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Quasilinear elliptic systems with critical Sobolev exponents in RN
Nonlinear Analysis 66 (2007) 1485–1497 www.elsevier.com/locate/na
Quasilinear elliptic systems with critical Sobolev exponents in R N Ali Djellit ∗ , Saˆadia Tas University of Annaba, Faculty of Sciences, Department of Mathematics, BP 12, 23000, Annaba, Algeria Received 21 July 2005; accepted 2 February 2006
Abstract We study here a class of quasilinear elliptic systems involving the p-Laplacian operator; the right hand sides of systems are closely related to the critical Sobolev exponents. Under some additional assumptions on the nonlinearities, the corresponding functional verifies the Palais–Smale condition (P S)c for c belonging to a specified range. So, we can use the Mountain Pass Theorem to prove the existence of at least one nontrivial solution. c 2006 Elsevier Ltd. All rights reserved. MSC: 35P65; 35P30 Keywords: p-Laplacian operator; Critical Sobolev exponent; Palais–Smale condition; Mountain Pass Theorem; Concentration–compactness principle
1. Introduction In the present paper we investigate the existence of nontrivial weak solutions of the vector valued quasilinear elliptic system ⎧ ∂F ∗ ⎪ − p u = f (x)|u| p −2 u + λ (x, u, v) in R N ⎪ ⎪ ⎨ ∂u (Sc ) ∂F ∗ ⎪ (x, u, v) in R N −q v = g(x)|v|q −2 v + λ ⎪ ⎪ ∂v ⎩ u, v|x| → +∞ → 0 for all λ belonging to ]0, λ1 [. Of course p is the so-called p-Laplacian operator, i.e. p u = Np div(|∇u| p−2 ∇u), 1 < p, q < N. The real number p∗ = N− p designates the critical Sobolev ∗ Corresponding author. Fax: +213 08 87 25 26.
E-mail address: a [email protected] (A. Djellit). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.02.005
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exponent of p and f, g, F are real valued functions satisfying some assumptions. The unknown real valued functions u and v stay in appropriate spaces. Critical problems appear in many areas including differential geometry, quantum field theory and optimization problems related to Sobolev and isoperimetric inequalities (see for example [12]). In recent years there has been a marked increase of research interest in critical problems and a number of excellent works have been published. Finer details are elucidated in ∗ Brezis and Nirenberg [4]. The authors studied the semilinear equation −u = u 2 −1 + λu in a smooth bounded domain Ω ⊂ R N with u = 0 on ∂Ω . They proved the existence of positive solutions for N ≥ 3 and 0 < λ < λ1 . We refer to λ1 as the first eigenvalue of the Laplacian. It is well known that λ1 is simple and the associated eigenfunction u has one sign. In a natural way, the study has been extended by [2] to nonlinear elliptic systems. In his review article, Miyagaki [13] examines the equation −u + a(x)u = λ|u|q−1 u + |u| p−1 u in R N with 1 < q < p ≤ 2∗ − 1 and a(x) ≥ a0 > 0. The author obtains comparable results. Unlike this, Noussair et al. ∗ [15] inspected in detail the quasilinear problem of type −m u = p(x)u m −1 + q(x)u γ in a connected open set Ω of R N with m − 1 < γ < m ∗ . They proved also the existence of positive solutions. An additional work due to Swanson and Yu [16] deals with an equation of the form ∗ −m u = λa(x)u m−1 + f (x)u m −1 + g(x)u q in R N and u → 0 when |x| → +∞. The authors establish existence results. A delightful result was described by Drabek and Huang [7] ∗ concerning the problem − p u = λg(x) |u| p−2 u + f (x)|u| p −2 u in R N ; λ ∈]0, λ1 [. Here, the real number λ1 is the first principaleigenvalue of the problem − p u = λg(x)|u| p−2u in R N , under the following condition R N g (x) |u| p dx > 0 (see [1,5]). We will keep in mind a multiplicity result obtained by Noussair and Swanson [14] for some critical potential systems involving the Laplacian operator. We also mention existence and bifurcation results established for quasilinear elliptic systems in [17,18] and the comments of [20] related to external domain. This contribution comes within the continuation of results described in [8,9]. In this work, following the same ideas, we deal with the existence of nontrivial solutions of System (Sc ) for all λ ∈ ]0, λ1 [. However the real number λ1 is the first eigenvalue of the system ⎧ ∂F ⎪ − p u = λ (x, u, v) in R N ⎪ ⎪ ⎨ ∂u (Svp ) ∂F N ⎪ ⎪ ⎪−q v = λ ∂v (x, u, v) in R ⎩ u, v |x| → +∞ → 0, u > 0, v > 0. Assume that the nonlinearity F satisfies mixed and subcritical growth conditions, then the solutions are obtained invoking the Mountain Pass Theorem [10]. Because of the loss of compactness in Sobolev imbeddings, the Palais–Smale condition is only satisfied when the energy level of the corresponding functional remains below a certain critical level linked to the best Sobolev constants. In what follows the concentration–compactness principle of P.L. Lions [11] plays an important role in the study of the solvability of (Sc ). This article has the following structure: in Section 2, we introduce some notation and hypotheses; Section 3 is devoted to the proof of the main results. 2. Notation and hypotheses We denote by D 1,m (R N ) the completion of C0∞ (R N ) in the norm u 1,m
1 m N m |∇u| dx ≡ ∇u m = R ;
1 < m < N.
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It is well known that D 1,m (R N ) is a uniformly convex Banach space and may be written as
N ∗ D 1,m (R N ) = u ∈ L m (R N ) : ∇u ∈ L m (R N ) . Moreover, we have the following Sobolev inequality ∃C (N, m) > 0,
∀u ∈ D 1,m (R N ) : u m ∗ ≤ C (N, m) u 1,m .
