USA-Chile Workshop on Nonlinear Analysis,
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 343–357.
http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)
Nontrivial solutions of semilinear elliptic
systems in RN ∗
Jianfu Yang
Abstract
We establish an existence result for strongly indefinite semilinear elliptic systems in RN .
1
Introduction
The main objective of this paper is to establish existence results for the semilinear elliptic system
−∆u + u = g(x, v),
−∆v + v = f (x, u) inRN ,
(1.1)
u(x) → 0 and v(x) → 0 as |x| → ∞.
(1.2)
The existence of solutions of (1.1)-(1.2) is usually investigated by finding critical
points of a related functional. Typical features of the problem are that firstly,
the related functional is strongly indefinite; secondly, the growths of f in u and
g in v at infinity may not be ‘symmetric’; and lastly, Sobolev embeddings in
general are not compact, then the Palais - Smale condition may not be satisfied.
Existence results were recently obtained in [12] and [15] in bounded domains.
The arguments lie in the decomposition of Sobolev spaces by eigenfunctions of
Laplacian operator and a use of linking theorems. Using spectral family theory
of non-compact operator, the author and Figueiredo [13] find a suitable linking
structure for the functional associate to (1.1)-(1.2) and prove that problem (1.1)(1.2) possesses at least a positive solution if f and g depend on the variable
x radially. Furthermore, it is also shown in [13] that all positive solutions
of problem (1.1)-(1.2) are exponentially decaying. In this paper, we establish
existence results for general cases. Assume that
H1) f, g : RN × R → R are measurable in the first variable, continuous in the
Rt
Rt
second variable. Both F (x, t) = 0 f (x, s)ds and G(x, t) = 0 g(x, s)ds are
increasing and strictly convex in t.
H2) limt→0 f (x, t)/t = 0,
limt→0 g(x, t)/t = 0
∗ Mathematics Subject Classifications: 35J50, 35J55.
Key words: indefinite, semilinear, elliptic system.
c
2001
Southwest Texas State University.
Published January 8, 2001.
343
uniformly in
x ∈ RN .
344
Nontrivial solutions of semilinear elliptic systems
H3) There is a constant c > 0 such that |f (x, t)| ≤ c(|t|p + 1) and |g(x, t)| ≤
c(|t|q + 1), where 0 < p, q < (N + 2)/(N − 2), N ≥ 3.
H4) There are constants α, β > 2 such that 0 < αF (x, t) ≤ tf (x, t) and 0 <
βG(x, t) ≤ tg(x, t), for t 6= 0.
H5) f (x, t) → f¯(t) and g(x, t) → ḡ(t) uniformly for t bounded as |x| → ∞.
|f (x, t) − f¯(t)| ≤ (R)|t| and |g(x, t) − ḡ(t)| ≤ (R)|t| whenever |x| ≥ R,
|t| ≤ δ, where (R) → ∞ as R → ∞.
H6) F (x, t) ≥ F̄ (t) and G(x, t) ≥ Ḡ(t),
meas{x ∈ RN : f (x, t) 6≡ f¯(t)} > 0 or meas{x ∈ RN : g(x, t) 6≡ ḡ(t)} > 0.
H7) Both f¯(t)/t and ḡ(t)/t are increasing in t.
Our main result is as follows.
Theorem 1.1 Assume (H1)-(H7). Then problem (1.1)-(1.2) possesses at least
one nontrivial exponentially decaying solution.
The restriction of exponents in (H3) is due to the fact that we only know
the decaying law in the case.
We analyze the convergence of Palais-Smale sequence of associate functional
to (1.1)-(1.2) in Section 3. It is shown that the energy levels of solutions of the
related autonomous system
−∆u + u = ḡ(v),
u(x) → 0,
−∆v + v = f¯(u) in RN ,
v(x) → 0 as
|x| → 0 .
(1.3)
(1.4)
are obstacle levels preventing strong convergence of Palais-Smale sequences of
(1.1)-(1.2). The possible critical values to be found are between obstacle levels. To retain the compactness, we have to get control of critical values. It is
harder to handle critical values described by linking structure than that by the
Mountain Pass Theorem. We use dual variational method as [3], [4] and [11].
