mathbb {R}^{N-2}\) $(y^{\prime },y^{\prime \prime })\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$ and \(V(y) = V(|y'|,y'')\) $V(y)=V(|y^{\prime }|,y^{\prime \prime })$ and \(Q(y) = Q(|y'|,y'')\) $Q(y)=Q(|y^{\prime }|,y^{\prime \prime })$ are bounded nonnegative functions in \(\mathbb {R}^{+} \times \mathbb {R}^{N-2}\) ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N-2}$ . By using finite-dimensional reduction method and local Pohozaev-type identities, we show that if \(\frac{2+N-\sqrt{N^2+4}}{4}< s <\min \{\frac{N}{4}, 1\}\) $\frac{2+N-\sqrt{N^2+4}}{4}<s<min\{\frac{N}{4},1\}$ and \(Q(r,y'')\) $Q(r,y^{\prime \prime })$ has a stable critical point \((r_0,y_0'')\) $(r_0,y_0^{\prime \prime })$ with \(r_0>0,\; Q(r_0,y_0'') > 0\) $r_0>0,Q(r_0,y_0^{\prime \prime })>0$ and \( V(r_0,y_0'') > 0\) $V(r_0,y_0^{\prime \prime })>0$ , then the above problem has infinitely many solutions, whose energy can be arbitrarily large.