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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
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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
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[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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