The method is of the advantage avoiding the indefinite character of original
functional. We start with problem (1.1)-(1.2) in bounded domains. Although
existence result in the case is known, it has no control of critical values. We
establish in Section 2 an existence result by the Mountain Pass Theorem and
bound uniformly corresponding critical values by the energy level of ground
state of problem (1.3)-(1.4). Then we construct a Palais - Smale sequence for
the functional associated to problem (1.1)-(1.2). Theorem 1.1 is proved in Section 4.
2
Existence results in bounded domains
Let Ω be a bounded domain. We consider the problem
−∆u + u = g(x, v), −∆v + v = f (x, u) in Ω,
u = 0, v = 0 on ∂Ω .
(2.1)
(2.2)
Jianfu Yang
345
The solutions of (2.1)-(2.2) will be found by looking for critical points of associate functional. The main result in this section is as follows.
Theorem 2.1 Assume (H1) − (H4). Then problem (2.1)-(2.2) possesses at
least a nontrivial solution.
To prove Theorem 2.1 we will need the lemmas below. First we define the
dual functional associate to (2.1)-(2.2). It is well known that the inclusions
ir : Wo1,r (Ω) → Lp+1 (Ω),
is : Wo1,s (Ω) → Lq+1 (Ω)
sN
are compact if 2 < p + 1 < NrN
−r , N > r and 2 < q + 1 < N −s , N > s. The
0
r
operator −∆ + id : Wo1,r (Ω) → Wo−1,r (Ω) is an isomorphism, where r0 = r−1
.
Hence
T = i2 (−∆ + id)−1 i∗2 : L1+1/q (Ω) → Lp+1 (Ω).
is continuous. Denote by X = Lp+1 (Ω)× Lq+1 (Ω), X ∗ = L1+1/p (Ω)× L1+1/q (Ω)
and let
0 T
0
T −1
, K = A−1 =
.
A=
T 0
T −1
0
For each x, the Legendre-Fenchel transformations F ∗ (x, ·) of F (x, ·), and
G (x, ·) of G(x, ·) are defined by
∗
F ∗ (x, s) = sup{st − F (x, t)},
t∈R
G∗ (x, s) = sup{st − G(x, t)}
t∈R
(2.3)
respectively. Equivalently, we have
and
0
F ∗ (x, s) = st − F (x, t)
with s = f (x, t),
t = Fs∗ (x, s)
G∗ (x, s) = st − G(x, t)
with s = g(x, t),
t = G∗s (x, s).
0
(2.4)
(2.5)
In the same way, we define F̄ ∗ , Ḡ∗ for F̄ , Ḡ. By (H6) and properties of LegendreFenchel transformations, we have
F ∗ (x, s) ≤ F̄ ∗ (s),
G∗ (x, s) ≤ Ḡ∗ (s).
(2.6)
We may verify following properties of F ∗ , G∗ in Lemmas 2.2 and 2.3 as [3], [10]
and [16].
Lemma 2.2 F ∗ , G∗ ∈ C 1 and
F ∗ (x, s) ≥ (1 −
0
1
)sF ∗ (x, s),
α
G∗ (x, s) ≥ (1 −
F ∗ (s, x) ≥ C|s|1+1/p − C,
0
1
)sG∗ (x, s),
β
G∗ (x, s) ≥ C|s|1+1/q − C.
Lemma 2.3 There exist δ > 0, Cδ and Cδ0 > 0 such that
if |s| ≤ δ
Cδ |s|2 ,
Cδ |s|2 ,
∗
1
1
,
G
F ∗ (x, s) ≥
(x,
s)
≥
1+ p
0
Cδ |s|
Cδ0 |s|1+ q ,
, if |s| ≥ δ
if
if
(2.7)
(2.8)
|s| ≤ δ
.
|s| ≥ δ
Nontrivial solutions of semilinear elliptic systems
346
We may verify that the dual functional
Z
Z
1
∗
∗
hw, Kwi dx,
Ψ(w) = ΨΩ (w) = (F (x, w1 ) + G (x, w2 )) dx −
2 Ω
Ω
is well defined and C 1 on X ∗ . A critical point w of Ψ satisfies
0
0
(−∆ + id)−1 w2 = Fs∗ (x, w1 ) and (−∆ + id)−1 w1 = G∗s (x, w2 ).
Let
u = (−∆ + id)−1 w2 ,
v = (−∆ + id)−1 w1 .
