On Yamabe-type problems on Riemannian manifolds with boundary

Testo completo

(1)Pacific Journal of Mathematics. ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY M ARCO G HIMENTI , A NNA M ARIA M ICHELETTI AND A NGELA P ISTOIA. Volume 284. No. 1. September 2016.

(2) PACIFIC JOURNAL OF MATHEMATICS Vol. 284, No. 1, 2016 dx.doi.org/10.2140/pjm.2016.284.79. ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY M ARCO G HIMENTI , A NNA M ARIA M ICHELETTI AND A NGELA P ISTOIA Let (M, g) be an n-dimensional compact Riemannian manifold with boundary. We consider the Yamabe-type problem ( −1 g u + au = 0 on M, ∂ν u + n−2 bu = (n − 2)un/(n−2)±ε 2. on ∂ M,. where a ∈ C 1 (M), b ∈ C 1 (∂ M), ν is the outward pointing unit normal to ∂ M, 1 g u := div g ∇g u, and ε is a small positive parameter. We build solutions which blow up at a point of the boundary as ε goes to zero. The blowing-up behavior is ruled by the function b − Hg , where Hg is the boundary mean curvature.. 1. Introduction Let (M, g) be a smooth, compact Riemannian manifold of dimension n ≥ 3 with a boundary ∂M which is the union of a finite number of smooth closed compact submanifolds embedded in M. A well-known problem in differential geometry is whether (M, g) is necessarily conformally equivalent to a manifold of constant scalar curvature whose boundary is minimal. When the boundary is empty this is called the Yamabe problem (see Yamabe [1960]), which has been completely solved by Aubin [1976], Schoen [1984] and Trudinger [1968]. Cherrier [1984] and Escobar [1992a; 1992b] studied the problem in the context of manifolds with boundary and gave an affirmative solution to the question in almost every case. The remaining cases were studied by Marques [2005; 2007], by Almaraz [2010] and by Brendle and Chen [2014]. Once the problem is solvable, a natural question about compactness of the full set of solutions arises. Concerning the Yamabe problem, it was first raised by Schoen in a topics course at Stanford University in 1988. A necessary condition is that the manifold is not conformally equivalent to the standard sphere Sn , since the group of The research that lead to the present paper was partially supported by the group GNAMPA of Istituto Nazionale di Alta Matematica (INdAM). MSC2010: 35J20, 58J05. Keywords: Yamabe problem, blowing-up solutions, compactness. 79.

(3) 80. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. conformal transformations of the round sphere is not compact itself. The problem of compactness has been widely studied in recent years and has been completely solved by Brendle [2008], Brendle and Marques [2009] and Khuri, Marques and Schoen [Khuri et al. 2009]. In the presence of a boundary, a necessary condition is that M is not conformally equivalent to the standard ball Bn . The problem when the boundary of the manifold is not empty has been studied by V. Felli and M. Ould Ahmedou [2003; 2005], Han and Li [1999] and Almaraz [2011a; 2011b]. In particular, Almaraz studied the compactness property in the case of scalar-flat metrics. Indeed, the zero scalar curvature case is particularly interesting because it leads one to study a linear equation in the interior with a critical Neumann-type nonlinear boundary condition  n −2 −1g u + R u=0 on M, u > 0 in M, 4(n −1) g (1-1)  n −2 ∂ν u + Hg u = (n − 2)u n/(n−2) on ∂M, 2 where ν is the outward pointing unit normal to ∂M, Rg is the scalar curvature of M with respect to g, and Hg is the boundary mean curvature with respect to g. We note that in this case compactness of solutions is equivalent to establish a priori estimates for solutions to equation (1-1). Almaraz [2011b] proved that compactness holds for a generic metric g. On the other hand, in [Almaraz 2011a] it was proved that if the dimension of the manifold is n ≥ 25, compactness does not hold because it is possible to build blowing-up solutions to (1-1) for a suitable metric g. We point out that the problem of compactness in dimension n ≤ 24 is still not completely understood. An interesting issue, closely related to the compactness property, is the stability problem. One can ask whether or not the compactness property is preserved under perturbations of the equation, which is equivalent to having or not having uniform a priori estimates for solutions of the perturbed problem. Let us consider the more general problem −1g u + a(x)u = 0 in M, u > 0 in M, (1-2) n/(n−2) ∂ν u + b(x)u = (n − 2)u on ∂M. We say that the problem (1-2) is stable if for any sequences of C 1 functions aε : M → R and bε : ∂M → R converging in C 1 to functions a : M → R and b : ∂M → R, for any sequence of exponents pε := n/(n − 2) ± ε converging to the critical one n/(n − 2) and for any sequence of associated solutions u ε bounded in H 1 (M) of the perturbed problems ( −1g u + aε (x)u = 0 in M, u ε > 0 in M, (1-3) n −2 n/(n−2)±ε ∂ν u + b (x)u = (n − 2)u ε on ∂M, 2 ε.

(4) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. 81. there is a subsequence u εk which converges in C 2 to a solution to the limit problem (1-2). The stability of the Yamabe problem has been introduced and studied by Druet [2003; 2004] and by Druet and Hebey [2005a; 2005b]. Recently, Esposito, Pistoia and Vetois [Esposito et al. 2014], Micheletti, Pistoia and Vetois [Micheletti et al. 2009] and Esposito and Pistoia [2014] proved that a priori estimates fail for perturbations of the linear potential or of the exponent. In this paper, we investigate the question of stability of the problem (1-2). It is clear that it is not stable if it is possible to build solutions u ε to perturbed problems (1-3) which blow up at one or more points of the manifold as the parameter ε goes to zero. Here, we show that the behavior of the sequence u ε is dictated by the difference (1-4). ϕ(q) = b(q) − Hg (q) for q ∈ ∂M.. More precisely, we consider the problem  −1g u + a(x)u = 0 (1-5)  ∂ u + n −2 b(x)u = (n − 2)u n/(n−2)±ε ∂ν 2. on M, u > 0 in M, on ∂M.. We assume that a ∈ C 1 (M) and b ∈ C 1 (∂M) are such that the linear operator Lu := −1g u + au with Neumann boundary condition B u := ∂ν u + 12 (n − 2)bu is coercive; namely, there exists a constant c > 0 such that Z Z n −2 (1-6) |∇g u|2 + a(x)u 2 dµg + b(x)u 2 dσ ≥ ckuk2H 1 (M) . 2 M ∂M Here ε > 0 is a small parameter, 1g u := divg ∇g u, and the space H 1 (M) is the closure of C ∞ (M) with respect to the norm Z 1/2 2 2 kuk H 1 = |∇g u| + u dµg . M. The problem (1-5) turns out to be either slightly subcritical or slightly supercritical if the exponent in the nonlinearity is either n/(n−2)−ε or n/(n−2)+ε, respectively. Let us state our main result. Theorem 1. Assume (1-6) and n ≥ 7. (i) If q0 ∈ ∂M is a strict local minimum point of the function ϕ defined in (1-4) with ϕ(q0 ) > 0, then provided ε > 0 is small enough, there exists a solution u ε of (1-5) in the slightly subcritical case such that u ε blows up at a boundary point when ε → 0+ . (ii) If q0 ∈ ∂M is a strict local maximum point of the function ϕ defined in (1-4) with ϕ(q0 ) < 0, then provided ε < 0 is small enough, there exists a solution u ε of (1-5) in the slightly supercritical case such that u ε blows up at a boundary point when ε → 0+ ..

