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On bifurcation for semilinear elliptic Dirichlet problems on geodesic balls
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Doctopic: Partial Differential Equations
[m3L; v 1.125; Prn:4/02/2014; 9:21] P.1 (1-7)
J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On bifurcation for semilinear elliptic Dirichlet problems on geodesic balls Alessandro Portaluri a,1 , Nils Waterstraat b,∗,2 a Department of Agriculture, Forest and Food Sciences, Università degli studi di Torino, Via Leonardo da Vinci, 44, 10095 Grugliasco (TO), Italy b Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
a r t i c l e
i n f o
Article history: Received 15 November 2013 Available online xxxx Submitted by M. del Pino Keywords: Variational bifurcation Crossing forms Semilinear elliptic PDE Morse index theorem
a b s t r a c t We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale. © 2014 Elsevier Inc. All rights reserved.
1. Introduction Let (M, g) be an oriented Riemannian manifold of dimension n and let Δ = div grad : C ∞ (M ) → C ∞ (M ) denote the associated Laplace–Beltrami operator. Let V : M ×R → R be a C 1 -function such that V (p, 0) = 0 for all p ∈ M and V (p, ξ) C 1 + |ξ|α ,
∂V β ∂ξ (p, ξ) C 1 + |ξ| ,
(p, ξ) ∈ M × R,
(1)
for some C > 0 and constants α, β 0 depending on the dimension n of M (cf. [1, §1.2]). In this paper we deal with solutions of the semilinear equation −Δu(p) + V p, u(p) = 0,
p ∈ M,
(2)
* Corresponding author. E-mail addresses: [email protected] (A. Portaluri), [email protected] (N. Waterstraat). A. Portaluri was supported by the grant PRIN2009 “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”. 2 N. Waterstraat was supported by a postdoctoral fellowship of the German Academic Exchange Service (DAAD) and by the program “Professori Visitatori Junior” of GNAMPA-INdAM. 1
0022-247X/$ – see front matter © 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2014.01.064
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Doctopic: Partial Differential Equations
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A. Portaluri, N. Waterstraat / J. Math. Anal. Appl. ••• (••••) •••–•••
under Dirichlet boundary conditions. Note that many equations from geometric analysis are of the type (2). Let us refer to [2,3] and just mention as an example on compact manifolds of dimension n 3 the equation 4
n+2 n−1 Δu(p) + s(p)u(p) = μu(p) n−2 , n−2
p ∈ M,
(3)
where s : M → R denotes the scalar curvature function and μ the Yamabe invariant of the metric g on M . 4 Positive solutions u ∈ C ∞ (M ) of (3) give rise to metrics g˜ of constant scalar curvature on M by g˜ = u n−2 g. We now fix a point p0 ∈ M and assume that the unit ball B(0, 1) ⊂ Tp0 M is contained in the maximal domain on which the exponential map expp0 at p0 is an embedding. Let us denote by B(p0 , r) = expp0 (B(0, r)) the geodesic ball of radius 0 < r 1 around p0 and consider the Dirichlet boundary value problems −Δu(p) + V p, u(p) = 0, p ∈ B(p0 , r), (4) u(p) = 0, p ∈ ∂B(p0 , r). We call r∗ ∈ (0, 1] a bifurcation radius for the boundary value problems (4) if there exists a sequence of radii rn → r∗ and functions un ∈ H01 (B(p0 , rn )) such that un is a non-trivial weak solution of (4) on B(p0 , rn ) and un H01 (B(p0 ,rn )) → 0. Let us now consider the linearised boundary value problems −Δu(p) + f (p)u(p) = 0, p ∈ B(p0 , r), (5) u(p) = 0, p ∈ ∂B(p0 , r), ∗ where f (p) = ∂V ∂ξ (p, 0), p ∈ M . In what follows, we suppose that f : M → R is smooth. We call r ∈ (0, 1] a conjugate radius for (5) if
m r∗ := dim u ∈ C 2 B p0 , r∗ : u solves (5) > 0, and from now on we assume that m(1) = 0. Our main result reads as follows: Theorem 1.1. The bifurcation radii of (4) are precisely the conjugate radii of (5). Let us once again consider Eq. (3). By linearising (3) along the trivial solution u ≡ 0, we get ⎧ ⎨ 4 n − 1 Δu(p) + s(p)u(p) = 0, p ∈ B(p , r), 0 n−2 ⎩ u(p) = 0, p ∈ ∂B(p0 , r),
(6)
and so the bifurcation radii do not depend on the Yamabe invariant μ. Moreover, if g is already of constant scalar curvature, then the bifurcation radii are entirely determined by the spectrum of the Laplace–Beltrami operator. In particular, there is no bifurcation on manifolds of constant positive scalar curvature. We explain below in the proof of Theorem 1.1 that we obtain from our methods a new proof of the Morse–Smale index theorem [8] (cf. also [9]) for the linearised equations (5). As a consequence, we conclude that m(r) = 0 for all but a finite number of radii r ∈ (0, 1), and moreover, we derive from Theorem 1.1 the following corollary: Corollary 1.2. Let μ denote the Morse index of (5) on B(p0 , 1), i.e. the number of negative eigenvalues counted according to their multiplicities. If μ = 0, then there exist at least μ max0
