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We prove the existence of rotating solitary waves (vortices) for the nonlinear Klein-Gordon equation with nonnegative potential, by finding nonnegative cylindrical solutions to the standing equation where $x=(y,z)\in {ℝ}^k\times {ℝ}^{N-k}$, $N>k\ge 2$, $\mu >0$ and $\lambda \ge 0$. The nonnegativity of the potential makes the equation suitable for physical models and guarantees the wellposedness of the corresponding Cauchy problem, but it prevents the use of standard arguments in providing the functional associated to $(†)$ with bounded Palais-Smale sequences....

We prove the existence of cylindrical solutions to the semilinear elliptic problem $-\Delta u+\frac{u}{{\left|y\right|}^2}=f\left(u\right)$, $u\in H^1\left({ℝ}^N\right)$, $u\ge 0$, where $(y,z)\in {ℝ}^k\times {ℝ}^{N-k}$, $N>k\ge 2$ and $f$ has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schr¨odinger and Klein–Gordon equations.