Then (u, v) satisfies (2.1)-(2.2). Furthermore, denoting by
Z
Z
Z
F (x, u) dx −
G(x, v) dx
Φ(z) = (∇u∇v + uv) dx −
Ω
Ω
Ω
the functional of (2.1) -(2.2) defined on Ho1 (Ω) × Ho1 (Ω), we deduce by (2.4) and
(2.5) that Φ(z) = Ψ(w). Such a result is also valid for solutions of (1.1)-(1.2).
Now we use the Mountain Pass Theorem to find critical points of Ψ.
Following arguments of [6], we know that (H2) implies F ∗ (x, t)/t2 → ∞ and
∗
G (x, t)/t2 → ∞. Thus 0 is a local minmum of Ψ. Precisely,
Lemma 2.4 There exist constants α, ρ > 0, independent of Ω, such that
Ψ(w) ≥ α > 0
if
kwkX ∗ = ρ.
Lemma 2.5 There exist T > 0 and w ∈ E such that Ψ(tw) ≤ 0 whenever
t ≥ T.
Proof. Taking w ∈ X ∗ , w 6≡ 0 such that
Z
hw, Kwi dx > 0,
Ω
whence by (H4), for t > 0
Z
Z
Z
β
β
α
α
1 2
β−1
β−1
α−1
α−1
|w1 |
dx + t
|w2 |
dx − t
hw, Kwi dx.
Ψ(tw) ≤ t
2
Ω
Ω
Ω
β
α
, β−1
< 2, the assertion follows for t > 0 large.
Since α−1
Let
Γ = {g ∈ C([0, 1], X ∗ ) : g(0) = 0, g(1) = e},
where e = T w. We define
c = cΩ = inf g∈Γ sup Ψ(g(t)).
(2.9)
t∈[0,1]
The Mountain Pass Theorem will imlpy that c is a critical value of Ψ if the
Palais-Smale ((PS) for short) condition holds. It is known from Lemma 2.4 that
Jianfu Yang
347
corresponding critical points are nontrivial. Then the proof of Theorem 2.1 is
completed.
Now we verify the (PS) condition. By a (PS) condition for Ψ we mean that
any sequence {wn } ⊂ X ∗ such that |Ψ(wn )| is uniformly bounded in n and
Ψ0 (wn ) → 0 as n → ∞ possesses a convergent subsequence.
Lemma 2.6 The (PS) condition holds for Ψ.
Proof. Let {wn } be a (PS) sequence of Ψ, that is
|Ψ(wn )| ≤ C
Ψ0 (wn ) → 0
as n → ∞
for some constant C > 0. This inequality and lemma 2.2 yield
Z
[F ∗ (x, wn1 ) + G∗ (x, wn2 )] dx
Ω
Z
1
hwn , Kwn i dx + C
≤
2 Ω
Z
0
0
1
(Fs∗ (x, wn1 )wn1 + G∗s (x, wn2 )wn2 ) dx + o(1)kwn kX ∗ + C
≤
2 Ω
Z
Z
1 β
1 α
∗
1
F (x, wn ) dx +
G∗ (x, wn2 ) dx + o(1)kwn kX ∗ .
≤
2α−1 Ω
2β−1 Ω
That is
Z
Ω
[F ∗ (x, wn1 ) + G∗ (x, wn2 )] dx ≤ C + o(1)kwn kX ∗ .
By Lemma 2.3, we obtain
1+1/p
1+1/q
kwn1 kL1+1/p + kwn2 kL1+1/q ≤ C + o(1)kwn kX ∗ .
It implies that kwn kX ∗ is bounded. We may assume wn → w weakly in X ∗ as
n → ∞. Since the operator (−∆ + id)−1 is compact, it follows
un := (−∆ + id)−1 wn2 → (−∆ + id)−1 w2
−1
vn := (−∆ + id)
wn1
−1
→ (−∆ + id)
As a result, wn = (f (x, un ), g(x, vn )) → w
pletes the present proof.
3
w
1
in X ∗
as n → ∞,
∗
as n → ∞.
in
X
in X ∗
as
n → ∞ which com-
Palais-Smale sequence
In this section, we prove a global compact result for problem (1.1)-(1.2). Let
E = H 1 (RN ) × H 1 (RN ). By our assumptions, the functional
Z
Z
(∇u∇v + uv) dx −
(F (x, u) + G(x, v)) dx
Φ(z) =
RN
RN
is C 1 on E. The functional Φ∞ is defined with F̄ and Ḡ replacing F and G in
Φ respectively.
Nontrivial solutions of semilinear elliptic systems
348
Proposition 3.1 Asumme (H1)-(H6). Let {zn } be a (P S)c sequence of Φ, i.e.