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Doctopic: Partial Differential Equations
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Let us point out that a proof of Theorem 1.1 and Corollary 1.2 for the special case that M is a star-shaped domain in Rn can be found in [6]. The following section is devoted to the more general setting which we consider here. 2. The proof Our main reference for the Laplace–Beltrami operator on manifolds with boundary is [10, §2.4]. Let us recall at first that in local coordinates Δu =
n
|g|− 2 1
j,k=1
1 ∂u ∂ jk 2 g , |g| ∂xj ∂xk
where g jk , 1 j, k n, are the components of the inverse of the metric tensor g = {gjk } and |g| := | det{gjk }| is the absolute value of its determinant. Denoting by dvolg the volume form of g, we find for v ∈ H01 (B(p0 , r)), 0 < r 1,
−
(Δu)(p)v(p) dvolg +
B(p0 ,r)
V p, u(p) v(p) dvolg
B(p0 ,r)
=−
v(x)
j,k=1
B(0,r)
n
=
1 g(x) 2 V x, u(x) v(x) dx
B(0,r)
1 g(x) 2 V x, u(x) v(x) dx
B(0,r)
n
= rn
1 ∂ jk g(x) 2 ∂u (x) dx + g (x) ∂xj ∂xk
1 ∂u ∂v g jk (x)g(x) 2 k (x) j (x) dx + ∂x ∂x
j,k=1
B(0,r)
n
B(0,1) j,k=1
1 ∂u ∂v g jk (r · x)g(r · x) 2 k (r · x) j (r · x) dx ∂x ∂x
1 g(r · x) 2 V r · x, u(r · x) v(r · x) dx,
+ rn B(0,1)
and analogously
−
(Δu)(p)v(p) dvolg +
B(p0 ,r)
f (p)u(p)v(p) dvolg
B(p0 ,r)
n
= rn
B(0,1) j,k=1
+ rn
1 ∂u ∂v g jk (r · x)g(r · x) 2 k (r · x) j (r · x) dx ∂x ∂x
1 g(r · x) 2 f (r · x)u(r · x)v(r · x) dx.
B(0,1)
We now set B := B(0, 1) and define for 0 r 1 a functional qr : H01 (B) × H01 (B) → R by qr (u, v) =
n B j,k=1
1 ∂u ∂v g jk (r · x)g(r · x) 2 k (x) j (x) dx + r2 ∂x ∂x
B
1 g(r · x) 2 V r · x, u(x) v(x) dx
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as well as a quadratic form hr : H01 (B) → R by hr (u) =
n B j,k=1
1 ∂u ∂u g jk (r · x)g(r · x) 2 k (x) j (x) dx + r2 ∂x ∂x
1 g(r · x) 2 f (r · x)u(x)2 dx.
B
From the computations above we obtain: (i) r∗ ∈ (0, 1] is a bifurcation radius for (4), if and only if there exist a sequence {rn }n∈N ⊂ (0, 1], rn → r∗ , and a sequence of non-trivial functions {un }n∈N ⊂ H01 (B), un → 0, such that qrn (un , ·) = 0 ∈ (H01 (B))∗ for all n ∈ N; (ii) by using elliptic regularity arguments, r∗ ∈ (0, 1] is a conjugate radius for (5) if and only if hr is degenerate. We now define a function ψ : [0, 1] × H01 (B) → R by
ψ(r, u) =
n B j,k=1
1 ∂u ∂u g jk (r · x)g(r · x) 2 k (x) j (x) dx + r2 ∂x ∂x
1 g(r · x) 2 G r · x, u(x) dx,
B
where t G(x, t) =
V (x, ξ) dξ. 0
It is a standard result that ψ is C 2 -smooth under the growth conditions (1), and Du ψr = qr (u, ·), u ∈ H01 (B). Moreover, 0 ∈ H01 (B) is a critical point of all functionals ψr and the corresponding Hessians are given by D02 ψr = hr (cf. [7, App. B]). From the compactness of the inclusion H01 (B) → L2 (B) (cf. [10, Prop. 4.4]), we see at once that the Riesz representation of the quadratic form hr is a selfadjoint Fredholm operator. In particular, it is invertible if hr is non-degenerate. Let us now assume at first that r∗ ∈ (0, 1] is not a conjugate radius. Then hr∗ is non-degenerate and we conclude by the implicit function theorem [1, §2.2] that the equation qr (u, ·) = 0 has no other solutions than (r, 0) ∈ [0, 1] × H01 (B) in a neighbourhood of (r∗ , 0). Consequently, (r∗ , 0) is not a bifurcation radius, and we have shown that every bifurcation radius in (0, 1] is a conjugate radius. In order to prove the remaining implication of Theorem 1.1, we make use of the bifurcation theory for critical points of smooth functionals developed in [4]. Accordingly, we consider for r0 ∈ (0, 1) the quadratic forms Γ (h, r0 ) : ker hr0 → R,
Γ (h, r0 )[u] =
d hr u. dr r=r0
By [4, Thm. 1 & Thm. 4.1], r0 is a bifurcation radius if Γ (h, r0 ) is non-degenerate and has a non-vanishing signature (cf. also Section 2.1 in [6]). Consequently, we now assume that r0 ∈ (0, 1) is a conjugate radius and our aim is to compute Γ (h, r0 ). Let us write for simplicity of notation 1 ajk (x) = g jk (x)g(x) 2 , x ∈ B, 1 j, k n, 1 f˜(x) = g(x) 2 f (x), x ∈ B.
