Electronic Journal of Differential Equations, Vol. 2010(2010), No. 15, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ENTIRE SOLUTIONS FOR A CLASS OF p-LAPLACE
EQUATIONS IN R2
ZHENG ZHOU
Abstract. We study the entire solutions of the p-Laplace equation
− div(|∇u|p−2 ∇u) + a(x, y)W 0 (u(x, y)) = 0,
(x, y) ∈ R2
where a(x, y) is a periodic in x and y, positive function. Here W : R → R is
a two well potential. Via variational methods, we show that there is layered
solution which is heteroclinic in x and periodic in y direction.
1. Introduction
In this paper we consider the p-Laplacian Allen-Cahn equation
− div(|∇u|p−2 ∇u) + a(x, y)W 0 (u(x, y)) = 0,
lim u(x, y) = ±σ
x→±∞
(x, y) ∈ R2
uniformly w.r.t. y ∈ R.
(1.1)
where we assume 2 < p < ∞ and
(H1) a(x, y) is Hölder continuous on R2 , positive and
(i) a(x + 1, y) = a(x, y) = a(x, y + 1).
(ii) a(x, y) = a(x, −y).
(H2) W ∈ C 2 (R) satisfies
(i) 0 = W (±σ) < W (s) for any s ∈ R \ {±σ}, and W (s) = O(|s ∓ σ|p ) as
s → ±σ;
(ii) there exists R0 > σ such that W (s) > W (R0 ) for any |s| > R0 .
p−1 2
p |σ
− t2 |p . This is similar with case
Qk
p = 2, where the typical examples of W are given by W (t) = 14 i=1 (t − zi )2 , where
zi , i = 1, 2, . . . k < ∞ are zeros of W (t). The case p = 2 can be viewed as stationary
Allen-Cahn equation introduced in 1979 by Allen and Cahn. We recall that the
Allen-Cahn equation is a model for phase transitions in binary metallic alloys which
corresponds to taking a constant function a and the double well potential W (t). The
function u in these models is considered as an order parameter describing pointwise
the state of the material. The global minima of W represent energetically favorite
pure phases and different values of u depict mixed configurations.
For example, here we may take W (t) =
2000 Mathematics Subject Classification. 35J60, 35B05, 35B40.
Key words and phrases. Entire solution; p-Laplace Allen-Cahn equation;
Variational methods.
c
2010
Texas State University - San Marcos.
Submitted September 15, 2009. Published January 21, 2010.
1
2
Z. ZHOU
EJDE-2010/15
In 1978, De Giorgi [11] formulated the following question. Assume N > 1 and
consider a solution u ∈ C 2 (RN ) of the scalar Ginzburg-Laudau equation:
∆u = u(u2 − 1)
satisfying |u(x)| ≤ 1,
∂u
∂xN
0
> 0 for every x = (x , xN ) ∈ R
(1.2)
N
and
lim
xN →±∞
0
u(x , xN ) =
±1. Then the level sets of u(x) must be hyperplanes; i.e., there exists g ∈ C 2 (R)
such that u(x) = g(ax0 − xn ) for some fixed a ∈ RN −1 . This conjecture was first
proved for N = 2 by Ghoussoub and Gui in [13] and for N = 3 by Ambrosio and
Cabré in [5]. For 4 ≤ N ≤ 8 and assuming an additional limiting condition on u,
the conjecture has been proved by Savin in [25] .
Alessio, Jeanjean and Montecchiari [2] studied the equation −4u+a(x)W 0 (u) =
0 and obtained the existence of layered solutions based on the crucial condition that
there is some discrete structure of the solutions to the corresponding ODE.
In [3], when a(x, y) > 0 is periodic in x and y, the authors got the existence
of infinite multibump type solutions, where a(x, y) = a(x, −y) takes an important
role [3](see also [3, 20, 21, 22, 23, 24]).