Then, we consider Sm the best Sobolev constant defined by
u m 1,m 1,m N {0} Sm ≡ C −m (N, m) = inf , u ∈ D (R ) \ . u m m∗ We denote by Z the product space D 1, p (R N ) × D 1,q (R N ); Z ∗ is the dual space of Z equipped with the dual norm · ∗ and Iλ , J, K , L are functionals defined for all (u, v) ∈ Z by 1 1 p q u 1, p + v 1,q p q K (u, v) = R N F (x, u (x) , v (x)) dx 1 N 1 N ∗ p∗ L (u, v) = ∗ R f (x)|u| dx + ∗ R g(x)|u|q dx p q Iλ (u, v) = J (u, v) − λK (u, v) − L(u, v), ∀λ > 0. J (u, v) =
In order to show the existence result, we suppose that (H1) f and g are positive and bounded functions. (H2) F ∈ C 1 R N , R, R and F (x, 0, 0) = 0. (H3) For all U = (u, v) ∈ R2 and for almost every x ∈ R N ∂F ≤ a1 (x) |U | p1 −1 + a2 (x) |U | p2 −1 U (x, ) ∂u ∂F ≤ b1 (x) |U |q1 −1 + b2 (x) |U |q2 −1 U (x, ) ∂v where 1 < p1 , q1 < min ( p, q) , max ( p, q) < p2 , q2 < min p∗ , q ∗ ai ∈ L αi (R N ) ∩ L βi (R N ), bi ∈ L γi (R N ) ∩ L δi (R N ), i = 1, 2. p∗ q∗ p∗ q ∗ αi = ∗ , γi = ∗ , βi = ∗ ∗ , p − pi q − qi p q − p ∗ ( pi − 1) − q ∗ p∗ q ∗ δi = ∗ ∗ . p q − q ∗ (qi − 1) − p∗ (H4) ∃θ p , θq with
1 p∗
< θp <
1 p
,
1 q∗
< θq <
1 q
such that
∂F ∂F (x, u, v)u + θq (x, u, v)v ∂u ∂v 1 ∂F 1 ∂F (x, u, v)u + (x, u, v)v. ≤ p ∂u q ∂v p−N q−N 1 Np 1 Nq p ∗ S p f ∞ , Sq g ∞q Set c = min . N N 0 < F (x, u, v) ≤ θ p
If c∗ =
N
1 N
p−N
S pp f ∞p , then we assume that
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(H5) there exists x 0 ∈ R N such that f ∞ = x ∈ R N sup f (x) = f (x 0 ), (H6) the function f is such that f (x) − f (x 0 ) = O (|x − x 0 |) in a neighborhood of x 0 , (H7) there exist two positive constants α and c, p < α < p∗ , such that F (x, u, v) ≥ c |u|α . q−N
N
If c∗ = N1 Sqq g ∞q , we replace f, p, p∗ , c |u|α by g, q, q ∗ , c|v|α in (H5), (H6) and (H7) respectively. Notice that under assumptions (H1), (H2) and (H3), the functional Iλ is well defined and is of class C 1 . Moreover for all (u, v) , (w, z) ∈ Z |∇u| p−2 ∇u · ∇wdx + |∇v|q−2 ∇v · ∇zdx Iλ (u, v) (w, z) = N N R R ∂F ∂F (x, u, v)w + (x, u, v)z dx −λ N ∂u ∂v R ∗ ∗ − f (x)|u| p −2 uwdx − g(x)|v|q −2 vzdx. RN
RN
Hence, weak solutions of System (Sc ) are exactly critical points of the functional Iλ . We recall that Iλ satisfies (P S)c if each sequence (u n , vn ) of Z verifying Iλ (u n , vn ) → c and Iλ (u n , vn ) → 0 in Z ∗ possesses a convergent subsequence in Z . We recall also the following concentration–compactness principle of Lions (see [12, p. 158]). ∗ p p 1, p N Let (u n ) converge weakly to u in D (R ) such that |∇u n | , |u n | converge weakly to nonnegative bounded measures μ and ν respectively. Then, for some at most countable set E, we have ∗ (i) ν = |u| p + j ∈E ν j δx j , (ii) μ ≥ |∇u| p + j ∈E μ j δx j , p ∗
(iii) μ j ≥ S p υ jp , ∀ j ∈ E. Here δx j is the Dirac measure at x j , x j ∈ R N ; μ j , υ j ∈ ]0, +∞[. A crucial aspect in the theory of critical p-Laplacian problems consists in the fact that −1
C p = S p p , the best constant of Sobolev imbedding, is attained by the radial function u ε defined by ⎛ u ε (x) = M ⎝ Observe that M=
N
ε ε
p p−1
N−p p−1
⎞ N− p
1 p−1
+ |x − x 0 |
p
p p−1
⎠
for any ε > 0, x 0 , x ∈ R N .
N− p p−1 N p−2N+2 p
,
is chosen such that u ε solves the scalar critical equation − p u ε = u εp
∗ −1
in R N , (see [12]).
In the same way, we introduce the radial function vε corresponding to the constant Sq .
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3. Existence of solutions We need some lemmas to prove the main theorem. Lemma 1. Assume (H2) and (H3) hold. The functional K is weakly lower semicontinuous in Z and the operator K is compact from Z to Z ∗ . Proof. Let (u n , vn ) be a bounded sequence in Z so there exists a subsequence, denoted again as (u n , vn ), weakly convergent to (u, v) in Z . Let B R be the ball of R N centered at the origin with radius R. We denote by B R the complement of B R in R N and by K R the functional defined in D 1, p (B R ) × D 1,q (B R ) expressed by K R (u, v) = B R F (x, u (x) , v (x)) dx. It is well known that K R is weakly lower semicontinuous and K R is also compact (see for example [19]). Set K (u n , vn ) = K R (u n , vn ) + F(x, u n , vn ) dx. BR
Using both (H3) and properties of the weights ai and bi , we have for R sufficiently large F(x, u n , vn ) dx → 0 as n → +∞. BR
Hence, K is weakly lower semicontinuous. A similar argument yields the compactness of K . Lemma 2. Under hypotheses (H2)–(H4), the functional J is bounded from below on the set Λ = {(u, v) ∈ Z , K (u, v) = 1} and achieves its minimum λ1 . Proof. Set λ1 = (u, v) ∈ Λ inf J (u, v). It is clear that J is weakly lower semicontinuous and coercive in Z , then in Λ. The set Λ is weakly closed since K is weakly lower semicontinuous. On the other hand, K (u, v) = 0 means that ∂F ∂F (x, u, v)w + (x, u, v)z dx = 0, ∀ (w, z) ∈ Z . ∂u ∂v RN Substituting w for θ p u and z for θq v respectively, and taking into account (H4), we obtain K (u, v) = 0. According to Theorem 6.3.2 in Berger [3], the statement is proved. Lemma 3. Under hypotheses (H2)–(H4), for all λ ∈ ]0, λ1 [, ∃σ > 0 such that p q J (u, v) − λK (u, v) ≥ σ u 1, p + v 1,q , ∀ (u, v) ∈ Z . Proof. Suppose that there exists λ ∈]0, λ1 [ and a sequence (u n , vn ) of Z such that 1 p q u n 1, p + vn 1,q . J (u n , vn ) − λK (u n , vn ) < n Using the variational characterization of λ1 , i.e. J (u, v) ≥ λ1 K (u, v) , ∀ (u, v) ∈ Z . p q Consequently 1 − λλ1 J (u n , vn ) < n1 u n 1, p + vn 1,q .
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1 1 − − p q We set γn = max u n 1, p , vn 1,q , u n = γnp u n and vn = γnq v n . p − q − − − So, we obtain 1 − λλ1 J u n , v n < n1 u n + v n . 1,q 1, p − − − − Since u n , v n is bounded in Z , there exists a subsequence denoted again as u n , v n , − − − − weakly convergent to u, v . Apply the limiting process n → +∞, we get 1 − λλ1 J u, v ≤ 0, i.e. λ1 ≤ λ. This yields a contradiction. Lemma 4. Under hypotheses (H1)–(H4) and for all λ ∈ ]0, λ1 [, the functional Iλ satisfies the (P S)c condition for all c ∈ ]0, c∗ [. Proof. Let (u n , vn ) be a sequence of Z such that Iλ (u n , vn ) → c and Iλ (u n , vn ) → 0 in Z ∗ . We have 1 1 1 ∗ p q f (x) |u n | p dx Iλ (u n , vn ) = u n 1, p + vn 1,q − ∗ p q p RN 1 ∗ q − ∗ g(x) |u n | dx − λ F (x, u n , vn ) dx q RN RN = c + o (1) , as n is sufficiently large. On the other hand
|∇u n | p−2 ∇u n · ∇wdx + |∇vn |q−2 ∇vn · ∇zdx RN RN ∂F ∂F (x, u n , vn )w + (x, u n , vn )z dx −λ N ∂u ∂v R ∗ ∗ p −2 − f (x) |u n | u n wdx − g(x) |vn |q −2 vn z dx N N R R p q = o (1) u n 1, p + vn 1,q , as n is sufficiently large.
Iλ (u n , vn ) (w, z) =
Hence
1 1 p q − θ p u n 1, p + − θq vn 1,q Iλ (u n , vn ) − Iλ (u n , vn ) θ p u n , θq vn = p q 1 1 ∗ ∗ + θp − ∗ f (x) |u n | p dx + θq − ∗ g(x) |u n |q dx p q RN RN ∂F ∂F (x, u n , vn ) + θq vn (x, u n , vn ) − F (x, u n , vn ) dx. +λ θ p un ∂u ∂v RN
According to (H4), we get 1 1 p q p q − θ p u n 1, p + − θq vn 1,q . c + o (1) + o (1) u n 1, p + vn 1,q ≥ p q Hence the sequence (u n , vn ) is bounded in Z ; so there exists a subsequence, denoted again as (u n , vn ), weakly convergent to (u, v) in Z . Then, there exist positive and bounded measures μ, μ, ¯ ν and ν¯ defined in R N such that ∗
∗
|∇u n | p μ and |u n | p ν(respectively |∇vn |q μ¯ and |vn |q ν¯ ).