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Doctopic: Partial Differential Equations
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5
For u ∈ ker hr0 we have by definition Γ (h, r0 )[u] =
n
∇ajk (r0 · x), x
B j,k=1
∂u ∂u dx + ∂xk ∂xj
B
2 d r f˜(r · x) u(x)2 dx. dr r=r0
(7)
Let us now introduce a new function by vr (x) := u( rr0 · x), r ∈ (0, r0 ], x ∈ B, and denote d 1 ∇u(x), x . u(x) ˙ := vr (x) = dr r=r0 r0
(8)
It is readily seen that vr satisfies −
n
j,k=1
∂vr ∂ jk a (r · x) k + r2 f˜(r · x)vr (x) = 0, ∂xj ∂x
and by differentiating with respect to r at r = r0 we have n
0=−
j,k=1
d + dr
n
∂u ∂ u˙ ∂ jk ∂ jk ∇a (r0 · x), x − a (r0 · x) k ∂xj ∂xk ∂xj ∂x j,k=1
r2 f˜(r · x) u(x) + r02 f˜(r0 · x)u(x). ˙
(9)
r=r0
We multiply (9) by u and integrate over B: 0=−
n B j,k=1
d dr
+ B
n ∂u ∂ u˙ ∂ jk ∂ jk ∇a (r0 · x), x u(x) dx − a (r0 · x) k u(x) dx ∂xj ∂xk ∂xj ∂x 2 r f˜(r · x) u(x)2 dx +
r=r0
B j,k=1
r02 f˜(r0 · x)u(x)u(x) ˙ dx. B
Let ν(x) = (ν1 (x), . . . , νn (x)), x ∈ ∂B, denote the outward pointing unit normal to the boundary of B. Using u|∂B = 0, we obtain from integration by parts 0=
n B j,k=1
+
∂u ∂u ∇a (r0 · x), x dx − ∂xk ∂xj jk
n
B j,k=1
+
B j,k=1
+
∂ u˙ ∂u ajk (r0 · x) k j dx − ∂x ∂x
∂B
r2 f˜(r · x) u(x)2 dx +
j,k=1
ajk (r0 · x)νj (x)
B
n B j,k=1
∂u u(x) ˙ dS + ∂xk
r02 f˜(r0 · x)u(x)u(x) ˙ dx.
+
∂ u ˙ ajk (r0 · x)νj (x) k u(x) dS ∂x
r02 f˜(r0 · x)u(x)u(x) ˙ dx B
j,k=1
∂u ∇a (r0 x), ˙ x νj (x) k u(x) dS ∂x jk
∂u ∂u ∇ajk (r0 · x), x dx − ∂xk ∂xj
n
j,k=1
n
∂B
r=r0
B
=
∂B
d dr
n
n
B
∂u ∂ jk a (r0 · x) k u(x) ˙ dx ∂xj ∂x 2 d r f˜(r · x) u(x)2 dx dr r=r0
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Since u ∈ ker hr0 , −
n
j,k=1
∂ ∂u jk a (r0 · x) k + r02 f˜(r0 · x)u(x) = 0, ∂xj ∂x
x ∈ B,
(10)
and it follows from (7) and (8) that n
1 Γ (h, r0 )[u] = − r0
j,k=1
∂B
∂u a (r0 · x)νj (x) k ∇u(x), x dS. ∂x jk
(11)
If we set A(x) := {ajk (x)}, x ∈ B, and use that ν(x) = x for all x ∈ ∂B, we can rewrite (11) as
1 Γ (h, r0 )[u] = − r0
A(r0 · x)x, ∇u(x)
∇u(x), x dS.
∂B
Denoting by (A(r0 · x)x)T , x ∈ ∂B, the tangential component of the vector A(r0 · x)x, we have
T A(r0 · x)x, ∇u(x) = ∇u(x), x A(r0 · x)x, x + ∇u(x), A(r0 · x)x
and hence Γ (h, r0 )[u] = −
1 r0
∇u(x), x
2
A(r0 · x)x, x dS
∂B
1 − r0
∇u(x), x
T ∇u(x), A(r0 · x)x dS.
