Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

We consider the following nonlinear fractional scalar field equation$$(-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N,$$where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and$1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, weshow that this problem hasinfinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.