Φ(zn ) → c
and
Φ0 (zn ) → 0
as
n → 0.
(3.1)
Then there exists a subsequence (still denoted by {zn }) for which the following
holds: there exist an integer k ≥ 0, sequences {xin } ⊂ RN , |xin | → ∞ as n → ∞
for 1 ≤ i ≤ k, a solution z of (1.1)-(1.2) and solutions z i (1 ≤ i ≤ k) of
(1.3)-(1.4) such that
zn → z
weakly in E,
Pk
Φ(zn ) → Φ(z) + i=1 Φ∞ (z i ),
Pk
zn − (z + i=1 z i (x − xin )) → 0 in
(3.2)
(3.3)
E
(3.4)
as n → ∞, where we agree that in the case k = 0 the above holds without z i , xin .
Proof. The result can be derived from the arguments for one equation [5].
First we remark that the boundedness of {zn } in E can be deduced as [13] by
(3.1). Therefore we may assume
zn → z
weakly
zn → z
strongly
in
E,
q+1
N
N
in Lp+1
loc (R ) × Lloc (R ),
a.e. in RN
R
as n → ∞. Denote Q(z) = RN (∇u∇v + uv) dx, we have
zn → z
Q(zn ) = Q(zn − z) + Q(z) + o(1).
It follows from Brezis & Lieb’s lemma [8] that
Z
Z
Z
F (x, un ) dx =
F (x, un − u) dx +
RN
and
RN
RN
G(x, vn ) dx =
RN
F (x, u) dx + o(1)
(3.6)
G(x, v) dx + o(1).
(3.7)
RN
Z
Z
(3.5)
Z
G(x, vn − v) dx +
RN
Hence we obtain
Φ(zn ) = Φ(zn − z) + Φ(z) + o(1),
Φ0 (zn ) = Φ0 (zn − z) + Φ0 (z) + o(1) (3.8)
as n → ∞. Let zn1 = zn − z. We may deduce from (H5) as [17] and [19] that
Z
Z
u1n [f (x, u1n ) − f¯(u1n )] dx → 0 and
vn1 [g(x, vn1 ) − ḡ(vn1 )] dx → 0
RN
as well as
Z
[F (x, u1n ) − F̄ (u1n )] dx → 0 and
RN
RN
Z
RN
[G(x, vn1 ) − Ḡ(vn1 )] dx → 0
Jianfu Yang
349
as n → ∞. Therefore
Φ∞ (zn1 ) = Φ(zn1 ) + o(1) = Φ(zn ) − Φ(z) + o(1)
Φ
∞0
(zn1 )
=Φ
0
(zn1 )
0
0
+ o(1) = Φ (zn ) − Φ (z) + o(1).
(3.9)
(3.10)
Suppose zn1 = zn − z 6→ 0 strongly in E(otherwise we shall have finished). We
want to show that there exists x1n ⊂ RN such that |x1n | → +∞ and zn1 (x+x1n ) →
z 1 6≡ 0 weakly in E. We note that
Φ∞ (zn1 ) ≥ α > 0
because kzn1 kE 6→ 0. In fact, were it not true, we would have
0
Φ∞ (zn1 ) → 0,
< Φ∞ (zn1 ), η >= o(1)kηkE
as n → ∞.
(3.11)
β
α
u1n , α+β
vn1 ) =: ηn in (3.11), it follows
Taking η = ( α+β
Z
Z
β
α
o(1)kηn kE =
u1n f¯(u1n ) dx +
v 1 ḡ(v 1 ) dx
α + β RN
α + β RN n n
Z
Z
(3.12)
F̄ (u1n ) dx −
Ḡ(vn1 ) dx.
−
RN
RN
Using hypothesis (H4) we obtain
Z
(F̄ (u1n ) + Ḡ(vn1 )) dx = o(1)kηkE .
RN
This and (3.12) yield
Z
u1n f¯(u1n ) dx = o(1)kηn kE ,
RN
Z
RN
vn1 ḡ(vn1 ) dx = o(1)kηn kE .
(3.13)
It follows from assumptions (H2)-(H4) that
¯ 2 ≤ Ctf¯(t) if
|f(t)|
|t| ≤ 1,
0
|f¯(t)|(p+1) ≤ Ctf¯(t) if
|t| > 1.
