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Quantization effects for $$-\Delta u=u(1-| u|^ 2)$$ in $$\mathbb{R}^ 2$$. (English) Zbl 0809.35019
From the introduction: In the first part of this paper we study the problem $-\Delta u= u(1- | u|^ 2) \qquad \text{in } \mathbb{R}^ 2. \tag{1}$ Our main result is
Theorem 1. Assume $$u: \mathbb{R}^ 2\to \mathbb{C}$$ is a smooth function satisfying (1). Then $\int_{\mathbb{R}^ 2} (| u|^ 2 -1)^ 2= 2\pi d^ 2$ for some integer $$d=0,1,2, \dots,\infty$$.
In the second part we consider a sequence $$u_ n$$ of solutions of $$- \Delta u_ n= u_ n(1- | u_ n|^ 2)$$ in the disc $$B_{R_ n}$$ with $$R_ n\to \infty$$. Under some appropriate assumptions we prove that ${\textstyle {1\over 2}} \int_{B_{R_ n}} |\nabla u_ n|^ 2\geq \pi d^ 2\log R_ n- C.$

##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
##### Keywords:
quantization effects; Ginzburg-Landau energy
Full Text:
##### References:
 [1] F. Bethuel, H. Brezis & F. Hélein, Ginzburg-Landau Vortices, Birkhäuser (to appear). [2] Th. Cazenave, personal communication. [3] P. Hagan, Spiral waves in reaction diffusion equations, SIAM J. Applied Math. 42 (1982), p. 762-786. · Zbl 0507.35007 · doi:10.1137/0142054 [4] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), p. 135-148. · Zbl 0246.35025 · doi:10.1007/BF02760233 [5] L. Nirenberg, personal communication. [6] L. Shafrir, Remarks on solutions of ? ??u = u(1 ? |u|2) in ?2, C. R. Acad. Sci. Paris (to appear). · Zbl 0806.35030
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