Volumen 37 (2003), p´aginas 11–23
Nontrivial solitary waves of GKP equation in multi-dimensional spaces
Benjin Xuan
∗University of Science and Technology of China, Hefei Universidad Nacional de Colombia, Bogot´a
Abstract. In this paper, using the Mountain Pass Lemma without (PS) con- dition due to Ambrosetti and Rabinowitz, we obtain the existence of the non- trivial solitary waves of Generalized Kadomtsev-Petviashvili equation in multi- dimensional spaces and for superlinear nonlinear termf(u) which satisfies some growth condition. By the Pohozaev type variational identity, we obtain the nonexistence of the nontrivial solitary waves for power function nonlinear case, i.e. f(u) =upwherep≥2(2n−1)/(2n−3).
Keywords and phrases. Mountain Pass Lemma, Solitary wave, Generalized Kado- mtsev-Petviashvili equation.
2000 Mathematics Subject Classification. Primary: 35J60.
1. Introduction
In this paper, we shall investigate the existence and nonexistence of the non- trivial solitary waves of Generalized Kadomtsev-Petviashvili equation in multi- dimensional spaces
wt+wxxx+ (f(w))x=Dx−1∆yw, (1.1) where (t, x, y) ∈ R+ ×R×Rn−1, n ≥ 2, Dx−1h(x, y) = Rx
−∞h(s, y)ds and
∆y := ∂y∂22
1 +∂y∂22
2 +· · ·+∂y∂22
n−1.
Kadomtsev-Petviashvili equation and its generalization appear in many Phy- sic progress (cf. [3], [4], [5], [6], [7] and the references therein). A solitary wave
∗Supported by Grant 10071080 and 10101024 from the NNSF of China.
11
is a solution of the form
w(t, x, y) =u(x−ct, y), wherec >0 is fixed. Substituting in (1.1), there holds
−cux+uxxx+ (f(u))x=Dx−1∆yu, or
¡−uxx+D−2x ∆yu+cu−f(u)¢
x= 0. (1.2)
In [4] and [5], using constrained minimization, De Bouard and Saut obtained the existence and nonexistence of solitary waves in the case where power non- linearitiesf(u) =up, p=m/n, m, nare relatively prime,nis odd. In Chapter 7 of [7], Willem extended the results of [4] to the case where n= 2, f(u) is a continuous function satisfying some structure conditions.
In this paper we mainly deal with the case wheren≥2 and f(u) is a con- tinuous function. The rest of this paper is organized as: §2 gives the functional setting of the problem and some embedding theorems which will be used latter;
§3 deals with the existence of the nontrivial solitary waves. In§4, first we de- rive a variational identity and then use this identity to prove the nonexistence of the nontrivial solitary waves.
2. Preliminaries
In order to attack the existence and nonexistence of the nontrivial solitary waves of problem (1.1) we apply the following functional setting:
Definition 2.1. OnY :={gx|g∈ D(Rn)}, we define the inner product (u, v) :=
Z
Rn
£uxvx+D−1x ∇yu·D−1x ∇yv+cuv¤
dV, (2.1)
where∇y= ( ∂
∂y1
,· · ·, ∂
∂yn−1
), dV =dxdy, and the corresponding norm kuk:=³Z
Rn
£u2x+|Dx−1∇yu|2+cu2¤ dV´1/2
. (2.2)
A functionu:Rn→Rbelongs toX if there exists{um}+∞m=1⊂Y such that:
(a) um→ua.e. onRn;
(b) kuj−ukk →0 asj, k→ ∞.
Note that the spaceX with inner product (2.1) and norm (2.2) is a Hilbert space.
