Existence, stability and instability for Einstein-scalar field Lichnerowicz equations by Emmanuel Hebey

Transcription

1 Existence, stability and instability for Einstein-scalar field Lichnerowicz equations by Emmanuel Hebey Joint works with Olivier Druet and with Frank Pacard and Dan Pollack Two hours lectures IAS, October 2008 Augmented June 2009

2 Contents: Introduction and main results Further directions and comments - 1 Further directions and comments - 2 Proof of Theorem 1 Proof of the stability part in Theorem 2 Proof of the instability part in Theorem 2 The case a 0. Unpublished result.

3 Given Ψ scalar field, and V (Ψ) a potential, Einstein-scalar field equations are written as: G ij = i Ψ j Ψ 1 2 ( α Ψ α Ψ) γ ij V (Ψ)γ ij, where γ is the spacetime metric, and G = Rc γ 1 2 S γγ is the Einstein curvature tensor. In the massive Klein-Gordon field theory, V (Ψ) = 1 2 m2 Ψ 2.

4 The constraint equations, using the conformal method, are 4(n 1) n 2 g u + h(g, ψ)u = f (ψ, τ)u 2 1 a(σ,w,π) + u 2 +1, (1) div g (DW ) = n 1 n u2 τ π ψ, (2) where g = div g, 2 = 2n/(n 2), h = S g ψ 2, a = σ + DW 2 + π 2, f = 4V (ψ) n 1 n τ 2 and S g is the scalar curvature of g. Here, ψ, π and τ are functions connected to the physics setting (τ mean curvature of spacelike hypersurface), σ TT -tensor, W vector field, and D the conformal Killing operator given by (DW ) ij = ( i W ) j + ( j W ) i 2 n (div g W )g ij. The system (1) (2) is decoupled in the constant mean curvature setting, namely when τ = C te.

5 The free data are (g, σ, τ, ψ, π). The determined data are u and W. They satisfy 4(n 1) n 2 g u + h(g, ψ)u = f (ψ, τ)u 2 1 a(σ,w,π) + u 2 +1, (1) div g (DW ) = n 1 n u2 τ π ψ. (2) ore details available in the survey paper by Robert Bartnik and Jim Isenberg: The constraint equations, arxiv:gr-qc/ v1, 2004.

6 (, g) compact, =, n 3. Let h, a, and f be arbitrary smooth functions in. Assume a > 0. Consider g u + hu = fu a u 2 +1, (EL) where g = div g, and 2 = 2n n 2. Example: (Sub and supersolution method, Choquet-Bruhat, Isenberg, Pollack, 2006). Assume g + h is coercive and f 0. Let v > 0 and u 0 > 0 be such that g u 0 + hu 0 = v. For t > 0, let u t = tu 0. We have: (i) u t is a subsolution of (EL) when t 1, and (ii) u t is a supersolution of (EL) when t 1. Since u t u t when t t, the sub and supersolution method provides a solution u [u t, u t ] for (EL).

7 Question: What can we say when g + h is coercive and either f changes sign or f is everywhere positive, i.e. when max f > 0? Assume g + h is coercive. Define u 2 h = ( u 2 + hu 2) dv g, u H 1. Let S(h) to be the smallest constant such that ( for all u H 1. u 2 dv g ) 2/2 S(h) 2/2 ( u 2 + hu 2) dv g

8 Theorem 1: (H.-Pacard-Pollack, Comm. ath. Phys., 2008) Let (, g) be a smooth compact Riemannian manifold, n 3. Let h, a, and f be smooth functions in. Assume that g + h is coercive, that a > 0 in, and that max f > 0. There exists C = C(n), C > 0 depending only on n, such that if ϕ 2 h a ϕ 2 dv g < C(n) (S(h) max f ) n 1 and f ϕ2 dv g > 0 for some smooth positive function ϕ > 0 in, then the Einstein-scalar field Lichnerowicz equation (EL) possesses a smooth positive solution. Example: if fdv g > 0 then take ϕ 1 and the condition reads as C(n, g, h) adv g < (max f ) n 1, where C(n, g, h) > 0 depends on n, g and h.

9 A perturbation of (EL) is a sequence (EL α ) α of equations, α N, which are written as g u + h α u = f α u for all α. Here we require that h α h, a α a, k α 0 a α u k α (EL α ) in C 0 as α +, and that f α f in C 1,η as α +, where η > 1 2. If (EL) satisfies the assumption of Theorem 1, any perturbation of (EL) also satisfies the assumptions of Theorem 1. A sequence (u α ) α is a sequence of solutions of (EL α ) α if for any α, u α solves (EL α ).

