SIGN-CHANGING AND MULTIPLE SOLUTIONS
FOR THE p-LAPLACIAN
SIEGFRIED CARL AND KANISHKA PERERA
Received 25 June 2002
We obtain a positive solution, a negative solution, and a sign-changing solution for a class of p-Laplacian problems with jumping nonlinearities using variational and super-subsolution methods.
1. Introduction
Let Ω ⊂ RN , N ≥ 1, be a bounded domain with a smooth boundary ∂Ω. In this
paper, we consider the quasilinear elliptic boundary value problem
1,p
u ∈ W0 (Ω) : −∆ p u = f (x,u) in W −1,p (Ω),
(1.1)
where ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian, 1 < p < ∞. By W 1,p (Ω) we de1,p
note the usual Sobolev space with dual space (W 1,p (Ω))∗ , and W0 (Ω) denotes
its subspace whose elements have generalized homogeneous boundary values
and whose dual space is given by W −1,p (Ω). We assume the following growth
and asymptotic behaviour of the nonlinear right-hand side f of (1.1):
(H1) f : Ω × R → R is a Carathéodory function satisfying
f (x,t) ≤ C |t | p−1 + 1 ,
p −1
f (x,t) = a t +
p −1
− b t−
+ g(x,t),
(1.2)
where
lim
t →0
g(x,t)
= 0 uniformly in x.
|t | p−1
Here C denotes some generic positive constants.
Copyright © 2002 Hindawi Publishing Corporation
Abstract and Applied Analysis 7:12 (2002) 613–625
2000 Mathematics Subject Classification: 35J65, 35J20, 35B05
URL: http://dx.doi.org/10.1155/S1085337502207010
(1.3)
614
Sign-changing and multiple solutions
The set Σ p of those points (a,b) ∈ R2 for which the asymptotic problem
1,p
u ∈ W0 (Ω) : −∆ p u = a u+
p −1
p −1
− b u−
in W −1,p (Ω)
(1.4)
has a nontrivial solution, is called the Fučı́k spectrum of the p-Laplacian on
Ω, where u+ = max{u,0} and u− = max{−u,0}. The Fučı́k spectrum was introduced in the semilinear case p = 2 by Dancer [8] and Fučı́k [14] who recognized
its significance for the solvability of problems with jumping nonlinearities. In
the semilinear ODE case p = 2, N = 1, Fučı́k [14] showed that Σ2 consists of
a sequence of hyperbolic-like curves passing through the points (λl ,λl ), where
{λl }l∈N are the eigenvalues of −d 2 /dx2 , with one or two curves going through
each point. Drábek [12] has recently shown that Σ p has this same general shape
for all p > 1 in the ODE case.
In the PDE case N ≥ 2, much of the work to date on Σ p has been done
for the semilinear case p = 2. It is now known that Σ2 consists, at least locally, of curves emanating from the points (λl ,λl ) (see, e.g., [2, 7, 8, 10, 14,
25]). Schechter [28] has shown that Σ2 contains two continuous and strictly
decreasing curves through (λl ,λl ), which may coincide, such that the points in
the square (λl−1 ,λl+1 )2 that are either below the lower curve or above the upper
curve are not in Σ2 , while the points between them may or may not belong to Σ2
when they do not coincide.
In the quasilinear PDE case p = 2, N ≥ 2, it is known that the first eigenvalue λ1 of −∆ p is positive, simple, and admits a positive eigenfunction ϕ1 (see
Lindqvist [20]), so Σ p clearly contains the two lines λ1 × R and R × λ1 . In addition, σ(−∆ p ) has an unbounded sequence of variational eigenvalues {λl } satisfying a standard min-max characterization, and Σ p contains the corresponding
sequence of points {(λl ,λl )}. A first nontrivial curve Ꮿ in Σ p through (λ2 ,λ2 )
asymptotic to λ1 × R and R × λ1 at infinity was recently constructed and variationally characterized by a mountain-pass procedure by Cuesta et al. [6] (see
Figure 1.1). More recently, unbounded sequences of curves (analogous to the
lower and upper curves of Schechter) have been constructed and variationally
characterized by min-max procedures by Micheletti and Pistoia [26] for p ≥ 2
and by the second author [27] for all p > 1.