We will show that if estimate kukLq(Rn)≤C³Z
Rn
£u2x+|D−1x ∇yu|2¤ dV´1/2
(2.3)
holds for a certain constantC > 0 and all functions u∈Y, there is only one possibility: q= ¯p= 2(2n−1)2n−3 . In fact, letu∈Y, u6≡0, and define forλ >0 the rescaled function
uλ(x, y) =u(λx, λ2y), (x, y)∈R×Rn−1. Applying (2.3) touλ, there holds
kuλkLq(Rn)≤C³Z
Rn
£(uλ)2x+|Dx−1∇yuλ|2¤ dV´1/2
. (2.4)
But simple computation implies Z
Rn
|uλ|qdV = 1 λ2n−1
Z
Rn
|u|qdV, (2.5)
Z
Rn
(uλ)2xdV = 1 λ2n−3
Z
Rn
u2xdV, (2.6)
and
Z
Rn
|Dx−1∇yuλ|2dV = 1 λ2n−3
Z
Rn
|Dx−1∇yu|2dV. (2.7) Inserting these equalities into (2.4), there holds
1
λ(2n−1)/qkukLq(Rn)≤C 1 λ(2n−3)/2
³Z
Rn
£u2x+|D−1x ∇yu|2¤ dV´1/2
. That is
kukLq(Rn)≤Cλ2n−q1−2n−23³Z
Rn
£u2x+|D−1x ∇yu|2¤ dV´1/2
(2.8) But then if 2n−1q −2n−32 6= 0, upon sendingλto either 0 or∞in (2.8), we can obtain a contradiction. Thus the only possibility is that 2n−1q −2n−32 = 0, i.e, q= ¯p=2(2n−1)2n−3 .
Actually, from the embedding theorems for anisotropic Sobolev spaces(cf.
[2], p. 323), the following lemma asserts that (2.3) holds if and only ifq= ¯p.
Lemma 2.2. Ifq= ¯p=2(2n−1)2n−3 , there exists a constantC >0such that (2.3) holds for all functionsu∈X.
From the interpolation theorem and estimate (2.3), there is an embedding theorem aboutX as follows:
Lemma 2.3. The following embeddings are continuous:
X ,→Lp(Rn),2≤p≤p.¯ Lemma 2.4. The following embeddings are compact:
X ,→,→Lploc(Rn),2≤p <p.¯
Proof. Suppose that{um}∞m=1⊂X is bounded in norm (2.2). Without loss of generality, assume that there exists{gm}∞m=1⊂L2loc(Rn) such thatum=∂xgm. Letvm= (vm,1, vm,2, · · ·, vm,n−1) =∇ygm∈(L2(Rn))n−1.
Multiplying gm by ψ ∈ D(Rn) such that 0 ≤ ψ ≤ 1, ψ ≡ 1 on B(0, R) and suppψ ⊂ B(0,2R), we may assume that suppgm ⊂ B(0,2R). Selecting if necessary to a subsequence, we may assume that um * u=∂xg in X and replacinggm bygm−g, we may assume thatg = 0. Denote byF[u](r, s) the Fourier transform ofu(x, y).
Let
Q−1={(r, s)∈Rn ¯
¯|r| ≤ρ, |si| ≤ρ2, i= 1,2,· · ·, n−1}, Q0={(r, s)∈Rn ¯
¯|r|> ρ}, Q1={(r, s)∈Rn ¯
¯|r|< ρ, |s1|> ρ2}, ...
Qi={(r, s)∈Rn ¯
¯|r|< ρ, |s1|< ρ2,· · · , |si−1|< ρ2, |si|> ρ2}, ...
Qn−1={(r, s)∈Rn ¯
¯|r|< ρ, |s1|< ρ2,· · · , |sn−2|< ρ2, |sn−1|> ρ2}.
ThenRn=
n−1
S
i=−1
Qi andQi∩Qj =∅, i6=j. Forρ >0, there holds
Z
B(0,2R)
|um|2dV = Z
Rn
|F[um]|2drds=
n−1
X
i=−1
Z
Qi
¯¯F[um]¯
¯
2drds. (2.9) It is clear that
Z
Q0
¯¯F[um]¯
¯
2drds= Z
Q0
1 4π2r2
¯¯F[∂xum]¯
¯
2drds≤ 1
4π2ρ2|∂xum|22, and fori= 1,· · · , n−1, there holds
Z
Qi
¯¯F[um]¯
¯
2dxdy= Z
Qi
r2
|si|2
¯¯F[vm,i]¯
¯
2drds≤ 1 ρ2|vm|22. For anyε >0, there existsρ >0 large enough, such that
n−1
X
i=0
Z
Qi
¯¯F[um]¯
¯
2drds≤ε/2.