10 Definition: (Elliptic stability) The Einstein-scalar field Lichnerowicz equation (EL) is said to be: (i) stable if for any perturbation (EL α ) α of (EL), and any H 1 -bounded sequence (u α ) α of smooth positive solutions of (EL α ) α, there exists a smooth positive solution u of (EL) such that, up to a subsequence, u α u in C 1,θ () for all θ (0, 1), and (ii) bounded and stable if for any perturbation (EL α ) α of (EL), and any sequence (u α ) α of smooth positive solutions of (EL α ) α, the sequence (u α ) α is bounded in H 1 and there exists a smooth positive solution u of (EL) such that, up to a subsequence, u α u in C 1,θ () for all θ (0, 1).

11 Remark 1: Assuming stronger convergences for the h α s, f α s, etc., then we get stronger convergences for the u α s. E.g., if h α h, f α f, a α a and k α 0 in C p,θ, p N and θ (0, 1), then u α u in C p+2,θ, θ < θ. Remark 2: Stability means that if you slightly perturb h, a, and f, and even if you add to the equation a small background noise represented by k, then, in doing so, you do note create solutions which stand far from a solution of the original equation. Remark 3: Say (EL) is compact if any H 1 -bounded sequence (u α ) α of solutions of (EL) does possess a subsequence which converges in C 2. Say (EL) is bounded and compact if any sequence (u α ) α of solutions of (EL) does possess a subsequence which converges in C 2. Stability implies compactness. Bounded stability implies bounded compactness.

12 Let D = C () 4 and D be given by D D = 3 f i C 0,1 + f 4 C 1,1 i=1 for all D = (f 1, f 2, f 3, f 4 ) D. For D = (h, a, k, f ) in D consider g u + hu = fu a u k. (EL ) If D = (h, a, 0, f ), then (EL ) = (EL). Let Λ > 0, D = (h, a, k, f ) in D, and define S D,Λ = { u solution of (EL ) s.t. u H 1 Λ }, and S D = { u solution of (EL ) }. When D = (h, a, 0, f ) we recover solutions of (EL).

13 For X, Y C 2 define dc (X ; Y ) = sup inf v u 2 C u X v Y 2. By convention, d C 2 (X ; ) = + if X, and d C 2 ( ; Y ) = 0 for all Y, including Y =. Let D = (h, a, 0, f ) be given. Stability (EL) is compact and ε > 0, Λ > 0, δ > 0 s.t. D = (h, a, k, f ) D, D D D < δ d C 2 (S D,Λ; S D,Λ ) < ε. Bounded stability (EL) is bounded compact and ε > 0, δ > 0 s.t. D = (h, a, k, f ) D, D D D < δ d C 2 (S D ; S D ) < ε.

14 Theorem 2: (Druet-H., ath. Z., 2008) Let (, g) be a smooth compact Riemannian manifold of dimension n 3, and h, a, f C () be smooth functions in with a > 0. Assume n = 3, 4, 5. Then the Einstein-scalar field Lichnerowicz equation g u + hu = fu a u 2 +1 is stable. The equation is even bounded and stable assuming in addition that f > 0 in. On the contrary, (EL) is not anymore stable a priori when n 6. (EL)

15 I. Further directions and comments - 1 Lemma 1: (H.-Pacard-Pollack, Comm. ath. Phys., 2008) Assume a 0, f > 0, and n n ( (n 1) n 1 a n+2 3n 2 4n f 4n dvg ) 4n n+2 > ( ) (h + ) n+2 4n n+2 4 dv g f n 2 4. Then the Einstein-scalar field Lichnerowicz equation (EL) does not possess solutions. In particular, for any h, and any f > 0, there exist a positive constante C = C(n, g, h, f ) such that if a n+2 4n dvg C, then (EL) does not possess solutions.

16 Proof: Integrating (EL), fu 2 1 dv g + By Hölder s inequalities, hudv g a n+2 3n 2 4n f 4n ( ( dvg (h + ) n+2 4 dv g f n 2 4 adv g u 2 +1 = hudv g. ) 4 n+2 ( ) 3n 2 fu 2 1 4n dv g fu 2 1 dv g ) n 2 n+2 ( ) n+2 adv g 4n u 2 +1, and.