Inherited from the above results, I wonder under what condition p-Laplace type
equation (1.1) would have two dimensional layered solutions periodical in y. Adapting the renormalized variational introduced in [2, 3] (see also [21, 22]) to the pLaplace case, we prove
Theorem 1.1. Assume (H1)–(H2). Then there exists entire solution for (1.1),
which behaves heteroclinic in x and periodic in y direction.
2. The periodic problem
To prove Theorem 1.1, we first consider the equation
− div(|∇u|p−2 ∇u) + a(x, y)W 0 (u(x, y)) = 0,
(x, y) ∈ R2
u(x, y) = u(x, y + 1)
lim u(x, y) = ±σ
x→±∞
(2.1)
uniformly w.r.t. y ∈ R.
The main feature of this problem is that it has mixed boundary conditions, requiring
the solution to be periodic in the y variable and of the heteroclinic type in the x
variable.
Letting S0 = R × [0, 1], we look for minima of the Euler-Lagrange functional
Z
1
|∇u(x, y)|p + a(x, y)W (u(x, y)) dx dy
I(u) =
p
S0
on the class
1,p
Γ = {u ∈ Wloc
(S0 ) : ku(x, ·) ∓ σkLp (0,1) → 0. x → ±∞}
R1
where ku(x1 , ·) − u(x2 , ·)kpLp (0,1) = 0 |u(x1 , y) − u(x2 , y)|p dy. Setting
Γp = {u ∈ Γ : u(x, 0) = u(x, 1)for a.e. x ∈ R}
cp = inf I
Γp
and Kp = {u ∈ Γp : I(u) = cp }
Then we use the reversibility assumption (H1)-(ii) to show that the minima c on
Γ equals minima cp on Γp , and so solutions of (2.1).
Note the assumptions on a and W are sufficient to prove that I is lower semicon1,p
tinuous with respect to the weak convergence in Wloc
(S0 ); i.e., if un → u weakly
EJDE-2010/15
ENTIRE SOLUTIONS
3
1,p
in Wloc
(Ω) for any Ω relatively compact in S0 , then I(u) ≤ lim inf n→∞ I(un ).
Moreover we have
1,p
1,p
Lemma 2.1. If (un ) ⊂ Wloc
(S0 ) is such that un → u weakly in Wloc
(S0 ) and
I(un ) → I(u), then I(u) ≤ lim inf n→∞ un and
Z
Z
a(x, y)W (un ) dx dy →
a(x, y)W (u) dx dy
S0
S0
Z
Z
|∇un |p dx dy →
|∇u|p dx dy
S0
S0
1,p
Proof. Since un → u weakly in Wloc
(S0 ), k∇ukLp (S0 ) ≤ lim inf n→∞ k∇un kLp (S0 )
by the lower semicontinuous of the norm. By compact embedding theorem, we
p
convergence and Fatou lemma, we have
Rhave un → u in Lloc (S0 ), using pointwise
R
a(x,
y)W
(u)
dx
dy
≤
lim
inf
a(x,
y)W (un ) dx dy, then
n→∞
S0
S0
Z
Z
a(x, y)W (u) dx dy ≤ lim sup
a(x, y)W (un ) dx dy
Z
i
h
1
|∇un |p dx dy
= lim sup I(un ) −
p
n→∞
Z S0
1
= I(u) − lim inf
|∇un |p dx dy
n→∞
S0 p
Z
≤
a(x, y)W (u) dx dy.
n→∞
S0
S0
S0
R
R
Thus, S0 a(x, y)W (un ) dx dy → S0 a(x, y)W (u) dx dy, and since I(un ) → I(u), we
R
R
have S0 |∇un |p dx dy → S0 |∇u|p dx dy.
1,p
By Fubini’s Theorem, if u ∈ Wloc
(S0 ), then u(x, ·) ∈ W 1,p (0, 1), and for all
x1 , x2 ∈ R, we have
Z
1
p
1
Z
|u(x1 , y) − u(x2 , y)| dy =
0
Z
x2
|
0
∂x u(x, y)dx|p dy
x1
≤ |x1 − x2 |p−1
Z
1
x2
Z
|∂x u(x, y)dx|p dx dy
0
x1
p−1
≤ pI(u)|x1 − x2 |
.
