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

Abstract

We consider the following fractional Schrödinger equation involving critical exponent:

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

where \(2_s^*=\frac{2N}{N-2s}\), \((y',y'') \in \mathbb {R}^{2} \times \mathbb {R}^{N-2}\) and \(V(y) = V(|y'|,y'')\) and \(Q(y) = Q(|y'|,y'')\) are bounded nonnegative functions in \(\mathbb {R}^{+} \times \mathbb {R}^{N-2}\). By using finite-dimensional reduction method and local Pohozaev-type identities, we show that if \(\frac{2+N-\sqrt{N^2+4}}{4}< s <\min \{\frac{N}{4}, 1\}\) and \(Q(r,y'')\) has a stable critical point \((r_0,y_0'')\) with \(r_0>0,\; Q(r_0,y_0'') > 0\) and \( V(r_0,y_0'') > 0\), then the above problem has infinitely many solutions, whose energy can be arbitrarily large.