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Applying the concentration–compactness principle, there exist x j ⊂ R N ; μ j , ν j ⊂ ]0, +∞[ such that (i), (ii) and (iii) hold. First, we show that the set E is finite (respectively E¯ corresponding to (vn )). For j ∈ E, we define φ j ∈ C0∞ R N as follows 1 if x ∈ B x j , ε φ j (x) = and ∇φ j ≤ C. N 0 if x ∈ R \ B x j , 2ε Substitute w for φ j u n and z for 0 in Iλ (u n , vn ) (w, z). Using the fact that Iλ (u n , vn ) → 0 in Z ∗ , we obtain |∇u n | p−2 u n ∇u n · ∇φ j dx lim n→+∞ R N |∇u n | p φ j dx = lim − n→+∞ N R ∂F ∗ + (x, u n , vn )φ j u n dx . f (x) |u n | p φ j dx + λ (1.1) RN R N ∂u Observe that u n u implies R N ∂∂uF (x, u n , vn )φ j u n dx → R N ∂∂uF (x, u, v)φ j udx. According to Lions’ principle, we have p |∇u n | φ j dx → φ j dμ and RN RN ∗ f (x) |u n | p φ j dx → f (x)φ j dν. RN
RN
Consequently p−2 |∇u n | lim u n ∇u n .∇φ j dx = f (x)φ j dμ n→+∞ R N RN ∂F +λ (x, u, v)φ j udx − φ j dμ. ¯ N R ∂u RN H¨older’s inequality gives p−2 |∇u n | 0 ≤ lim u n ∇u n · ∇φ j dx n→+∞ R N 1 p−1 ≤ C lim
n→+∞
RN
|∇u n | p dx
n→+∞
B(x j ,2ε)
|u n | p dx
1
≤ C lim
p
p
p
B(x j ,2ε)
|u n | p dx
.
1, p B(x , 2ε) , there exists a subsequence denoted as (u ) such that in D Since (u n ) is bounded j n u n → u in L p B(x j , 2ε) . So |∇u n | p−2 u n ∇u n · ∇φ j dx 0 ≤ lim n→+∞ R N 1 p
≤C
B(x j ,2ε)
|u| p dx
→ 0.
ε→0
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Letting ε tend to 0 in (*), we obtain f (x j )υ j = μ j . p ∗
p ∗
But μ j ≥ S p υ jp , thus f (x j )υ j ≥ S p υ jp , i.e. υ j ≥ Consequently N p Sp p∗ δx j υ ≥ |u| + f (x j ) j ∈E and hence RN
|u n |
p∗
dx →
RN
dυ ≥ |u|
p∗
+
Sp f (x j )
Sp f (x j )
N p
.
N p
Card E.
∗ If Card E = ∞ then R N |u n | p dx → +∞. A contradiction. n→+∞ −
Now, we prove that E (respectively E ) is empty. We have 1 u n vn 1 ∗ , − ∗ Iλ (u n , vn ) − Iλ (u n , vn ) f (x) |u n | p dx = N p q p p R 1 1 1 ∂F q∗ + − ∗ un (x, u n , vn ) g(x) |u n | dx + λ N q q p ∂u RN R 1 ∂F (x, u n , vn ) − F (x, u n , vn ) dx. + vn q ∂v Hypothesis (H4) gives Iλ (u n , vn ) − Iλ (u n , vn )
u n vn , p q
≥
1 ∗ f (x) |u n | p dx N RN 1 ∗ + g(x) |u n |q dx. N RN
When n → +∞, we obtain 1 1 1 ∗ p∗ c≥ f (x) |u| dx + f (x j )υ j + g(x) |u|q dx N RN N j ∈E N RN 1 − + g(x j )υ j . N − j∈ E
− f (x j )υ j + N1 − g(x j )υ j . j ∈E N N − p q S S and in a similar manner υ j ≥ g(xqj ) . But υ j ≥ f (xpj )
So c ≥
1 N
j ∈E
−
Suppose that E ∪ E = ∅ and thus − 1− N 1− N 1 Nq 1 N p Card E + q Card Sq g(x j ) c ≥ S pp f (x j ) E. N N p−N q−N N 1 1 Np c ≥ min S p f ∞p , Sqq g ∞q . N N
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−
We get a contradiction, hence E ∪ E = ∅, which means that u n p∗ → u p∗
and
vn q ∗ → v q ∗ .
∗ Taking this together with the fact that (u n , vn ) (u, v) in Z , we have u n → u in L p R N and ∗ vn → v in L q R N . On the other hand J (u n , vn ) − J (u m , vm ) (u n − u m , 0) = Iλ (u n , vn ) − Iλ (u m , vm ) (u n − u m , 0) + λ K (u n , vn ) − K (u m , vm ) (u n − u m , 0) + L (u n , vn ) − L (u m , vm ) (u n − u m , 0) . Iλ (u n , vn ) → 0, i.e. Iλ (u n , vn ) is a Cauchy sequence in Z ∗ . The compactness of K gives (u n , vn ) (u, v) ⇒ K (u n , vn ) → K (u, v) , i.e. K (u n , vn ) is a Cauchy sequence in Z ∗ . Moreover, using H¨older’s inequality L (u n , vn ) − L (u m , vm ) (u n − u m , 0) ∗ ∗ = f (x) |u n | p −2 u n − |u m | p −2 u m (u n − u m ) dx RN ∗ ∗ u n − u m L p∗ (R N ) . ≤ f ∞ |u n | p −2 u n − |u m | p −2 u m p∗ L p∗ −1 (R N ) ∗ Since (u n ) is a Cauchy sequence in L p R N , L (u n , vn ) is a Cauchy sequence in Z ∗ . We note according to [6] that ∀λ, μ ∈ R N ⎧ ⎨ |λ| p−2 λ − |μ| p−2 μ . (λ − μ) if p ≥ 2 p |λ − μ| p ≤ ⎩ |λ| p−2 λ − |μ| p−2 μ . (λ − μ) 2 (|λ| + |μ|) (2−2p) p if 1 < p < 2. Replacing λ and μ by ∇u n , ∇u m respectively and integrating over R N , we obtain p u n − u m 1, p ≤ J (u n , vn ) − J (u m , vm ) (u n − u m , 0) if p ≥ 2 2− p 2 p p u n − u m 21, p ≤ J (u n , vn ) − J (u m , vm ) (u n − u m , 0) u n 1, p + u m 1, p if 1 < p < 2. Taking into account the fact that (u n ) is bounded in D 1, p (R N ) and J (u n , vn ) − J (u m , vm ) (u n − u m , 0) → 0, n,m→+∞
we find that (u n ) is a Cauchy sequence in D 1, p (R N ) and so converges in D 1, p (R N ). We proceed similarly for (vn ) with J (u n , vn ) − J (u m , vm ) (0, vn − vm ). Lemma 5. Under the hypotheses (H1)–(H3), ∃ρ, ω > 0 such that ∀λ ∈]0, λ1 [: p
q
u 1, p + v 1,q = ρ ⇒ Iλ (u, v) ≥ ω.
