zbMATH — the first resource for mathematics

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 \[ {\textstyle {1\over 2}} \int_{B_{R_ n}} |\nabla u_ n|^ 2\geq \pi d^ 2\log R_ n- C. \]

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.