Differential and Integral Equations

Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in $\mathbb{R}^K$

Alberto Farina

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the finite-energy solutions of a generalization of the classical Ginzburg-Landau system (problem (0.1)). In the first part of this paper we consider the solutions of (0.1) satisfying $K=M=2$ and $\int_{\mathbb{R}^2} P_n(\vert u \vert^2) <\infty$. We establish a phenomenon of quantization for the "mass" $\int_{\mathbb{R}^2} P_n(\vert u \vert^2)$ that genera\-lizes a well-known result of Brezis, Merle and Rivière for the classic Ginzburg-Landau system of equations in $\mathbb{R}^2$. The second part is devoted to solutions of (0.1) satisfying $\int_{\mathbb{R}^K} \vert\nabla u\vert^2 < \infty$. We establish some Liouville-type results and in particular we prove that any locally $L^3$ solution of the Ginzburg-Landau system $-\Delta u = u(1-\vert u \vert^2)$ in $\mathbb{R}^K$ (K$>$1) satisfying $\int_{\mathbb{R}^K}\vert \nabla u\vert^2 <\infty$, is a constant function. We also obtain that any solution of (0.1) with finite energy (0.4) is a constant. In the last section we prove that any solution $u$ of $(0.1)$ is smooth and satisfies $\vert u \vert^2 \le k_n $ on $\mathbb{R}^K$.

Article information

Source
Differential Integral Equations, Volume 11, Number 6 (1998), 875-893.

Dates
First available in Project Euclid: 30 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367329481

Mathematical Reviews number (MathSciNet)
MR1659256

Zentralblatt MATH identifier
1074.35504

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 58E50: Applications

Citation

Farina, Alberto. Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in $\mathbb{R}^K$. Differential Integral Equations 11 (1998), no. 6, 875--893. https://projecteuclid.org/euclid.die/1367329481


Export citation