CURVATURE EQUATION
K.C. CHANG AND TAN ZHANG
In memory of Professor S.S. Chern
Abstract. We combine heat flow method with Morse theory, super- and sub-solution method with Schauder’s fixed point theorem to show the existence of multiple solutions of the prescribed mean curvature equation under some special circumstances.
In this paper, we study the existence of multiple solutions of the following equation: (0.1)
(
M(u) = nH(x, u) in Ω ⊂ Rn u|∂Ω= ϕ,
where
M(u) := div ∇u
(1 + |∇u|2)1/2.
It is well known in [?], that equation (0.1) has a unique solution if (0.2) H0:= sup(x,u)∈ ¯Ω×R|H(x, u)| <
n − 1
n infx∈∂ΩH 0(x),
where H0(x) stands for the mean curvature of ∂Ω at the point x, and if
(0.3) Hu(x, u) ≥ 0.
For simplicity, we shall henceforth consider the boundary condition u|∂Ω = 0
in-stead. In order to find multiple solutions of equation (0.1), we must avoid assump-tion (0.3). However, to our knowledge, this condiassump-tion has always played a crucial role in the a priori estimate for the solution of equation (0.1). It is quite difficult to study equation (0.1) for general Ω, so we shall only focus on the following special cases:
(1) For n = 1, we set up the Morse theory for the functional
I(u) := Z Ω (p1 + ˙u2+ H(x, u)) dx, with H(x, u) = Z u 0
H(x, t) dt, and provide a multiple solution result under
assump-tions (H1), (H2), (H3).
Because of the special feature of equation (0.1), it is well known that, critical point theory can not be set up on the W1,p spaces for p > 1 using the above functional.
But for p = 1 or other function spaces, the lack of Palais Smale Condition is again a major difficulty. However, as the first author has noticed in [3], sometimes the heat flow, to which the Palais Smale Condition is irrelevant, can be used as a replacement 1991 Mathematics Subject Classification. Primary 53A10, 58E05; Secondary 35A15, 58C30.
Key words and phrases. Mean curvature equation, Morse Theory, multiple solutions. 1
of the pseudo-gradient flow in critical point theory. We shall study the related heat equation, and use the heat flow to set up the Morse theory of isolated critical points for the above functional. Critical groups for isolated critical points are counted, and Morse relation is applied. The main result of the first section is Theorem 1.2, which asserts the existence of three nontrivial solutions.
(2) For n > 1, we shall further assume H = H(r, u) is rotationally symmetric. Consequently, equation (0.1) can be reduced to the following O.D.E.:
¨
u +n − 1
r ˙u (1 + ˙u
2) = nH(r, u)(1 + ˙u2)3/2.
We then use the sub- and super- solution method to prove that there are at least two nontrivial solutions if assumptions (H1’), (H2’) hold, and a degree argument to prove the existence of the third nontrivial solution under the additional assumption (H3’) and (H4). The main result of the second section is Theorem 2.2.
This paper is divided into two parts. In part 1, we deal with case (1), using the heat flow method instead of the traditional pseudo-gradient flow method on some H¨older space. This approach will thereby enable us to bypass the Palais Smale Condition. In part 2, we deal with case (2). We first make some simple estimates, and then comes the crucial point in which, we construct a positive small sub-solution and a negative small super-solution of equation (0.1).
1 In the case n = 1, we have:
M(u) = u¨
(1 + ˙u2)3/2.
Let R > 0 and let J = [−R, R]. Equation (0.1) is now reduced to the O.D.E.:
(1.1)
( ¨
u = H(x, u)(1 + ˙u2)3/2 for x ∈ (−R, R) u(±R) = 0.
We assume that H ∈ C1(J × R1, R1) satisfies:
(H1) H0:= sup(x,u)∈J×R|H(x, u)| <
1 2R. It is easy to verify that
w±λ(x) = ±(−λ +pR2+ λ2− x2), ∀λ > 0, |x| ≤ R
is a pair of super- and sub- solutions of equation (1.1), provided λ <√3R. If we also assume: (H2) H(x, 0) = 0 and − Hu(x, 0) > ( π 2R) 2, ∀x ∈ J then u±(x) = ±² cos(πx 2R), |x| ≤ R
are positive sub- and negative super- solutions of (1.1) respectively, provided ² > 0 is sufficiently small.