17 ( X + a n+2 3n 2 4n f 4n ) 4n ( n+2 dvg X 1 n ) (h + ) n+2 4 n+2 4 dv g f n 2 4, where This implies ( X = fu 2 1 dv g ) 4 n+2. n n ( (n 1) n 1 a n+2 3n 2 4n f 4n dvg ) 4n n+2 ( ) (h + ) n+2 4n n+2 4 dv g f n 2 4.

18 II. Further directions and comments - 2 Fix h, a and f. Assume g + h is coercive, and a, f > 0. Let t > 0 and consider g u + hu = fu According to Theorem 1 and the Lemma: ta u (EL t) (i) (Theorem 1) for t 1, (EL t ) possesses a solution, (ii) (Lemma 1) for t 1, (EL t ) does not possess any solution. Assuming n = 3, 4, 5, (iii) (Theorem 2) (EL t ) t is bounded and stable for t [t 0, t 1 ], where 0 < t 0 < t 1.

19 Let Λ > 0. Define { } Ω Λ = u C 2,θ s.t. u C 2,θ < Λ and min u > Λ 1. Fix t 0 1 such that (EL t0 ) possesses a solution. Fix t 1 1 such that (EL t1 ) does not possess any solution. Assume n = 3, 4, 5. Define F t : Ω Λ C 2,θ by ( F t u = u L 1 fu ta ), u 2 +1 where L = g + h, and t [t 0, t 1 ]. By (iii), there exists Λ 0 > 0 such that F 1 t (0) Ω Λ0 for all t [t 0, t 1 ]. Then, by (ii), deg(f t0, Ω Λ, 0) = 0 for all Λ 1. In particular, assuming that the solutions of the equations are nondegenerate, the solution in Theorem 1 needs to come with another solution.

20 III. Proof of Theorem 1 We aim in proving: Let (, g) be a smooth compact Riemannian manifold, n 3. Let h, a, and f be smooth functions in. Assume that g + h is coercive, that a > 0 in, and that max f > 0. There exists C = C(n), C > 0 depending only on n, such that if ϕ 2 h a ϕ 2 dv g < C(n) (S(h) max f ) n 1 and f ϕ2 dv g > 0 for some smooth positive function ϕ > 0 in, then the Einstein-scalar field Lichnerowicz equation g u + hu = fu a u 2 +1 (EL) possesses a smooth positive solution. ethod: approximated equations, mountain pass analysis.

21 Fix ε > 0. Define I (1) (u) = 1 2 and where u H 1. Let ( u 2 + hu 2) dv g 1 2 I ε (2) (u) = 1 2 adv g (ε + (u + ) 2 ) 2 /2, I ε = I (1) + I (2) ε. f (u + ) 2 dv g, Let ϕ > 0 be as in Theorem 1. Assume ϕ h = 1. The conditions in the theorem read as and f ϕ2 dv g > 0. a ϕ 2 dv g < C(n) (S(h) max f ) n 1 (1)

22 Let Φ, Ψ : R + R be the functions given by Φ(t) = 1 2 t2 max f 2 S(h)t 2, and Ψ(t) = 1 2 t2 + max f 2 S(h)t 2. These functions satisfy Φ ( u h ) I (1) (u) Ψ ( u h ) (2) for all u H 1. Let t 1 > 0 be such that Φ is increasing up to t 1 and decreasing after: ( ) (n 2)/4 t 1 = S(h) max f. Let t 0 > 0 be given by 1 t 0 = 2(n 1) t 1.

23 Then Ψ(t 0 ) 1 2 Φ(t 1) (3) and for C 1 the condition in the theorem translates into 1 a 2 dv g < 1 (t 0 ϕ) 2 2 Φ(t 1). (4) Let ρ = Φ(t 1 ). Then, by (3) and (4), and by we can write that for all u s.t. u h = t 1. I ε (t 0 ϕ) < ρ Φ ( u h ) I (1) (u) Ψ ( u h ), (2) I ε (u) ρ

24 We got that there exists ρ > 0 such that I ε (t 0 ϕ) < ρ and I ε (u) ρ for all u s.t. u h = t 1. Also t 1 > t 0. Since f ϕ2 dv g > 0, I ε (tϕ) as t +. We can apply the mountain pass lemma.