(3.14)
Taking η = (φ, 0) in (3.11) and using (3.14) and Hölder’s inequality, we obtain
Z
(∇φ∇vn1 + φvn1 ) dx|
|
RN
Z
Z
+
φf¯(u1n ) dx|
(3.15)
≤ |
{|u1n |≤1}
Z
≤ C(
RN
{|u1n |>1}
Z
0
0
dx) kφkL2 + C(
|f¯(u1n )|(p+1) dx)1/(p+1) kφkLp+1
N
R
Z
0
1
1 ¯ 1
un f (un ) dx) 2 + C(
u1n f¯(u1n ) dx)1/(p+1) ].
|f (u1n )|2
Z
≤ CkφkH s [(
RN
1
2
RN
which with (3.13) imply that
kvn1 kH 1 = o(1).
(3.16)
Nontrivial solutions of semilinear elliptic systems
350
Similarly, we show that
ku1n kH 1 = o(1).
(3.17)
kzn1 kE
→ 0 , we get a contradiction.
(3.16) and (3.17) yield
We decompose RN into N-dimensional unit hypercubes Qj with vertices
having integer coordinates and put
dn = maxj (ku1n kLp+1 (Qj ) + kvn1 kLq+1 (Qj ) ).
We claim that there is a β > 0 such that
dn ≥ β > 0
∀n ∈ N.
(3.18)
Suppose, by contradiction, that dn → 0 as n → ∞. Since
0
Φ∞ (zn1 ) → 0
(3.19)
is bounded, we have by (H2) and (H3) that
Z
Z
∞ 1
1 ¯ 1
≤ Φ (zn ) ≤
un f (un ) dx +
vn1 ḡ(vn1 ) dx + o(1)
noting that
0
as n → ∞,
kzn1 kE
RN
RN
≤ C (ku1n kp+1
+ kvn1 kq+1
) + (ku1n k2L2 (RN ) + kvn1 k2L2 (RN ) )
Lp+1 (RN )
Lq+1 (RN )
X
1 q+1
1 2
1 2
≤ C
(ku1n kp+1
Lp+1 (Qj ) + kvn kLq+1 (Qj ) ) + (kun kL2 (RN ) + kvn kL2 (RN ) )
j
≤ C dp−1
n
X
j
≤ C dp−1
n
X
j
ku1n k2Lp+1 (Qj ) + C dq−1
n
ku1n k2H 1 (Qj ) + C dq−1
n
X
j
X
j
kvn1 k2Lq+1 (Qj ) + C
kvn1 k2H 1 (Qj ) + C
1 2
q−1
1 2
≤ C dp−1
n kun kH 1 + C dn kvn kH 1 + C.
Let n → ∞ and then → 0, we obtain Φ∞ (zn1 ) → 0 as n → ∞. This and
(3.19) imply as above that kzn1 kE → 0 as n → ∞, a contradiction, hence we
have (3.18).
Let {x1n } be the center of a hypercube Qj in which
dn = ku1n kLp+1 (Qj ) + kvn1 kLq+1 (Qj ) .
Now we show that
|x1n | → ∞ as n → ∞.
(3.20)
{x1n }
were bounded, by passing to a subsequence if necessary we should
If
find that x1n would be in the same Qj and so they should coincide. Therefore
in that Qj , for every n > no , no fixed and large enough, we should have
Z
Z
(∇ū1n ∇v̄n1 + ū1n v̄n1 ) dx −
(F̄ (ū1n ) + Ḡ(v̄n1 )) dx + o(1)
Φ∞ |E(Qj ) (z̄n1 ) =
Qj
Z
Qj
F̄ (ū1n ) dx + (β − 1)
Z
Ḡ(v̄n1 ) dx + o(1)
≥
(α − 1)
≥
1 β
C(kū1n kα
Lα (Qj ) + kv̄n kLβ (Qj ) ) + o(1)
≥
1 β
C(kū1n kα
Lp+1 (Qj ) + kv̄n kLq+1 (Qj ) ) + o(1),
RN
RN
Jianfu Yang
and
351
0
Φ∞ (z̄n1 ) → 0 as n → 0,
where
z̄n1 (x)
=
zn1 (x)
0
z ∈ Qj
x ∈ RN \Qj .
Hence z̄n1 should converge strongly in E(Qj ) to a nonzero function, contradicting
to zn1 → 0 weakly in E, so we have (3.20). Let
zn1 (· + x1n ) → z 1
weakly
in E.