(5) 82. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. We say that u ε blows up at a point q0 of the boundary if there exists a family of points qε ∈ ∂M such that qε → q0 as ε → 0 and, for any neighborhood U ⊂ M of q0 , we have that supq∈U u ε (q) → +∞ as ε → 0. Our result does not concern the stability of the geometric Yamabe problem (1-1). Indeed, the function ϕ in (1-4) turns out to be identically zero. It would be interesting to discover the function which rules the behavior of blowing-up sequences in this case. We expect that it depends on the trace-free second fundamental form as it is suggested by Almaraz [2011b], where a compactness result in the subcritical case is established. The case of low dimension also remains open, where we expect that the function ϕ in (1-4) should be replaced by a function which depends on the Weyl tensor of the boundary, as suggested by Escobar [1992a; 1992b]. The proof of our result relies on a very well known Ljapunov–Schmidt procedure. In Section 2 we set up the problem, and in Section 3 we reduce the problem to a finite dimensional one, which is then studied in Section 4. 2. Setting of the problem Let us rewrite problem (1-5) in a more convenient way. First of all, assumption (1-6) allows us to endow the Hilbert space H := H 1 (M) with the scalar product Z Z n −2 hhu, viiH := (∇g u∇g v + a(x)uv) dµg + b(x)uv dσ 2 ∂M M and the induced norm kuk2H := hhu, uiiH . We define the exponent  2(n −1)  in the subcritical case,  n −2 sε =   2(n −1) + nε in the supercritical case, n −2 and the Banach space H := H 1 (M) ∩ L sε (∂M) endowed with the norm kukH = kuk H + |u| L sε (∂M) . Notice that in the subcritical case H is identical to the Hilbert space H . By trace theorems, we have the inclusion W 1,τ (M) ⊂ L t (∂M) for any t and τ satisfying t ≤ τ (n − 1)/(n − τ ). We consider i : H 1 (M) → L 2(n−1)/(n−2) (∂M) and its adjoint with respect to hh · , · iiH , namely i ∗ : L 2(n−1)/n (∂M) → H 1 (M) defined by hhϕ, i (g)iiH = ∗. Z ∂M. ϕg dσ. for all ϕ ∈ H 1 ,.

(6) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. 83. so that u = i ∗ (g) is the weak solution of the problem  −1g u + a(x)u = 0 on M, (2-1) ∂ n −2  u+ b(x)u = g on ∂M. ∂ν 2 We recall that by [Nittka 2011], if u ∈ H 1 (M) is a solution of (2-1), then for 2n/(n + 2) ≤ q ≤ n/2 and r > 0 we have kuk L (n−1)q/(n−2q) (∂M) = ki ∗ (g)k L (n−1)q/(n−2q) (∂M) ≤ kgk L (n−1)q/(n−q)+r (∂M) .. (2-2). By this result, we can choose q and r such that (n −1)q 2(n −1) = + nε n −2q n −2. (2-3). and. (n −1)q 2(n −1)+n(n −2)ε +r = , n −q n +(n −2)ε. that is, q=. n−2 n−1 ε n + 2 + 2n n−2 n−1 ε. 2n + n 2. . and r =. 2(n − 1) + n(n − 2)ε 2(n − 1) + n(n − 2)ε . − n n + (n − 2)ε ε n + (n − 2) n−1. So, if u ∈ L 2(n−1)/(n−2)+nε (∂M), then n +ε. |u| n−2. ∈L. 2(n−1)+n(n−2)ε n+ε(n−2) (∂M). and, in light of (2-2), also i ∗ (|u|n/(n−2)+ε ) ∈ L 2(n−1)/(n−2)+nε (∂M). Finally, we rewrite problem (1-5) — both in the subcritical and the supercritical case — as (2-4). u = i ∗ ( f ε (u)),. u ∈ H,. where the nonlinearity f ε (u) is defined as f ε (u) := (n − 2)(u + )n/(n−2)+ε in the supercritical case or f ε (u) := (n − 2)(u + )n/(n−2)−ε in the subcritical case. Here u + (x) := max{0, u(x)}. By assumption (1-6), a solution to problem (2-4) is strictly positive and actually is a solution to problem (1-5). Therefore, we are led to build solutions to problem (2-4) which blow-up at a boundary point as ε goes to zero. The main ingredient to cook up our solutions are the standard bubbles Uδ,ξ (x, t) :=. δ (n−2)/2 , ((δ+t)2 +|x −ξ |2 )(n−2)/2. (x, t) ∈ Rn−1 × R+ , δ > 0, ξ ∈ Rn−1 ,. which are all the solutions to the limit problem −1U = 0 on Rn−1 × R+ , (2-5) n/(n−2) ∂ν U = (n − 2)U on Rn−1 × {t = 0}. We set Uδ (x, t) := Uδ,0 (x, t)..

(7) 84. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. We also need to introduce the linear problem −1V = 0 on Rn−1 × R+ , (2-6) 2/(n−2) ∂ν V = nU1 V on Rn−1 × {t = 0}. In [Almaraz 2011b] it has been proved that the n-dimensional space of solutions of (2-6) is generated by the functions xi ∂U1 = (2 − n) for i = i, . . . , n − 1, 2 ∂ xi ((1 + t) + |x|2 )n/2 n/2

(8) ∂Uδ

(9) n −2 1 V0 = = t 2 + |x|2 − 1 .

(10) 2 2 ∂δ δ=1 2 (1+t) +|x| Vi =. Next, for a point q ∈ ∂M and the (n − 1)-dimensional unitary ball B n−1 (0, R) in Rn−1 , we introduce the Fermi coordinates ψq∂ : B n−1 (0, R) × [0, R) → M. We read the bubble on the manifold as the function Wδ,q (ξ ) = Uδ (ψq∂ )−1 ξ χ (ψq∂ )−1 ξ , and the functions Vi on the manifold as the functions 1 1 i for i = 0, . . . , n − 1, Z δ,q (ξ ) = (n−2)/2 Vi (ψq∂ )−1 ξ χ (ψq∂ )−1 ξ δ δ where χ(x, t) = χ(|x|) ˜ χ˜ (t), for χ˜ a smooth cut off function, χ˜ (s) ≡ 1 for 0 ≤ s < R/2 and χ(s) ˜ ≡ 0 for s ≥ R. Then, it is necessary to split the Hilbert space H into the sum of the orthogonal spaces. 0. n−1 K δ,q = Span Z δ,q , . . . , Z δ,q and . ⊥ i K δ,q = ϕ ∈ H 1 (M) | hhϕ, Z δ,q iiH = 0 for all i = 0, . . . , n − 1 . Finally, we can look for a solution to problem (2-4) in the form u ε (x) = Wδ,q (x) + φ(x) where the blow-up point q is in ∂M, the blowing-up rate δ satisfies (2-7). δ := dε. for some d > 0. ⊥ and the remainder term φ belongs to the infinite dimensional space K δ,q ∩ H of codimension n. We are led to solve the system ∗ (2-8) 5⊥ f ε (Wδ,q (x) + φ(x)) = 0, δ,q Wδ,q (x) + φ(x) − i (2-9) 5δ,q Wδ,q (x) + φ(x) − i ∗ f ε (Wδ,q (x) + φ(x)) = 0, ⊥ 5⊥ δ,q and 5δ,q being the projections on K δ,q and K δ,q , respectively..