If I(u) < +∞, the function x → u(x, ·) is Hölder continuous from a dense subset
of R with values in Lp (0, 1) and so it can be extended to a continuous function on
1,p
R. Thus, any function u ∈ Wloc
(S0 ) ∩ {I < +∞} defines a continuous trajectory
in Lp (0, 1) verifying
d(u(x1 , ·), u(x2 , ·))p =
Z
1
|u(x1 , y) − u(x2 , y)|p dy
0
p−1
≤ pI(u)|x1 − x2 |
, ∀x1 , x2 ∈ R.
(2.2)
4
Z. ZHOU
EJDE-2010/15
1,p
Lemma 2.2. For all r > 0, there exists µr > 0, such that if u ∈ Wloc
(S0 ) satisfies
min ku(x, ·) ± σkW 1,p (0,1) ≥ r for a.e. x ∈ (x1 , x2 ), then
Z x2 h Z 1
i
1
|∇u|p + a(x, y)W (u(x, y))dy dx
x1
0 p
p
1
p − 1 p−1
(2.3)
d(u(x1 , ·), u(x2 , ·))p +
≥
µr (x2 − x1 )
p−1
p(x2 − x1 )
p
≥ µr d(u(x1 , ·), u(x2 , ·))
Proof. We define the functional
Z
F (u(x, ·)) =
0
1
1
|∂y u(x, y)|p + aW (u(x, y))dy
p
on W 1,p (0, 1), where a = minR2 a(x, y) > 0. To prove the lemma, we first to claim
that:
For any r > 0, there exists µr > 0, such that if q(y) ∈ W 1,p (0, 1) is such that
p
p−1
min kq(y) ± σkW 1,p (0,1) ≥ r, thenF (q(y)) ≥ p−1
. Namely, if qn (·) ∈ W 1,p (0, 1)
p µr
and F (qn ) → 0, then min kqn ± σkW 1,p (0,1) → 0.
Assume by contradiction that if F (qn ) → 0 and min kqn ± σkL∞ (0,1) ≥ ε0 > 0.
Then there exists a sequence (yn1 ) ⊂ [0, 1] such that min |qn (yn1 ) ± σ| ≥ ε0 . Since
R1
aW (qn )dy → 0 there exists a sequence (yn2 ) ⊂ [0, 1] such that |qn (yn2 ) ± σ| < ε20 .
0
Then
ε0
≤ |qn (yn2 ) − qn (yn1 )|
2
Z yn2
≤|
|q̇n (t)|dt |
1
yn
1
≤ |yn2 − yn1 |1− p
hZ
1
|q̇n (t)|p dt
i1/p
0
1
p
1/p
≤ p (F (qn ))
→ 0.
It is a contradiction.
R1
Since min kqn ± σkL∞ (0,1) → 0 as F (qn ) → 0, then 0 |q̇n (y)|p dy → 0, and it
follows that kqn − σkW 1,p (0,1) → 0 as F (qn ) → 0.
1,p
Observe that if (x1 , x2 ) ⊂ R and u ∈ Wloc
(S0 ) are such that F (u(x, ·)) ≥
p
p−1 p−1
p µr
for a.e. x ∈ (x1 , x2 ), by Hölder’s and Yung’s inequalities we have
Z x2 h Z 1
i
1
|∇u|p + a(x, y)W (u(x, y))dy dx
x1
0 p
Z x2 Z 1
Z x2 Z 1
1
1
≥
|∂x u|p dy dx +
|∂y u|p + aW (u) dy dx
p
p
x1
0
x1
0
Z Z
Z x2
1 1 x2
=
|∂x u|p dx dy +
F (u(x, ·))dx
p 0 x1
x1
p
1
p − 1 p−1
p
≥
d(u(x
,
·),
u(x
,
·))
+
µ
(x2 − x1 )
r
1
2
p(x2 − x1 )p−1
p
≥ µr d(u(x1 , ·), u(x2 , ·)).