The main goal of this paper is to identify the set of points (a,b) relative to
the Fučı́k spectrum which ensure the existence of sign-changing solutions of
(1.1). More precisely, assuming the existence of a positive supersolution u and
a negative subsolution u of (1.1) and (a,b) located above the curve Ꮿ, we prove
the existence of at least three nontrivial solutions within the order interval [u,u];
a positive solution, a negative solution, and a sign-changing solution.
There are many existence and multiplicity results for (1.1) in the literature
(see, e.g., [5, 6, 9, 13, 23, 27]). However, to the best of our knowledge, the first
results on sign-changing solutions were obtained only recently by Li and Zhang
[17]. In their paper the authors assume that p > N and f is independent of x
and locally Lipschitz in t. All these assumptions can be relaxed by our approach
S. Carl and K. Perera 615
b
(a,b)
(λ2 ,λ2 )
Ꮿ
(λ1 ,λ1 )
a
Figure 1.1
which is very different from that of Li and Zhang. Our main result, which will
be proved in Section 3 (Theorem 3.1), improves upon their results.
2. Preliminaries
We denote the norm in W 1,p (Ω) and L p (Ω) by · and · p , respectively, and
recall the notion of sub- and supersolutions.
Definition 2.1. A function u ∈ W 1,p (Ω) is a supersolution of (1.1) if the following
holds:
(i) u ≥ 0 on ∂Ω,
1,p
p
(ii) Ω |∇u| p−2 ∇u · ∇ϕdx ≥ Ω f (x,u)ϕdx for all ϕ ∈ W0 (Ω) ∩ L+ (Ω).
Similarly, u is a subsolution of (1.1) if the reversed inequalities of Definition
p
2.1 hold with u replaced by u. Here L+ (Ω) stands for the positive cone of L p (Ω).
Consider the boundary value problem
1,p
u ∈ W0 (Ω) : −∆ p u = h in W −1,p (Ω).
(2.1)
Besides the hypothesis (H1) we will assume the following hypotheses to hold
throughout the rest of the paper.
(H2) There exist a positive supersolution u and a negative subsolution u of
(1.1), and the point (a,b) ∈ R2 is above the curve Ꮿ of the Fučı́k spectrum.
(H3) Any solution u of (2.1) with h ∈ L∞ (Ω) belongs to C 1 (Ω).
Remark 2.2. (i) Assuming the existence of super- and subsolutions as in hypothesis (H2) is a weaker assumption than the usual condition on the jumping
nonlinearity at infinity.
616
Sign-changing and multiple solutions
(ii) By (H3) we impose C 1 (Ω)-regularity of the solution of (2.1). As for regularity results up to the boundary (C 1,α -regularity) we refer, for example, to Giaquinta and Giusti [16], Giaquinta [15], Liu and Barrett [21, 22], Lieberman
[18, 19], or Manfredi [24].
Lemma 2.3. If u ≥ (resp., ≤) 0 is a solution of (1.1), then either u > (resp., <) 0 or
u ≡ 0. Moreover, if u > 0, then there is an ε > 0 such that u ≥ εϕ1 , where ϕ1 is the
eigenfunction of the first eigenvalue of −∆ p .
Proof. First, we note that by the results of Anane [1] and DiBenedetto [11] any
solution u of (1.1) belongs to L∞ (Ω) ∩ C 1 (Ω), and thus the right-hand side of
(1.1) yields a function h ∈ L∞ (Ω), which by (H2) implies that u ∈ C 1 (Ω). If
u ≥ 0 is a solution of (1.1) which is not identically zero, then by means of the
Harnack inequality (Trudinger [29, Theorem 1.1]) u must be positive in Ω. For
> 0, let Ω = {x ∈ Ω : dist(x,∂Ω) ≤ }. Then for sufficiently small, we have
f (x,u(x)) ≥ 0 for all x ∈ Ω by (H1) and (H2). This allows us to apply the strong
maximum principle due to Vázquez [30] to get the strict inequality ∂u/∂ ν (x) >
0 for all x ∈ ∂Ω, where ν is the interior normal at x. The eigenfunction ϕ1 of
the first eigenvalue of −∆ p is positive, is of class C 1,α (Ω) for α ∈ (0,1), and also
satisfies ∂ϕ1 /∂ ν (x) > 0 (see [1, 20]). Therefore, for ε sufficiently small, we obtain
u ≥ εϕ1 in Ω.