(11) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. 85. 3. The finite dimensional reduction In this section we perform the finite dimensional reduction. We rewrite the auxiliary equation (2-8) in the equivalent form L(φ) = N (φ) + R,. (3-1). ⊥ ⊥ where L = L δ,q : K δ,q ∩ H → K δ,q ∩ H is the linear operator. . ∗ 0 L(φ) = 5⊥ δ,q φ(x) − i ( f ε (Wδ,q )[φ]) , N (φ) is the nonlinear term ∗ . (3-2) N (φ) = 5⊥ f ε (Wδ,q (x)+φ(x)) −i ∗ f ε (Wδ,q (x)) −i ∗ ( f ε0 (Wδ,q )[φ]) δ,q i and the error term R is defined by ∗ . f ε (Wδ,q (x)) − Wδ,q (x) . (3-3) R = 5⊥ δ,q i 3.1. The invertibility of the linear operator L. Lemma 2. For a, b ∈ R with 0 < a < b, there exists a positive constant C0 = C0 (a, b) ⊥ such that, for ε small, for any q ∈ ∂M, for any d ∈ [a, b] and for any φ ∈ K δ,q ∩ H, we have kL δ,q (φ)kH ≥ C0 kφkH . Proof. We argue by contradiction. Suppose that there exist two sequences of real numbers εm → 0 and dm ∈ [a, b], a sequence of points qm ∈ ∂M and a sequence of functions φεm dm ,qm ∈ K ε⊥m dm ,qm ∩ H such that kφεm dm ,qm kH = 1. and. kL εm dm ,qm (φεm dm ,qm )kH → 0 as m → +∞.. For the sake of simplicity, we set δm = εm dm and define n φ˜ m := δm(n−2)/2 φδm ,qm(ψq∂m (δm η))χ (δm η) for η = (z, t) ∈ R+ , z ∈ Rn−1 , t ≥ 0.. Since kφεm dm ,qm k H ≤ 1, by a change of variables we easily get that {φ˜ m }m is bounded n ) (but not in H 1 (Rn )). Therefore, there exists φ n ) such that ˜ ∈ D 1,2 (R+ in D 1,2 (R+ + n ), in L 2n/(n−2) (Rn ) and strongly φ˜ m * φ˜ almost everywhere, weakly in D 1,2 (R+ + 2(n−1)/(n−2) n in L loc (∂ R+ ). Since φδm ,qm ∈ K δ⊥m ,qm , and taking (2-6) into account, for i = 0, . . . , n − 1 we get Z Z 2/(n−2) ˜ ˜ 0) dz. (3-4) o(1) = ∇ φ∇Vi dz dt = n U1 (z, 0)Vi (z, 0)φ(z, n R+. Rn−1.

(12) 86. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. Indeed, by a change of variables we have. 0 = φδm ,qm , Z δi m ,qm H Z = ∇g φδm ,qm ∇g Z δi m ,qm + a(x)φδm ,qm Z δi m ,qm dµg M Z n −2 b(x)φδm ,qm Z δi m ,qm dσ + 2 ∂M Z ∂ ∂ = |gqm (δη)|1/2 δ (n−2)/2 gqαβm (δη) V (η)χ (δη) φ (ψ ∂ (δ η)) δη n ∂ηα i ∂ηα δm ,qm qm m R+ Z + |gqm (δη)|1/2 δ (n+2)/2 a(ψq∂m (δη))Vi (η)φδm ,qm (ψq∂m (δm η)) δη n R+. Z + n ∂ R+. Z =. |gqm (δz, 0)|1/2 δ n/2 b(ψq∂m (δη))φδm ,qm (ψq∂m (δm z, 0))Vi (δm z, 0) dz. ∇Vi (η)∇ φ˜ m (η) + δ 2 a(qm )Vi (η)φ˜ m (η) δη. n R+. +δ Z =. Z n ∂ R+. b(qm )Vi (z, 0)φ˜ m (z, 0) δη + O(δ). ∇Vi (η)∇ φ˜ m (η) + O(δ) =. n R+. Z. ˜ ∇Vi (η)∇ φ(η) + o(1),. n R+. By definition of L δm ,qm we have (3-5). φδm ,qm − i ∗ ( f ε0 (Wδm ,qm )[φδm ,qm ]) − L δm ,qm (φδm ,qm ) =. n−1 X. i cm Z δi m ,qm .. i=0 i → 0 as m → ∞. Multiplying We want to prove that, for all i = 0, . . . , n − 1, cm j (3-5) by Z δm ,qm we obtain, by definition of i ∗ , n−1 X. i . j j i cm Z δm ,qm , Z δm ,qm H = i ∗ ( f ε0m (Wδm ,qm )[φδm ,qm ]), Z δm ,qm H. i=0. Z = ∂M. j. f ε0m (Wδm ,qm )[φδm ,qm ]Z δm ,qm dσ.. Moreover, by multiplying (3-5) by φδm ,qm we obtain that Z kφδm ,qm k H − f ε0m (Wδm ,qm )φδ2m ,qm dσ → 0. ∂M. Thus ( f ε0m (Wδm ,qm ))1/2 φδm ,qm is bounded and weakly convergent in L 2 (∂M). With this consideration we easily get.

(13) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. Z ∂M. 87. j. f ε0m (Wδm ,qm )[φδm ,qm ]Z δm ,qm dσ Z j = ( f ε0m (Wδm ,qm ))1/2 φδm ,qm ( f ε0m (Wδm ,qm ))1/2 Z δm ,qm dσ ∂M Z 2/(n−2) ˜ 0)Vi (z, 0) dz + o(1) = o(1), =n U1 (z, 0)φ(z, Rn−1. once we take (3-4) into account. Now, it is easy to prove that. i j Z δm ,qm , Z δm ,qm H = Cδi j + o(1), i → 0 as m → ∞ for each i = 0, . . . , n − 1. This, hence we can conclude that cm combined with (3-5) and using kL εm dm ,qm (φεm dm ,qm )kH → 0, gives us that. (3-6). n−1 X. i φδ ,q − i ∗ ( f 0 (Wδ ,q )[φδ ,q ]) = cm kZ i kH + o(1) = o(1). m m m m m m ε H i=0. n ) and define Choose a smooth function ϕ ∈ C0∞ (R+ −1 −1 1 1 ϕm (x) = (n−2)/2 ϕ ψq∂m (x) χ ψq∂m (x) δm δm. for x ∈ M.. We have that kϕm kH is bounded and, by (3-6), that hhφδm ,qm , ϕm iiH Z . = f ε0m (Wδm ,qm )[φδm ,qm ]ϕm dσ + φδm ,qm − i ∗ ( f ε0m (Wδm ,qm )[φδm ,qm ]), ϕm H Z∂M = f ε0m (Wδm ,qm )[φδm ,qm ]ϕm dσ + o(1) ∂M Z 1 2/(n−2)±εm = (n ± εm (n − 2)) U (z, 0)φ˜ m (z, 0)ϕ dz + o(1) ±εn/(n−2) 1 n−1 δm R Z 2/(n−2) ˜ 0)ϕ(z, 0) dz + o(1), =n U1 (z, 0)φ(z, Rn−1. 2(n−1)/(n−2). by the strong L loc. n ) convergence of φ ˜ m . On the other hand, (∂ R+ Z ˜ hhφδm ,qm , ϕm iiH = ∇ φ∇ϕ δη + o(1), n R+. so φ˜ is a weak solution of (2-5) and we conclude that φ˜ ∈ Span{V0 , V1 , . . . , Vn }..