∂B
Since
T ∂u div u(x) (x) A(r0 · x)x ∂ν
and
∂u ∂ν (x)
T T ∂u ∂u (x) ∇u(x), A(r0 · x)x = u(x) ∇ (x), A(r0 · x)x + ∂ν ∂ν T ∂u + u(x) (x) div A(r0 · x)x ∂ν
= ∇u(x), x , we obtain from the assumed Dirichlet boundary conditions T T ∇u(x), x ∇u(x), A(r0 · x)x = div u(x) x, ∇u(x) A(r0 · x)x ,
and hence we finally get by Stokes’ theorem 2 1 Γ (h, r0 )[u] = − ∇u(x), x A(r0 · x)x, x dS 0, r0
x ∈ ∂B,
(12)
∂B
where we use that A(x) is positive definite for all x ∈ B. Moreover, we obtain from (12) that if Γ (h, r0 )[u] = 0 for some u ∈ ker hr0 , then
∂u (x) = 0 ∇u(x), x = ∇u(x), ν(x) = ∂ν
for all x ∈ ∂B which implies u ≡ 0 by the unique continuation property (cf. [11, Def. 3.3 and Prop. 3.4] and [5]).
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Doctopic: Partial Differential Equations
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In summary, we have shown that Γ (h, r0 ) is negative definite, and so in particular non-degenerate with the non-vanishing signature sgn Γ (h, r0 ) = m(r0 ).
(13)
Consequently, r0 is a bifurcation radius and Theorem 1.1 is proven. Let us now prove Corollary 1.2. We note at first that the Morse index μ of (5) on the full domain B(p0 , 1) is given by the Morse index μ(h1 ) of the quadratic form h1 . Moreover, since h0 is positive, we see that μ(h0 ) = 0. It is shown in [4, Prop. 3.9 & Thm. 4.1] that if Γ (h, r) is non-degenerate for all r ∈ (0, 1), then ker hr = 0 for all but a finite number of r ∈ (0, 1) and
μ(h1 ) − μ(h0 ) =
sgn Γ (h, r).
0
Consequently, we conclude from (13) that m(r) = dim ker hr = 0 for all but a finite number of r ∈ (0, 1) and μ=
m(r).
(14)
0
Let us point out that (14) was obtained by Smale in [8] by studying the monotonicity of eigenvalues under shrinking of domains. Hence we have obtained a new proof of Smale’s theorem for the boundary value problem (5), and moreover, Corollary 1.2 is now an immediate consequence of (14) and Theorem 1.1. References [1] A. Ambrosetti, G. Prodi, A Primer of Nonlinear Analysis, Cambridge Stud. Adv. Math., vol. 34, Cambridge University Press, 1993. [2] T. Aubin, Nonlinear Analysis on Manifolds. Monge–Ampère Equations, Grundlehren Math. Wiss., vol. 252, SpringerVerlag, 1988. [3] A.L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., Springer, 1987. [4] P.M. Fitzpatrick, J. Pejsachowicz, L. Recht, Spectral flow and bifurcation of critical points of strongly-indefinite functionals, Part I: General theory, J. Funct. Anal. 162 (1999) 52–95. [5] L. Hörmander, Linear Partial Differential Operators, Grundlehren Math. Wiss., Bd. 116, Academic Press/Springer-Verlag, New York/Berlin, Göttingen, Heidelberg, 1963. [6] A. Portaluri, N. Waterstraat, On bifurcation for semilinear elliptic Dirichlet problems and the Morse–Smale index theorem, J. Math. Anal. Appl. 408 (2013) 572–575, arXiv:1301.1458 [math.AP]. [7] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, American Mathematical Society, Providence, RI, 1986, Published for the Conference Board of the Mathematical Sciences, Washington, DC. [8] S. Smale, On the Morse index theorem, J. Math. Mech. 14 (1965) 1049–1055. [9] S. Smale, Corrigendum: “On the Morse index theorem”, J. Math. Mech. 16 (1967) 1069–1070. [10] M.E. Taylor, Partial Differential Equations I—Basic Theory, Appl. Math. Sci., vol. 115, Springer-Verlag, 1996. [11] K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Differential Geom. 8 (1973) 555–564.
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[m3L; v 1.125; Prn:4/02/2014; 9:21] P.1 (1-7)
J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On bifurcation for semilinear elliptic Dirichlet problems on geodesic balls Alessandro Portaluri a,1 , Nils Waterstraat b,∗,2 a Department of Agriculture, Forest and Food Sciences, Università degli studi di Torino, Via Leonardo da Vinci, 44, 10095 Grugliasco (TO), Italy b Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
a r t i c l e
i n f o
Article history: Received 15 November 2013 Available online xxxx Submitted by M. del Pino Keywords: Variational bifurcation Crossing forms Semilinear elliptic PDE Morse index theorem
a b s t r a c t We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale. © 2014 Elsevier Inc. All rights reserved.