25 Let t 2 1. Define c ε = inf γ Γ max u γ I ε(u) where Γ is the set of continuous paths joining t 0 ϕ to t 2 ϕ. The PL provides a Palais-Smale sequence (u ε k ) k such that I ε (u ε k ) c ε and I ε(u ε k ) 0 as k +. The sequence (u ε k ) k is bounded in H 1. Up to a subsequence, u ε k u ε in H 1. Then u ε satisfies g u ε + hu ε = fu 2 1 ε + In particular, u ε is positive and smooth. au ε (ε + u 2 ε)

26 We can prove that the c ε s are bounded independently of ε. In particular the family (u ε ) ε is bounded in H 1. Now we can pass to the limit as ε 0 because u ε will never approach zero. Take x ε such that u ε (x ε ) = min u ε. Then g u ε (x ε ) 0 and h(x ε ) + f (x ε ) u ε (x ε ) 2 2 This implies that there exists δ 0 > 0 such that min u ε δ 0 a(x ε ) (ε + u ε (x ε ) 2 ) for all ε. If u ε u in H 1, then u δ 0 and u solves (EL).

27 IV. Proof of the stability part in Theorem 2 We aim in proving: Let (, g) be a smooth compact Riemannian manifold of dimension n 3, and h, a, f C () be smooth functions in with a > 0. Assume n = 3, 4, 5. Then the Einstein-scalar field Lichnerowicz equation g u + hu = fu a u 2 +1 is stable, and even bounded and stable if f > 0 in. ethod: blow-up analysis, sharp pointwise estimates. (EL)

28 Let (EL α ) α be a perturbation of (EL). Let also (u α ) α be a sequence of solutions of (EL α ). Consider (H1A) f > 0 in, (H1B) (u α ) α is bounded in H 1, (H2) ε 0 > 0 s.t. u α ε 0 in for all α. We claim that: Stability Theorem: (Druet-H., ath. Z., 2008) Let n 5. Let (EL α ) α be a perturbation of (EL) and (u α ) α a sequence of solutions of (EL α ) α. Assume (H1A) or (H1B), and we also assume (H2). Then the sequence (u α ) α is uniformly bounded in C 1,θ, θ (0, 1). By (H2), g u α Cu 2 1 α, where C > 0 does not depend on α.

29 Proof of stability theorem: By contradiction. We assume that u α + as α +. We also assume (H1A) or (H1B), and (H2). Let (x α ) α and (ρ α ) α be such that (i) x α is a critical point of u α for all α, (ii) ρ n 2 2 α sup Bxα (ρ α) u α + as α +, and (iii) d g (x α, x) n 2 2 uα (x) C for all x B xα (ρ α ) and all α. Then : ain Estimate: Assume (i) (iii). Then we have that ρ α 0, ρ n 2 2 α u α (x α ) +, and u α (x α )ρ n 2 ( α u α expxα (ρ α x) ) λ + H(x) x n 2 in C 2 loc (B 0(1)\{0}) as α +, where λ > 0 and H is a harmonic function in B 0 (1) which satisfies that H(0) = 0.

30 There exist C > 0, a sequence (N α ) α of integers, and for any α, critical points x 1,α,..., x Nα,α of u α such that ( min i=1,...,n α d g (x i,α, x) ) n 2 2 for all x and all α. We have N α 2. Define d α = min d g (x i,α, x j,α ) 1 i<j N α u α (x) C (1) and let the x i,α s be such that d α = d g (x 1,α, x 2,α ). We have d α 0 as α +. oreover, as α +. d n 2 2 α u α (x 1,α ) + (2)

31 Define ũ α by ) ũ α (x) = d n 2 2 α u α (exp x1,α (d α x), where x R n. Let ṽ α = ũ α (0)ũ α. Then gα ṽ α where g α δ as α +. Because of C ũ α (0) 2 2 ṽ 2 1 α, (3) d n 2 2 α u α (x 1,α ) +, (2) ũ α (0) + as α +. Independently, by elliptic theory, for any R > 0, ṽ α G in C 1 loc (B 0(R)\{ x i } i=1,...,p ) as α +, where, because of (3), G is nonnegative and harmonic in B 0 (R)\{ x i } i=1,...,p.

32 Then, G(x) = p i=1 λ i + H(x), x x i n 2 where λ i > 0 and H is harmonic without singularities. In particular, in a neighbourhood of 0, G(x) = λ 1 x n 2 + H(x). By ( min d g (x i,α, x) i=1,...,n α ) n 2 2 u α (x) C (1) d n 2 2 α u α (x 1,α ) + (2) we can apply the main estimate with x α = x 1,α and ρ α = dα 10. In particular, H(0) = 0.