The proof is complete.
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5
As a direct consequence of Lemma 2.2, we have the following result.
1,p
Lemma 2.3. If u ∈ Wloc
(S0 ) ∩ {I < +∞}, then d u(x, ·), ±σ → 0 as x → ±∞.
Proof. Note that since
Z
I(u) =
S0
1
|∇u|p + a(x, y)W (u(x, y)) dx dy < +∞,
p
W (u(x, y)) → 0 as |x| → +∞. Then by Lemma 2.2, lim inf x→+∞ d u(x, ·), σ =
0. Next we show that lim supx→+∞ d u(x, ·), σ = 0 by contradiction. We assume that there exists r ∈ (0, σ/4) such that lim supx→+∞ d(u(x, ·), σ) > 2r,
by (2.2) there
exists infinite intervals (pi , si ), i ∈ N such that d u(pi , ·), σ = r,
d u(si , ·), σ = 2r and r ≤ d u(x, ·), σ ≤ 2r for x ∈ ∪i (pi , si ), i ∈ N by Lemma
2.2, this implies I(u) = +∞, it’s a contradiction. Similarly, we can prove that
limx→−∞ d u(x, ·), −σ = 0.
Now we consider the functional on the class
1,p
Γ = {u ∈ Wloc
(S0 ) : I(u) < +∞, d u(x, ·), ±σ → 0 as x → ±∞}
Let
c = inf I
Γ
and K = {u ∈ Γ : I(u) = c}
(2.4)
We will show that K is not empty, and we start noting that the trajectory in Γ
with action close to the minima has some concentration properties.
For any δ > 0, we set
λδ =
1 p
δ + max
a(x, y) ·
max W (s).
p
R2
|s±σ|≤p1/p δ
(2.5)
Lemma 2.4. There exists δ̄0 ∈ (0, σ/2) such that for any δ ∈ (0, δ̄0 ) there exists
ρδ > 0 and lδ > 0, for which, if u ∈ Γ and I(u) ≤ c + λδ , then
(i) min ku(x, ·) ± σkW 1,p (0,1) ≥ δ for a.e. x ∈ (s, p) then p − s ≤ lδ .
(ii) if ku(x− , ·) + σkW 1,p (0,1) ≤ δ, then d(u(x− , ·), −σ) ≤ ρδ for any x ≤ x− ,
and if ku(x+ , ·) − σkW 1,p (0,1) ≤ δ, then d(u(, ·), σ) ≤ ρδ for any x ≥ x+ .
Proof. By Lemma 2.2, as in this case, there exists µδ > 0 such that
Z pZ 1
1
|∇u|p + a(x, y)W (u) dx dy ≥ µδ (p − s).
p
s
0
Since I(u) ≤ c + λδ there exists lδ < +∞ such that p − s < lδ .
To prove (ii), we first do some preparation, µrδ ≥ p−1
p λδ , ρδ = max{δ, rδ } +
p−1
3( pµ
)
r
δ
p−1
p
λδ . Let δ̄0 ∈ (0, σ/2) be such that ρδ < σ/2 for all δ ∈ (0, δ̄0 ). Let
δ ∈ (0, δ̄0 ), u ∈ Γ, I(u) ≤ +∞ and x− ∈ R be such that
Define


−1
u− (x, y) = x − x− + (x − x− + 1)u(x− , y)


u(x, y)
ku(x− , ·) + σkW 1,p (0,1) ≤ δ.
if x < x− − 1,
if x− − 1 ≤ x− ,
if x ≥ x− .