In fact, we have:
−M(u+) = (2Rπ )2u+
(1 + ( ˙u+)2)3/2 < −Hu(x, 0)u +.
It follows that −M(u+) ≤ −H(x, u+) for ² > 0 small, i.e., u+ is a sub-solution.
Hence, (u+, w+
λ) and (u−, wλ−) are two pairs of sub- and super- solutions of (1.1).
Now we consider the heat equation related to (1.1):
(1.2) vt− M(v) = −H(x, v) in (0, T ) × (−R, R), v(t, ±R) = 0, ∀t ∈ [0, T ], v(0, x) = v0, ∀x ∈ J,
where the initial data v0∈ C1+α(J) with v(±R) = 0 and α ∈ (0, 1).
We begin by introducing the weighted parabolic H¨older space H(−1−α)2+α (Ω) on Ω :=
[0, T ] × J, for 0 < α < 1, as follows: ½ v ∈ C(Ω) ¯ ¯ ¯ ¯ P β+2j≤2d(X)max{β+2j−1−α,0}|∂xβ∂tjv(X)| + d(X, Y )(Σβ+2j=2 |∂β x∂tjv(X)−∂βx∂jtv(Y )| |X−Y |α + |∂xv(X)−∂xv(Y )| |X−Y |1+α ) < +∞, ∀X, Y ∈ Ω, X 6= Y ¾
where X = (t, x), Y = (τ, y), |X−Y | = (|x−y|2+|t−τ |)1
2, d(X, Y ) = min{d(X), d(Y )},
and d(X0) = dist{X0, (∂Ω\{t = T }) ∩ {t < t0}}, for X0= (t0, x0).
The norm of H(−1−α)2+α is defined by ||v|| = supX,Y ∈Ω,X6=Y
µ P β+2j≤2d(X)max{β+2j−1−α,0}|∂xβ∂tjv(X)| +d(X, Y )(Σβ+2j=2|∂ β x∂tjv(X)−∂βx∂jtv(Y )| |X−Y |α + |∂xv(X)−∂xv(Y )| |X−Y |1+α ) ¶ .
According to [?], the solution v ∈ H(−1−α)2+α (Ω) exists for any T > 0. Applying the
Maximum Principle, we have:
u+(x) ≤ v(t, x) ≤ w+(x), ∀(t, x) ∈ Ω, if u+(x) ≤ v0(x) ≤ w+(x), ∀x ∈ J. Similarly, w−(x) ≤ v(t, x) ≤ u−(x), ∀(t, x) ∈ Ω, if w−(x) ≤ v0(x) ≤ u−(x), ∀x ∈ J.
We now view the solution v(t, x) as a flow, and consider the functional
I(u) :=
Z
J
with H(x, u) = Z u
0
H(x, t) dt, then the Euler-Lagrange equation for I is exactly
equation (1.1).
Along the flow, we have:
d dtI(v(t, ·)) = − Z J ∂v ∂t(t, x)[M(v(t, x)) − H(x, v(t, x))] dx = − Z J µ ∂v ∂t(t, x) ¶2 dx ≤ 0
Hence, the functional I is nonincreasing. We denote [u, v] = {w ∈ C2+α(J) ∩ C
0(J) | u(x) ≤ w(x) ≤ v(x), ∀x ∈ J}, and
notice that, if the initial data v0falls into the ordered interval [u+, w+] (or [w−, u−]
resp.), then v(t, x) is bounded, and I(v(t, ·)) is therefore bounded from below. Thus,
c := lim
t→∞I(v(t, ·)) exists, and
Z ∞ 0 Z J µ ∂v ∂t(t, x) ¶2 dx dt = I(v0) − c
There must be a sequence tj → +∞ such that
vj(x) := ∂v
∂t(tj, x) → 0 in L 2(J).
Let uj(x) := v(tj, x) for all j. Substituting these into equation (1.2), we obtain a
sequence of equations:
(1.3) M(uj) − H(x, uj) = vj, ∀j
We want to show that {uj} subconverges to a solution of (1.1). To this end, we
shall prove that ||¨uj||2 is bounded as follows.
Let
zj= ˙uj
(1 + ˙u2 j)1/2
.
