Existence of solutions for elliptic systems with critical Sobolev exponent ∗

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Existence of solutions for elliptic systems with critical Sobolev exponent ^∗

Pablo Amster, Pablo De N´ apoli, & Maria Cristina Mariani

Abstract

We establish conditions for existence and for nonexistence of nontrivial solutions to an elliptic system of partial differential equations. This system is of gradient type and has a nonlinearity with critical growth.

1 Introduction

The purpose of this work is to extend some results known for the quasilinear elliptic equation

−∆u=u^p−1+λu in Ω

u= 0 on∂Ω (1.1)

to the general system

−∆ui=fi(u) +

n

X

j=1

aijuj in Ω ui= 0 on∂Ω.

(1.2)

First we recall some results for the single equation (1.1) on a bounded domain Ω⊂R^N. If 2< p <2^∗= 2N/(N−2) (the critical Sobolev exponent), then (1.1) has a nontrivial solution if and only if λ < λ₁(Ω), the first eigenvalue of −∆.

This is proved by applying the Mountain Pass Theorem for finding nontrivial critical points for the following functional in the Sobolev space H₀¹(Ω).

ϕ(u) = 1 2

Z

Ω

|∇u|²−λ 2 Z

Ω

u²−1 p

Z

Ω

F(u) (1.3)

where F(u) = |u|^p. Then by the compact imbedding H₀¹(Ω) ,→ L^p(Ω), ϕ satisfies the Palais-Smale condition (P S). However whenp = 2^∗, ϕ may not satisfy the Palais-Smale condition (PS) due to the lack of compactness of the above imbedding. For λ ≤ 0, a Pohozaev identity shows that there are no

∗Mathematics Subject Classifications: 35J50.

Key words: Elliptic Systems, Critical Sobolev exponent, variational methods.

2002 Southwest Texas State University.c

Submitted January 2, 2002. Published June 2, 2002.

1

nontrivial solutions when Ω is star shaped. For the case 0< λ < λ1(Ω), Brezis a nd Nirenberg [1] proved the existence of at least one nontrivial solution when N ≥4. Their proof relies in the fact that ϕ satisfies (P S)c (Palais-Smale at levelc) ifc < c^∗=S^N/2/N, where

S= inf

u∈D^1,2₀ (R^N),kuk2∗=1k∇uk²2

which is the best constant in the Sobolev inequality. Moreover, when the value ofSand the optimal functions are explicitly known, it is possible to prove that if

S_λ= inf

u∈H₀¹(Ω),kuk^2∗=1k∇uk²2+λkuk²2

thenSλ< Sforλ >0. Then, using the Mountain Pass Theorem a critical value c < c^∗ is obtained. For a detailed exposition see [12].

Quasilinear elliptic systems have been studied by several authors [4, 5, 6].

For gradient type systems such as (1.2), Boccardo and de Figueiredo [2] used variational arguments to show the existence of nontrivial solutions. They proved existence of solutions for the problem

−∆pu=Fu(x, u, v) in Ω

−∆qv=Fv(x, u, v) in Ω u=v= 0 on∂Ω,

(1.4)

whereF is superlinear and subcritical. In this article, we study the critical case p=q= 2^∗.

The general problem of finding a condition on the matrixA= (aij) for which (1.2) admits a nontrivial solution is still an open question. In this paper, we present some results toward the solution of this question. ForAsymmetric with kAk< λ1(Ω), we prove that the method presented in [1] can be applied. More precisely, we define appropriate numbers SF,A andSF such that ifSF,A < SF

then (1.2) admits a solution. Furthermore, we show cases where this inequality holds. We prove also that in some particular cases the condition kAk< λ1(Ω) is necessary. We conclude this paper by showing that Pohoazev’s nonexistence result may be generalized to problem (1.2) when Ais symmetric and negative definite. We remark that the symmetry of A can be considered as a natural condition, since the proof of existence is based on the variational structure of the problem.

Before we state our results, we recall the following definitions [8].

D^1,2(R^N,Rⁿ) ={u= (u₁, . . . , u_n)∈L^2∗(R^N,Rⁿ) :∇u_i∈L²(R^N,R^N)} LetA= (aij)∈R^n×n. We shall say that:

i)Ais nonnegative (A≥0) ifaij ≥0 for all i, j.

ii)Ais reducible if by a simultaneous permutation of rows and columns, it may be written in the form

B 0

C D

where B and D are square matrices. Throughout this article, the Euclidean norm inRⁿ will be denoted by| · |.