(14) 88. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. This, combined with (3-4), gives that φ˜ = 0. Proceeding as before we have hhφδm ,qm , φδm ,qm iiH Z = f ε0m (Wδm ,qm )[φδm ,qm ]φδm ,qm dσ + o(1) ∂M Z 1 2/(n−2)±εm = (n ± εm (n − 2)) U1 (z, 0)φ˜ m2 (z, 0)ϕ dz + o(1) = o(1). ±εn/(n−2) Rn−1 δm In a similar way, by (3-6) we have

(15)

(16) |φδm ,qm | L sε =

(17) i ∗ ( f ε0 (Wδm ,qm )[φδm ,qm ])

(18) L sε + o(1) = o(1), which gives kφδm ,qm kH → 0, which is a contradiction.. . 3.2. The estimate of the error term R. Lemma 3. For a, b ∈ R with 0 < a < b, there exists a positive constant C1 = C1 (a, b) such that, for ε small, for any q ∈ ∂M and for any d ∈ [a, b] we have kRε,δ,q kH ≤ C1 ε|ln ε| Proof. We estimate ∗. i f ε (Wδ,q (x)) − Wδ,q (x) H. ≤ i ∗ f ε (Wδ,q (x)) − i ∗ f 0 (Wδ,q (x)) H + i ∗ f 0 (Wδ,q (x)) − Wδ,q (x) H . By definition of i ∗ there exists 0 which solves the equation  −1g 0 + a(x)0 = 0 on M, (3-7)  ∂ 0 + n −2 b(x)0 = f 0 (Wδ,q ) on ∂M, ∂ν 2 so, by (3-7), we have ∗. i f 0 (Wδ,q (x)) − Wδ,q (x) H = k0(x) − Wδ,q (x)k2H Z = [−1g (0 − Wδ,q ) + a(0 − Wδ,q )](0 − Wδ,q ) dµg M Z h i ∂ (n −2) + (0 − Wδ,q ) + b(x)(0 − Wδ,q ) (0 − Wδ,q ) dµg 2 ∂ M ∂ν Z = [1g Wδ,q − aWδ,q ](0 − Wδ,q ) dµg M Z h i ∂ + f 0 (Wδ,q ) − Wδ,q (0 − Wδ,q ) dµg ∂ν ∂M Z n −2 − b(x)Wδ,q (0 − Wδ,q ) dµg := I1 + I2 + I3 . 2 ∂M.

(19) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. 89. We obtain (3-8). I1 = k0 − Wδ,q k H O(δ).. In fact, I1 ≤ |1g Wδ,q − aWδ,q | L 2n/(n+2) (M) |0 − Wδ,q | L 2n/(n−2) (M) ≤ |1g Wδ,q − aWδ,q | L 2n/(n+2) (M) k0 − Wδ,q k H . We easily have that |Wδ,q | L 2n/(n+2) = O(δ 2 ). For the other term we have, in coordinates, (3-9). k 1g Wδ,q = 1[Uδ χ] + (g ab − δab )∂ab [Uδ χ] − g ab 0ab ∂k [Uδ χ ],. k 0ab being the Christoffel symbols. Using the expansion of the metric g ab given by (4-2) and (4-3) we have that

(20) ab

(21)

(22) (g − δab )∂ab [Uδ χ ]

(23) 2n/(n+2) = O(δ), L (M) (3-10)

(24) ab k

(25)

(26) g 0 ∂k [Uδ χ ]

(27) 2n/(n+2) = O(δ 2 ). ab L (M). Since Uδ is a harmonic function we deduce (3-11). |1[Uδ χ ]| L 2n/(n+2) (M) = |Uδ 1χ + 2∇Uδ ∇χ | L 2n/(n+2) (M) = O(δ 2 ).. For the second integral I2 we have I2 = k0 − Wδ,q k H O(δ 2 ),. (3-12) since.

(28)

(29)

(30)

(31) ∂ I2 ≤

(32) f 0 (Wδ,q ) − Wδ,q

(33) 2(n−1)/n |0 − Wδ,q | L 2(n−1)/n−2 (∂M) ∂ν L (∂M)

(34)

(35)

(36)

(37) ∂ ≤ C

(38) f 0 (Wδ,q ) − Wδ,q

(39) 2(n−1)/n k0 − Wδ,q k H , ∂ν L (∂M) and, using the boundary condition for (2-5), we have

(40)

(41)

(42)

(43) ∂ (3-13)

(44) f 0 (Wδ,q ) − Wδ,q

(45) 2(n−1)/n ∂ν L (∂M) Z h 1 = n/2 |g(δz, 0)|1/2 (n − 2)U n/(n−2) (z, 0)χ n/(n−2) (δz, 0) δ n−1 R n i 2(n−1) 2(n−1) n ∂U n−1 − χ (δz, 0) (z, 0) δ dz ∂t Z ≤C (n − 2)U n/(n−2) (z, 0)[χ n/(n−2) (δz, 0) Rn−1. − χ (δz, 0)]. 2(n−1) n. n 2(n−1). dz. = O(δ 2 )..

(46) 90. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. Lastly, (3-14). I3 ≤ |Wδ,q | L 2(n−1)/n (∂M) |0 − Wδ,q | L 2(n−1)/(n−2) (∂M) = k0 − Wδ,q k H O(δ).. By (3-8), (3-12) and (3-14) we conclude that ∗. i f 0 (Wδ,q (x)) − Wδ,q (x) = k0(x) − Wδ,q (x)k = O(δ). H H. To conclude the proof we estimate the term i ∗ f ε (Wδ,q (x)) −i ∗ f 0 (Wδ,q (x)) H . We have, by the properties of i ∗ , that ∗ i f ε (Wδ,q (x)) − i ∗ f 0 (Wδ,q (x)) H

(47)

(48) n/(n−2) ≤

(49) Wδ,q (x)n/(n−2)±ε − Wδ,q (x)

(50) L 2(n−1)/n (∂M) . Z =. 1 δ ±ε(n−2)/2. Rn−1. U. ±ε. ! n 2(n−1) 2(n−1) n n/(n−2) dz (z, 0) − 1 U (z, 0) + O(δ 2 ).. To estimate the last integral, we first recall two Taylor expansions with respect to ε: 1 U ±ε = 1 ± ε ln U + ε2 ln2 U + o(ε2 ), 2 n −2 (n −2)2 2 δ ∓ε(n−2)/2 = 1 ∓ ε ln δ + ε2 ln δ + o(ε2 ln2 δ). 2 8. (3-15) (3-16). In light of (3-15) and (3-16) we have (3-17). ∗. i ( f ε (Wδ,q )) − i ∗ ( f 0 (Wδ,q )) H Z

(51)

(52) n −2 ε ln δ ± ε ln U (z, 0) + O(ε 2 ) ≤

(53) ∓ 2 Rn−1 n

(54) 2(n−1) 2(n−1)

(55) n 2 n/(n−2) + O(ε ln δ) U (z, 0)

(56) dz + O(δ 2 ) =.

(57)

(58) n/(n−2) n −2 ε ln δ

(59) U (z, 0)

(60) L 2(n−1)/(n−2) (Rn−1 ) 2 Z n 2(n−1) 2(n−1)/(n−2) +ε U (z, 0) ln U (z, 0) dz Rn−1. + O(ε2 ) + O(ε2 |ln δ|) + O(δ 2 ) = O(ε) + O(ε|ln δ|) + O(δ 2 ). Choosing δ = dε concludes the proof of Lemma 3 for the subcritical case..