Since uj(±R) = 0, there is ξ ∈ J such that ˙uj(ξ) = 0, i.e., zj(ξ) = 0. According to
equation (1.3), we have: |zj(x)| ≤ Z x ξ (|H(x, uj(x))| + |vj(x)|) dx ≤ 2H0R + (2R)1/2( Z J |vj|2dx)1/2
It follows, from ||vj||2 → 0 and (H1), that there is ² > 0, which depends only on H0 and R, such that ||zj||∞≤ 1 − ². Hence, || ˙uj||∞≤1 − ²
² .
By setting M² := (1 + (1² − 1)2)3/2, we have that ||¨uj||2 ≤ (H0+ 1)M²; hence, ∃ u∗
Finally, ∀ϕ ∈ C∞ 0 (J), from Z J · ¨ uj(x) (1 + ˙u2 j(x))3/2 − H(x, uj(x)) ¸ ϕ(x) dx → 0, it follows that Z J [M(u∗ +) − H(x, u∗+)]ϕ(x) dx = 0, i.e., M(u∗ +(x)) = H(x, u∗+(x)) a.e., thus u∗+∈ C3(J).
In summary, along the heat flow, there is a subsequence {tj} % +∞ such that v(tj, ·) * u∗+ in W2,2(J), where u∗+ is a solution of (1.1). Let K be the set of
all solutions of (1.1) in the order interval [u+, w+],. According to the previous
estimates, it is W2,2(J) bounded, so is compact in C1(J). Since inf{I(u) | u ∈
[u+, w+]} = inf{I(u), k u ∈ K} , therefore the functional I has a minimizer, again
denoted by u∗
+, in the ordered interval [u+, w+]. A similar argument shows that we
also have a solution u∗
− ∈ C3( ¯J) which minimizes the functional I in the ordered
interval [w−, u−]. Since [w−, u−] ∩ [u+, w+] = ∅, u∗
+ 6= u∗−; they are two distinct
nontrivial solutions of (1.1).
In light of the above result, we shall next seek a third nontrivial solution of (1.1). This is based on a Morse-theoretic approach, we refer to [?] and [?] for further details.
When v0 ∈ C1+α(J) ∩ C0(J), 0 < α < 12, it is known ([?]) that the solution v(t, x) ∈ H(−1−α)2+α (Ω), which defines a deformation η : [0, +∞) × (C1+α(J) ∩ C0(J)) → C1+α(J) ∩ C0(J) by η(t, v0) = v(t, ·). It is easy to verify the continuity
of the mapping η by standard arguments. In particular, η : [0, +∞) × X → X, where X = [w−, w+] ∩ C1+α(J) ∩ C
0(J) is a closed convex set in the Banach space C1+α(J) ∩ C
0(J). Now, ∀a ∈ R1, we denote Ia := {u ∈ X | I(u) ≤ a}. The
following deformation lemma holds:
Lemma 1.1. (Deformation lemma) If there exists no critical point of the functional
I in the energy interval I−1[a, b], except perhaps some isolated critical points at the level a, then Ia is a deformation retract of Ib.
Proof. It is sufficient to prove: If the orbit O(v0) = {η(t, v0) | t ∈ R1+} ⊂ I−1(a, b],
and if the limiting set ω(v0) is isolated; then η(t, v0) has a weak W2,2 limit w, (and
then limt→+∞η(t, v0) = w in C1+α, 0 < α < 12) on the level I−1(a).
Indeed, by the previous argument, ∀w ∈ ω(v0), we have a sequence tj → +∞ such
that v(tj, ·) * w in W2,2(J). Thus,
kv(tj, ·) − wkC1 → 0.
Suppose the conclusion is not true, then ∃²0 > 0, ∃t01 < t001 < t02 < t002 < · · · , such
that kv(t0 j, ·) − wkC1 = ²0 2 ≤ kv(t, ·) − wkC1 ≤ kv(t 00 j, ·) − wkC1= ²0, ∀t ∈ Ij:= [t0j, t00j], ∀ j = 1, 2, · · · , and ω(v0) ∩ B²0(w)\{w} = ∅.
The above inequalities imply the C1− boundedness of v(t, ·) on I
kv(t, ·)kC1 ≤ kwkC1+ ²0.
After simple estimates, we have
kv(t, ·)kC2 ≤ C, ∀t ∈ Ij,
where C is a constant independent of j.