1. Introduction Let (M, g) be an oriented Riemannian manifold of dimension n and let Δ = div grad : C ∞ (M ) → C ∞ (M ) denote the associated Laplace–Beltrami operator. Let V : M ×R → R be a C 1 -function such that V (p, 0) = 0 for all p ∈ M and V (p, ξ) C 1 + |ξ|α ,
∂V β ∂ξ (p, ξ) C 1 + |ξ| ,
(p, ξ) ∈ M × R,
(1)
for some C > 0 and constants α, β 0 depending on the dimension n of M (cf. [1, §1.2]). In this paper we deal with solutions of the semilinear equation −Δu(p) + V p, u(p) = 0,
p ∈ M,
(2)
* Corresponding author. E-mail addresses: [email protected] (A. Portaluri), [email protected] (N. Waterstraat). A. Portaluri was supported by the grant PRIN2009 “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”. 2 N. Waterstraat was supported by a postdoctoral fellowship of the German Academic Exchange Service (DAAD) and by the program “Professori Visitatori Junior” of GNAMPA-INdAM. 1
0022-247X/$ – see front matter © 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2014.01.064
JID:YJMAA 2
AID:18236 /FLA
Doctopic: Partial Differential Equations
[m3L; v 1.125; Prn:4/02/2014; 9:21] P.2 (1-7)
A. Portaluri, N. Waterstraat / J. Math. Anal. Appl. ••• (••••) •••–•••
under Dirichlet boundary conditions. Note that many equations from geometric analysis are of the type (2). Let us refer to [2,3] and just mention as an example on compact manifolds of dimension n 3 the equation 4
n+2 n−1 Δu(p) + s(p)u(p) = μu(p) n−2 , n−2
p ∈ M,
(3)
where s : M → R denotes the scalar curvature function and μ the Yamabe invariant of the metric g on M . 4 Positive solutions u ∈ C ∞ (M ) of (3) give rise to metrics g˜ of constant scalar curvature on M by g˜ = u n−2 g. We now fix a point p0 ∈ M and assume that the unit ball B(0, 1) ⊂ Tp0 M is contained in the maximal domain on which the exponential map expp0 at p0 is an embedding. Let us denote by B(p0 , r) = expp0 (B(0, r)) the geodesic ball of radius 0 < r 1 around p0 and consider the Dirichlet boundary value problems −Δu(p) + V p, u(p) = 0, p ∈ B(p0 , r), (4) u(p) = 0, p ∈ ∂B(p0 , r). We call r∗ ∈ (0, 1] a bifurcation radius for the boundary value problems (4) if there exists a sequence of radii rn → r∗ and functions un ∈ H01 (B(p0 , rn )) such that un is a non-trivial weak solution of (4) on B(p0 , rn ) and un H01 (B(p0 ,rn )) → 0. Let us now consider the linearised boundary value problems −Δu(p) + f (p)u(p) = 0, p ∈ B(p0 , r), (5) u(p) = 0, p ∈ ∂B(p0 , r), ∗ where f (p) = ∂V ∂ξ (p, 0), p ∈ M . In what follows, we suppose that f : M → R is smooth. We call r ∈ (0, 1] a conjugate radius for (5) if
m r∗ := dim u ∈ C 2 B p0 , r∗ : u solves (5) > 0, and from now on we assume that m(1) = 0. Our main result reads as follows: Theorem 1.1. The bifurcation radii of (4) are precisely the conjugate radii of (5). Let us once again consider Eq. (3). By linearising (3) along the trivial solution u ≡ 0, we get ⎧ ⎨ 4 n − 1 Δu(p) + s(p)u(p) = 0, p ∈ B(p , r), 0 n−2 ⎩ u(p) = 0, p ∈ ∂B(p0 , r),
(6)
and so the bifurcation radii do not depend on the Yamabe invariant μ. Moreover, if g is already of constant scalar curvature, then the bifurcation radii are entirely determined by the spectrum of the Laplace–Beltrami operator. In particular, there is no bifurcation on manifolds of constant positive scalar curvature. We explain below in the proof of Theorem 1.1 that we obtain from our methods a new proof of the Morse–Smale index theorem [8] (cf. also [9]) for the linearised equations (5). As a consequence, we conclude that m(r) = 0 for all but a finite number of radii r ∈ (0, 1), and moreover, we derive from Theorem 1.1 the following corollary: Corollary 1.2. Let μ denote the Morse index of (5) on B(p0 , 1), i.e. the number of negative eigenvalues counted according to their multiplicities. If μ = 0, then there exist at least μ max0
JID:YJMAA
AID:18236 /FLA
Doctopic: Partial Differential Equations
[m3L; v 1.125; Prn:4/02/2014; 9:21] P.3 (1-7)
A. Portaluri, N. Waterstraat / J. Math. Anal. Appl. ••• (••••) •••–•••
3