and note that u− ∈ Γ and I(u− ) ≥ c, then ku− +σkW 1,p (0,1) = |x−x− +1|·ku(x− , ·)+
σkW 1,p (0,1) ≤ δ when x− − 1 ≤ x ≤ x− . Recall that kqkL∞ (0,1) ≤ p1/p kqkW 1,p (0,1)
6
Z. ZHOU
EJDE-2010/15
for any q ∈ W 1,p (0, 1), then ku− + σkL∞ (0,1) ≤ p1/p ku− + σkW 1,p (0,1) ≤ p1/p δ, by
definition (2.5) of λδ , we have
Z x− Z 1
1
[
|∇u− |p + a(x, y)W (u− )dy]dx ≤ λδ .
x− −1 0 p
Since
Z
x−
Z
I(u− ) = I(u) −
x−
−∞
Z 1
x− −1
0
Z
+
we obtain
Z
x−
Z
0
1
1
|∇u|p + a(x, y)W (u) dy dx
p
1
|∇u− |p + a(x, y)W (u− ) dy dx
p
1
1
|∇u|p + a(x, y)W (u) dy dx ≤ 2λδ .
(2.6)
p
−∞ 0
Now, assume by contradiction that there exists x1 < x− such that d(u(x1 , ·), −σ) ≥
ρδ , by (2.2) there exists x2 ∈ (x1 , x− ) such that d(u(x, ·), −σ) ≥ max{δ, rδ } for
x ∈ (x1 , x2 ) and d(u(x1 , ·), u(x1 , ·)) ≥ ρδ − max{δ, rδ }. By Lemma 2.2, we have
Z x− Z 1
pµrδ p−1
1
|∇u|p + a(x, y)W (u) dy dx ≥ (
) p ρδ − max{δ, rδ } ≥ 3λδ
p−1
−∞ 0 p
which contradicts (2.6). Thus d(u(x, ·), −σ) ≤ ρδ for any x ≤ x− . Analogously, we
can prove if ku(x+ , ·) − σkW 1,p (0,1) ≤ δ, then d(u(x, ·), σ) ≤ ρδ as x ≥ x+ .
1,p
To exploit the compactness of I on Γ, we set the function X : Wloc
(S0 ) →
R ∪ {+∞} given by
X(u) = sup{x : d(u(x, ·), σ)} ≥ σ/2.
Setting χ(s) = min |s ± σ|, by(H3 ), there exist 0 < w1 < w2 such that
w1 χp (s) ≤ W (s) ≤ w2 χp (s) when χ(s) ≤ σ/2.
(2.7)
Now, we can get the compactness of the minimizing sequence of I in Γ.
Lemma 2.5. If (un ) ⊂ Γ is such that I(un ) → c and X(un ) → X0 ∈ R, then there
exists u0 ∈ K such that, along a sequence, un → u0 weakly in W 1,p (S0 ).
1,p
Proof. We now show that (un ) is bounded in Wloc
(S0 ), i.e., (u
R n ) is bounded in
p
p
Lloc (S0 ), (∇un ) is bounded in Lloc (S0 ). Since I(un ) → cand S0 |∇un |p dx dy ≤
pI(un ), we have that (∇un ) is bounded in Lploc (S0 ). If we can prove that un (x, ·)
is bounded in Lp (0, 1) for a.e. x ∈ R, then (un ) is bounded in Lploc (S0 ).
Let Br = {q ∈ Lp (0, 1)/kqkLp (0,1) ≤ r}, we assume by contradiction that for any
R > 2σ, there exists x̄ ∈ R such that u(x̄, ·) ∈
/ BR for u ∈ Γ ∩ {I(u) ≤ c + λ}, λ > 0,
such that ku(x̄, ·)kLp (0,1) ≥ R, then d(u(x̄, ·), σ) ≥ ku(x̄, ·)kLp (0,1) − kσkLp (0,1) ≥
R − σ. Since d(u(x, ·), ±σ) → 0 as x → ±∞, by continuity there exists x1 > x̄ such
that d(u(x1 , ·), σ) ≤ σ/2 and d(u(x, ·), σ) ≥ σ/2 for x ∈ (x̄, x1 ). Using Lemma 2.2,
we get
c + λ ≥ I(u) ≥ µσ/2 d(u(x1 , ·), u(x̄, ·)) ≥ µσ/2 (R − 3σ/2).