1,p
Lemma 2.4. Given a bounded sequence {un } ⊂ W0 (Ω) and a sequence of positive
reals {εn } with εn → 0 as n → ∞, for a subsequence,
1
p −1
εn
Ω
g x,εn un (x) dx −→ 0
Further, if G is the primitive of g, that is, G(x,t) =
1
p
εn
Ω
G x,εn un (x) dx −→ 0
as n −→ ∞.
t
0 g(x,s)ds,
(2.2)
then
as n −→ ∞
(2.3)
for a subsequence.
Proof. Passing to a subsequence (again denoted by (un )), we may assume that
un → u a.e. and in L p (Ω). By Egoroff ’s theorem, for any µ > 0 there is a measurable subset Ωµ of Ω such that |Ω \ Ωµ | ≤ µ and un → u uniformly on Ωµ . Thus
εn un → 0 a.e. in Ωµ . We have
1
p −1
εn
Ω
g x,εn un (x) dx
g x,εn un (x) p −1
=
p−1 un (x) dx
p −1 Ωµ εn un (x)
g x,εn un (x) un (x) p−1 dx.
+
p −1 p
−
1
Ω\Ωµ εn
un (x)
(2.4)
S. Carl and K. Perera 617
By (1.2) and (1.3),
g(x,t)
|t | p−1
≤ C.
(2.5)
The first integral on the right-hand side of (2.4) tends to zero by the asymptotic
behavior (1.3) of g, (2.5), and Lebesgue’s dominated convergence theorem (observe that the integrand is majorized by C(|u(x)| + δ) p−1 for any δ > 0 due to the
uniform convergence in Ωµ ). The second integral is bounded by
1/ p (p−1)/ p
u n ≤ Cµ1/ p −→ 0
C Ω \ Ωµ as µ −→ 0,
(2.6)
which proves (2.2). Observing that the elementary inequality
G x,εn un (x) ≤ εn un (x)g x,τn (x)εn un (x) (2.7)
holds, where 0 ≤ τn (x) ≤ 1, which yields
1
p
εn
Ω
G x,εn un (x) dx ≤
g x,τn (x)εn un (x) p
p−1 un (x) dx,
p
−
1
Ω τn (x) p−1 εn un (x)
(2.8)
we see that (2.3) follows similarly.
Lemma 2.5. Problem (1.1) has a positive solution u > 0 within the order interval
[0,u] and a negative solution u < 0 within the order interval [u,0].
Proof. In the proof we focus on the existence of a positive solution only since the
existence of a negative solution can be shown in a similar way.
As is well known, solutions of (1.1) are the critical points of
Φ(u) =
where F(x,t) =
t
0
Ω
|∇u| p − p F(x,u) dx,
1,p
u ∈ W0 (Ω),
(2.9)
f (x,s)ds. Let f be the following truncated nonlinearity:



0,

t ≤ 0,
f (x,t) =  f (x,t),
0 < t < u(x),


 f x,u(x) , t ≥ u(x),
(2.10)
and F its associated primitive given by
F(x,t) =
Consider the functional
Φ(u) =
Ω
t
f (x,s)ds.
(2.11)
|∇u| p − pF(x,u) dx
(2.12)
0
618
Sign-changing and multiple solutions
whose critical points are the solutions of the auxiliary boundary value problem
1,p
u ∈ W0 (Ω) : −∆ p u = f (x,u) in W −1,p (Ω).
(2.13)
1,p
Obviously, Φ : W0 (Ω) → R is bounded from below, weakly lower semicontinuous, and coercive. Thus, there is a global minimizer, that is, a critical point u of
Φ
0 = Φ (u),ϕ =
Ω
|∇u| p−2 ∇u · ∇ϕ − f (x,u)ϕ dx.
(2.14)
We will show that this global minimizer is in fact a positive solution of (1.1)
within [0,u]. Taking in (2.14) the special test function ϕ = u− := max(−u,0) we
get in view of the definition of f the equation
0=
Ω
p
|∇u| p−2 ∇u · ∇u− − f (x,u)u− dx = u− ,
(2.15)
which shows that u− = 0, and thus u ≥ 0. Since u is a supersolution, ϕ = (u −
1,p
p
u)+ ∈ W0 (Ω) ∩ L+ (Ω), so by Definition 2.1 and (2.14) we obtain
0≥
|∇u| p−2 ∇u − |∇u| p−2 ∇u · ∇(u − u)+
Ω
− f (x,u) − f (x,u) (u − u)+ dx
=
|∇u| p−2 ∇u − |∇u| p−2 ∇u · (∇u − ∇u)dx ≥ 0,
(2.16)
{u>u}
which implies that ∇(u − u)+ = 0, and thus u ≤ u. This shows that the global
minimizer u of the functional Φ satisfies u ∈ [0,u], and thus u is a solution of
(1.1) due to the definition of f . Since a > λ1 , we get by hypothesis (H1) that
Φ εϕ1 < 0,
ε > 0 small.