Denote by Q̄ the unit hypercube centered at the origin, we have kzn1 kE(Q̄) ≥
β > 0, thus z 1 6≡ 0 and
0
hΦ∞ (z 1 ), ηi = 0,
∀η ∈ E.
(3.21)
Iterating the procedure, we obtain sequences xln , |xln | → ∞ and
znl (x) = znl−1 (x + xm ) − z l−1 (x), j ≥ 2
znl (x + xln ) → z l (x) weakly in E
as n → 0, where each z l satisfying (3.21) and by induction
kznl kE = kznl−1 k2E − kz l−1 k2E = kzn k2E − kzk2E −
l−1
X
kz i k2E + o(1).
i=1
Φ∞ (znl ) = Φ∞ (znl−1 ) − Φ∞ (z l−1 ) + o(1) = Φ(zn ) − Φ(z) −
l−1
X
Φ(z i ) + o(1).
i=1
Since z l is a solution of (1.3)-(1.4) and z l 6≡ 0, we may prove as Lemma 4.1
below that kz l kE ≥ C > 0. Thus the iteration will terminate at some index
k ≥ 0. The assertion follows.
4
Uniform bounds and proof of Theorem 1.1
We shall bound critical values defined in (2.9) by the energy of the ground state
of problem (1.3)-(1.4). By a ground state of problem (1.3)-(1.4) we mean a
minimizer of the variational problem
Φ∞ = inf{Φ∞ (u, v) : (u, v) ∈ E is a solution of (1.3)-(1.4), (u, v) 6≡ (0, 0)}.
(4.1)
It is shown in [13] that problem (1.3)-(1.4) has a positive radially decaying
solution, so the variational problem (4.1) is well defined.
Lemma 4.1 Variational problem (4.1) is assumed by a nontrivial solution of
(1.3)-(1.4).
Nontrivial solutions of semilinear elliptic systems
352
Proof. Let zn = (un , vn ) be a minimizing sequence of Φ∞ . It is obvious that
{zn } is a (PS) sequence of Φ∞ . We deduce by Proposition 3.1 that
Φ∞ = Φ(zn ) + o(1) =
k
X
Φ∞ (zj ) + o(1),
j=1
where zj is a solution of (1.3)-(1.4). By the definition of Φ∞ , k = 1. The
proof will be completed if we show z1 6= 0. To this end, we bound solutions of
(1.3)-(1.4) in H 1 norm below by a positive constant.
Suppose z = (u, v) is a solution of (1.3)-(1.4), we have
Z
Z
2
2
uḡ(v) dx, kvkH 1 =
v f¯(u) dx,
(4.2)
kukH 1 =
RN
and
RN
Z
Z
(∇u∇v + uv) dx =
RN
Z
¯ dx.
uf(u)
vḡ(v) dx =
RN
(4.3)
RN
By assumptions (H2), (H3) and (H5), we obtain
N +2
f¯(u) ≤ C |u| N −2 + u,
N +2
ḡ(v) ≤ C |v| N −2 + v.
(4.4)
We deduce by (4.2)-(4.4) and Hölder’s inequality that
∗
2 −1
∗
kuk2H 1 ≤ C kvkL
2∗ kukL2 + kukL2 kvkL2 ,
where 2∗ =
2N
N −2 .
Using Young’s inequality and Sobolev embedding, we obtain
∗
∗
∗
∗
kuk2H 1 ≤ C (kuk2H 1 + kvk2H 1 ) + kvk2H 1 .
Similarly,
kvk2H 1 ≤ C (kuk2H 1 + kvk2H 1 ) + kuk2H 1 .
So for small, we have
∗
∗
kuk2H 1 + kvk2H 1 ≤ C(kuk2H 1 + kvk2H 1 ).
It yields that kukH 1 or kvkH 1 ≥ C > 0, uniformly for solutions of (1.3)-(1.4),
and where C > 0 is independent of z = (u, v). Consequently, z1 = (u1 , v1 ) 6≡ 0.
Let Rn → ∞, Bn = BRn (0). Taking Ω = Bn in problem (2.1)-(2.2), we infer
from Theorem 2.1 that there exists a solution zn of problem (2.1)- (2.2) defined
on Bn for each n. Moreover,
Φ(zn ) = Ψ(wn ) = cn ≥ α > 0,
(4.5)
where zn = Kwn , Φ = ΦRN and Ψ = ΨRN . We have extended zn to RN by
letting zn = 0 outside Bn .
