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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
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