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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) Zbl 1074.35504
Finite energy solutions of the problem $\begin{gathered} u \in L_{loc}^{4n-1}(\mathbb R^K,\mathbb R^M),\;(K,M) \in N^+ \times N^+, \\ \Delta u = u P'_n(| u| ^2) \;\text{ in } \mathcal D'(\mathbb R^K,\mathbb R^M) \end{gathered}$ are studied where $$P_n$$ is a polynomial of the type $$P_n(t) = \frac 12 \Pi _{j=1}^n (t-k_j)^2,\;0<k_1<\dots <k_n<\infty$$. First, the case $$K=M=2$$ and solutions satisfying $$\int _{\mathbb R^2} P_n(| u| ^2) < \infty$$ are considered. A phenomenon of quantization of the mass $$\int _{\mathbb R^2} P_n(| u| ^2)$$ is established which generalizes a well-known result of Brezis, Merle and Riviere for the classical Ginzburg-Landau system in $$\mathbb R^2$$. Further, solutions satisfying $$\int _{\mathbb R^K} | \nabla u| ^2 < \infty$$ are considered and some Liouville-type results are established. Particularly, it is proved 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 constant. Finally, it is shown that any solution $$u$$ of the system is smooth and $$| u| ^2 \leq k_n$$ on $$\mathbb R^K$$.

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J60 Nonlinear elliptic equations