Proposition 4.1 cn < Φ∞ for n large.
Jianfu Yang
353
Proof. Since each element w in Xn∗ = L1+1/p (Bn ) × L1+1/q (Bn ) can be extended to an element of X ∗ by letting w = 0 outside Bn , we shall denote ΨBn
as Ψ in brief. By Lemma 4.1, Φ∞ is assumed. Let zo = (uo , vo ) be a minimizer
of Φ∞ . Choosing
w1o = f¯(uo ), w2o = ḡ(vo )
R
and using (H4)-(H5) and equations (1.3)-(1.4), one has RN < wo , Kwo > dx >
0, where wo = (w1o , w2o ). Moreover, we know as Lemma 2.5 that there are
t1 , t2 > 0 such that
maxt≥0 Ψ(two ) = maxt1 ≤t≤t2 Ψ(two ).
Suppose that to ∈ (t1 , t2 ) and
Ψ(to wo ) = maxt1 ≤t≤t2 Ψ(two ).
Because F (x, t) ≥ F̄ (t) and G(x, t) ≥ Ḡ(t), one has F ∗ (x, s) ≤ F̄ ∗ (s) and
G∗ (x, s) ≤ Ḡ(s). By the assumption (H6),
Ψ(to wo ) < Ψ∞ (to wo ),
it follows
sup Ψ(two ) < sup Ψ∞ (two ).
t≥0
t≥0
(4.6)
The density of real number field implies that there exists > 0 such that
sup Ψ(two ) + 2 < sup Ψ∞ (two ).
t≥0
t≥0
(4.7)
Let φ ∈ Co∞ (RN ), 0 ≤ φ ≤ 1 and φ ≡ 1 if |x| ≤ 12 ; φ ≡ 0 if |x| > 1; φn (x) =
φ( Rxn ). Then zn := (φn uo , φn vo ) converges to (uo , vo ) in E. Let
w1n = f¯(φn uo ),
w2n = ḡ(φn vo ).
We also have wn → wo in X ∗ . Suppose
Ψ(tn wn ) = sup Ψ(twn ),
t≥0
then {tn } is bounded. Indeed, if tn → ∞, arguments in Lemma 2.5 would yield
supt≥0 Ψ(twn ) → −∞. It is not possible because the value is not negative.
Suppose tn → t̄o , the continuity of the functional Ψ gives
Ψ(tn wn ) → Ψ(t̄o wo ).
We claim that Ψ(t̄o wo ) = supt≥0 Ψ(two ). In fact, for every > 0 there exists
δ > 0 such that
Ψ(to wo ) − ≤ Ψ(two )
whenever |t − to | < δ. By the continuity of Ψ, we may find no > 0 such that if
n ≥ no
Ψ(two ) ≤ Ψ(twn ) + , Ψ(tn wn ) ≤ Ψ(t̄o wo ) + .
Nontrivial solutions of semilinear elliptic systems
354
Therefore if n ≥ no we have
Ψ(to wo ) − ≤ Ψ(tn wn ) + ≤ Ψ(t̄o wo ) + 2 ≤ Ψ(to wo ) + 2.
Since is arbitrary, the conclusion holds. By the same arguments, we find that
there exists sn such that sn → s̄o and
Ψ∞ (sn wn ) = sup Ψ∞ (twn ) → Ψ∞ (s̄o wo ) = sup Ψ∞ (two )
t≥0
t≥0
as n → ∞. By (4.7), we obtain Ψ(tn wn ) + < Ψ∞ (sn wn ) for n large enough.
We may assume sn > 0, and then
dΨ∞ (twn )
|t=sn = 0,
dt
that is
Z
Z
0
0
[F̄s∗ (sn w1n )w1n + Ḡ∗s (sn w2n )w2n ] dx − sn
RN
RN
< wn , Kwn > dx = 0. (4.8)
By the definition of Legendre - Fenchel transformation, we obtain
Z
[F̄ ∗ (sn w1n ) + Ḡ∗ (sn w2n )] dx
RN
Z
0
0
[F̄s∗ (sn w1n )sn w1n + Ḡ∗s (sn w2n )sn w2n ] dx
=
RN
Z
[F̄ (f¯−1 (sn w1n )) + Ḡ(ḡ −1 (sn w2n ))] dx
(4.9)
−
RN
Z
Z
< wn , Kwn > dx −
[F̄ (f¯−1 (sn w1n )) + Ḡ(ḡ −1 (sn w2n ))] dx.