Sinceum*0 inL2(Rn), there holds F[um](r, s) =
Z
B(0,2R)
um(x, y)e−2iπ(xr+y·s)dV →0, asm→ ∞ and
¯¯F[um](r, s)¯
¯≤c0|um|2≤c1.
Lebesgue’s dominated convergence theorem implies that Z
Q−1
¯¯F[um]¯
¯
2drds→0, as m→ ∞.
Thus we have proved thatum→0 inL2loc(Rn). By Lemma 2.3 and interpola- tion theorem, there holdsum→0 inLploc(Rn) if 2≤p <p.¯ ¤X Lemma 2.5. If{um}+∞m=1is bounded inX and if
sup
(x,y)∈Rn
Z
B(x,y;r)
|um|2dV →0, asn→ ∞. (2.10) Thenum→0 inLp(Rn)for2< p <p.¯
Proof. Let 2< s <p¯andu∈X. By H¨older inequality and Lemma 2.3, there holds
|u|Ls(B(x,y;r))≤ |u|1−λL2(B(x,y;r))|u|λLp¯(B(x,y;r))
≤c0|u|1−λL2(B(x,y;r))
¡ Z
B(x,y;r)
£u2x+|D−1x ∇yu|2+cu2]dV¢λ2
, (2.11) where 1
s = 1−λ 2 +λ
¯
p. Choosingssuch that λs
2 = 1, i.e.,s=2(2n+ 1) 2n−1 , there holds
Z
B(x,y;r)
|u|sdV ≤cs0|u|(1−λ)sL2(B(x,y;r))
Z
B(x,y;r)
£u2x+|D−1x ∇yu|2+cu2]dV. (2.12) Now, covering Rn by balls of radius rin such a way that each point of Rn is contained in at most 3 balls, then there holds
Z
Rn
|u|sdV ≤3cs0 sup
(x,y)∈Rn
|u|(1−λ)sL2(B(x,y;r))
Z
Rn
£u2x+|Dx−1∇yu|2+cu2]dV. (2.13) Under assumption (2.10), (2.13) implies um → 0 in Ls(Rn). By H¨older in- equality and Lemma 2.3, there holdsum→0 inLp(Rn) for all 2< p <p.¯ ¤X We recall the following Mountain Pass Lemma without (PS) condition as our Lemma 2.6 (cf. [1]).
Lemma 2.6 (Mountain Pass Lemma). Suppose X is a Banach space and E∈C1(X, R)satisfies the following geometrical properties:
(1) E(0) = 0, and there existsρ >0, such thatE¯
¯
¯∂B
ρ(0) ≥α >0;
(2) There existse∈X\Bρ(0), such thatE(e)≤0.
LetΓbe the set of all passes which connects0ande, i.e., Γ ={g∈C([0,1], E)¯
¯g(0) = 0, g(1) =e}, (2.14) and
c= inf
g∈Γ max
t∈[0,1]E(g(t)). (2.15)
Thenc≥αandEpossesses a (PS)c sequence at levelc defined by (2.15), i.e., there exists a sequence {um}+∞m=1 such thatE(um)→c and DE(um)→ 0 as m→ ∞.
3. Existence of nontrivial solitary waves The solitary waves of problem (1.1) satisfies:
(¡−uxx+Dx−2∆yu+cu−f(u)¢
x= 0,
u∈X, (3.1)
wherec >0. The weak solutions of (3.1) are the critical points of the functional E defined onX as
E(u) :=
Z
Rn
¡1
2[u2x+|D−1x ∇yu|2+cu2]−F(u)¢ dV,
whereF(u) = Z u
0
f(s)ds. Assume:
(f1) f ∈C0(R,R), f(0) = 0 and for some 2< p <p¯= 2(2n−1)2n−3 , 0 < c0 <
c, c1>0, there holds
|f(u)| ≤c0|u|+c1|u|p−1; (f2) There existsv∈X such that
f(λv)
λ →+∞, asλ→+∞;
(f3) There existsα >2 such that, foru∈R, there holds αF(u)≤uf(u).
