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By Makoto Namba (auth.)

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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.

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