# Download Families of Meromorphic Functions on Compact Riemann by Makoto Namba (auth.) PDF

By Makoto Namba (auth.)

Similar mathematics books

Extra info for Families of Meromorphic Functions on Compact Riemann Surfaces

Sample text

This is in fact independent of λ. , Mε := u ∈ E: distλ (u, M) ε . 17. Assume (a1 ), (a2 ) and (f0 ). Then there exists ε0 > 0 such that Kλ P ± ε ⊂ int P ± for all 0 < ε ε ε0 , λ 0, so P ± is Kλ -attractive uniformly in λ. Consequently, ηλt P ± ε ⊂ int P ± ε for all t > 0, 0 < ε ε0 and λ 0. P ROOF. We write Vλ (x) = λa(x) + 1. For u ∈ E, we denote v = Kλ (u) and u+ = max{0, u}, u− = min{0, u}. Note that, for any u ∈ E and 2 p 2∗ , u− Lp v− 2 λ = inf w∈P + u−w Lp . 3) Since = (v, v − )λ = RN (∇v · ∇v − + Vλ vv − ) dx = RN f (x, u)v − dx, the fact that v + ∈ P + and v − v + = v − implies distλ v, P + · v − λ v− 2 λ RN f (x, u− )v − dx.

Then vij is a Dirichlet eigenfunction of − − f ′ (u) in Bi with eigenvalue µij < 0. Since vij changes sign, there exists a positive eigenfunction vi0 ∈ H01 (Bi ) ⊂ H01 (Ω) of − − f ′ (u) with eigenvalue µi0 < µij < 0. It follows that the quadratic form v → (− − f ′ (u))v, v L2 is negative on span{vij : i = 1, . . , k − 1, j = 0, . . , N}. Since the vij , i = 1, . . , k − 1, j = 0, . . , N , are linearly independent by construction, the negative eigenspace of − − f ′ (u) in H01 (Ω) has dimension at least (k − 1)(N + 1) = (nod(u) − 1)(N + 1).

Clearly Λ is closed in (0, a). Let us prove that Λ is open. Let µ ∈ Λ and let K be a smooth compact subset of Ωµ such that |Ωλ \ K| is sufficiently small for λ near µ. From wµ c > 0 in K, it follows that wλ 0 in K for λ near µ. 27 implies that wλ 0 in Ωλ \ K. Thus wλ 0 in Ωλ for λ near µ. Hence Λ is open in (0, a) and Λ = ]0, a[. It follows immediately that, on Ω, u(−x1 , y) u(x1 , y). 16), one finds that u(x1 , y) = u(−x1 , y). It is easy to conclude that ∂u/∂x1 < 0 for x1 > 0 using Hopf’s lemma.