But Z Ij k∂tv(t, ·)k22dt = I(v(t0j, ·)) − I(v(t00j, ·)) → 0. Since ²0 2 ≤ kv(t00j, ·) − v(t0j, ·)k2 ≤Rt00j t0 j k∂tv(t, ·)k2dt ≤ (t00 j − t0j) 1 2(Rt 00 j t0 j k∂v(t, ·)k 2 2dt) 1 2, it follows t00
j − t0j→ +∞, and then ∃t∗j ∈ Ij such that k∂tv(t∗j, ·)k2→ 0. Obviously, kv(t∗
j, ·)kC2≤ C, and v(t∗j, ·) ∈ B²0\B²0 2 (w).
Again, by the previous argument, ∃z ∈ ω(v0) such that v(t∗j, ·) → z. Then we have: z ∈ ω(v0) ∩ B²0\B²02(w). This is impossible, since ω(v0) ∩ B²0(w)\{w} = ∅. ¤
From lemma 1.1, the Morse relation holds for I on X. In context, critical groups
Cq(I, u0) = Hq(U ∩ Ic, (U \ {u0}) ∩ Ic) are defined for an isolated critical point u0
of I, where U is an isolated neighborhood of u0, c = I(u0), and Hq(Y, Z) are the
graded singular relative homology groups for q = 0, 1, · · · .
In the estimation of number of solutions, we can assume, without loss of generality, that there are only finitely many critical points {u1, u2, · · · , uN} of I. Noticing that
both u∗
± are local minimizers of I, it follows that Cq(I, u∗±) = δq0.
On the other hand, let βq be the qth Betti number of X, q = 0, 1, · · · . Since the
set X is contractible, β0= 1, and βq = 0, ∀q ≥ 1. Let Mq be the qth Morse type
number of I:
Mq= ΣNj=1rank Cq(I, uj), ∀q.
The Morse relation reads as Σ∞
0 (Mq− βq)tq = (1 + t)P (t),
where P is a formal power series with nonnegative coefficients.
Therefore, we must have at least one more critical point u∗of I. If there are critical
points other than u∗ and u∗
±, the conclusion follows, so we may assume u∗ is the
unique critical point other than u∗
±. Then we have
rank C0(I, u∗) + rank C1(I, u∗) 6= 0,
according to the Morse relation.
In order to distinguish u∗ from θ (the trivial solution), we assume further
(H3) −Hu(x, 0) > (π
R) 2.
Since d2I(θ, ϕ) = Z J ( ˙ϕ2+ H u(x, 0)ϕ2) dx,
under the assumption (H3), we have Cq(I, θ) = 0, for q = 0 and 1. Again, this
will be a contradiction, if besides u∗
±, I has only the critical point θ. Thus we have
indeed established the following:
Theorem 1.2. Assume that H ∈ C1(J × R1, R1) satisfies (H1) and (H2), then equation (1.1) has at least two distinct nontrivial solutions, one positive and one negative. If in addition, (H3) is satisfied, then (1.1) will have at least three distinct nontrivial solutions.
2
In the case n ≥ 1, we assume H = H(r, u) is rotationally symmetric. Let Ω =
BR(0) ⊂ Rn, the ball centered at the origin of radius R. Equation (0.1) is then
reduced to: (2.1) 1 rn−1drd µ rn−1 du dr (1+(du dr)2) 1/2 ¶ = nH(r, u) for r ∈ (0, R) du dr(0) = u(R) = 0,
and assumption (0.2) then becomes
(2.2) H0:= sup(r,u)∈[0,R]×R|H(r, u)| < n − 1 n · 1 R, if n ≥ 2, and (2.3) H0< 1 R, if n = 1
We shall use a fixed point argument in conjunction with the super- and sub- solution method to tackle (2.1). We begin by delivering the following a priori estimate for solutions of (2.1):
Lemma 2.1. There is a constant C, depending only on n, H0, and R, such that all solutions u of (2.1) satisfy
(2.4) ||u||C2 ≤ C.
Proof. For n ≥ 2, it suffices to show
(1) |ur(r)| ≤ √ nH0r, and (2) |ur(r)| ≤ √ n(1 − 1 n)
For then, u(r) is bounded by (2) and the boundary condition u(R) = 0; in the mean time, urr is bounded by (1) and the alternative expression of (2.1):
urr+n − 1
r ur(1 + u 2
r) = nH(r, u)(1 + u2r)3/2.
For simplicity, we let
v = r n−1u r (1 + u2 r)1/2 .
It follows that (2.5) ur= ± v rn−1(1 − ( v rn−1)2)1/2 , and (2.6) vr= nrn−1H.
