zbMATH — the first resource for mathematics

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 \(\mathbb R^K\).

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