Badiale, Marino and Benci, Vieri and Rolando, Sergio:
Three Dimensional Vortices in the Nonlinear Wave Equation
Bollettino dell'Unione Matematica Italiana Serie 9 2 (2009), fasc. n.1, p. 105-134, (English)
pdf (396 Kb), djvu (293 Kb). | MR 2493647 | Zbl 1178.35263
Sunto
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.
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