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Finite-energy solutions, quantization effects, and Liouville-type results for a variant of the Ginzburg-Landau systems in $$\mathbb{R}^ K$$. (English. Abridged French version) Zbl 0885.35032
Summary: We study the finite-energy solutions $$u\in L^{8n-2}_{\text{loc}}(\mathbb{R}^K,\mathbb{R}^M)$$ of the problem $\Delta u= uP_n'(|u|^2),\quad n\in\mathbb{N}^+,\quad P_n(t)= {1\over 2}\prod^n_{j= 1} (t-k_j)^2,\quad 0<k_1<\cdots< k_n<\infty.\tag{1}$ In the first part of this note, we consider the solutions of (1) satisfying $$K= M=2$$ and $$\int_{\mathbb{R}^2} P_n(|u|^2)<+\infty$$. We establish a phenomenon of quantization for the “mass” $$\int_{\mathbb{R}^2} P_n(|u|^2)$$, that generalizes a well-known result of Brézis, Merle, and Rivière for the classic Ginzburg-Landau system of equations in $$\mathbb{R}^2$$. The second part is devoted to solutions of (1) satisfying $$\int_{\mathbb{R}^K}|\nabla u|^2<+\infty$$. We establish some Liouville-type results and, in particular, we prove that any locally $$L^3$$ solution of the Ginzburg-Landau system $$-\Delta u= u(1-|u|^2)$$ in $$\mathbb{R}^K$$ $$(K>1)$$ satisfying $$\int_{\mathbb{R}^K}|\nabla u|^2<+\infty$$, is a constant function. We also prove that any solution of (1) with finite energy $$E_4(u):= E_2(u)+ E_3(u)<+\infty$$ is a constant.

##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
##### Keywords:
phenomenon of quantization for the mass
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