It now remains to show |v| ≤ (1 −1
n)rn−1. If so, from (2.5), we have: |ur| ≤ (1 − 1 n)[1 − (1 − 1 n) 2]−1/2≤√n(1 − 1 n), which proves (2). Since ur(0) = 0, v(0) = 0, by (2.6), we have: |v(r)| ≤ Z r 0 |vr(t)| dt ≤ H0rn, hence, (2.7) |v(r)| rn−1 ≤ H0r. (2.7) subsequently implies |v(r)| rn−1 ≤ H0R ≤ 1 − 1 n, and |ur| = ( v rn−1) (1 − ( v rn−1)2)1/2 ≤√nH0r.
For n = 1, assumption (2.2) is replaced by assumption (2.3), and we have |v| ≤
H0r ≤ H0R. Let ² := 1 − H0R, we then have: |ur| ≤ 1 − ²√
² . This completes the
proof. ¤
Next, we construct two pairs of sub- and super- solutions of (2.1) as in §1. However, for n > 1, the construction of the second pair of such solutions is a bit more complicated. Again, we let
wλ(r) = ±(−λ +
p
R2+ λ2− r2), ∀λ > 0, 0 ≤ r ≤ R.
The following properties hold:
(w1) wλ(R) = 0, ˙wλ(0) = 0, (w2) M(wλ) = −M(−wλ) = −n (R2+ λ2)1/2, (w3) w˙λ(R) = −R λ, (w4) |wλ(r)| ≤R 2− r2 2λ , | ˙wλ| ≤ r λ.
−M(wλ) = n (R2+ λ2)1/2 ≥ n( n − 1 n ) 1 R ≥ −nH(r, wλ), provided λ ≤ λ0:= √ 2n − 1 n − 1 R, for n > 1, ²R, for n = 1, where 0 < ² <q 1 (H0R)2 − 1.
This verifies (−wλ, wλ) is a pair of sub- and super- solutions of (2.1), when λ ≤ λ0.
Next, we consider the functions
zn(r) := r1− n 2Jn 2−1(r), for n = 1, 2, · · · , where Jn 2−1(r) denotes the ( n
2− 1)-order Bessel function.
Let µn be the first zero of Jn
2−1(r), and let vn(r) := zn(µn Rr), n = 1, 2, · · · , we then have: (v1) vn(R) = 0, ˙vn(0) = 0, (v2) ¨vn+n − 1 r ˙vn= −( µn R) 2v n, (v3) ∃ Mn > 0 such that | ˙vn(r)| ≤ Mn.
We introduce assumption (H2’) in place of (H2) for n > 1:
(H2’) H(r, 0) = 0 and − Hu(r, 0) > ( µn R) 2, ∀r ∈ [0, R]. and define u+:= ( δvn− wλ, for n > 1 δv1, for n = 1,
We compute to see that
M(u+) = 1 (1 + (u+)2r)3/2 ½ δ(¨vn+n − 1 r ˙vn) − [ ¨wλ+ n − 1 r ( ˙wλ+ ˙w 3 λ)] + n − 1 r (δ 3˙v3 n− 3δ2˙v2nw˙λ+ 3δ ˙vnw˙2λ) ¾ = −( µn R)2 (1 + (u+)2r)3/2 δvn+ µ 1 + ˙w2 λ 1 + (u+)2r ¶3/2 n (R2+ λ2)1/2 + Q(n, δ, λ)(r) (1 + (u+)2r)3/2 ,
where the remainder
Q(n, δ, λ)(r) = n − 1 r (δ
3˙v3
n− 3δ2˙vn2w˙λ+ 3δ ˙vnw˙2λ).
−M(u+) = (µn R)2 (1 + (u+)2r)3/2 u++ (µn R)2 (1 + (u+)2r)3/2 wλ − µ 1 + ˙w2 λ (1 + (u+)2r) ¶3/2 n (R2+ λ2)1/2 − Q(n, δ, λ)(r) (1 + (u+)2r)3/2 .