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Proof. We have ∗ p∗ p∗ p∗ f (x)|u| p dx ≤ f ∞ u p∗ ≤ C p f ∞ u 1, p . N R ∗ q∗ q∗ q∗ g(x)|v|q dx ≤ f ∞ v q ∗ ≤ Cq g ∞ v 1,q . RN
Taking into account Lemma 3, we get p q p∗ q∗ Iλ (u, v) ≥ σ u 1, p + v 1,q − C u 1, p + v 1,q ∗ p q∗ with C = max C p f ∞ , Cq g ∞ . Thus for some ρ > 0 sufficiently small, we have Iλ (u, v) ≥ ω > 0.
c∗ ,
We have seen that Iλ satisfies the (P S)c condition for all 0 < c < in order to apply the Mountain Pass Theorem, we will estimate Iλ on a critical path, i.e. finding a critical point with an energy level in the compactness interval requires the construction of a special path ξ (t) joining the origin to a point e of Z with Iλ (e) < 0
and such that 0 < Sup Iλ (ξ (t)) < c∗ . t ≥0
N p
p−N p
Suppose that c∗ = N1 S p f ∞ (in the other case, we proceed similarly taking the radial function vε ). Let R > 0, φ R ∈ C0∞ (R N ) verifying φ R ≡ 1 in B(x 0 , R) and φ R ≡ 0 in R N \ B(x 0 , 2R). Put Uε = φ R u ε and wε = Uε p∗ . Uε Then, we have the following lemma. Lemma 6. Under hypotheses (H1)–(H6), there exist ε > 0 and t0 > 0 such that ∀λ ∈ ]0, λ1 [, Iλ (t0 wε , 0) < 0 and 0 < Supt ≥0 Iλ (twε , 0) < c∗ . Proof. We have
∗ tp tp p p∗ wε 1, p − ∗ Iλ (twε , 0) = f (x) |wε | dx − λ F (x, twε , 0) dx. p p RN RN Since F (x, u, v) ≥ 0, it is clear that ∗ tp tp ∗ p wε 1, p − ∗ Iλ (twε , 0) ≤ f (x) |wε | p dx. p p RN But p < p∗ and f is positive, so Iλ (twε , 0) → −∞. t →+∞ Thus, there exists t0 sufficiently large for which Iλ (t0 wε , 0) < 0. By virtue of Lemma 5 and for t sufficiently small, we obtain Iλ (twε , 0) ≥ ω > 0.
So Iλ (twε , 0) achieves its supremum at a point tε > 0. Then, we write p p∗ tε tε ∗ p |wε | p dx − λ wε 1, p − ∗ f (x 0 ) Iλ (tε wε , 0) = F (x, tε wε , 0) dx p p RN RN p∗ tε ∗ + ∗ ( f (x 0 ) − f (x)) |wε | p dx. p RN
A. Djellit, S. Tas / Nonlinear Analysis 66 (2007) 1485–1497
1495 ∗
For positive reals A and B, the maximum of the function θ (t) = Ap t p − pB∗ t p for t ≥ 0 is N− p attained at t = BA p2 . ∗ p Putting A ≡ wε 1, p = S p , B ≡ f (x 0 ) R N |wε | p dx = f (x 0 ), we obtain N− p N p p∗ p p Sp Sp tε 1 1 tε p p∗ |wε | dx ≤ wε 1, p − ∗ f (x 0 ) − ∗ f (x 0 ) N p p p f (x 0 ) p f (x 0 ) R p−N 1 N = S pp f ∞p . N On the other hand tε is bounded above, indeed, since Iλ (tε wε , 0) = Supt ≥0 Iλ (twε , 0), we have Iλ (tε wε , 0) = 0, in particular Iλ (tε wε , 0) (tε wε , 0) = 0, i.e. ∂F ∗ ∗ p tεp wε 1, p − tεp f (x) |wε | p dx − λ (x, tε wε , 0) tε wε dx = 0. RN R N ∂u ∗ p∗ p p So tε R N f (x) |wε | p dx ≤ tε wε 1, p or ∗ ∗ ∗ |wε | p dx ≤ S p . tεp − p ( f (x) − f (x 0 )) |wε | p dx + f (x 0 ) RN
RN
Hypothesis (H5) gives N− p tε ≤ cS p p2 . We have also ∗ ( f (x 0 ) − f (x)) |wε | p dx RN
≤
⎛
cR p∗
u ε p∗
⎝ B(x 0,R)
ε
⎞ (N− p) p∗
1 p−1
p
p
p
ε p−1 + |x − x 0 | p−1
⎠
dx.
Making the change of variables r = |x − x 0 | and s = rε , we obtain
∗
RN
( f (x 0 ) − f (x)) |wε | p dx ≤ C
R ε
s
Np N−1− p−1
N
ds ≤ Cε p−1 .
0
Moreover, by virtue of (H6), we get ⎛
RN
F (x, tε wε , 0) dx ≥ C
⎝ B(x 0,R)
ε p
⎞ α(N− p)
1 p−1
p
p
ε p−1 + |x − x 0 | p−1
⎠
Putting again r = |x − x 0 | and s = rε , we find RN
F (x, tε wε , 0) dx ≥ Cε
p) N− α( N− p
R ε
0
s N−1 ds. α(N− p) p p p−1 1+s
dx.
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For ε > 0 sufficiently small such that Rε > 1, p) p) N− α( N− − α(N− p p F (x, tε wε , 0) dx ≥ Cε 2 RN
R ε
s
N− p) N−1− α( p−1
ds.
1
p) γ = N − 1 − α(N− p−1 , we have ⎧ 1 ⎨ if γ = −1 C1 ε N−β ln F (x, tε wε , 0) dx ≥ ε ⎩C ε N−β−γ −1 if γ = −1. RN 2
Defining β =
α(N− p) , p
Hence RN
F (x, tε wε , 0) dx ≥ cεδ
⎧ ⎪ ⎨N − β
αp α− p+1 with δ = αp ⎪ . ⎩ N − β − γ − 1 if N = α− p+1 if N =
Finally Iλ (tε wε , 0) ≤
p−N N 1 Np S p f ∞p − Cεδ + Cε p−1 N
and the lemma follows because δ <
N p−1 .
Now, we state our main result Theorem. Assume (H1)–(H6) are satisfied; the System (Sc ) possesses at least a nontrivial solution. Proof. We define c = Infϕ∈Γ Sup0≤t ≤1 Iλ (ϕ (t)) where Γ denotes the class of all continuous paths ϕ in Z joining the origin of Z to t0 wε . Lemma 6 implies that 0 < c < c∗ and hence Iλ satisfies (P S)c by Lemmas 5 and 6 ensuring the geometric conditions. Consequently, the Mountain Pass Theorem can be applied to conclude that Iλ has a critical point with corresponding critical value c. So, System (Sc ) possesses a nontrivial weak solution. Acknowledgment The authors wish to express their gratitude to ANDRU for providing support through the grant CU39904. References [1] W. Allegretto, Y.X. Huang, Eigenvalues of the indefinite weight p-Laplacian in weighted R N spaces, Funkcial. Ekvac. 38 (1995) 233–242. [2] P. Amster, P. De N´apoli, M.C. Mariani, Existence of solutions for elliptic systems with critical Sobolev exponent, Electron. J. Differen. Equ. (49) (2002) 13. [3] S.M. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, San Francisco, London, 1977. [4] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure. Appl. Math. 36 (1983) 437–477. [5] P. Drabek, Nonlinear eigenvalues problem for the p-Laplacian in R N , Math. Nah. 173 (1995) 131–138. [6] G. Dinca, P. Jebelean, Some existence results for a class of nonlinear equations involving a duality mapping, Nonlinear Anal. 46 (2001) 347–363. [7] P. Drabek, Y.X. Huang, Multiplicity of positive solutions for some quasilinear elliptic equations in R N with critical Sobolev exponent, J. Difference Equ. 140 (1997) 106–132.
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[8] A. Djellit, S. Tas, Existence of solutions for a class of elliptic systems in R N involving the p-Laplacian, EJDE 2003 (56) (2003) 1–8. [9] A. Djellit, S. Tas, On some nonlinear elliptic systems, Nonlinear Anal. 59 (2004) 695–706. [10] O. Kavian, Introduction a` la th´eorie des points critiques et applications aux probl`emes elliptiques, Springer-Verlag, Paris, 1993. [11] P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case, Parts 1 and 2, Ann. Inst. H. Poincar´e: Anal. Non lin´eaire 1 (1984) 109–145, 223–283. [12] P.L. Lions, The concentration–compactness principle in the calculus of variations. The limit case, Parts 1 and 2, Rev. Mat. Iberoamericana 1 (1985) 145–201; 1(2) (1985) 45–120. [13] O.H. Miyagaki, On a class of semilinear elliptic problems in R N with critical growth, Nonlinear Anal. 29 (7) (1997) 773–781. [14] E.S. Noussair, C.A. Swanson, Multiple finite energy states for critical potential systems, Nonlinear Anal. 26 (2) (1996) 217–240. [15] E.S. Noussair, C.A. Swanson, Y. Jianfu, Quasilinear elliptic problems with critical exponents, Nonlinear Anal. 20 (3) (1993) 285–301. [16] C.A. Swanson, L.S. Yu, Critical p-Laplacian problems in R N , Ann. Matematica pura ed applicata (IV) CLXIX (1995) 233–250. [17] N.M. Stavrakakis, N.B. Zographopoulos, Existence results for quasilinear elliptic systems in R N , Electron. J. Differ. Equ. (39) (1999) 15. [18] N.M. Stavrakakis, N.B. Zographopoulos, Bifurcation results for quasilinear elliptic systems on R N , Adv. Differ. Equ. 8 (3) (2003) 315–336. [19] P.H. Rabinowitz, On a class of nonlinear Schr¨odinger equations, Z. Angew. Math. Phys. 43 (1992) 270–291. [20] L.S. Yu, Nonlinear p-Laplacian problems on unbounded domains, Proc. Amer. Math. Soc. 115 (4) (1992) 1037–1045.