(2.17)
As u is a global minimizer of Φ, it follows Φ(u) ≤ Φ(εϕ) < 0, and thus u must be
a positive solution of (1.1).
Definition 2.6. A solution u+ is called the least positive solution of (1.1) if any
other positive solution u of problem (1.1) satisfies u ≥ u+ . Similarly, u− is the
greatest negative solution of (1.1) if any other negative solution u satisfies u ≤ u− .
Lemma 2.7. Problem (1.1) has a least positive solution u+ and a greatest negative
solution u− .
Proof. We are going to prove the existence of the least positive solution only,
since the proof of the existence of the greatest negative solution is analogous.
In view of Lemma 2.5, there exists a positive solution u ∈ [0,u], and applying
S. Carl and K. Perera 619
Lemma 2.3 there is a ε > 0 small enough such that εϕ1 ≤ u, where ϕ1 is the eigenfunction that belongs to the first eigenvalue λ1 of −∆ p . Since a > λ1 , one readily
verifies that εϕ1 is a subsolution of problem (1.1) for sufficiently small ε > 0.
Thus there is an ε0 > 0 such that ε0 ϕ1 and u forms an ordered pair of sub- and
supersolutions. Applying [3, Corollary 5.1.2] on the existence of extremal solutions for general quasilinear elliptic problems, we obtain the existence of a least
and greatest solution of (1.1) with respect to the order interval [ε0 ϕ1 ,u]. We denote the least solution within this interval by u0 . Now let (εn )∞
n=0 be a decreasing
sequence with εn → 0 as n → ∞, and denote by un the corresponding least solution of (1.1) with respect to the order interval [εn ϕ1 ,u]. Then obviously (un ) is
a decreasing sequence of least positive solutions of (1.1) which converges to its
nonnegative pointwise limit u∗ in L p (Ω). We will show that u∗ is in fact the least
positive solution, that is, u∗ = u+ . First we verify that u∗ is a solution of (1.1).
Since the un are solutions of (1.1) we get from the equation
p
u n =
Ω
∇un p−2 ∇un · ∇un dx =
f x,un un dx,
Ω
(2.18)
which by the growth condition (H1) and the boundedness in L p (Ω) of the se1,p
quence (un ) implies its boundedness in W0 (Ω), that is, un ≤ c. Thus there
1,p
exists a subsequence weakly convergent in W0 (Ω), and due to the strong convergence of (un ) in L p (Ω) even the entire sequence is weakly convergent in
1,p
W0 (Ω) with weak limit u∗ . From (1.1) with the test function un − u∗ , we obtain
− ∆ p un ,un − u∗ =
Ω
∇un p−2 ∇un · ∇ un − u∗ dx
(2.19)
limsup − ∆ p un ,un − u∗ ≤ 0.
(2.20)
=
Ω
f x,un un − u∗ dx,
which implies
n
The weak convergence of (un ) and (2.20) along with the S+ -property of the oper1,p
ator −∆ p (see, e.g., [3, Chapter D]) yield its strong convergence in W0 (Ω). This
allows the passage to the limit in (1.1) with u replaced by un , and hence u∗ is a
solution of problem (1.1). To show that u∗ > 0, our argument is by contradic1,p
tion. Suppose u∗ = 0, that is, un → 0 in W0 (Ω). Since un > 0 we may consider
ũn := un / un which satisfies
Ω
∇ũn p−2 ∇ũn · ∇ϕdx =
Ω
p −1
aũn +
g x,un
p−1 ϕdx.