33 However, and λ 1 G(x) = x n 2 + λ 2 + Ĥ(x) x x 2 n 2 0 H(x) = By the maximum principle, and we get that λ 2 + Ĥ(x). x x 2 n 2 Ĥ(0) min Ĥ B 0 (R) H(0) λ 2 x 2 n 2 λ 1 R n 2 λ 2 (R x 2 ) n 2. By construction, x 2 = 1. Choosing R 1 sufficiently large, H(0) > 0. A contradiction.

34 It remains to prove the stability part in theorem 2. We introduced (H1A) f > 0 in, (H1B) (u α ) α is bounded in H 1, (H2) ε 0 > 0 s.t. u α ε 0 in for all α, and we proved that (H1A) or (H1B), and (H2) C 1,θ convergences for the u α s solutions of perturbations of (EL). Let (EL α ) α be any perturbation of (EL), and (u α ) α be any sequence of solution of (EL α ) α. It suffices to prove (H2). Let x α be such that u α (x α ) = min u α. Then g u α (x α ) 0 and we get that h α (x α ) 1 u α (x α ) ( aα (x α ) u α (x α ) k α(x α ) ) + f α (x α )u α (x α ) 2 2. In particular, u α ε 0 > 0 and (H2) is satisfied. We can apply the stability theorem. This proves the stability part of Theorem 2.

35 V. Proof of the instability part in Theorem 2 We aim in proving: When n 6 the Einstein-scalar field Lichnerowicz equation is not a priori stable. g u + hu = fu a u 2 +1 (EL) ethod: explicit constructions of examples.

36 A first construction. Lemma 2: (Druet-H., ath. Z., 2008) Let (S n, g 0 ) be the unit sphere, n 7. Let x 0 S n. Let a and u 0 be smooth positive functions such that g0 u 0 + n(n 2) u 0 = 4 n(n 2) u a u There exist sequences (h α ) α and (Φ α ) α such that h α n(n 2) 4 in C 0 (S n ), max Φ α + and Φ α 0 in Cloc 2 (S n \{x 0 }) as α +. In addition g0 u α + h α u α = for all α, where u α = u 0 + Φ α. n(n 2) u 2 1 α + a 4 u 2 +1 α

37 Proof of Lemma 2: Let ϕ α be given by ϕ α (x) = ( β 2 α 1 β α cos d g0 (x 0, x) ) n 2 2, where β α > 1 for all α and β α 1 as α +. The ϕ α s satisfy Let g0 ϕ α + where ψ α is such that n(n 2) ϕ α = 4 u α = u 0 + ϕ α + ψ α, n(n 2) ϕ 2 1 α. 4 g0 u 0 + g0 ϕ α + g0 ψ α ( ) n(n 2) n(n 2) = (u 0 + ϕ α ) ε α (u 0 + ϕ α ) 4 4 a + (u 0 + ϕ α ) We have ε α 0 as α +.

38 For any sequence (x α ) α of points in S n, ψ α (x α ) = o (β α 1) (n 2) n 4 2 2(n 4) (β α 1) + d g0 (x 0, x α ) 2 + o(1). (1) Thanks to (1), ψ α u α 0 and u 2 3 α ψ α 0 (2) in C 0 (S n ) as α +. For instance, either ψ α (x α ) 0 and ψ α (x α )/u α (x α ) 0, or ψ α (x α ) 0. In that case, because of (1), d g0 (x 0, x α ) 0. Then, ψ α (x α )/u α (x α ) 0 since ψ α (x α ) ϕ α (x α ) C te( (β α 1) + d g0 (x 0, x α ) 2).

39 Let h α be such that g0 u α + h α u α = n(n 2) u 2 1 α + a 4 u 2 +1 α for all α. Write g0 u α = g0 u 0 + g0 ϕ α + g0 ψ α. By the equation satisfied by ψ α, ( ) n(n 2) ( ) h α u α = O u 2 2 α ψ α + O (ψ α ) + ε α u α. 4 Divide by u α, and conclude thanks to ψ α u α 0 and u 2 3 α ψ α 0 (2) that h α n(n 2) 4 in C 0 as α +. This proves Lemma 2.

40 Say that (, g) has a conformally flat pole at x 0 if g is conformally flat around x 0. Thanks to Lemma 2 we get: Lemma 3: (Druet-H., ath. Z., 2008) Let (, g) be a smooth compact Riemannian manifold with a conformally flat pole, n 7. There exists δ > 0 such that the Einstein-scalar Lichnerowicz equation g u + n 2 4(n 1) S g u = u a u 2 +1 is not stable on (, g) and possesses smooth positive solutions for all smooth functions a > 0 such that a 1 < δ.