Quasilinear elliptic systems with critical Sobolev exponents in R N Ali Djellit ∗ , Saˆadia Tas University of Annaba, Faculty of Sciences, Department of Mathematics, BP 12, 23000, Annaba, Algeria Received 21 July 2005; accepted 2 February 2006
Abstract We study here a class of quasilinear elliptic systems involving the p-Laplacian operator; the right hand sides of systems are closely related to the critical Sobolev exponents. Under some additional assumptions on the nonlinearities, the corresponding functional verifies the Palais–Smale condition (P S)c for c belonging to a specified range. So, we can use the Mountain Pass Theorem to prove the existence of at least one nontrivial solution. c 2006 Elsevier Ltd. All rights reserved. MSC: 35P65; 35P30 Keywords: p-Laplacian operator; Critical Sobolev exponent; Palais–Smale condition; Mountain Pass Theorem; Concentration–compactness principle
1. Introduction In the present paper we investigate the existence of nontrivial weak solutions of the vector valued quasilinear elliptic system ⎧ ∂F ∗ ⎪ − p u = f (x)|u| p −2 u + λ (x, u, v) in R N ⎪ ⎪ ⎨ ∂u (Sc ) ∂F ∗ ⎪ (x, u, v) in R N −q v = g(x)|v|q −2 v + λ ⎪ ⎪ ∂v ⎩ u, v|x| → +∞ → 0 for all λ belonging to ]0, λ1 [. Of course p is the so-called p-Laplacian operator, i.e. p u = Np div(|∇u| p−2 ∇u), 1 < p, q < N. The real number p∗ = N− p designates the critical Sobolev ∗ Corresponding author. Fax: +213 08 87 25 26.
E-mail address: a [email protected] (A. Djellit). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.02.005
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exponent of p and f, g, F are real valued functions satisfying some assumptions. The unknown real valued functions u and v stay in appropriate spaces. Critical problems appear in many areas including differential geometry, quantum field theory and optimization problems related to Sobolev and isoperimetric inequalities (see for example [12]). In recent years there has been a marked increase of research interest in critical problems and a number of excellent works have been published. Finer details are elucidated in ∗ Brezis and Nirenberg [4]. The authors studied the semilinear equation −u = u 2 −1 + λu in a smooth bounded domain Ω ⊂ R N with u = 0 on ∂Ω . They proved the existence of positive solutions for N ≥ 3 and 0 < λ < λ1 . We refer to λ1 as the first eigenvalue of the Laplacian. It is well known that λ1 is simple and the associated eigenfunction u has one sign. In a natural way, the study has been extended by [2] to nonlinear elliptic systems. In his review article, Miyagaki [13] examines the equation −u + a(x)u = λ|u|q−1 u + |u| p−1 u in R N with 1 < q < p ≤ 2∗ − 1 and a(x) ≥ a0 > 0. The author obtains comparable results. Unlike this, Noussair et al. ∗ [15] inspected in detail the quasilinear problem of type −m u = p(x)u m −1 + q(x)u γ in a connected open set Ω of R N with m − 1 < γ < m ∗ . They proved also the existence of positive solutions. An additional work due to Swanson and Yu [16] deals with an equation of the form ∗ −m u = λa(x)u m−1 + f (x)u m −1 + g(x)u q in R N and u → 0 when |x| → +∞. The authors establish existence results. A delightful result was described by Drabek and Huang [7] ∗ concerning the problem − p u = λg(x) |u| p−2 u + f (x)|u| p −2 u in R N ; λ ∈]0, λ1 [. Here, the real number λ1 is the first principaleigenvalue of the problem − p u = λg(x)|u| p−2u in R N , under the following condition R N g (x) |u| p dx > 0 (see [1,5]). We will keep in mind a multiplicity result obtained by Noussair and Swanson [14] for some critical potential systems involving the Laplacian operator. We also mention existence and bifurcation results established for quasilinear elliptic systems in [17,18] and the comments of [20] related to external domain. This contribution comes within the continuation of results described in [8,9]. In this work, following the same ideas, we deal with the existence of nontrivial solutions of System (Sc ) for all λ ∈ ]0, λ1 [. However the real number λ1 is the first eigenvalue of the system ⎧ ∂F ⎪ − p u = λ (x, u, v) in R N ⎪ ⎪ ⎨ ∂u (Svp ) ∂F N ⎪ ⎪ ⎪−q v = λ ∂v (x, u, v) in R ⎩ u, v |x| → +∞ → 0, u > 0, v > 0. Assume that the nonlinearity F satisfies mixed and subcritical growth conditions, then the solutions are obtained invoking the Mountain Pass Theorem [10]. Because of the loss of compactness in Sobolev imbeddings, the Palais–Smale condition is only satisfied when the energy level of the corresponding functional remains below a certain critical level linked to the best Sobolev constants. In what follows the concentration–compactness principle of P.L. Lions [11] plays an important role in the study of the solvability of (Sc ). This article has the following structure: in Section 2, we introduce some notation and hypotheses; Section 3 is devoted to the proof of the main results. 2. Notation and hypotheses We denote by D 1,m (R N ) the completion of C0∞ (R N ) in the norm u 1,m
1 m N m |∇u| dx ≡ ∇u m = R ;
1 < m < N.
A. Djellit, S. Tas / Nonlinear Analysis 66 (2007) 1485–1497
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It is well known that D 1,m (R N ) is a uniformly convex Banach space and may be written as
N ∗ D 1,m (R N ) = u ∈ L m (R N ) : ∇u ∈ L m (R N ) . Moreover, we have the following Sobolev inequality ∃C (N, m) > 0,
∀u ∈ D 1,m (R N ) : u m ∗ ≤ C (N, m) u 1,m .