By assumption (f1) and Lemma 2.3,E∈C1(X,R).
Lemma 3.1. Under assumptions (f1) and (f2), there existse∈X and r >0 such thatkek ≥rand
b:= inf
kuk=rE(u)> E(0) = 0≥E(e).
Proof. From (f1), there holds
|F(u)|=| Z u
0
f(s)ds| ≤c0
|u|2 2 +c1
p|u|p. Then from the definition of the norm (2.2) inX, there holds
E(u)≥kuk2
2 −
Z
Rn
¡c0
2|u|2+c1
p|u|p¢
dV ≥(1 2 −c0
2c)kuk2−c1|u|pp. By Lemma 2.3, there exists r >0 such that
b:= inf
kuk=rE(u)> E(0) = 0.
It follows from assumption (f2) that
E(λv)→ −∞, as λ→+∞.
Hence there existsλ0>0 such thate=λ0v satisfieskek> r, E(e)≤0. ¤X Define
d:= inf
γ∈Γ max
t∈[0,1]E(γ(t)),
Γ :={γ∈C([0,1];X) : γ(0) = 0, γ(1) =e}.
Clearly, d ≥ b > 0. Applying Lemma 2.6, there exists a (PS)c sequence {um}+∞m=1 at level c=dsuch that
E(um)→dand DE(um)→0 asm→ ∞.
Theorem 3.2. Under assumptions (f1)–(f3), problem (3.1) possesses a non- trivial solution.
Proof. 1. Boundness of (PS)c sequence.
Let{um}+∞m=1 be the sequence derived by Lemma 2.6, i.e.,E(um)→dand DE(um)→0 asm→ ∞. Asm→ ∞, from assumption (f3), there holds
d+o(1) +o(1)kumk ≥E(um)−α−1(DE(um), um)
= (12−α1)kumk2+ Z
Rn
£α−1umf(um)−F(um)¤ dV
≥(12−α1)kumk2. Hence{um}+∞m=1 is bounded inX.
2.δ:= lim
m→∞ sup
(x,y)∈Rn
Z
B(x,y; 1)
|um|2dV 6= 0.
Otherwise, by Lemma 2.5, there holds um → 0 in Ls(Rn) for 2 < s <
2(2n−1)
2n−3 . It follows that
0< d =E(um)−12(DE(um), um) +o(1)
= Z
Rn
[1
2umf(um)−F(um)]dV +o(1) = 0(1), which is a contradiction.
3.Existence of a nontrivial solution of problem (3.1).
Selecting if necessary a subsequence, we can assume that there existes a sequence (xm, ym)⊂Rn such that
Z
B(xm,ym;1)
|um|2dV > δ/2.
Definevm(x, y) :=um(x+xm, y+ym) so that Z
B(0;1)
|vm|2dV > δ/2.
Selecting if necessary a subsequence, we can assume that there existes av∈X such that
vm* v inX, as m→ ∞.
By Lemma 2.4,vm→v inL2loc(Rn) and sov6= 0, and for everyw∈X, there holds
Z
Rn
(f(vm)−f(v))w dV = Z
B(0,R)
(f(vm)−f(v))w dV +
Z
Rn\B(0,R)
(f(vm)−f(v))w dV.
Since w ∈ X, then w ∈ Lp(Rn) and {vm} is bounded in X, hence {vm} is bounded in Lp(Rn), thus for any ε > 0, there exists R = R(ε) > 0 large enough and independent onmsuch that
Z
Rn\B(0,R)
(f(vm)−f(v))w dV < ε, ∀m On the other hand, for thisR >0, from Lemma 2.4, there holds
Z
B(0,R)
(f(vm)−f(v))w dV →0, asm→ ∞.
Thus, there holds Z
Rn
f(vm)w dV → Z
Rn
f(v)w dV, asm→ ∞, which implies
(DE(v), w) = lim
m→∞(DE(vm), w) = 0
HenceDE(v) = 0 andv is a nontrivial solution of problem (3.1). ¤X
4. Nonexistence of nontrivial solitary waves
In this section, we derive a Pohozaev type variational identity of the solitary wave of problem:
¡−uxx+D−2x ∆yu−g(u)¢
x= 0, whereg∈C1(R,R) such thatg(0) = 0 and defineG(u) :=
Z u 0
g(s)ds.