(61) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. 91. For the supercritical case, we have to control |Rε,δ,q | L sε (∂M) . As in the previous case we consider

(62)

(63) |Rε,δ,q | L sε (∂M) ≤

(64) i ∗ f ε (Wδ,q (x)) − i ∗ f 0 (Wδ,q (x))

(65) L sε (∂M)

(66)

(67) +

(68) i ∗ f 0 (Wδ,q (x)) − Wδ,q (x)

(69) L sε (∂M) . As before, set 0 = i ∗ ( f 0 (Wδ,q (x)). Since 0 solves (3-7), 0 − Wδ,q solves   −1g (0 − Wδ,q ) + a(x)(0 − Wδ,q ) = −1g Wδ,q + a(x)Wδ,q     n −2 ∂ (0 − Wδ,q ) + b(x)(0 − Wδ,q )  ∂ν 2    ∂ n −2  = f 0 (0) + Wδ,q + b(x)Wδ,q ∂ν 2. on M, on ∂M.. We choose q as in (2-3), and r = ε. Thus, by Theorem 3.14 in [Nittka 2011], we have |0 − Wδ,q | L sε (∂M) ≤ | − 1g Wδ,q + a(x)Wδ,q | L q+ε (M)

(70)

(71)

(72)

(73) n −2 ∂ b(x)Wδ,q

(74) (n−1)q/(n−q)+ε +

(75) f 0 (0) + Wδ,q + . ∂ν 2 L (∂M) We remark that q=. n−2 n−1 ε n + 2 + 2n n−2 n−1 ε. 2n + n 2. . =. 2n + O + (ε) with 0 < O + (ε) < Cε n +2. for some positive constant C. By direct computation we have + (ε). ,. 1−O + (ε). .. |a(x)Wδ,q | L q+ε (M) ≤ Cδ 2−O |b(x)Wδ,q | L (n−1)q/(n−q)+ε (∂M) ≤ Cδ. Moreover, proceeding as in (3-9), (3-10), (3-11) and (3-13) we get |1g Wδ,q | L q+ε (M) ≤ Cδ 2−O (ε) ,

(76)

(77) +

(78)

(79) ∂ ≤ Cδ 1−O (ε) .

(80) f 0 (0) + Wδ,q

(81) (n−1)q/(n−q)+ε ∂ν L (∂M) +. Since i ∗ ( f ε (Wδ,q )) solves (1-5), and i ∗ ( f ε |u|n/(n−2)+ε (Wδ,q )) solves (1-5), we again use Theorem 3.14 in [Nittka 2011]. Taking (3-15) and (3-16) into account,.

(82) 92. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. we finally get

(83)

(84) (3-18)

(85) i ∗ ( f ε (Wδ,q )) − i ∗ ( f 0 (Wδ,q ))

(86) L sε (∂M) ≤ | f ε (Wδ,q ) − f 0 (Wδ,q )| L 2(n−1)/n+O + (ε) (∂M) Z h 1 ε −O + (ε) U (z, 0) − 1 ≤δ δ ε(n−2)/2 Rn−1 ·U = δ −O. + (ε). n/(n−2). 1 + i 2(n−1) 2(n−1)/n+O + (ε) n +O (ε) dz (z, 0) + O(δ 2 ). (O(ε|ln δ|) + O(ε)) + O(δ 2 ).. Now, choosing δ = dε, we can conclude the proof, since δ −O. + (ε). = 1 + O + (ε)|ln(εd)| = 1 + O + (ε|ln ε|) = O(1).. . 3.3. Solving (2-8): the remainder term φ. Proposition 4. For a, b ∈ R with 0 < a < b, there exists a positive constant C = C(a, b) such that, for ε small, for any q ∈ ∂M and for any d ∈ [a, b] there exists a unique φδ,q which solves (2-8). This solution satisfies kφδ,q kH ≤ Cε|ln ε|. Moreover the map q 7 → φδ,q is a C 1 (∂M, H) map. Proof. First of all, we point out that N is a contraction mapping. We remark that the conjugate exponent of sε is  2(n −1)  in the subcritical case,  0 n sε =   2(n −1)+εn(n −2) in the supercritical case. n +εn(n −2) By the properties of i ∗ and using the expansion of f ε (Wδ,q + φ1 ) centered in Wδ,q + φ2 we have. kN (φ1 )− N (φ2 )kH ≤ f ε (Wδ,q +φ1 )− f ε (Wδ,q +φ2 )− f ε0 (Wδ,q )[φ1 −φ2 ] L sε0 (∂M). ≤ f ε0 (Wδ,q +θ φ1 +(1−θ )φ2 )− f ε0 (Wδ,q ) [φ1 −φ2 ] L sε0 (∂M) and, since |φ1 −φ2 |sε ∈ L sε /sε (∂M) and | f ε0 (·)|sε ∈ L (sε /sε ) (∂M) as f ε0 (·) ∈ L sε (∂M), we have 0. 0. 0. 0 0. kN (φ1 ) − N (φ2 )kH. ≤ f ε0 (Wδ,q + θ φ1 + (1 − θ )φ2 ) − f ε0 (Wδ,q ) L sε (∂M) kφ1 − φ2 k L sε (∂M) = γ kφ1 − φ2 kH ,.

(87) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. 93. where. γ = f ε0 (Wδ,q + θ φ1 + (1 − θ )φ2 ) − f ε0 (Wδ,q ) L sε (∂M) < 1 provided kφ1 kH and kφ2 kH are sufficiently small. In the same way we can prove that kN (φ)kH ≤ γ kφkH with γ < 1 if kφkH is sufficiently small. Next, by Lemmas 2 and 3 we have kL −1 (N (φ) + Rε,δ,q )kH ≤ C(γ kφkH + ε|ln ε|), where C = max{C0 , C0 C1 } > 0, for the constants C0 , C1 which appear in Lemmas 2 and 3. Notice that, given C > 0, it is possible (up to a choice of kφkH sufficiently small) to choose 0 < Cγ < 12 . Now, if kφkH ≤ 2Cε|ln ε|, then the map T (φ) := L −1 (N (φ) + Rε,δ,q ) is a contraction from the ball kφkH ≤ 2Cε|ln ε| in itself, so, by the fixed point theorem, there exists a unique φδ,q with kφδ,q kH ≤ 2Cε|ln ε| solving (3-1), and hence (2-8). The regularity of the map q 7→ φδ,q can be proven via the implicit function theorem. 4. The reduced problem Problem (1-5) has a variational structure. Weak solutions to (1-5) are critical points of the energy functional Jε : H → R given by Z 1 Jε (u) = (|∇u|2 + a(x)u 2 ) dµg 2 M Z Z n −2 (n −2)2 2 + b(x)u dσ − u (2n−2)/(n−2)±ε dσ. 4 ∂M 2n −2±ε(n −2) ∂M Let us introduce the reduced energy Iε : (0, +∞) × ∂M → R by (4-1). Iε (d, q) := Jε (Wεd,q + φεd,q ),. where the remainder term φεd,q has been found in Proposition 4. 4.1. The reduced energy. Here we use the following expansion for the metric tensor on M: (4-2). g i j (y) = δi j + 2h i j (0)yn + O(|y|2 ) for i, j = 1, . . . , n − 1,. (4-3). g in (y) = δin for i = 1, . . . , n − 1, √ g(y) = 1 − (n − 1)H (0)yn + O(|y|2 ),. (4-4).