which is a contradiction for R large enough. We conclude that (un ) is bounded in
1,p
1,p
Wloc
(S0 ), thus there exists u0 ∈ Wloc
(S0 ) such that up to a sequence, un → u0
1,p
weakly in Wloc
(S0 ). We shall prove that u0 ∈ Γ; i.e., d(u0 (x, ·), ±σ) → 0 as
x → ±∞. First we claim that:
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7
For any small ε > 0, there exists λ(ε) ∈ (0, λδ̄ ) and l(ε) > lδ̄ such
that if u ∈ Γ ∩ {I(u) ≤ c + λ(ε)} then
Z
Z 1
W (u(x, y)) dy dx ≤ ε.
|x−X(u)|≥l(ε)
(2.8)
0
Indeed, let δ < δ̄ be such that 3λδ ≤ aw1 ε where a = minR2 a(x, y). Given any
u ∈ Γ ∩ {I(u) ≤ c + λδ }, by Lemma 2.4, there exists x− ∈ (X(u) − lδ , X(u))
and x+ ∈ (X(u), X(u) + lδ ) such that ku(x− , ·) + σ)kW 1,p (0,1) ≤ δ and ku(x+ , ·) −
σkW 1,p (0,1) ≤ δ. We define the function


−σ
if x < x− − 1,





σ(x − x− ) + (x − x− + 1)u(x− , y) if x− − 1 ≤ x− ,
ũ(x, y) = u(x, y)
if x− ≤ x ≤ x+ ,



(x+ − x + 1)u(x+ , y) + σ(x − x+ ) if x+ ≤ x < x+ + 1,



σ
if x > x+ + 1
which belongs to Γ, and I(ũ) ≥ c,
Z
Z 1
1
|∇u|p + a(x, y)W (u) dy dx
p
|x−X(u)|≥lδ 0
x−
≤ I−∞
(u) + Ix+∞
(u)
+
x
= I(u) − I(ũ) + Ix−−−1 (ũ)
+ Ixx++ +1 (ũ)
≤ 3λδ
then (2.8) follows setting l(ε) = lδ̄ and λ(ε) = λδ .
From (2.8) it is easy to see that u(x, y) → σ as x → +∞. Combining (2.8) and
(2.7) we obtain
Z
Z 1
Z
Z 1
w1 |u(x, y) − σ|p dx dy ≤
W (u(x, y)) dy dx ≤ ε;
|x−X(u)|≥l(ε)
|x−X(u)|≥l(ε)
0
0
i.e., d u(x, ·), σ → 0 as x → +∞. Analogously, we can get that d u(x, ·), −σ → 0
as x → −∞, it follows that u0 ∈ Γ.
As a consequence, we get the following existence result.
Proposition 2.6. K 6= ∅ and any u ∈ K satisfies u ∈ C 1,α (R2 ) is a solution of
− div(|∇u|p−2 ∇u) + a(x, y)W 0 (u(x, y)) = 0 on S0 with ∂y u(x, 0) = ∂y u(x, 1) = 0
for all x ∈ R, and kukL∞ (S0 ) ≤ R0 . Finally, u(x, y) → ±σ as x → ±∞ uniformly
in y ∈ [0, 1].
