Three Dimensional Vortices in the Nonlinear Wave Equation

Badiale, Marino ; Benci, Vieri ; Rolando, Sergio

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 105-134

Résumé

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 \begin{equation} \tag{\dag} -\Delta u + \frac{\mu}{|y|^{2}} u + \lambda u = g(u), \quad u \in H^{1}(\mathbb{R}^{N}), \quad \int_{\mathbb{R}^{N}} \frac{u^{2}}{|y|^{2}} \, dx \infty,\end{equation} where $x=(y,z) \in \mathbb{R}^{k} \times \mathbb{R}^{N-k}$, $N > k \ge 2$, $\mu > 0$ 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 $(\dag)$ with bounded Palais-Smale sequences.