(88) 94. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. where (y1 , . . . , yn ) are the Fermi coordinates and, by definition of h i j , n−1. 1 X H= h ii . n −1. (4-5). i. We also recall that on ∂M the Fermi coordinates coincide with the exponential ones, so we have that √ g(y1 , . . . , yn−1 , 0) = 1 + O(|y|2 ). (4-6) To improve the readability of this paper, hereafter we write z = (z 1 , . . . , z n−1 ) to indicate the first n − 1 Fermi coordinates and t to indicate the last one, so (y1 , . . . , yn−1 , yn ) = (z, t). Moreover, indices i, j conventionally refer to sums from 1 to n − 1, while l, m usually refer to sums from 1 to n. Proposition 5. (i) If (d0 , q0 ) ∈ (0, +∞) × ∂M is a critical point for the reduced energy Iε defined in (4-1), then Wεd0 ,q0 + φεd0 ,q0 ∈ H solves problem (1-5). (ii) It holds true that Iε (d, q) = cn (ε) + ε[αn dϕ(q) − βn ln d] + o(ε) in the subcritical case, Iε (d, q) = cn (ε) + ε[αn dϕ(q) + βn ln d] + o(ε) in the supercritical case, C 0 -uniformly with respect to d in compact subsets of (0, +∞) and q ∈ ∂M. Here cn (ε) is a constant which only depends on ε and n, αn and βn are positive constants which only depend on n, and ϕ(q) = h(q) − Hg (q) is the function defined in (1-4) . Proof. (i) Set q := q(y) = ψq∂0 (y). Since (d0 , q0 ) is a critical point, we have, for any h ∈ 1, . . . , n − 1,

(89)

(90) ∂ 0= Iε d, ψq∂0 (y)

(91) ∂ yh y=0 DD = Wεd,q(y) + φεd,q(y) − i ∗ ( f ε (Wεd,q(y) + φεd,q(y) )), EE

(92)

(93) ∂ ∂ W + φ

(94) ∂ yh εd,q(y) ∂ yh εd,q(y) H y=0 n−1 DD EE

(95) X

(96) ∂ ∂ i = cεi Z εd,q(y) , Wεd,q(y) + φεd,q(y)

(97) ∂ yh ∂ yh H y=0 i=0. =. n−1 X i=0. n−1 DD DD EE

(98) EE

(99) X

(100)

(101) ∂ ∂ i i cεi Z εd,q(y) , Wεd,q(y)

(102) Z εd,q(y) , φεd,q(y)

(103) , − cli ∂ yh ∂ yh H y=0 H y=0 i=0. using that φεd,q(y) is a solution of (2-8) and that EE DD EE DD ∂ ∂ i i Z εd,q(y) , φεd,q(y) = − Z εd,q(y) , φεd,q(y) ∂ yh ∂ yh H H.

(104) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. 95. ⊥ since φεd,q(y) ∈ K εd,q(y) for all y. Now it is enough to observe that. ∂ i. ≤ Z εd,q(y) kφεd,q(y) kH = o(1), ∂ yh H H DD EE. ∂ 1 1 i i h Z εd,q(y) , Wεd,q(y) = Z εd,q(y) , Z εd,q(y) = δ i h + o(1), H ∂ yh εd εd H DD. ∂ i Z ,φ ∂ yh εd,q(y) εd,q(y). EE. to conclude that 0=. n−1 1 X i ih cε (δ + o(1)), εd i=0. and so cεi = 0 for all i = 0, . . . , n − 1. This concludes the proof of (i). (ii) We prove (ii) in two steps. Step 1. We prove that for ε small enough and for any q ∈ ∂M, |Jε (Wδ,q + φδ,q ) − Jε (Wδ,q )| ≤ kφδ,q k2H + Cε|ln ε|kφδ,q kH = o(ε). We have |Jε (Wδ,q + φδ,q ) − Jε (Wδ,q )|

(105) Z

(106)

(107)

(108) 1

(109) =

(110) [−1g Wδ,q + a(x)Wδ,q ]φδ,q dµg

(111)

(112) + kφδ,q k2H 2 M

(113)

(114) Z h i

(115)

(116) n −2 ∂

(117) Wδ,q + b(x)Wδ,q − f 0 (Wδ,q ) φδ,q dσ

(118)

(119) +

(120) 2 ∂M ∂ν

(121)

(122) Z

(123)

(124)

(125) +

(126) f 0 (Wδ,q ) − f ε (Wδ,q ) φδ,q dσ

(127)

(128) ∂M.

(129) Z

(130) +

(131)

(132). ∂M. (n −2)2 (2n−2)/(n−2)±ε (Wδ,q + φδ,q )(2n−2)/(n−2)±ε − Wδ,q 2n −2±ε(n −2)

(133)

(134) − f ε (Wδ,q )φδ,q dσ

(135)

(136) .. With the same estimate of I1 in Lemma 3 we obtain that

(137) Z

(138)

(139)

(140)

(141) [−1g Wδ,q + a(x)Wδ,q ]φδ,q dµg

(142) = O(δ)kφδ,q k , H

(143)

(144) M. and in light of the estimate of I2 and I3 in Lemma 3 we get

(145) Z h

(146) i

(147)

(148) ∂ n −2

(149)

(150) = O(δ)kφδ,q k . W + b(x)W − f (W ) φ dσ δ,q δ,q 0 δ,q δ,q H

(151)

(152) 2 ∂M ∂ν.

(153) 96. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. In the subcritical case, following the computation in (3-17) we obtain

(154) Z

(155)

(156)

(157)

(158) [ f 0 (Wδ,q ) − f ε (Wδ,q )]φδ,q dσ

(159)

(160)

(161) ∂M. ≤ C| f 0 (Wδ,q ) − f ε (Wδ,q )| L 2(n−1)/n (∂M) |φδ,q | L 2(n−1)/(n−2) (∂M) = [O(ε) + O(ε ln δ)]kφδ,q k H = O(ε|ln ε|)kφδ,q k H , and in a similar way, for the supercritical case, in light of (3-18) we get

(162) Z

(163)

(164)

(165)

(166) [ f 0 (Wδ,q ) − f ε (Wδ,q )]φδ,q dσ

(167)

(168)

(169) ∂M. ≤ C| f 0 (Wδ,q ) − f ε (Wδ,q )| L 2(n−1)/n+O + (ε) (∂M) |φδ,q | L 2(n−1)/(n−2)−O + (ε) (∂M) + ≤ δ −O (ε) (O(ε ln δ) + O(ε)) + O(δ 2 ) kφδ,q k H = O(ε|ln ε|)kφδ,q k H . Finally, by the Taylor expansion formula, for some θ ∈ (0, 1) we immediately have

(170) Z

(171) 2n−2 ±ε

(172)

(173) 2n−2 ±ε (n −2)2 n−2

(174)

(175) n−2 W + φ − W − f (W )φ dσ δ,q δ,q ε δ,q δ,q δ,q

(176)

(177) ∂M 2n −2±ε(n −2)

(178)

(179) Z

(180)

(181) n ±ε(n −2) 2 ±ε 2 Wδ,q + θ φδ,q n−2 φδ,q dσ

(182)

(183) =

(184)

(185) 2 ∂M sεs−2 Z s2 Z

(186)

(187) 2 ±ε sε ε ε s

(188) Wδ,q + θ φδ,q

(189) n−2 sε −2 dσ |φδ,q | ε dσ ≤C ∂M. ∂M. 2 ≤ C|Wδ,q + θ φδ,q |sLεs−2 ε kφδ,q kH. ≤ Ckφδ,q k2H .. Choosing δ = dε, and recalling that, by Proposition 4, kφδ,q kH = O(ε| ln ε|) concludes the proof. Step 2. We prove that Jε (Wδ,q ). n −2 (n −2)3 (n −3) = C(ε) + ε d [b(q) − H (q)] ± ln d ω I n−2 + o(ε) 4 4(n −2)(2n −2) n−1 n−2. C 0 -uniformly with respect to d in compact subsets of (0, +∞) and q ∈ ∂M, where Z 1 C(ε) = |∇U (y)|2 dy 2 R+n Z Z 2n−2 2n−2 (n −2)3 (n −2)2 n−2 − U (z, 0) dz ± ε U n−2 (z, 0) dz 2n −2 Rn−1 2n −2 Rn−1 Z 2 2n−2 (n −2) ∓ε U n−2 (z, 0) ln U (z, 0) dz 2n −2 Rn−1 Z 2n−2 (n −2)3 ∓ ε|ln ε| U n−2 (z, 0) dz, 2(2n −2) Rn−1.