First, we give a formal argument explaining the variational identity. For any λ >0, define a transformationT(λ) :X →X as
T(λ)u(x, y) :=u(x/λ, y/λ2), (x, y)∈R×Rn−1.
ThenT(1) = idX. Ifu∈X is a critical point of functionalE(u), we conjecture that
∂
∂λ
¯
¯
¯
¯
¯λ=1
E(T(λ)u) = 0. (4.1)
A simple computation shows that E(T(λ)u) = λ2n−3
2 Z
Rn
¡u2x+|D−1x ∇yu|2¢
dV −λ2n−1 Z
Rn
G(u)dV. (4.2) and
∂
∂λ
¯
¯λ=1E(T(λ)u) = 2n−3
2 Z
Rn
¡u2x+|D−1x ∇yu|2¢
dV −(2n−1) Z
Rn
G(u)dV,
(4.3)
which implies that Z
Rn
¡u2x+|D−1x ∇yu|2¢
dV =2(2n−1) 2n−3
Z
Rn
G(u)dV. (4.4) In fact, we have the following Theorem:
Theorem 4.1. Any solution of
¡−uxx+D−2x ∆yu−g(u)¢
x= 0, u∈X∩Hloc2 (Rn),
G(u), g(u)u∈L1(Rn), g(u)D−1x ∇yu∈(L1(Rn))n−1,
(4.5)
satisfies (4.4).
Proof. 1. Let
J(u) :=
Z
Rn
¡1
2[u2x+|Dx−1∇yu|2]−G(u)¢ dV.
Then a weak solution of problem (4.5) is a critical point of operator J. Let ψ ∈ D(R) be such that 0 ≤ ψ ≤ 1, ψ(r) = 1 for r = 1 and ψ(r) = 0 for r≥2, |ψ0(r)| ≤2, |ψ00(r)| ≤4. Define a sequence of functions onRn as:
ψm(x, y) :=ψ(x2+|y|2
m2 ), ∀(x, y)∈Rn. 2.For any solution of problem (4.5), there holds
3 2 Z
Rn
u2xdV −1 2
Z
Rn
|D−1x ∇yu|2dV + Z
Rn
¡G(u)−g(u)u¢
dV = 0. (4.6) For every integerm, there holds
Z
Rn
¡−uxx+Dx−2∆yu−g(u)¢¡
ψmxu¢
xdV = 0. (4.7)
Integrating by parts, there holds
− Z
Rn
uxx¡ ψmxu¢
xdV =− Z
Rn
uxx¡
ψm,xxu+ψmu+ψmxux¢ dV
= Z
Rn
·3
2u2x(ψm,xx+ψm) + 2ψm,xuux+ψm,xxxuux
¸ dV.
Lebesgue dominated convergence theorem implies that, asm→ ∞, there holds
− Z
Rn
uxx¡ ψmxu¢
xdV = 3 2
Z
Rn
u2xdV +o(1). (4.8) Similarly, there hold
Z
Rn
Dx−2∆yu¡ ψmxu¢
xdV
=− Z
Rn
¡D−1x ∆yu¢¡
ψmxu¢ dV
=− Z
Rn n−1
X
i=1
∂
∂yi
¡Dx−1uyi
¢¡ψmxu¢ dV
= Z
Rn n−1
X
i=1
D−1x uyi
∂
∂yi
¡ψmxu¢ dV
= Z
Rn
³n−1X
i=1
Dx−1uyiψm,yixu+
n−1
X
i=1
Dx−1uyiψmx∂
∂xD−1x uyi
´dV
= Z
Rn
³n−1X
i=1
Dx−1uyiψm,yixu−1 2
n−1
X
i=1
|D−1x uyi|2(ψm,xx+ψm)´ dV
=−1 2
Z
Rn
|D−1x ∇yu|2dV +o(1),
(4.9)
and
− Z
Rn
g(u)¡ ψmxu¢
xdV
=− Z
Rn
g(u)¡
ψm,xxu+ψmu+ψmxux¢ dV
=− Z
Rn
¡g(u)ψmu+g(u)ψm,xxu+dG(u) dx ψmx¢
dV
= Z
Rn
(G(u)−g(u)u)dV +o(1).