u n (2.21)
620
Sign-changing and multiple solutions
By definition ũn = 1, so there is a subsequence (ũn ) that converges weakly in
1,p
1,p
W0 (Ω) and strongly in L p (Ω) to ũ due to the compact embedding of W0 (Ω)
⊂ L p (Ω). Taking in (2.21) as special test function ϕ = ũn − ũ, we get for the righthand side of (2.21)
p −1
aũn +
Ω
g x,un
un (x) p−1
ũn (x) p−1 ũn − ũ dx −→ 0,
(2.22)
as n → ∞, because the terms in parentheses are Lq (Ω)-bounded. Hence (2.21)
implies
limsup − ∆ p ũn , ũn − ũ ≤ 0,
(2.23)
n
which due to the S+ -property of −∆ p implies the strong convergence of ũn → ũ
1,p
in W0 (Ω). Moreover, the third integral term on the right-hand side of (2.21)
converges to zero by Lemma 2.4, so we may pass to the limit to get
Ω
|∇ũ| p−2 ∇ũ · ∇ϕdx =
Ω
aũ p−1 ϕdx
∀ϕ ∈ C0∞ (Ω),
(2.24)
that is, ũ satisfies the boundary value problem
1,p
ũ ∈ W0 (Ω) : −∆ p ũ = aũ p−1
in W −1,p (Ω).
(2.25)
Since ũn = 1 and ũn > 0, by Lemma 2.3 we have the same properties for ũ,
which, however, contradicts the fact that a nontrivial solution of (2.25) changes
sign. So far we have shown that the limit u∗ of the least solutions un ∈ [εn ϕ1 ,u] is
a positive solution of (1.1). Finally, to prove that u∗ is the least positive solution,
let w be any positive solution of (1.1). Then by Lemma 2.3 there is a εn > 0 for n
sufficiently large such that εn ϕ1 ≤ w which by definition of the sequence of least
solutions (un ) yields u∗ ≤ un ≤ w, which proves that u∗ = u+ is in fact the least
positive one.
3. Main result
Theorem 3.1. Let hypotheses (H1), (H2), and (H3) be satisfied. Then the boundary value problem (1.1) has at least three nontrivial solutions: a positive solution, a
negative solution, and a sign-changing solution.
Proof. Let
f+ (x,t) =



0,

t ≤ 0,
t
f (x,t),
0 < t < u+ (x), F+ (x,t) = 0 f+ (x,s)ds,



 f x,u+ (x) , t ≥ u+ (x),
(3.1)
S. Carl and K. Perera 621
and consider
+ (u) =
Φ
Ω
|∇u| p − pF+ (x,u) dx.
(3.2)
Arguments similar to those in the proof of Lemma 2.5 show that critical points
+ are solutions of (1.1) in the order interval [0,u+ ], so 0 and u+ are the only
of Φ
+ by Lemmas 2.3 and 2.7. Now, Φ
+ is bounded from below
critical points of Φ
and coercive, and
+ εϕ1 < 0,
Φ
ε > 0 small
(3.3)
+ has a global minimizer at a negative critical level. It follows
since a > λ1 , so Φ
+ and Φ
+ (u+ ) < 0.
that u+ is the (strict) global minimizer of Φ
Now let
f(x,t) =
 


 f x,u− (x) ,
t ≤ u− (x),
t
u− (x) < t < u+ (x), F(x,t)
= 0 f(x,s)ds,
t ≥ u+ (x),
f (x,t),



 f x,u+ (x) ,
Φ(u)
=
dx.
|∇u| p − pF(x,u)
(3.4)
Ω
are solutions of (1.1) in the order interval [u− ,u+ ],
As before, critical points of Φ
so it follows from Lemmas 2.3 and 2.7 that any nontrivial critical point other
than u± is a sign-changing solution.
and Φ(u
± ) < 0.
Lemma 3.2. The solutions u± are strict local minimizers of Φ,
Proof. We only consider u+ as the argument for u− is similar. Suppose that there
1,p
+ ). By (1.2) and
j ) ≤ Φ(u
is a sequence u j → u+ in W0 (Ω), u j = u+ with Φ(u
(1.3) we have
F(x,t)
≤ C |t | p ,
(3.5)
so
uj =
Φ
+p
−p
∇u − pF x,u+ dx +
∇u − pF x,u− dx
Ω
j
+
+ uj
≥Φ
j
j
j
j
Ω
p
p
+ u− − C u− .
j
(3.6)
p
If u−j = 0, then u+j = u+ and
uj ≤ Φ
u+ = Φ
+ u+ ,
+ u+j ≤ Φ
Φ
(3.7)
622
Sign-changing and multiple solutions
+ , so u−j = 0.
contradicting the fact that u+ is the unique global minimizer of Φ
We will show that
−p
u > C u − p ,
j
j
j large.
p
(3.8)
+ (u+j ) < Φ
+ (u+ ).