(190) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. 97. and n−2 In−2. ∞. Z = 0. s n−2 dz, (1+s 2 )n−2. and ωn−1 is the volume of the (n − 1)-dimensional unit ball. We compute each term separately. First, we have, by a change of variables and by (4-2), (4-3) and (4-4), Z. n Z X. √ ∂ ∂ U (y) U (y) g(δy) dy + o(δ) ∂ yl ∂ ym l,m=1 Z Z = |∇U (y)|2 dy − δ(n − 1)H (q) yn |∇U (y)|2 dy. |∇Wδ,q |2 dµg = M. n R+. glm (δy). n R+. + 2δ. n R+. n−1 Z X i, j=1. n R+. yn h i j (q). ∂ ∂ U (y) U (y) dy + o(δ). ∂ yi ∂yj. By a symmetry argument we can simplify the last integral to obtain, in a more compact form, Z Z Z 1 1 (n −1)H (q) 2 2 |∇Wδ,q | dµg = |∇U | − δ yn |∇U |2 n 2 M 2 R+n 2 R+ Z n−1 2 X ∂U +δ h ii (q) (y) + o(δ). yn n ∂ yi R+ i=1. Since. ∂U ∂U = for all i, l = 1, . . . , n − 1, by (4-9) we get ∂ yi ∂ yl. n−1 X i=1. h ii (q). Z n R+. yn. . Z n−1 n−1 2 2 X ∂U 1 X ∂U (y) dy = h ii (q) yn (y) dy n ∂ yi n −1 ∂ yl R+ i=1 l=1 Z H (q) = U 2 (z, 0) dz, 4 n−1 R. and in light of (4-7) we conclude that Z Z Z 1 1 (n −2)H (q) 2 2 |∇Wδ,q | dµg = |∇U | − δ U 2 (z, 0) dz + o(δ). 2 M 2 R+n 4 n−1 R By a change of variables, we immediately obtain Z Z √ 1 δ2 2 a(x)|Wδ,q | dµg = a(x)U 2 (y) g(δy) dy + o(δ 2 ) = O(δ 2 ). n 2 M 2 R+.

(191) 98. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. Coming to the boundary integral, we get, by a change of variables, by (4-6), and by expanding b, Z Z √ n −2 n −2 2 b(z)|Wδ,q | dσ = δ b(δz)U 2 (z, 0) g(δz) dz + O(δ 2 ) 4 ∂M 4 Rn−1 Z n −2 = δb(q) U 2 (z, 0) dz + O(δ 2 ). 4 Rn−1 Introducing the abbreviation Un (z) = U (2n−2)/(n−2) (z, 0), by (3-15), (3-16) and (4-6), we have Z |Wδ,q |(2n−2)/(n−2)±ε dσ ∂M Z √ = δ ∓ε(n−2)/2Un (z)U ±ε (z, 0) g(δz) dz + o(δ) Rn−1. Z. Z. n −2 ε ln δ = Un (z) dz ± ε Un (z) ln U (z, 0) dz ∓ 2 Rn−1 Rn−1. Z Rn−1. Un (z) dz. + o(δ) + O(ε2 ) + O(ε 2 ln δ), (n −2)2 (n −2)2 (n −2)3 = ∓ε , we get 2n −2±ε(n −2) 2n −2 2n −2 Z (n −2)2 − |W |(2n−2)/(n−2)−ε dσ 2n −2±ε(n −2) ∂M δ,q Z Z (n −2)3 (n −2)2 Un (z) dz ± ε U (z) dz =− 2n −2 Rn−1 2n −2 Rn−1 n Z Z (n −2)2 (n −2)3 ∓ε U (z) ln U (z, 0) dz ± ε ln δ Un (z) dz 2n −2 Rn−1 n 2(2n −2) Rn−1 and, since. + o(δ) + O(ε2 ) + O(ε 2 ln δ). Notice that, with the choice δ = dε it holds that o(δ) + O(ε2 ) + O(ε 2 ln δ) = o(ε) and ε ln δ = ε ln d − ε|ln ε|. At this point we have Z n −2 Jε (Wδ,q ) = C(ε) + εd [b(q) − H (q)] U 2 (z, 0) dz 4 n−1 R Z (n −2)3 ±ε ln d Un (z) dz + o(ε|ln ε|). 2(2n −2) Rn−1 To conclude, observe that Z n−2 U 2 (z, 0) dz = ωn−1 In−2 Rn−1. Z and Rn−1. n−2 Un (z) dz = ωn−1 In−1 ,.

(192) ON YAMABE-TYPE PROBLEMS ON RIEMANNIAN MANIFOLDS WITH BOUNDARY. where Iβα. Z. 99. sα ds. (1+s 2 )β. ∞. = 0. n −3 n−2 n−2 The conclusion follows after we observe that In−1 = I (for a proof, see 2(n −2) n−2 [Almaraz 2011b, Lemma 9.4(b)]). 4.2. Proof of Theorem 1. Let us introduce Iˆ(d, q) = αn dϕ(q) − βn ln d. If q0 is a local minimizer of ϕ(q) with ϕ(q0 ) > 0, set d0 = βn /(αn ϕ(q0 )) > 0. Thus the pair (d0 , q0 ) is a critical point for Iˆ. Moreover, since there exists a neighborhood B such that ϕ(q) > ϕ(q0 ) on ∂B, it is possible to find a neighborhood e B ⊂ [a, b] × ∂M, (d0 , q0 ) ∈ e B such that Iˆ(d, q) > Iˆ(d0 , q0 ) for (d, q) ∈ ∂ e B. Since, in the subcritical case, by (i) of Proposition 5 we have Iε (d, q) = cn (ε) + ε Iˆ(d, q) + o(ε), we get that for ε sufficiently small there is a (d ∗, q ∗ ) ∈ e B such that Wεd ∗,q ∗ +φεd ∗,q ∗ is a critical point for Iε . Then, by (i) of Proposition 5, Wεd ∗,q ∗ + φεd ∗,q ∗ ∈ H is a solution for problem (1-5) in the subcritical case. The proof for the supercritical case follows in a similar way. 4.3. Some technicalities. If U is a solution of (2-5), then the following hold: Z Z 1 2 t|∇U | dz dt = (4-7) U 2 (z, 0) dz, n 2 n−1 R+ R Z Z (4-8) t|∇U |2 dz dt = 2 t|∂t U |2 dz dt, n R+. Z (4-9). t n R+. n R+. n−1 X. |∂zi U |2 dz dt =. i=1. 1 4. Z Rn−1. U 2 (z, 0) dz.. Proof. To simplify the notation, we set n η = (z, t) ∈ R+. where z ∈ Rn−1 and t ≥ 0.. The first estimate can be obtained by integration by parts, taking into account that 1U = 0. Indeed, Z Z Z n Z X 2 ηn |∇U | δη = − U ∂l [ηn ∂l U ] δη = − U ∂n U δη − ηn U 1U δη n R+. l=1. =−. 1 2. n R+. Z n R+. ∂n [U 2 ] δη =. n R+. 1 2. Z Rn−1. U 2 (z, 0) dz.. n R+.