(4.10)
Substituting (4.8)–(4.10) into (4.7) yields (4.6)
3.On the other hand, sinceuis a weak solution of problem (4.5), i.e.,DJ(u) = 0, then from (DJ(u), u) = 0, there holds
Z
Rn
¡u2x+|Dx−1∇yu|2¢ dV =
Z
Rn
g(u)u dV. (4.11) 4.For any solution of problem (4.5), there holds
−n−1 2
Z
Rn
u2xdV−n−3 2
Z
Rn
|Dx−1∇yu|2dV+(n−1) Z
Rn
G(u)dV = 0. (4.12) For every integerm, there also holds
Z
Rn
¡−uxx+D−2x ∆yu−g(u)¢¡
ψmy·Dx−1∇yu¢
xdV = 0. (4.13) Integrating by parts and applying Lebesgue dominated convergence theorem imply that, asm→ ∞, there hold
− Z
Rn
uxx¡
ψmy·Dx−1∇yu¢
xdV
=− Z
Rn
uxx¡
ψm,xy·D−1x ∇yu+ψmy· ∇yu¢ dV
= Z
Rn
ux¡
ψm,xy·D−1x ∇yu+ψmy· ∇yu¢
xdV
= Z
Rn
ux¡
ψm,xxy·D−1x ∇yu+ 2ψm,xy· ∇yu+ψmy· ∇yux¢ dV
=−n−1 2
Z
Rn
u2xdV +o(1),
(4.14)
Z
Rn
¡Dx−2∆yu¢¡
ψmy·D−1x ∇yu¢
xdV
=− Z
Rn
¡Dx−1∆yu¢¡
ψmy·D−1x ∇yu¢ dV
=− Z
Rn
¡
n−1
X
i=1
∂
∂yi
(Dx−1uyi)¢¡
ψmy·Dx−1∇yu¢ dV
= Z
Rn n−1
X
i=1
(D−1x uyi)¡
ψmy·D−1x ∇yu¢
yidV
=−n−3 2
Z
Rn
|D−1x ∇yu|2dV +o(1)
(4.15)
and
− Z
Rn
g(u)¡
ψmy·D−1x ∇yu¢
xdV
=− Z
Rn
g(u)¡
ψm,xy·D−1x ∇yu+ψmy· ∇yu¢ dV
=− Z
Rn
¡g(u)ψm,xy·D−1x ∇yu+
n−1
X
i=1
dG(u) dyi
yiψm¢ dV
= (n−1) Z
Rn
G(u)dV +o(1).
(4.16)
Thus, from equations (4.13)–(4.16) (4.12) holds. Equations (4.6), (4.11) and
(4.12) imply equation (4.4). ¤X
Theorem 4.2. (Nonexistence of nontrivial solitary wave) If g ∈ C1(R;R) satisfiesg(0) = 0and
2(2n−1)
2n−3 G(u)−g(u)u <0, ∀u6= 0, (4.17) then0is the only solution of problem (4.5).
Proof. Ifu6≡0 is a solution of problem (4.5), then (4.4)-(4.11), there holds Z
Rn
£2(2n−1)
2n−3 G(u)−g(u)u¤ dV = 0
which contradicts (4.17). ¤X
Corollary 4.3. Let c > 0, and p ≥ 2(2n−1)2n−3 , then 0 is the only solution of problem:
¡−uxx+D−2x ∆yu+cu− |u|p−2u¢
x= 0, u∈X∩Hloc2 (Rn),
|u|p−2uDx−1∇yu∈(L1(Rn))n−1.
(4.18)
Proof. Sinceg(u) =|u|p−2u−cu, thenG(u) =1
p|u|p−c
2u2, thus (4.17) holds.
¤X
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(Recibido en diciembre de 2002)
Department of Mathematics University of Science and Technology of China Anhui, Hefei, China Departamento de Matem´aticas Universidad Nacional de Colombia Bogot´a Colombia e-mail: [email protected] e-mail:[email protected]