Then, we consider Sm the best Sobolev constant defined by
u m 1,m 1,m N {0} Sm ≡ C −m (N, m) = inf , u ∈ D (R ) \ . u m m∗ We denote by Z the product space D 1, p (R N ) × D 1,q (R N ); Z ∗ is the dual space of Z equipped with the dual norm · ∗ and Iλ , J, K , L are functionals defined for all (u, v) ∈ Z by 1 1 p q u 1, p + v 1,q p q K (u, v) = R N F (x, u (x) , v (x)) dx 1 N 1 N ∗ p∗ L (u, v) = ∗ R f (x)|u| dx + ∗ R g(x)|u|q dx p q Iλ (u, v) = J (u, v) − λK (u, v) − L(u, v), ∀λ > 0. J (u, v) =
In order to show the existence result, we suppose that (H1) f and g are positive and bounded functions. (H2) F ∈ C 1 R N , R, R and F (x, 0, 0) = 0. (H3) For all U = (u, v) ∈ R2 and for almost every x ∈ R N ∂F ≤ a1 (x) |U | p1 −1 + a2 (x) |U | p2 −1 U (x, ) ∂u ∂F ≤ b1 (x) |U |q1 −1 + b2 (x) |U |q2 −1 U (x, ) ∂v where 1 < p1 , q1 < min ( p, q) , max ( p, q) < p2 , q2 < min p∗ , q ∗ ai ∈ L αi (R N ) ∩ L βi (R N ), bi ∈ L γi (R N ) ∩ L δi (R N ), i = 1, 2. p∗ q∗ p∗ q ∗ αi = ∗ , γi = ∗ , βi = ∗ ∗ , p − pi q − qi p q − p ∗ ( pi − 1) − q ∗ p∗ q ∗ δi = ∗ ∗ . p q − q ∗ (qi − 1) − p∗ (H4) ∃θ p , θq with
1 p∗
< θp <
1 p
,
1 q∗
< θq <
1 q
such that
∂F ∂F (x, u, v)u + θq (x, u, v)v ∂u ∂v 1 ∂F 1 ∂F (x, u, v)u + (x, u, v)v. ≤ p ∂u q ∂v p−N q−N 1 Np 1 Nq p ∗ S p f ∞ , Sq g ∞q Set c = min . N N 0 < F (x, u, v) ≤ θ p
If c∗ =
N
1 N
p−N
S pp f ∞p , then we assume that
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(H5) there exists x 0 ∈ R N such that f ∞ = x ∈ R N sup f (x) = f (x 0 ), (H6) the function f is such that f (x) − f (x 0 ) = O (|x − x 0 |) in a neighborhood of x 0 , (H7) there exist two positive constants α and c, p < α < p∗ , such that F (x, u, v) ≥ c |u|α . q−N
N
If c∗ = N1 Sqq g ∞q , we replace f, p, p∗ , c |u|α by g, q, q ∗ , c|v|α in (H5), (H6) and (H7) respectively. Notice that under assumptions (H1), (H2) and (H3), the functional Iλ is well defined and is of class C 1 . Moreover for all (u, v) , (w, z) ∈ Z |∇u| p−2 ∇u · ∇wdx + |∇v|q−2 ∇v · ∇zdx Iλ (u, v) (w, z) = N N R R ∂F ∂F (x, u, v)w + (x, u, v)z dx −λ N ∂u ∂v R ∗ ∗ − f (x)|u| p −2 uwdx − g(x)|v|q −2 vzdx. RN
RN
Hence, weak solutions of System (Sc ) are exactly critical points of the functional Iλ . We recall that Iλ satisfies (P S)c if each sequence (u n , vn ) of Z verifying Iλ (u n , vn ) → c and Iλ (u n , vn ) → 0 in Z ∗ possesses a convergent subsequence in Z . We recall also the following concentration–compactness principle of Lions (see [12, p. 158]). ∗ p p 1, p N Let (u n ) converge weakly to u in D (R ) such that |∇u n | , |u n | converge weakly to nonnegative bounded measures μ and ν respectively. Then, for some at most countable set E, we have ∗ (i) ν = |u| p + j ∈E ν j δx j , (ii) μ ≥ |∇u| p + j ∈E μ j δx j , p ∗
(iii) μ j ≥ S p υ jp , ∀ j ∈ E. Here δx j is the Dirac measure at x j , x j ∈ R N ; μ j , υ j ∈ ]0, +∞[. A crucial aspect in the theory of critical p-Laplacian problems consists in the fact that −1
C p = S p p , the best constant of Sobolev imbedding, is attained by the radial function u ε defined by ⎛ u ε (x) = M ⎝ Observe that M=
N
ε ε
p p−1
N−p p−1
⎞ N− p
1 p−1
+ |x − x 0 |
p
p p−1
⎠
for any ε > 0, x 0 , x ∈ R N .
N− p p−1 N p−2N+2 p
,
is chosen such that u ε solves the scalar critical equation − p u ε = u εp
∗ −1
in R N , (see [12]).
In the same way, we introduce the radial function vε corresponding to the constant Sq .
A. Djellit, S. Tas / Nonlinear Analysis 66 (2007) 1485–1497
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3. Existence of solutions We need some lemmas to prove the main theorem. Lemma 1. Assume (H2) and (H3) hold. The functional K is weakly lower semicontinuous in Z and the operator K is compact from Z to Z ∗ . Proof. Let (u n , vn ) be a bounded sequence in Z so there exists a subsequence, denoted again as (u n , vn ), weakly convergent to (u, v) in Z . Let B R be the ball of R N centered at the origin with radius R. We denote by B R the complement of B R in R N and by K R the functional defined in D 1, p (B R ) × D 1,q (B R ) expressed by K R (u, v) = B R F (x, u (x) , v (x)) dx. It is well known that K R is weakly lower semicontinuous and K R is also compact (see for example [19]). Set K (u n , vn ) = K R (u n , vn ) + F(x, u n , vn ) dx. BR
Using both (H3) and properties of the weights ai and bi , we have for R sufficiently large F(x, u n , vn ) dx → 0 as n → +∞. BR
Hence, K is weakly lower semicontinuous. A similar argument yields the compactness of K . Lemma 2. Under hypotheses (H2)–(H4), the functional J is bounded from below on the set Λ = {(u, v) ∈ Z , K (u, v) = 1} and achieves its minimum λ1 . Proof. Set λ1 = (u, v) ∈ Λ inf J (u, v). It is clear that J is weakly lower semicontinuous and coercive in Z , then in Λ. The set Λ is weakly closed since K is weakly lower semicontinuous. On the other hand, K (u, v) = 0 means that ∂F ∂F (x, u, v)w + (x, u, v)z dx = 0, ∀ (w, z) ∈ Z . ∂u ∂v RN Substituting w for θ p u and z for θq v respectively, and taking into account (H4), we obtain K (u, v) = 0. According to Theorem 6.3.2 in Berger [3], the statement is proved. Lemma 3. Under hypotheses (H2)–(H4), for all λ ∈ ]0, λ1 [, ∃σ > 0 such that p q J (u, v) − λK (u, v) ≥ σ u 1, p + v 1,q , ∀ (u, v) ∈ Z . Proof. Suppose that there exists λ ∈]0, λ1 [ and a sequence (u n , vn ) of Z such that 1 p q u n 1, p + vn 1,q . J (u n , vn ) − λK (u n , vn ) < n Using the variational characterization of λ1 , i.e. J (u, v) ≥ λ1 K (u, v) , ∀ (u, v) ∈ Z . p q Consequently 1 − λλ1 J (u n , vn ) < n1 u n 1, p + vn 1,q .
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1 1 − − p q We set γn = max u n 1, p , vn 1,q , u n = γnp u n and vn = γnq v n . p − q − − − So, we obtain 1 − λλ1 J u n , v n < n1 u n + v n . 1,q 1, p − − − − Since u n , v n is bounded in Z , there exists a subsequence denoted again as u n , v n , − − − − weakly convergent to u, v . Apply the limiting process n → +∞, we get 1 − λλ1 J u, v ≤ 0, i.e. λ1 ≤ λ. This yields a contradiction. Lemma 4. Under hypotheses (H1)–(H4) and for all λ ∈ ]0, λ1 [, the functional Iλ satisfies the (P S)c condition for all c ∈ ]0, c∗ [. Proof. Let (u n , vn ) be a sequence of Z such that Iλ (u n , vn ) → c and Iλ (u n , vn ) → 0 in Z ∗ . We have 1 1 1 ∗ p q f (x) |u n | p dx Iλ (u n , vn ) = u n 1, p + vn 1,q − ∗ p q p RN 1 ∗ q − ∗ g(x) |u n | dx − λ F (x, u n , vn ) dx q RN RN = c + o (1) , as n is sufficiently large. On the other hand
|∇u n | p−2 ∇u n · ∇wdx + |∇vn |q−2 ∇vn · ∇zdx RN RN ∂F ∂F (x, u n , vn )w + (x, u n , vn )z dx −λ N ∂u ∂v R ∗ ∗ p −2 − f (x) |u n | u n wdx − g(x) |vn |q −2 vn z dx N N R R p q = o (1) u n 1, p + vn 1,q , as n is sufficiently large.