Proof. By Lemma 2.5, the set K is not empty. By (H2 ), kukL∞ (S0 ) ≤ R0 . Indeed,
ũ = max{−R0 , min{R0 , u}} is a fortiori minimizer. Let η ∈ C0∞ (S0 ) and τ ∈ R,
then u + τ η ∈ Γ and since u ∈ K, I(u + τ η) is a C 1 function of τ with a local
minima at τ = 0. Therefore,
Z
I 0 (u)η =
|∇u|p−2 ∇u∇η + aW 0 (u)η dx dy = 0
S0
for all such η, namely u is a weak solution of the equation − div(|∇u|p−2 ∇u) +
a(x, y)W 0 (u(x, y)) = 0 on S0 . Standard regularity arguments show that u ∈
C 1,α (S0 ) for some α ∈ (0, 1) and satisfies the Neumann boundary condition (see
8
Z. ZHOU
EJDE-2010/15
[14][17][27]). Since kukL∞ (S0 ) ≤ R0 , there exists C > 0 such that kukC 1,α (S0 ) ≤ C,
which guarantees that u satisfies the boundary conditions. Indeed, assume by contradiction that u does not verify u(x, y) → −σ as x → −∞ uniformly with respect
to y ∈ [0, 1]. Then there exists δ > 0 and a sequence (xn , yn ) ∈ S0 with xn → −∞
and |u(xn , yn ) + σ| ≥ 2δ for all n ∈ N. The C 1,α estimate of u implies that there
exists ρ > 0 such that |u(x, y) + σ| ≥ δ for ∀ (x, y) ∈ Bρ (xn , yn ), n ∈ N. Along a
subsequence xn → −∞, yn → y0 ∈ [0, 1], |u(x, y) + σ| ≥ δ for (x, y) ∈ Bρ/2 (xn , y0 ),
which contradicts with the fact that d(u(x, ·), −σ) → 0 as x → −∞ since u ∈ Γ.
The other case is similar.
We shall explore the reversibility condition of (H1)-(ii), and we will prove that
the minimizer on Γ is in fact a solution of (2.1).
Lemma 2.7. cp = c.
Proof. Since Γp ⊂ Γ, cp ≥ c. Assume by contradiction that cp > c, then there
exists u ∈ Γ such that I(u) < cp . Writing
I(u) =
1/2
Z hZ
R
0
i
1
|∇u|p + aW (u)dy dx +
p
Z hZ
R
1
1/2
i
1
|∇u|p + aW (u)dy dx
p
= I1 + I2
c
it follows that min{I1 , I2 } < 2p . Suppose for example I1 < cp /2, define
(
u(x, y)
if x ∈ R and 0 ≤ y ≤ 21 ,
v(x, y) =
u(x, 1 − y) if x ∈ R and 21 ≤ y ≤ 1.
Then v ∈ Γp , by condition (H1)-(ii), I(v) = 2I1 < cp , this is a contradiction.
We shall prove that any u ∈ K is periodic in y.
Lemma 2.8. If u ∈ K then u(x, 0) = u(x, 1) for all x ∈ R.
Proof. Suppose u ∈ K and v as above, then v(x, y) = u(x, y) for y ∈ [0, 1/2]. By
(H1 )-(ii), I(u) = c = cp = I(v), so v ∈ K. Then u and v are solutions of
− div(|∇u|p−2 ∇u) + aW 0 (u(x, y)) = 0,
∂y u(x, 0) = ∂y u(x, 1) = 0
on S0 ,
for all x ∈ R.
(2.9)
Since u = v for y ∈ [0, 1/2], by the principle of unique continuation (see [8]), we
have u = v in R × [0, 1]. i.e. u(x, 0) = u(x, 1).
Remark 2.9. It is an open problem for the principle for p-harmonic functions in
case n ≥ 3 and p 6= 2. When p = ∞, the principle of unique continuation does not
hold.
Proof of Theorem 1.1. We now extend u periodically in y direction to the entire
space R2 , and write it as U (x, y). As a consequence of the above lemmas and
proposition 2.6, U (x, y) is an entire solution of (1.1), which is heteroclinic in x and
1-periodic in y direction.
EJDE-2010/15
ENTIRE SOLUTIONS
9
References
[1] G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi:
symmetry in 3D for general nonlinearities and a local minimality property, Acta Applicandae
Mathematicae, 65 (2001), 9–33.
[2] F. Alessio, L. Jeanjean and P. Montecchiari, Stationary layered solutions in R2 for a class of
non autonomous Allen-Cahn equations, Calc. Var. Partial Differ. Equ., 11 (2000), 177–202.