Let us point out that a proof of Theorem 1.1 and Corollary 1.2 for the special case that M is a star-shaped domain in Rn can be found in [6]. The following section is devoted to the more general setting which we consider here. 2. The proof Our main reference for the Laplace–Beltrami operator on manifolds with boundary is [10, §2.4]. Let us recall at first that in local coordinates Δu =
n
|g|− 2 1
j,k=1
1 ∂u ∂ jk 2 g , |g| ∂xj ∂xk
where g jk , 1 j, k n, are the components of the inverse of the metric tensor g = {gjk } and |g| := | det{gjk }| is the absolute value of its determinant. Denoting by dvolg the volume form of g, we find for v ∈ H01 (B(p0 , r)), 0 < r 1,
−
(Δu)(p)v(p) dvolg +
B(p0 ,r)
V p, u(p) v(p) dvolg
B(p0 ,r)
=−
v(x)
j,k=1
B(0,r)
n
=
1 g(x) 2 V x, u(x) v(x) dx
B(0,r)
1 g(x) 2 V x, u(x) v(x) dx
B(0,r)
n
= rn
1 ∂ jk g(x) 2 ∂u (x) dx + g (x) ∂xj ∂xk
1 ∂u ∂v g jk (x)g(x) 2 k (x) j (x) dx + ∂x ∂x
j,k=1
B(0,r)
n
B(0,1) j,k=1
1 ∂u ∂v g jk (r · x)g(r · x) 2 k (r · x) j (r · x) dx ∂x ∂x
1 g(r · x) 2 V r · x, u(r · x) v(r · x) dx,
+ rn B(0,1)
and analogously
−
(Δu)(p)v(p) dvolg +
B(p0 ,r)
f (p)u(p)v(p) dvolg
B(p0 ,r)
n
= rn
B(0,1) j,k=1
+ rn
1 ∂u ∂v g jk (r · x)g(r · x) 2 k (r · x) j (r · x) dx ∂x ∂x
1 g(r · x) 2 f (r · x)u(r · x)v(r · x) dx.
B(0,1)
We now set B := B(0, 1) and define for 0 r 1 a functional qr : H01 (B) × H01 (B) → R by qr (u, v) =
n B j,k=1
1 ∂u ∂v g jk (r · x)g(r · x) 2 k (x) j (x) dx + r2 ∂x ∂x
B
1 g(r · x) 2 V r · x, u(x) v(x) dx
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AID:18236 /FLA
Doctopic: Partial Differential Equations
[m3L; v 1.125; Prn:4/02/2014; 9:21] P.4 (1-7)
A. Portaluri, N. Waterstraat / J. Math. Anal. Appl. ••• (••••) •••–•••
4
as well as a quadratic form hr : H01 (B) → R by hr (u) =
n B j,k=1
1 ∂u ∂u g jk (r · x)g(r · x) 2 k (x) j (x) dx + r2 ∂x ∂x
1 g(r · x) 2 f (r · x)u(x)2 dx.
B
From the computations above we obtain: (i) r∗ ∈ (0, 1] is a bifurcation radius for (4), if and only if there exist a sequence {rn }n∈N ⊂ (0, 1], rn → r∗ , and a sequence of non-trivial functions {un }n∈N ⊂ H01 (B), un → 0, such that qrn (un , ·) = 0 ∈ (H01 (B))∗ for all n ∈ N; (ii) by using elliptic regularity arguments, r∗ ∈ (0, 1] is a conjugate radius for (5) if and only if hr is degenerate. We now define a function ψ : [0, 1] × H01 (B) → R by
ψ(r, u) =
n B j,k=1
1 ∂u ∂u g jk (r · x)g(r · x) 2 k (x) j (x) dx + r2 ∂x ∂x
1 g(r · x) 2 G r · x, u(x) dx,
B
where t G(x, t) =
V (x, ξ) dξ. 0
It is a standard result that ψ is C 2 -smooth under the growth conditions (1), and Du ψr = qr (u, ·), u ∈ H01 (B). Moreover, 0 ∈ H01 (B) is a critical point of all functionals ψr and the corresponding Hessians are given by D02 ψr = hr (cf. [7, App. B]). From the compactness of the inclusion H01 (B) → L2 (B) (cf. [10, Prop. 4.4]), we see at once that the Riesz representation of the quadratic form hr is a selfadjoint Fredholm operator. In particular, it is invertible if hr is non-degenerate. Let us now assume at first that r∗ ∈ (0, 1] is not a conjugate radius. Then hr∗ is non-degenerate and we conclude by the implicit function theorem [1, §2.2] that the equation qr (u, ·) = 0 has no other solutions than (r, 0) ∈ [0, 1] × H01 (B) in a neighbourhood of (r∗ , 0). Consequently, (r∗ , 0) is not a bifurcation radius, and we have shown that every bifurcation radius in (0, 1] is a conjugate radius. In order to prove the remaining implication of Theorem 1.1, we make use of the bifurcation theory for critical points of smooth functionals developed in [4]. Accordingly, we consider for r0 ∈ (0, 1) the quadratic forms Γ (h, r0 ) : ker hr0 → R,
Γ (h, r0 )[u] =
d hr u. dr r=r0
By [4, Thm. 1 & Thm. 4.1], r0 is a bifurcation radius if Γ (h, r0 ) is non-degenerate and has a non-vanishing signature (cf. also Section 2.1 in [6]). Consequently, we now assume that r0 ∈ (0, 1) is a conjugate radius and our aim is to compute Γ (h, r0 ). Let us write for simplicity of notation 1 ajk (x) = g jk (x)g(x) 2 , x ∈ B, 1 j, k n, 1 f˜(x) = g(x) 2 f (x), x ∈ B.
