  In this paper, we study the existence of radial and nonradial solutions to
the scalar field equations with fractional operators. For radial solutions, we
prove the existence of infinitely many solutions under $N \geq 2$. We also show
the existence of least energy solution (with the Pohozaev identity) and its
mountain pass characterization. For nonradial solutions, we prove the existence
of at least one nonradial solution under $N \geq 4$ and infinitely many
nonradial solutions under either $N =4$ or $N \geq 6$. We treat both of the
zero mass and the positive mass cases.