[3] F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered
solutions in R2 for a class of periodic Allen-Cahn Equations, Comm. Partial Differ. Equations,
27 (2002), 1537–1574.
[4] F. Alessio, and P. Montecchiari, Brake orbits type solutions to some class of semilinear elliptic
equations, Calc.Var.Partial Differ.Equ., 30 (2007), 51–83.
[5] L. Ambrosio and X. Cabré, Entire solutions of semilinear ellipticequations in R3 and a conjecture of De Giorgi, J. Amer. Math.Soc, 13 (2000), 725–739.
[6] M. T. Barlow, R. R. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi,
Comm. Pure Appl. Math., 53 (2000), 1007–1038.
[7] H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry for some bounded
entire solutions of some elliptic equations, Duke Math.J., 103 (2000), 375–396.
[8] B. Bojarski, T. Iwaniec, p-harmonic equation and quasiregular mappings, Partial Diff.
Equs(Warsaw 1984), 19 (1987), 25–38.
[9] L. Caffarelli, N. Garofalo, and F. Segala, A gradient bound for entire solutions of quasi-linear
equations and its consequences, Comm. Pure Appl. Math., 47 (1994),1457–1473.
[10] L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetryof positive solutions of
m-Laplace equations, J. Differential Equations, 206 (2004), 483–515.
[11] E. DeGiorgi, Convergence problems for functionals and operators, Pro.Int.Meet. on Recent Methods in Nonlinear Analysis, Rome,1978, E.De Giorge, E.Magenes, U,Mosco, eds.,
Pitagora Bologna, (1979), 131–188.
[12] A. Farina, Some remarks on a conjecture of De Giorgi, Calc.Var.,8,(1999), 233–245.
[13] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math.
Ann., 311 (1998), 481–491.
[14] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,
Springer.
[15] T. Iwaniec and J. Manfredi, Regularity of p-harmonic functions in the plane, Revista Matematica Iberoamericana, 5 (1989), 1–19.
[16] P. Lindqvist, On the growth of the solutions of the equation div(|∇u|p−2 ∇u) = 0 in ndimensional space, Journal of Differential Equations, 58 (1985), 307–317.
[17] P. Lindqvist , Regularity for the gradient of the solution to a nonlinear obstacle problem with
degenerate ellipticity, Nonlinear Analysis, 12 (1998), 1245–1255.
[18] J. Manfredi, p-harmonic functions in the plane, Proceedings of the American Mathematical
Society, 103 (1988), 473–479.
[19] J. Manfredi, Isolated singularities of p-harmonic functions in the plane, SIAM Journal on
Mathematical Analysis, 22 (1991), 424–439.
[20] P. H. Rabinowitz, Multibump solutions for an almost periodically forced singular hamiltonian
system, Electronic Journal of Differential Equations, 12 (1995), 1–21.
[21] P. H. Rabinowitz, Solutions of heteroclinic type for some classes of semilinear elliptic partial
differential equations, J.Math.Sci.Univ.Tokio, 1 (1994), 525–550.
[22] P. H. Rabinowitz, Heteroclinic for reversible Hamiltonian system, Ergod.Th.and Dyn.Sys.,
14 (1994), 817–829.
[23] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, Commun. Pure Appl. Math. 56, No.8 (2003), 1078-1134.
[24] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation II, Calc.
Var. Partial Differ. Equ. 21, No. 2 (2004), 157-207.
[25] O. Savin, Phase Transition: Regularity of Flat Level Sets, PhD. Thesis, University of Texas
at Austin, (2003).
[26] J. Serrin and H. Zou, Symmetry of ground states of quasilinear elitic equations, Arch.Rational
Mech.Anal., 148 (1999), 265–290.
[27] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential
Equations, 51 (1984), 126–150.
10
Z. ZHOU
EJDE-2010/15
Zheng Zhou
College of Mathematics and Econometrics, Hunan University, Changsha, China
E-mail address: zzzzhhhoou@yahoo.com.cn