Statement of results

Theorem 1.1 Let p= 2^∗ = 2N/(N−2). Let [·] be a norm on Rⁿ such that F(u) = [u]^p is differentiable. Define fi = ¹_p∂iF, and assume that A∈R^n×n is symmetric, with kAk< λ₁(Ω). Set

SF = inf

u∈D^1,2(R^N,Rⁿ),R

RNF(u)=1 n

X

i=1

Z

R^N

|∇ui|², SF,A(Ω) = inf

u∈H¹(Ω,Rⁿ),R

ΩF(u)=1 n

X

i=1

Z

Ω

|∇ui|²− Z

Ω

hAu, ui Then: 1)S_F is attained by a function u∈D^1,2(R^N,Rⁿ).

2) If S_F,A(Ω)< S_F then (1.2) admits at least one nontrivial weak solution.

As a consequence of this theorem, we have the existence of solutions for the following case.

Corollary 1.2 Let p,A andfi satisfy the conditions of the Theroem 1.1 with [u] =|u|q = (Pn

i=1|ui|^q)^1/q for some q≥2. Moreover, assume that N≥4 and that aii>0 for somei. Then (1.2) has a nontrivial weak solution.

Theorem 1.3 Let us assume that (1.2) admits a nonnegative nontrivial solu- tionu∈H₀¹(Ω,Rⁿ), and thatfi(u)≥0, withfi(u)>0foru >0. We denote by µmin andµmaxthe smallest and the largest eigenvalues ofA, respectively. Then

1) IfA is symmetric and positive definite, thenµmin< λ1(Ω).

2) IfA≥0 is irreducible, then µmax< λ1(Ω).

3) Ifaij >0 for everyi, j, andAis symmetric, then kAk< λ1(Ω).

Using a Pohozaev-type identity [10] we shall prove as in [11] the following nonexistence result.

Theorem 1.4 LetF ∈C¹(Rⁿ)be homogeneous of degreep= 2^∗= 2N/(N−2) and definef_i= ¹_p∂_iF. Assume that Ais symmetric and negative definite, and that Ωis star shaped. Then u= 0is the unique classical solution of (1.2).

2 The Brezis-Lieb Lemma

We shall use the following version of the Brezis-Lieb lemma [3].

Lemma 2.1 Assume that F ∈ C¹(Rⁿ) with F(0) = 0 and

∂F

∂u_i

≤ C|u|^p−1. Let (u_k)⊂L^p(Ω), (1≤p <∞). If(u_k) is bounded inL^p(Ω) andu_k →ua.e.

onΩ, then

lim

k→∞

Z

Ω

F(u_k)−F(u_k−u)

= Z

Ω

F(u)

Proof We first remark that u∈ L^p(Ω) and kukp ≤ lim infkukkp < ∞. We claim that for a fixedε >0 there existsc(ε) such that fora, b∈Rⁿ, it holds

|F(a+b)−F(a)| ≤ε|a|^p+c(ε)|b|^p (2.1) Indeed, writing

|F(a+b)−F(a)|=

n

X

i=1

Z 1 0

∂F

∂ui

(a+bt)bidt

≤C

n

X

i=1

Z 1 0

|a+bt|^p−1|bi|dt

and using thatxy≤c(ε)xe ^p+eεy^p0(x, y >0) we obtain

|F(a+b)−F(a)| ≤C

n

X

i=1

Z 1 0

(εe|a+bt|^p+c(eε)|bi|^p)dt

Moreover, as (x+y)^p ≤2^p−1(x^p+y^p) (x, y >0), we obtain:

|F(a+b)−F(a)| ≤2^p−1C

n

X

i=1

Z 1

0 ε(e|a|^p+t^p|b|^p) +c(eε)|bi|^pdt and (2.1) follows. Lettinga=u_k(x)−u(x),b=u(x) we obtain

|F(uk)−F(uk−u)| ≤ε|uk−u|^p+c(ε)|u|^p We introduce the functions:

f_k^ε= (|F(u_k)−F(u_k−u)−F(u)| −ε|u_k−u|^p)⁺ As|F(u)| ≤K|u|^p, then|f_k^ε| ≤(K+c(ε))|u|^p. By Lebesgue theoremR

Ωf_k^ε→0.

