Three Dimensional Vortices in the Nonlinear Wave Equation

Marino Badiale; Vieri Benci; Sergio Rolando

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 1, page 105-134
  • ISSN: 0392-4041

Abstract

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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 - Δ ⁢ u + μ | y | 2 ⁢ u + λ ⁢ u = g ⁢ ( u ) , u ∈ H 1 ⁢ ( ℝ N ) , ∫ ℝ N u 2 | y | 2 ⁢ 𝑑 x < ∞ , where x = ( y , z ) ∈ ℝ k × ℝ N - k , N > k ≥ 2 , μ > 0 and λ ≥ 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.

How to cite

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Badiale, Marino, Benci, Vieri, and Rolando, Sergio. "Three Dimensional Vortices in the Nonlinear Wave Equation." Bollettino dell'Unione Matematica Italiana 2.1 (2009): 105-134. <http://eudml.org/doc/290553>.

@article{Badiale2009,
abstract = {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.},
author = {Badiale, Marino, Benci, Vieri, Rolando, Sergio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {105-134},
publisher = {Unione Matematica Italiana},
title = {Three Dimensional Vortices in the Nonlinear Wave Equation},
url = {http://eudml.org/doc/290553},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Badiale, Marino
AU - Benci, Vieri
AU - Rolando, Sergio
TI - Three Dimensional Vortices in the Nonlinear Wave Equation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/2//
PB - Unione Matematica Italiana
VL - 2
IS - 1
SP - 105
EP - 134
AB - 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.
LA - eng
UR - http://eudml.org/doc/290553
ER -

References

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