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AID:18236 /FLA
Doctopic: Partial Differential Equations
[m3L; v 1.125; Prn:4/02/2014; 9:21] P.5 (1-7)
A. Portaluri, N. Waterstraat / J. Math. Anal. Appl. ••• (••••) •••–•••
5
For u ∈ ker hr0 we have by definition Γ (h, r0 )[u] =
n
∇ajk (r0 · x), x
B j,k=1
∂u ∂u dx + ∂xk ∂xj
B
2 d r f˜(r · x) u(x)2 dx. dr r=r0
(7)
Let us now introduce a new function by vr (x) := u( rr0 · x), r ∈ (0, r0 ], x ∈ B, and denote d 1 ∇u(x), x . u(x) ˙ := vr (x) = dr r=r0 r0
(8)
It is readily seen that vr satisfies −
n
j,k=1
∂vr ∂ jk a (r · x) k + r2 f˜(r · x)vr (x) = 0, ∂xj ∂x
and by differentiating with respect to r at r = r0 we have n
0=−
j,k=1
d + dr
n
∂u ∂ u˙ ∂ jk ∂ jk ∇a (r0 · x), x − a (r0 · x) k ∂xj ∂xk ∂xj ∂x j,k=1
r2 f˜(r · x) u(x) + r02 f˜(r0 · x)u(x). ˙
(9)
r=r0
We multiply (9) by u and integrate over B: 0=−
n B j,k=1
d dr
+ B
n ∂u ∂ u˙ ∂ jk ∂ jk ∇a (r0 · x), x u(x) dx − a (r0 · x) k u(x) dx ∂xj ∂xk ∂xj ∂x 2 r f˜(r · x) u(x)2 dx +
r=r0
B j,k=1
r02 f˜(r0 · x)u(x)u(x) ˙ dx. B
Let ν(x) = (ν1 (x), . . . , νn (x)), x ∈ ∂B, denote the outward pointing unit normal to the boundary of B. Using u|∂B = 0, we obtain from integration by parts 0=
n B j,k=1
+
∂u ∂u ∇a (r0 · x), x dx − ∂xk ∂xj jk
n
B j,k=1
+
B j,k=1
+
∂ u˙ ∂u ajk (r0 · x) k j dx − ∂x ∂x
∂B
r2 f˜(r · x) u(x)2 dx +
j,k=1
ajk (r0 · x)νj (x)
B
n B j,k=1
∂u u(x) ˙ dS + ∂xk
r02 f˜(r0 · x)u(x)u(x) ˙ dx.
+
∂ u ˙ ajk (r0 · x)νj (x) k u(x) dS ∂x
r02 f˜(r0 · x)u(x)u(x) ˙ dx B
j,k=1
∂u ∇a (r0 x), ˙ x νj (x) k u(x) dS ∂x jk
∂u ∂u ∇ajk (r0 · x), x dx − ∂xk ∂xj
n
j,k=1
n
∂B
r=r0
B
=
∂B
d dr
n
n
B
∂u ∂ jk a (r0 · x) k u(x) ˙ dx ∂xj ∂x 2 d r f˜(r · x) u(x)2 dx dr r=r0
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AID:18236 /FLA
Doctopic: Partial Differential Equations
[m3L; v 1.125; Prn:4/02/2014; 9:21] P.6 (1-7)
A. Portaluri, N. Waterstraat / J. Math. Anal. Appl. ••• (••••) •••–•••
6
Since u ∈ ker hr0 , −
n
j,k=1
∂ ∂u jk a (r0 · x) k + r02 f˜(r0 · x)u(x) = 0, ∂xj ∂x
x ∈ B,
(10)
and it follows from (7) and (8) that n
1 Γ (h, r0 )[u] = − r0
j,k=1
∂B
∂u a (r0 · x)νj (x) k ∇u(x), x dS. ∂x jk
(11)
If we set A(x) := {ajk (x)}, x ∈ B, and use that ν(x) = x for all x ∈ ∂B, we can rewrite (11) as
1 Γ (h, r0 )[u] = − r0
A(r0 · x)x, ∇u(x)
∇u(x), x dS.