Since|F(uk)−F(uk−u)−F(u)| ≤f_k^ε+ε|uk−u|^p, we obtain lim sup

k→∞

Z

Ω

|F(uk)−F(uk−u)−F(u)| ≤εc withc= sup_kku_k−uk^pp <∞. Letting ε→0, the result follows.

Remark In particular, this result holds forF is homogeneous of degreep.

3 Proofs of results

For the proof of part 1) of Theorem 1.1 we shall use the Lemma 3.1 below, which is a version of the concentration compactness lemma in [9].

LetF :Rⁿ→Rbe aC¹ function homogeneous of degreep= 2^∗, such that F(u)>0 ifu6= 0. By homogeneity, it is easy to see that

S_F = inf

u∈D^1,2(R^N,Rⁿ),u6=0

Pn k=1

R

R^N|∇u_k|² R

R^NF(u)2/2^∗

Lemma 3.1 Let(u⁽ⁱ⁾)⊂D₀^1,2(R^N,Rⁿ) be a sequence such that:

i) u⁽ⁱ⁾→uweakly inD^1,2(Ω)

ii) |∇(u⁽ⁱ⁾_k −u_k)| →µ_k inM(Rⁿ)weak^∗ fork= 1, ..., n.

iii) F(u⁽ⁱ⁾−u)→ν inM(Rⁿ)weak^∗ iv) u⁽ⁱ⁾→ua.e. onR^N

and define: µ=Pn k=1µk,

ν^∞= lim

R→∞

lim sup

i→∞

Z

|x|≥R

F(u⁽ⁱ⁾)dx , µ^∞_k = lim

R→∞

lim sup

i→∞

Z

|x|≥R

|∇u⁽ⁱ⁾_k |²dx

Then:

kνk^2/2∗ ≤ 1

S_F kµk (3.1)

(ν^∞)^2/2∗ ≤ 1 S_F

n

X

k=1

µ^∞_k (3.2)

lim sup

i→∞ |∇u⁽ⁱ⁾_k |²2=|∇u|²2+kµ_kk+µ^∞_k fork= 1, ..., n (3.3) lim sup

i→∞

Z

Ω

F(u⁽ⁱ⁾) = Z

Ω

F(u) +kνk+ν_∞ (3.4) Moreover, ifu= 0 and equality holds in (3.1), thenµ= 0or µis concentrated at a single point.

Proof of Theorem 1.1 Part 1) Let (u⁽ⁱ⁾)⊂D^1,2(R^N,Rⁿ) be a minimizing sequence forSF, i.e.,

Z

R^N

F(u⁽ⁱ⁾) = 1,

n

X

k=1

Z

R^N

|∇u⁽ⁱ⁾_k |²→SF

Using (3.1)-(3.3), we deduce, as in [12, Theorem 1.41], the existence of a se- quence (yi, λi)∈R^N ×R such thatλ^(N_i ^−2)/2u(λix+yi) has a convergent sub- sequence. In particular there exists a minimizer forSF.

To prove the second part of Theorem 1, we shall use the following version of the Mountain Pass Lemma [12].

Theorem 3.2 (Ambrosetti-Rabinowitz) Let X be a Hilbert space,ϕbe an element of C¹(X,R),e∈X andr >0 such that kek> r,b= inf_kuk=rϕ(u)>

ϕ(0)≥ϕ(e). Then for each ε >0 there exists u∈X such that c−ε≤ϕ(u)≤ c+εandkϕ⁰(u)k ≤εwhere

c= inf

γ∈Γ max

t∈[0,1]ϕ(γ(t)), withΓ ={γ∈C([0,1], X) :γ(0) = 0, γ(1) =e}.

Letting ε= 1/k, we get a Palais-Smale sequence at level c; i.e., a sequence (u^(k))⊂X such that

ϕ(u^(k))→c, ϕ⁰(u^(k))→0 We shall apply this result to the functional

ϕ(u) =1 2

Z

Ω n

X

i=1

|∇ui|²− 1 2^∗

Z

Ω

F(u)−1 2 Z

Ω n

X

i,j=1

aijuiuj

in the Sobolev spaceX =H₀¹(Ω,Rⁿ). AskAk< λ1(Ω), we may define onX the norm

kuk=



 Z

Ω n

X

i=1

|∇ui|²− Z

Ω n

X

i,j=1

aijuiuj





1/2

which is equivalent to the usual norm. By standard argumentsϕ∈C¹(X)and hϕ⁰(u), hi=

n

X

i=1

Z

Ω

∇u_i· ∇h_i−

n

X

i=1

Z

Ω

f_i(u)h_i− Z

Ω n

X

i,j=1

a_ijh_iu_j

It follows that the critical points of ϕare weak solutions of the system.

