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

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 $\mathbb{R}^K$.