Iλ (u n , vn ) (w, z) =
Hence
1 1 p q − θ p u n 1, p + − θq vn 1,q Iλ (u n , vn ) − Iλ (u n , vn ) θ p u n , θq vn = p q 1 1 ∗ ∗ + θp − ∗ f (x) |u n | p dx + θq − ∗ g(x) |u n |q dx p q RN RN ∂F ∂F (x, u n , vn ) + θq vn (x, u n , vn ) − F (x, u n , vn ) dx. +λ θ p un ∂u ∂v RN
According to (H4), we get 1 1 p q p q − θ p u n 1, p + − θq vn 1,q . c + o (1) + o (1) u n 1, p + vn 1,q ≥ p q Hence the sequence (u n , vn ) is bounded in Z ; so there exists a subsequence, denoted again as (u n , vn ), weakly convergent to (u, v) in Z . Then, there exist positive and bounded measures μ, μ, ¯ ν and ν¯ defined in R N such that ∗
∗
|∇u n | p μ and |u n | p ν(respectively |∇vn |q μ¯ and |vn |q ν¯ ).
A. Djellit, S. Tas / Nonlinear Analysis 66 (2007) 1485–1497
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Applying the concentration–compactness principle, there exist x j ⊂ R N ; μ j , ν j ⊂ ]0, +∞[ such that (i), (ii) and (iii) hold. First, we show that the set E is finite (respectively E¯ corresponding to (vn )). For j ∈ E, we define φ j ∈ C0∞ R N as follows 1 if x ∈ B x j , ε φ j (x) = and ∇φ j ≤ C. N 0 if x ∈ R \ B x j , 2ε Substitute w for φ j u n and z for 0 in Iλ (u n , vn ) (w, z). Using the fact that Iλ (u n , vn ) → 0 in Z ∗ , we obtain |∇u n | p−2 u n ∇u n · ∇φ j dx lim n→+∞ R N |∇u n | p φ j dx = lim − n→+∞ N R ∂F ∗ + (x, u n , vn )φ j u n dx . f (x) |u n | p φ j dx + λ (1.1) RN R N ∂u Observe that u n u implies R N ∂∂uF (x, u n , vn )φ j u n dx → R N ∂∂uF (x, u, v)φ j udx. According to Lions’ principle, we have p |∇u n | φ j dx → φ j dμ and RN RN ∗ f (x) |u n | p φ j dx → f (x)φ j dν. RN
RN
Consequently p−2 |∇u n | lim u n ∇u n .∇φ j dx = f (x)φ j dμ n→+∞ R N RN ∂F +λ (x, u, v)φ j udx − φ j dμ. ¯ N R ∂u RN H¨older’s inequality gives p−2 |∇u n | 0 ≤ lim u n ∇u n · ∇φ j dx n→+∞ R N 1 p−1 ≤ C lim
n→+∞
RN
|∇u n | p dx
n→+∞
B(x j ,2ε)
|u n | p dx
1
≤ C lim
p
p
p
B(x j ,2ε)
|u n | p dx
.
1, p B(x , 2ε) , there exists a subsequence denoted as (u ) such that in D Since (u n ) is bounded j n u n → u in L p B(x j , 2ε) . So |∇u n | p−2 u n ∇u n · ∇φ j dx 0 ≤ lim n→+∞ R N 1 p
≤C
B(x j ,2ε)
|u| p dx
→ 0.
ε→0
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A. Djellit, S. Tas / Nonlinear Analysis 66 (2007) 1485–1497
Letting ε tend to 0 in (*), we obtain f (x j )υ j = μ j . p ∗
p ∗
But μ j ≥ S p υ jp , thus f (x j )υ j ≥ S p υ jp , i.e. υ j ≥ Consequently N p Sp p∗ δx j υ ≥ |u| + f (x j ) j ∈E and hence RN
|u n |
p∗
dx →
RN
dυ ≥ |u|
p∗
+
Sp f (x j )
Sp f (x j )
N p
.
N p
Card E.
∗ If Card E = ∞ then R N |u n | p dx → +∞. A contradiction. n→+∞ −
Now, we prove that E (respectively E ) is empty. We have 1 u n vn 1 ∗ , − ∗ Iλ (u n , vn ) − Iλ (u n , vn ) f (x) |u n | p dx = N p q p p R 1 1 1 ∂F q∗ + − ∗ un (x, u n , vn ) g(x) |u n | dx + λ N q q p ∂u RN R 1 ∂F (x, u n , vn ) − F (x, u n , vn ) dx. + vn q ∂v Hypothesis (H4) gives Iλ (u n , vn ) − Iλ (u n , vn )
u n vn , p q
≥
1 ∗ f (x) |u n | p dx N RN 1 ∗ + g(x) |u n |q dx. N RN
When n → +∞, we obtain 1 1 1 ∗ p∗ c≥ f (x) |u| dx + f (x j )υ j + g(x) |u|q dx N RN N j ∈E N RN 1 − + g(x j )υ j . N − j∈ E
− f (x j )υ j + N1 − g(x j )υ j . j ∈E N N − p q S S and in a similar manner υ j ≥ g(xqj ) . But υ j ≥ f (xpj )
So c ≥
1 N
j ∈E
−
Suppose that E ∪ E = ∅ and thus − 1− N 1− N 1 Nq 1 N p Card E + q Card Sq g(x j ) c ≥ S pp f (x j ) E. N N p−N q−N N 1 1 Np c ≥ min S p f ∞p , Sqq g ∞q . N N
A. Djellit, S. Tas / Nonlinear Analysis 66 (2007) 1485–1497
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−
We get a contradiction, hence E ∪ E = ∅, which means that u n p∗ → u p∗
and
vn q ∗ → v q ∗ .
∗ Taking this together with the fact that (u n , vn ) (u, v) in Z , we have u n → u in L p R N and ∗ vn → v in L q R N . On the other hand J (u n , vn ) − J (u m , vm ) (u n − u m , 0) = Iλ (u n , vn ) − Iλ (u m , vm ) (u n − u m , 0) + λ K (u n , vn ) − K (u m , vm ) (u n − u m , 0) + L (u n , vn ) − L (u m , vm ) (u n − u m , 0) . Iλ (u n , vn ) → 0, i.e. Iλ (u n , vn ) is a Cauchy sequence in Z ∗ . The compactness of K gives (u n , vn ) (u, v) ⇒ K (u n , vn ) → K (u, v) , i.e. K (u n , vn ) is a Cauchy sequence in Z ∗ . Moreover, using H¨older’s inequality L (u n , vn ) − L (u m , vm ) (u n − u m , 0) ∗ ∗ = f (x) |u n | p −2 u n − |u m | p −2 u m (u n − u m ) dx RN ∗ ∗ u n − u m L p∗ (R N ) . ≤ f ∞ |u n | p −2 u n − |u m | p −2 u m p∗ L p∗ −1 (R N ) ∗ Since (u n ) is a Cauchy sequence in L p R N , L (u n , vn ) is a Cauchy sequence in Z ∗ . We note according to [6] that ∀λ, μ ∈ R N ⎧ ⎨ |λ| p−2 λ − |μ| p−2 μ . (λ − μ) if p ≥ 2 p |λ − μ| p ≤ ⎩ |λ| p−2 λ − |μ| p−2 μ . (λ − μ) 2 (|λ| + |μ|) (2−2p) p if 1 < p < 2. Replacing λ and μ by ∇u n , ∇u m respectively and integrating over R N , we obtain p u n − u m 1, p ≤ J (u n , vn ) − J (u m , vm ) (u n − u m , 0) if p ≥ 2 2− p 2 p p u n − u m 21, p ≤ J (u n , vn ) − J (u m , vm ) (u n − u m , 0) u n 1, p + u m 1, p if 1 < p < 2. Taking into account the fact that (u n ) is bounded in D 1, p (R N ) and J (u n , vn ) − J (u m , vm ) (u n − u m , 0) → 0, n,m→+∞
we find that (u n ) is a Cauchy sequence in D 1, p (R N ) and so converges in D 1, p (R N ). We proceed similarly for (vn ) with J (u n , vn ) − J (u m , vm ) (0, vn − vm ). Lemma 5. Under the hypotheses (H1)–(H3), ∃ρ, ω > 0 such that ∀λ ∈]0, λ1 [: p
q
u 1, p + v 1,q = ρ ⇒ Iλ (u, v) ≥ ω.