Assuming this for the moment, we have the contradiction Φ
To see that (3.8) holds, we first note that the measure of the set Ω j = {x ∈ Ω :
u j (x) < 0} goes to zero. To see this, given ε > 0, take a compact subset Ωε of Ω
such that |Ω \ Ωε | < ε and let Ωεj = Ωε ∩ Ω j . Then
u j − u + p ≥
p
Ωεj
u j − u+ p dx ≥
u+ dx ≥ c p Ωεj ,
ε
p
Ωj
(3.9)
where c = minΩε u+ > 0, so |Ωεj | → 0. Since Ω j ⊂ Ωεj ∪ (Ω \ Ωε ) and ε > 0 is arbitrary, the claim follows.
u j is bounded for a subseIf (3.8) does not hold, setting u j = u−j / u−j p , p
u
p = 1
quence, so u j → u in L (Ω) and a.e. for a further subsequence, where and u ≥ 0. But then Ωµ = {x ∈ Ω : u(x) ≥ µ} has positive measure for all sufficiently small µ > 0 and
p
u
≥
j − u
p
Ωµ \Ω j
p
u
dx =
j − u
u p dx ≥ µ p Ωµ − Ω j ,
Ωµ \Ω j
(3.10)
a contradiction.
Now a standard deformation argument gives a mountain-pass point u1 at the
critical value
c := inf
max
γ∈Γ u∈γ([−1,1])
u± ,
Φ(u)
>Φ
(3.11)
1,p
where Γ = {γ ∈ C([−1,1],W0 (Ω)) : γ(±1) = u± } is the class of paths join 0 := {u ∈
ing u± . To show that u1 = 0, we will construct a path that lies in Φ
1,p
W0 (Ω) : Φ(u)
< 0 }.
First we show that, for all sufficiently small ε > 0, ±εϕ1 can be joined by a path
0 . We have
γε in Φ
Φ(u)
= I(a,b) (u) −
Ω
G(x,u)dx,
(3.12)
S. Carl and K. Perera 623
where
I(a,b) (u) =
Ω
p
p
dx
|∇u| p − a u+ − b u−
(3.13)
is the functional associated with (1.4) and
p
G(x,t)
= F(x,t)
− a t+
p
− b t − = o |t | p
as t −→ 0.
(3.14)
1,p
Since (a,b) is above Ꮿ, there is a path γ0 in {u ∈ W0 (Ω) : I(a,b) (u) < 0, u
p =
1} joining ±ϕ1 by the construction of Ꮿ [6]. For u ∈ γ0 ([−1,1]),
Φ(εu)
≤ε
p
max I(a,b) γ0 [−1,1]
+
G(x,εu)
Ω
εp
dx ,
(3.15)
and the last integral goes to 0 uniformly on the compact set γ0 ([−1,1]) as ε → 0
by Lemma 2.4, so we can take γε = εγ0 .
We complete the proof by showing that ±εϕ1 and u± can be joined by paths
+ (u+ ) and
0 . Again we only consider εϕ1 and u+ . Setting α = inf Φ
+ = Φ
in Φ
β = Φ+ (εϕ1 ) = Φ(εϕ1 ) < 0, by the second deformation lemma (see, e.g., Chang
α+ := {u ∈ W01,p (Ω) : Φ
+ (u) ≤ α} = {u+ } is a strong defor[4]), the sublevel set Φ
β
β+ , Φ
β+ ) such that
mation retract of Φ+ , that is, there is an η ∈ C([0,1] × Φ
β
+,
(i) η(0,u) = u for all u ∈ Φ
(ii) η(t,u+ ) = u+ for all t ∈ [0,1],
β+ .
(iii) η(1,u) = u+ for all u ∈ Φ
β
+ joining εϕ1 and u+ . Now the path γ+
In particular, γ = η(·,εϕ1 ) is a path in Φ
+
defined by γ+ (t) = γ(t) also joins εϕ1 and u+ , and
+ (γ(t)) −
γ+ (t) = Φ
Φ
Ω
∇γ(t)− p ≤ β.
(3.16)
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Siegfried Carl: Fachbereich Mathematik und Informatik, Institut für Analysis, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle, Germany
E-mail address: carl@mathematik.uni-halle.de
Kanishka Perera: Department of Mathematical Sciences, Florida Institute of
Technology, Melbourne, FL 32901, USA
E-mail address: kperera@winnie.fit.edu
URL: http://winnie.fit.edu/∼kperera/