# 1400 mathematische Abiturientenaufgaben by Schaeffer A., Fuchs A.

By Schaeffer A., Fuchs A.

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5, where rλ = |zλ − xλ |. 5, using the Green’s representation formula we have ζλ (x) = 1 µ4,λ 2π log Ωλ 1 eu˜λ (y) v˜4,λ (y)dy |x − rδλλ y| H(rλ x + xλ , δλ y + xλ )eu˜λ (y) v˜4,λ (y)dy. 12) by log λ getting −ζλ (x) log λ = 1 1 log 4πb + 4πbR(x0 ) + o(1). 13) Repeating the same argument for the first derivatives we can show that the 1 (IR2 \ {0}). This gives us a contradiction since we have a convergence is Cloc λ sequence of points zˆλ = zλr−x such that ∇ζλ (ˆ zλ ) = 0 and which converges to a λ point zˆ with |ˆ z | = 1.

This proves the Theorem for i = 5, . . , 9. 1 Appendix Uniform boundedness of ∂u ˜λ ∂xi in Ωλ . 1. Let u ˜λ be as defined before. We have ∂u ˜λ C ≤ ∂xi |x| in Ωλ . 1) Proof Let Ω be such that B(x0 , 1) ⊂ Ω. 2) that uλ (x) → ¯ \ B(x0 , 1) . Hence we have 8πG(x, x0 ) in C 1 Ω C ∂u ˜λ ∂G(x, xλ ) ≤ C sup ≤ ∂xi ∂xi |x − xλ | Ω\B(x0 , 1 ) for x ∈ Ω \ B(x0 , 2). 2 Recalling that u ˜λ (y) = uλ (δλ y + xλ ) − uλ ∞ a simple computation shows us that ∂u ˜λ C ∂uλ (y) = δλ (δλ y + xλ ) ≤ ∂yj ∂xj |y| for y ∈ Ωλ \ B(0, δ2λ ).

31] G. Tarantello, Multiple condensate solutions for the Chern-SimonsHiggs theory. J. Math. Phys. 37 (1996), 3769-3796. [32] V. H. Weston On the asymptotic solution of a partial differential equation with an exponential nonlinearity. SIAM J. Math. Anal. 9 (1978), 1030-1053.