(193) 100. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. To obtain (4-8), we proceed in a similar way: since 1U = 0 we have Z n Z X 2 0=− 1U ηn ∂n U δη = ∂l U ∂l [ηn2 ∂n U ] δη n R+. Z = n R+. n R+. l=1. 2ηn |∂n U |2 δη +. n Z X l=1. n R+. 2 ηn2 ∂l U ∂ln U δη. Z 1 η2 ∂ |∇U |2 δη = 2ηn |∂n U | δη + n 2 R+n n n R+ Z Z 2 = 2ηn |∂t U | δη − ηn |∇U |2 δη, Z. 2. n R+. n R+. so (4-8) is proved. Now (4-9) is a direct consequence of the first two equalities. In fact, by (4-8) we have Z Z Z n−1 X 2 2 ηn |∇U | δη = ηn |∂i U | δη + ηn |∂n U |2 δη n R+. n R+. Z = n R+. n R+. i=1. ηn. n−1 X i=1. 1 |∂i U | δη + 2 2. Z n R+. ηn |∇U |2 δη.. Thus, Z n R+. ηn. n−1 X. |∂i U |2 δη =. i=1. 1 2. Z n R+. and in light of (4-7) we get the proof.. ηn |∇U |2 δη, . References [Almaraz 2010] S. d. M. Almaraz, “An existence theorem of conformal scalar-flat metrics on manifolds with boundary”, Pacific J. Math. 248:1 (2010), 1–22. MR Zbl [Almaraz 2011a] S. d. M. Almaraz, “Blow-up phenomena for scalar-flat metrics on manifolds with boundary”, J. Differential Equations 251:7 (2011), 1813–1840. MR Zbl [Almaraz 2011b] S. d. M. Almaraz, “A compactness theorem for scalar-flat metrics on manifolds with boundary”, Calc. Var. Partial Differential Equations 41:3-4 (2011), 341–386. MR Zbl [Aubin 1976] T. Aubin, “Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire”, J. Math. Pures Appl. (9) 55:3 (1976), 269–296. MR Zbl [Brendle 2008] S. Brendle, “Blow-up phenomena for the Yamabe equation”, J. Amer. Math. Soc. 21:4 (2008), 951–979. MR Zbl [Brendle and Chen 2014] S. Brendle and S.-Y. S. Chen, “An existence theorem for the Yamabe problem on manifolds with boundary”, J. Eur. Math. Soc. (JEMS) 16:5 (2014), 991–1016. MR Zbl [Brendle and Marques 2009] S. Brendle and F. C. Marques, “Blow-up phenomena for the Yamabe equation, II”, J. Differential Geom. 81:2 (2009), 225–250. MR Zbl.

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(195) 102. MARCO GHIMENTI, ANNA MARIA MICHELETTI AND ANGELA PISTOIA. Received May 29, 2015. Revised January 17, 2016. M ARCO G HIMENTI D IPARTIMENTO DI M ATEMATICA U NIVERSITÀ DI P ISA V IA F. B UONARROTI 1/ C 56127 P ISA I TALY marco.ghimenti@dma.unipi.it A NNA M ARIA M ICHELETTI D IPARTIMENTO DI M ATEMATICA U NIVERSITÀ DI P ISA V IA F. B UONARROTI 1/ C 56127 P ISA I TALY a.micheletti@dma.unipi.it A NGELA P ISTOIA D IPARTIMENTO S CIENZE DI BASE E A PPLICATE PER L’I NGEGNERIA U NIVERSTÀ DI ROMA “L A S APIENZA” V IA A NTONIO S CARPA 16 00161 ROMA I TALY angela.pistoia@uniroma1.it.

(196) PACIFIC JOURNAL OF MATHEMATICS Founded in 1951 by E. F. Beckenbach (1906–1982) and F. Wolf (1904–1989). msp.org/pjm EDITORS Don Blasius (Managing Editor) Department of Mathematics University of California Los Angeles, CA 90095-1555 blasius@math.ucla.edu Paul Balmer Department of Mathematics University of California Los Angeles, CA 90095-1555 balmer@math.ucla.edu. Vyjayanthi Chari Department of Mathematics University of California Riverside, CA 92521-0135 chari@math.ucr.edu. Daryl Cooper Department of Mathematics University of California Santa Barbara, CA 93106-3080 cooper@math.ucsb.edu. Robert Finn Department of Mathematics Stanford University Stanford, CA 94305-2125 finn@math.stanford.edu. Kefeng Liu Department of Mathematics University of California Los Angeles, CA 90095-1555 liu@math.ucla.edu. Jiang-Hua Lu Department of Mathematics The University of Hong Kong Pokfulam Rd., Hong Kong jhlu@maths.hku.hk. Sorin Popa Department of Mathematics University of California Los Angeles, CA 90095-1555 popa@math.ucla.edu. Igor Pak Department of Mathematics University of California Los Angeles, CA 90095-1555 pak.pjm@gmail.com. Jie Qing Department of Mathematics University of California Santa Cruz, CA 95064 qing@cats.ucsc.edu. Paul Yang Department of Mathematics Princeton University Princeton NJ 08544-1000 yang@math.princeton.edu PRODUCTION Silvio Levy, Scientific Editor, production@msp.org SUPPORTING INSTITUTIONS UNIV. OF CALIF., SANTA CRUZ. ACADEMIA SINICA , TAIPEI. STANFORD UNIVERSITY. CALIFORNIA INST. OF TECHNOLOGY. UNIV. OF BRITISH COLUMBIA. UNIV. OF MONTANA. INST. DE MATEMÁTICA PURA E APLICADA. UNIV. OF CALIFORNIA , BERKELEY. UNIV. OF OREGON. KEIO UNIVERSITY. UNIV. OF CALIFORNIA , DAVIS. UNIV. OF SOUTHERN CALIFORNIA. MATH . SCIENCES RESEARCH INSTITUTE. UNIV. OF CALIFORNIA , LOS ANGELES. UNIV. OF UTAH. NEW MEXICO STATE UNIV.. UNIV. OF CALIFORNIA , RIVERSIDE. UNIV. OF WASHINGTON. OREGON STATE UNIV.. UNIV. OF CALIFORNIA , SAN DIEGO. WASHINGTON STATE UNIVERSITY. UNIV. OF CALIF., SANTA BARBARA. These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its contents or policies. See inside back cover or msp.org/pjm for submission instructions. The subscription price for 2016 is US $440/year for the electronic version, and $600/year for print and electronic. Subscriptions, requests for back issues and changes of subscriber address should be sent to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163, U.S.A. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, Zentralblatt MATH, PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and Web of Knowledge (Science Citation Index). The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 798 Evans Hall #3840, Berkeley, CA 94720-3840, is published twelve times a year. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163. PJM peer review and production are managed by EditF LOW® from Mathematical Sciences Publishers. PUBLISHED BY. mathematical sciences publishers nonprofit scientific publishing http://msp.org/ © 2016 Mathematical Sciences Publishers.

(197) PACIFIC JOURNAL OF MATHEMATICS Volume 284. No. 1. September 2016. Bitwist manifolds and two-bridge knots JAMES W. C ANNON , W ILLIAM J. F LOYD , L EE R L AMBERT , WALTER R. PARRY and J ESSICA S. P URCELL. 1. Recognizing right-angled Coxeter groups using involutions C HARLES C UNNINGHAM , A NDY E ISENBERG , A DAM P IGGOTT and K IM RUANE. 41. On Yamabe-type problems on Riemannian manifolds with boundary M ARCO G HIMENTI , A NNA M ARIA M ICHELETTI and A NGELA P ISTOIA. 79. Quantifying separability in virtually special groups M ARK F. H AGEN and P RIYAM PATEL. 103. Conformal designs and minimal conformal weight spaces of vertex operator superalgebras T OMONORI H ASHIKAWA. 121. Coaction functors S. K ALISZEWSKI , M AGNUS B. L ANDSTAD and J OHN Q UIGG. 147. Cohomology and extensions of braces V ICTORIA L EBED and L EANDRO V ENDRAMIN. 191. Noncommutative differentials on Poisson–Lie groups and pre-Lie algebras S HAHN M AJID and W EN -Q ING TAO. 213.

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