41 VI. The case a 0. Unpublished result. When a > 0 in : let u > 0 be a solution of (EL). Let x 0 be such that u(x 0 ) = min u. Then g u(x 0 ) 0 and h(x 0 ) u(x 0 ) + f (x 0 ) u(x 0 ) 2 1 a(x 0) u(x 0 ) 2 +1 there exists ε 0 = ε 0 (h, f, a), ε 0 > 0, such that u ε 0 in. Question:Assume g + h is coercive, a 0 and max f > 0. What can we say when Zero(a)? In physics a = σ + DW 2 + π 2, where σ and π are free data, and W is the determined data given by the second equation in the system.

42 Recall (EL) is compact if any H 1 -bounded sequence (u α ) α of solutions of (EL) does possess a subsequence which converges in C 2. Recall (EL) is bounded and compact if any sequence (u α ) α of solutions of (EL) does possess a subsequence which converges in C 2. Theorem 3: (Druet, Esposito, H., Pacard, Pollack, Collected works - Unpublished, 2009) Assume g + h is coercive, a 0, a 0, and max f > 0. Theorem 1 remains true without any other assumptions than those of Theorem 1. Assuming that n = 3, 4, 5, the equation is compact and even bounded and compact when f > 0. Existence follows from a combination of Theorem 1 and the sub and supersolution method. Compactness follows from the stability theorem in the proof of Theorem 2 together with an argument by Pierpaolo Esposito.

43 Proof of the existence part in Theorem 3: Assume the assumptions of Theorem 1 are satisfied: there exists ϕ > 0 such that a C(n) ϕ 2 h dv g < ϕ 2 (S(h) max f ) n 1 (1) and f ϕ2 dv g > 0. Changing a into a + ε 0 for 0 < ε 0 1, (1) is still satisfied, and since a + ε 0 > 0 we can apply Theorem 1. In particular, (i) a a + ε 0, 0 < ε 0 1, and Theorem 1 u 1 a supersolution of (EL). Now let δ > 0 and let u 0 solve g u 0 + hu 0 = a δf For δ > 0 sufficiently small, u 0 is close to the solution with δ = 0, and since this solution is positive by the maximum principle, we get that u 0 > 0 for 0 < δ 1. Fix such a δ > 0.

44 Given ε > 0, let u ε = εu 0. Then g u ε + hu ε = εa δεf fu 2 1 ε + a u 2 +1 ε provided 0 < ε 1. In particular, (ii) u ε = εu 0, 0 < ε 1, is a subsolution of (EL). Noting that u ε u 1 for ε > 0 sufficiently small, we can apply the sub and supersolution method and get a solution u to (EL) such that u ε u u 1. The compactness part in Theorem 3 follows from the stability theorem in the proof of Theorem 2 together with the following result by Esposito which establishes the (H2) property of Druet and Hebey under general conditions. We do not need in what follows the C 1,η -convergence of the f α s. A C 0 -convergence (and even less) is enough.

45 Lemma 4: (Esposito, Unpublished, 2009) Let n 5. Let (EL α ) α be a perturbation of (EL) and (u α ) α a sequence of solutions of (EL α ) α. Assume a α 0 in for all α, and a 0. The (H2) property holds true: ε 0 > 0 such that u α ε 0 in for all α. Proof of the lemma: Let K > 0 be such that K + h α 1 in for all α. Define h α = K + h α and h = K + h. Let δ > 0 and v δ α, v δ, and r α be given by g v δ α + h α v δ α = a α δf α, g v δ + hv δ = a δf, g r α + h α r α = k α. There holds that v δ α v δ in C 0 () as α + and that v δ v 0 in C 0 () as δ 0. By the maximum principle, v 0 > 0 in. It follows that there exists δ > 0 sufficiently small, and ε 0 > 0, such that v δ α ε 0 in for all α 1. Fix such a δ > 0. Let t > 0

46 and define w α = tv δ α + r α. We have that r α 0 in C 0 () as α +. There exists t 0 > 0 such that g w α + h α w α = ta α tδf α + k α fα w 2 1 α + a α + k w 2 +1 α α for all 0 < t < t 0 and all α 1. As a consequence, since a α 0 in, g (u α w α ) + h α (u α w α ) f α u 2 1 α + fα w 2 1 α + a α a α u 2 +1 α w 2 +1 α 0 for all α 1, at any point such that u α w α 0. The maximum principle then gives that w α u α in for all α 1. Since w α ε 0 in for α 1, this ends the proof of the lemma.