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Proof. We have ∗ p∗ p∗ p∗ f (x)|u| p dx ≤ f ∞ u p∗ ≤ C p f ∞ u 1, p . N R ∗ q∗ q∗ q∗ g(x)|v|q dx ≤ f ∞ v q ∗ ≤ Cq g ∞ v 1,q . RN
Taking into account Lemma 3, we get p q p∗ q∗ Iλ (u, v) ≥ σ u 1, p + v 1,q − C u 1, p + v 1,q ∗ p q∗ with C = max C p f ∞ , Cq g ∞ . Thus for some ρ > 0 sufficiently small, we have Iλ (u, v) ≥ ω > 0.
c∗ ,
We have seen that Iλ satisfies the (P S)c condition for all 0 < c < in order to apply the Mountain Pass Theorem, we will estimate Iλ on a critical path, i.e. finding a critical point with an energy level in the compactness interval requires the construction of a special path ξ (t) joining the origin to a point e of Z with Iλ (e) < 0
and such that 0 < Sup Iλ (ξ (t)) < c∗ . t ≥0
N p
p−N p
Suppose that c∗ = N1 S p f ∞ (in the other case, we proceed similarly taking the radial function vε ). Let R > 0, φ R ∈ C0∞ (R N ) verifying φ R ≡ 1 in B(x 0 , R) and φ R ≡ 0 in R N \ B(x 0 , 2R). Put Uε = φ R u ε and wε = Uε p∗ . Uε Then, we have the following lemma. Lemma 6. Under hypotheses (H1)–(H6), there exist ε > 0 and t0 > 0 such that ∀λ ∈ ]0, λ1 [, Iλ (t0 wε , 0) < 0 and 0 < Supt ≥0 Iλ (twε , 0) < c∗ . Proof. We have
∗ tp tp p p∗ wε 1, p − ∗ Iλ (twε , 0) = f (x) |wε | dx − λ F (x, twε , 0) dx. p p RN RN Since F (x, u, v) ≥ 0, it is clear that ∗ tp tp ∗ p wε 1, p − ∗ Iλ (twε , 0) ≤ f (x) |wε | p dx. p p RN But p < p∗ and f is positive, so Iλ (twε , 0) → −∞. t →+∞ Thus, there exists t0 sufficiently large for which Iλ (t0 wε , 0) < 0. By virtue of Lemma 5 and for t sufficiently small, we obtain Iλ (twε , 0) ≥ ω > 0.
So Iλ (twε , 0) achieves its supremum at a point tε > 0. Then, we write p p∗ tε tε ∗ p |wε | p dx − λ wε 1, p − ∗ f (x 0 ) Iλ (tε wε , 0) = F (x, tε wε , 0) dx p p RN RN p∗ tε ∗ + ∗ ( f (x 0 ) − f (x)) |wε | p dx. p RN
A. Djellit, S. Tas / Nonlinear Analysis 66 (2007) 1485–1497
1495 ∗
For positive reals A and B, the maximum of the function θ (t) = Ap t p − pB∗ t p for t ≥ 0 is N− p attained at t = BA p2 . ∗ p Putting A ≡ wε 1, p = S p , B ≡ f (x 0 ) R N |wε | p dx = f (x 0 ), we obtain N− p N p p∗ p p Sp Sp tε 1 1 tε p p∗ |wε | dx ≤ wε 1, p − ∗ f (x 0 ) − ∗ f (x 0 ) N p p p f (x 0 ) p f (x 0 ) R p−N 1 N = S pp f ∞p . N On the other hand tε is bounded above, indeed, since Iλ (tε wε , 0) = Supt ≥0 Iλ (twε , 0), we have Iλ (tε wε , 0) = 0, in particular Iλ (tε wε , 0) (tε wε , 0) = 0, i.e. ∂F ∗ ∗ p tεp wε 1, p − tεp f (x) |wε | p dx − λ (x, tε wε , 0) tε wε dx = 0. RN R N ∂u ∗ p∗ p p So tε R N f (x) |wε | p dx ≤ tε wε 1, p or ∗ ∗ ∗ |wε | p dx ≤ S p . tεp − p ( f (x) − f (x 0 )) |wε | p dx + f (x 0 ) RN
RN
Hypothesis (H5) gives N− p tε ≤ cS p p2 . We have also ∗ ( f (x 0 ) − f (x)) |wε | p dx RN
≤
⎛
cR p∗
u ε p∗
⎝ B(x 0,R)
ε
⎞ (N− p) p∗
1 p−1
p
p
p
ε p−1 + |x − x 0 | p−1
⎠
dx.
Making the change of variables r = |x − x 0 | and s = rε , we obtain
∗
RN
( f (x 0 ) − f (x)) |wε | p dx ≤ C
R ε
s
Np N−1− p−1
N
ds ≤ Cε p−1 .
0
Moreover, by virtue of (H6), we get ⎛
RN
F (x, tε wε , 0) dx ≥ C
⎝ B(x 0,R)
ε p
⎞ α(N− p)
1 p−1
p
p
ε p−1 + |x − x 0 | p−1
⎠
Putting again r = |x − x 0 | and s = rε , we find RN
F (x, tε wε , 0) dx ≥ Cε
p) N− α( N− p
R ε
0
s N−1 ds. α(N− p) p p p−1 1+s
dx.
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A. Djellit, S. Tas / Nonlinear Analysis 66 (2007) 1485–1497
For ε > 0 sufficiently small such that Rε > 1, p) p) N− α( N− − α(N− p p F (x, tε wε , 0) dx ≥ Cε 2 RN
R ε
s
N− p) N−1− α( p−1
ds.
1
p) γ = N − 1 − α(N− p−1 , we have ⎧ 1 ⎨ if γ = −1 C1 ε N−β ln F (x, tε wε , 0) dx ≥ ε ⎩C ε N−β−γ −1 if γ = −1. RN 2
Defining β =
α(N− p) , p
Hence RN
F (x, tε wε , 0) dx ≥ cεδ
⎧ ⎪ ⎨N − β
αp α− p+1 with δ = αp ⎪ . ⎩ N − β − γ − 1 if N = α− p+1 if N =
Finally Iλ (tε wε , 0) ≤
p−N N 1 Np S p f ∞p − Cεδ + Cε p−1 N
and the lemma follows because δ <
N p−1 .
Now, we state our main result Theorem. Assume (H1)–(H6) are satisfied; the System (Sc ) possesses at least a nontrivial solution. Proof. We define c = Infϕ∈Γ Sup0≤t ≤1 Iλ (ϕ (t)) where Γ denotes the class of all continuous paths ϕ in Z joining the origin of Z to t0 wε . Lemma 6 implies that 0 < c < c∗ and hence Iλ satisfies (P S)c by Lemmas 5 and 6 ensuring the geometric conditions. Consequently, the Mountain Pass Theorem can be applied to conclude that Iλ has a critical point with corresponding critical value c. So, System (Sc ) possesses a nontrivial weak solution. Acknowledgment The authors wish to express their gratitude to ANDRU for providing support through the grant CU39904. References [1] W. Allegretto, Y.X. Huang, Eigenvalues of the indefinite weight p-Laplacian in weighted R N spaces, Funkcial. Ekvac. 38 (1995) 233–242. [2] P. Amster, P. De N´apoli, M.C. Mariani, Existence of solutions for elliptic systems with critical Sobolev exponent, Electron. J. Differen. Equ. (49) (2002) 13. [3] S.M. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, San Francisco, London, 1977. [4] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure. Appl. Math. 36 (1983) 437–477. [5] P. Drabek, Nonlinear eigenvalues problem for the p-Laplacian in R N , Math. Nah. 173 (1995) 131–138. [6] G. Dinca, P. Jebelean, Some existence results for a class of nonlinear equations involving a duality mapping, Nonlinear Anal. 46 (2001) 347–363. [7] P. Drabek, Y.X. Huang, Multiplicity of positive solutions for some quasilinear elliptic equations in R N with critical Sobolev exponent, J. Difference Equ. 140 (1997) 106–132.
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