Vortices with Prescribed L2 Norm in the Nonlinear Wave Equation

Abstract

Abstract We prove the existence of rotating solitary waves (vortices) with enough largely prescribed L2 norm for the nonlinear Klein-Gordon equation with nonnegative potential, which makes the equation suitable for physical models and guarantees the well-posedness of the corresponding Cauchy problem. This is done by finding nonnegative cylindrical solutions to the standing equation where x = (y, z) ∈ ℝk × RN–k, N > k ≥ 0 and μ > 0 is the fundamental eigenvalue, namely, the Lagrange multiplier of the minimization problem (with infimum over a suitable subspace of H1(ℝN)).