= s2n
RN
RN
Consider
(−∆ + id)−1 w2n = uo + σn ,
(−∆ + id)−1 w1n = vo + µn
in RN ,
we obtain
(−∆ + id)σn = ḡ(φn vo ) − ḡ(vo ),
(−∆ + id)µn = f¯(φn uo ) − f¯(uo ).
In terms of Lp −estimates, σn → 0 and µn → 0 in H 2,2 as n → ∞. Furthermore,
we infer from (4.8) that
Z
f¯−1 (sn w1n ) f¯−1 (w1n )
sn (w1n )2 [
−
] dx
sn w1n
w1n
RN
Z
ḡ −1 (sn w2n ) ḡ −1 (w2n )
sn (w2n )2 [
−
] dx
+
sn w2n
w2n
RN
Z
=
[w1n σn + w2n µn + (1 − φn )(w1n + w2n )] dx = o(1)
RN
Jianfu Yang
355
as n → ∞. The equality and assumption (H7) imply sn → 1 as n → ∞. hence
we deduce by (4.8) and (4.9) that
sup Ψ∞ (twn )
t≥0
Z
Z
1
(uo f¯(uo ) + vo ḡ(vo )) dx −
(F̄ (uo ) + Ḡ(vo )) dx + n
≤
2 RN
RN
= Ψ ∞ + n ,
where
n
=
Z
1 2
¯ o ) + vo ḡ(vo )) dx
(sn − 1)
(uo f(u
2
RN
Z
[(F̄ (φn uo ) − F̄ (uo )) + (Ḡ(φn vo ) − Ḡ(vo ))] dx
−
N
ZR
[(F̄ (φn uo ) − F̄ (f¯−1 (sn w1n )) + (Ḡ(φn vo ) − Ḡ(ḡ −1 (sn w2n ))] dx.
+
RN
The above estimates imply n = o(1) as n → ∞. Therefore
sup Ψ(twn ) < sup Ψ(twn )∞ − ≤ Ψ∞ − + o(1),
t≥0
t≥0
the assertion follows for n large.
Lemma 4.2 zn is a (PS) sequence of Φ in E.
Proof. It is readily to verify that cn = Φ(zn ) ≤ cn−1 = Φ(zn−1 ), thus by
Proposition 4.2
α ≤ c n ≤ c1 < Φ∞ ,
(4.10)
we obtain
cn = Φ(zn ) → c,
α ≤ c ≤ c1 < Φ∞ .
(4.11)
as n → ∞.
(4.12)
Now we show that
Φ0 (zn ) → 0,
In fact, ∀(φ, ψ) ∈ Co∞ (RN )×Co∞ (RN ), there is no > 0 such that suppφ, suppψ ⊂
Bn whenever n ≥ no and
Φ0 (zn )(φ, ψ) = 0,
if
n ≥ no .
This implies that Φ0 (zn )z → 0 as n → ∞ for all z ∈ Co∞ (RN ) × Co∞ (RN ). Hence
(4.12) follows because Co∞ (RN ) × Co∞ (RN ) is dense in H 1 (RN ) × H 1 (RN ). 356
Nontrivial solutions of semilinear elliptic systems
Completion of the proof of Theorem 1.1 We may prove that the (PS)
sequence zn of Φ is bounded in E as [13], and assume zn → zo weakly in E.
Obviously, zo solves (1.1)-(1.2). We claim that zo is nontrivial. In fact, Lemma
2.4, Proposition 3.1 and Proposition 4.2 give that
X
α ≤ Φ(zn ) = Φ(zo ) +
Φ∞ (zj ) + o(1) < Φ∞ .
j
If j = 0, Φ(zo ) ≥ α > 0, zo is a nontrivial solution; if j ≥ 1, then Φ(zo ) < 0, also
implying zo 6≡ 0. The decaying law of zo at infinity was proved in [13].
ACKNOWLEDGEMENTS
The work was partially supported by 21CSPJ, NSFJ and NSFC in China.
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Jianfu Yang
Department of Mathematics, IMECC-Unicamp
Caixa Postal 6065
Campinas 13081-970, SP, Brazil
e-mail: jfyang@ime.unicamp.br
and
Department of Mathematics, Nanchang University
Nanchang, Jiangxi 330047
P. R. of China