∂B
Denoting by (A(r0 · x)x)T , x ∈ ∂B, the tangential component of the vector A(r0 · x)x, we have
T A(r0 · x)x, ∇u(x) = ∇u(x), x A(r0 · x)x, x + ∇u(x), A(r0 · x)x
and hence Γ (h, r0 )[u] = −
1 r0
∇u(x), x
2
A(r0 · x)x, x dS
∂B
1 − r0
∇u(x), x
T ∇u(x), A(r0 · x)x dS.
∂B
Since
T ∂u div u(x) (x) A(r0 · x)x ∂ν
and
∂u ∂ν (x)
T T ∂u ∂u (x) ∇u(x), A(r0 · x)x = u(x) ∇ (x), A(r0 · x)x + ∂ν ∂ν T ∂u + u(x) (x) div A(r0 · x)x ∂ν
= ∇u(x), x , we obtain from the assumed Dirichlet boundary conditions T T ∇u(x), x ∇u(x), A(r0 · x)x = div u(x) x, ∇u(x) A(r0 · x)x ,
and hence we finally get by Stokes’ theorem 2 1 Γ (h, r0 )[u] = − ∇u(x), x A(r0 · x)x, x dS 0, r0
x ∈ ∂B,
(12)
∂B
where we use that A(x) is positive definite for all x ∈ B. Moreover, we obtain from (12) that if Γ (h, r0 )[u] = 0 for some u ∈ ker hr0 , then
∂u (x) = 0 ∇u(x), x = ∇u(x), ν(x) = ∂ν
for all x ∈ ∂B which implies u ≡ 0 by the unique continuation property (cf. [11, Def. 3.3 and Prop. 3.4] and [5]).
JID:YJMAA
AID:18236 /FLA
Doctopic: Partial Differential Equations
[m3L; v 1.125; Prn:4/02/2014; 9:21] P.7 (1-7)
A. Portaluri, N. Waterstraat / J. Math. Anal. Appl. ••• (••••) •••–•••
7
In summary, we have shown that Γ (h, r0 ) is negative definite, and so in particular non-degenerate with the non-vanishing signature sgn Γ (h, r0 ) = m(r0 ).
(13)
Consequently, r0 is a bifurcation radius and Theorem 1.1 is proven. Let us now prove Corollary 1.2. We note at first that the Morse index μ of (5) on the full domain B(p0 , 1) is given by the Morse index μ(h1 ) of the quadratic form h1 . Moreover, since h0 is positive, we see that μ(h0 ) = 0. It is shown in [4, Prop. 3.9 & Thm. 4.1] that if Γ (h, r) is non-degenerate for all r ∈ (0, 1), then ker hr = 0 for all but a finite number of r ∈ (0, 1) and
μ(h1 ) − μ(h0 ) =
sgn Γ (h, r).
0
Consequently, we conclude from (13) that m(r) = dim ker hr = 0 for all but a finite number of r ∈ (0, 1) and μ=
m(r).
(14)
0
Let us point out that (14) was obtained by Smale in [8] by studying the monotonicity of eigenvalues under shrinking of domains. Hence we have obtained a new proof of Smale’s theorem for the boundary value problem (5), and moreover, Corollary 1.2 is now an immediate consequence of (14) and Theorem 1.1. References [1] A. Ambrosetti, G. Prodi, A Primer of Nonlinear Analysis, Cambridge Stud. Adv. Math., vol. 34, Cambridge University Press, 1993. [2] T. Aubin, Nonlinear Analysis on Manifolds. Monge–Ampère Equations, Grundlehren Math. Wiss., vol. 252, SpringerVerlag, 1988. [3] A.L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., Springer, 1987. [4] P.M. Fitzpatrick, J. Pejsachowicz, L. Recht, Spectral flow and bifurcation of critical points of strongly-indefinite functionals, Part I: General theory, J. Funct. Anal. 162 (1999) 52–95. [5] L. Hörmander, Linear Partial Differential Operators, Grundlehren Math. Wiss., Bd. 116, Academic Press/Springer-Verlag, New York/Berlin, Göttingen, Heidelberg, 1963. [6] A. Portaluri, N. Waterstraat, On bifurcation for semilinear elliptic Dirichlet problems and the Morse–Smale index theorem, J. Math. Anal. Appl. 408 (2013) 572–575, arXiv:1301.1458 [math.AP]. [7] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, American Mathematical Society, Providence, RI, 1986, Published for the Conference Board of the Mathematical Sciences, Washington, DC. [8] S. Smale, On the Morse index theorem, J. Math. Mech. 14 (1965) 1049–1055. [9] S. Smale, Corrigendum: “On the Morse index theorem”, J. Math. Mech. 16 (1967) 1069–1070. [10] M.E. Taylor, Partial Differential Equations I—Basic Theory, Appl. Math. Sci., vol. 115, Springer-Verlag, 1996. [11] K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Differential Geom. 8 (1973) 555–564.