From (w4) and (v3), we obtain: 1 (1 + (u+)2r)3/2 = 1 + o(1) as δ → 0, λ → +∞, Q(n, δ, λ)(r) = O(δ3+ δ2λ−1+ δλ−2), and (µn R) 2w λ− (1 + ˙w2λ)3/2 n (R2+ λ2)1/2 ≤ µ2 n 2λ µ 1 − (r R) 2 ¶ − n (R2+ λ2)1/2 Let ² := Hu(r, 0) − (µRn)2, we have: −M(u+) < Hu(r, 0)u+− 1 (1 + (u+)2r)3/2 · ² 2u+− µ2 n 2λ µ 1 − (r R) 2 ¶ + n (R2+ λ2)1/2 ¸ + O(δ3), with λ = δ−2 as δ → 0.
If one can choose δ > 0 so small that
(2.8) 1 2 · (µn R) 2+ ² 2 ¸ (R2− r2)δ2≤ δ²vn(r) + n (R2+ δ−4)1/2,
then u+ is a sub-solution of (2.1) for n > 1.
This is possible, since vn(r) > 0 in [0, R), and since the last term in the right hand
side of (2.8) is of order δ2, furthermore, it is a positive constant on [0, R]. So, (2.8)
holds for small δ > 0.
As to n = 1, it has already been shown in section 1.
In summary, we obtained, as in the previous section, two pairs of sub- and super-solutions of (2.1), (u+, w+) and (w−, u−) where w± = ±wλ0 and u−= −u+.
In the next step, a fixed point argument is applied to obtain a third nontrivial solution of (2.1). Let X := {u ∈ C1([0, R]) | u
r(0) = u(R) = 0}. We define an
operator T : X → X ∩ C2([0, R]) via v := T u, ∀u ∈ X, and if v is a solution of the
linear O.D.E.: ( ¨ v +n−1 r ˙v = (1 + ˙u2)3/2(nH(r, u) −n−1r ˙u3) ˙v(0) = v(R) = 0
It is clear that u is a fixed point of T if and only if u is a solution of (2.1). According to the Maximum Principle, T maps the ordered interval [w−, u−]∩X and [u+, w+]∩ X into themselves. By the Schauder’s Fixed Point Theorem, there is a fixed point u∗
± in each of these ordered intervals.
We may assume, without loss of generality, that u∗
± is the only fixed point of T
of each of the corresponding set. For otherwise, there must be a third nontrivial solution.
Applying the Leray-Schauder’s degree theory, we have
ind (I − T, u∗ ±) = 1,
where ind (I −T, u∗
±) denotes the Leray-Schauder index of I −T at u∗±, respectively.
We can then use Amann’s Three Solutions Theorem [?] to confirm the existence of a third solution u∗. One may again assume, without loss of generality, that u∗ is
the only fixed point in [w−, w+] ∩ X other than u∗± . By a degree computation, ind (I − T, u∗) = 1,
provided deg (I − T, int ([w−, w+] ∩ X), θ) = 1.
As before, we should distinguish u∗ from θ, the trivial solution. For this reason,
stronger assumptions are imposed. Let µj
n be the jth zero of the Bessel function Jn2−1, and let m
j
n be the multiplicity
of µj n.
We assume that ∃j0≥ 2 satisfying
(H3’) µj0 n < p −Hu(r, 0)R < µjn0+1, ∀ r ∈ [0, R], and (H4) mn= Σjj=10 mjnis odd. Thus, ind (I − T, θ) = (−1)mn = −1. This implies u∗6= θ.
Lastly, we conclude our paper by stating the following:
Theorem 2.2. Assume that H ∈ C1([0, R]×R1, R1) satisfies (2.2) and (H2’), then equation (2.1) has at least two distinct nontrivial solutions, one positive and one negative. If in addition, (H3’) and (H4) are satisfied, then (2.1) will have at least three distinct nontrivial solutions.
Remark
In the case n = 1, by combining methods used in sections 1 and 2, we may obtain a result slightly different from both Theorem 1.2 and Theorem 2.2. Namely, if
H ∈ C1(J × R1, R1) satisfies H0< 1 R, H(x, u) = H(−x, u), ∀ x ∈ [−R, R], ∀ u ∈ R1, and H(x, 0) = 0 and − Hu(x, 0) > ( 3π 2R) 2 ∀ x ∈ [−R, R],
The proof is the same as that of Theorem 1.2, with an improvement of the a priori estimates of the solutions obtained in the first few paragraphs of §2, under the assumption that u is symmetric.
References
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LMAM, School of Math. Sci., Peking Univ., Beijing 100871, China1, kcchang@math.pku.edu.cn Dept. Math & Stats., Murray State Univ., Murray, KY 42071, USA, tan.zhang@murraystate.edu
