Three Dimensional Vortices in the Nonlinear Wave Equation

Badiale, Marino; Benci, Vieri; Rolando, Sergio

Badiale, Marino ; Benci, Vieri ; Rolando, Sergio

Bollettino dell'Unione Matematica Italiana, Tome 2 (2009), p. 105-134 / Harvested from Biblioteca Digitale Italiana di Matematica

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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 $-\Delta u+\frac{\mu}{|y|^{2}}u+\lambda u=g(u),\quad u\in H^{1}(\mathbb{R}^{N})% ,\quad\int_{\mathbb{R}^{N}}\frac{u^{2}}{|y|^{2}}\,dx<\infty,$ where $x=(y,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}$ , $N>k\geq 2$ , $\mu>0$ and $\lambda\geq 0$ . 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 $({\dagger})$ with bounded Palais-Smale sequences.

Publié le : 2009-02-01

MR 2493647 | Zbl 1178.35263

@article{BUMI_2009_9_2_1_105_0,
     author = {Marino Badiale and Vieri Benci and Sergio Rolando},
     title = {Three Dimensional Vortices in the Nonlinear Wave Equation},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {2},
     year = {2009},
     pages = {105-134},
     zbl = {1178.35263},
     mrnumber = {2493647},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_1_105_0}
}

Badiale, Marino; Benci, Vieri; Rolando, Sergio. Three Dimensional Vortices in the Nonlinear Wave Equation. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 105-134. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_1_105_0/

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