To ensure that the value c given by the mountain pass theorem is indeed a critical value we need to prove the following lemma.

Lemma 3.3 Let F be homogeneous of degree 2^∗. Then any (P S)c sequence withc < c^∗= ¹₂−₂¹∗

S_F^N/2

N has a convergent subsequence.

Proof Let (u^(k))⊂H₀¹(Ω,Rⁿ) be a (P S)c sequence. First we show that it is bounded.

hϕ⁰(u^(k)), u^(k)i=ku^(k)k²−

n

X

i=1

Z

Ω

fi(u^(k))u^(k)_i

SinceFis homogeneous of degree 2^∗,we have thatPn

i=1f_i(u^(k))u^(k)_i =F(u^(k)).

Then 1

2ku^(k)k²=ϕ(u^(k)) + 1 2^∗

Z

Ω

F(u^(k)) =ϕ(u^(k)) + 1

2^∗ ku^(k)k²− hϕ⁰(u^(k)), u^(k)i Hence, for k≥k0we have

1 2 − 1

2^∗

ku^(k)k²≤C+εku^(k)k and we conclude thatku^(k)kis bounded.

We may assume thatu^(k)→uweakly inH₀¹(Ω)ⁿ,u^(k)→uinL²(Ω)ⁿ, and u^(k)→ua.e..

Since (u^(k)) is bounded inL^2∗(Ω),f(u^(k)) is bounded inL^2N/(N+2)(Ω). So we may assume thatf(u^(k))→f(u) weakly inL^2N/(N+2). It follows thatuis a critical point ofϕ, i.e. uis a weak solution of the system. We deduce that

hϕ⁰(u), ui=kuk²− Z

Ω n

X

i=1

fi(u)ui= 0 Moreover,

ϕ(u) = kuk² 2 − 1

2^∗ Z

Ω

F(u) =kuk² 2 − 1

2^∗ Z

Ω n

X

i=1

f_i(u)u_i= (1 2− 1

2^∗)kuk²≥0 Writingv^(k)=u−u^(k), by Lemma 2.1 we have

Z

Ω

F(u^(k)) = Z

Ω

F(u) + Z

Ω

F(v^(k)) +o(1) and then

ϕ(u^(k)) = 1

2ku−v^(k)k²− 1 2^∗

Z

Ω

F(u^(k))− 1 2^∗

Z

Ω

F(v^(k)) +o(1) Asv^(k)→0 weakly,

ku^(k)k²=ku−v^(k)k²=kuk²+kv^(k)k²+o(1) and then we obtain

1

2kuk²+1

2kv^(k)k²− 1 2^∗

Z

Ω

F(u)− 1 2^∗

Z

Ω

F(v^(k))→c (3.5) On the other hand we also know that hϕ⁰(u^(k)), u^(k)i →0 and

hϕ⁰(u^(k)), u^(k)i=ku^(k)k²− Z

Ω n

X

i=1

fi(u^(k))u^(k)_i =ku^(k)k²−2^∗ Z

Ω

F(u^(k))

=kuk²+kv^(k)k²−2^∗ Z

Ω

F(u)−2^∗ Z

Ω

F(v^(k)) +o(1)→0

Hence,

kv^(k)k²−2^∗ Z

Ω

F(v^(k))→2^∗ Z

Ω

F(u)− kuk²= 0 We may therefore assume thatkv^(k)k²→b, 2^∗R

ΩF(v^(k))→b. From (3.5), we deduce that

ϕ(u) +1 2− 1

2^∗

b=c and sinceϕ(u)≥0,

1 2− 1

2^∗

b≤c

We claim that b = 0. Indeed, since u^(k) → 0 in L²(Ω,Rⁿ), it follows that Pn

i=1

R

Ω|∇(v^(k))i|²→b. On the other hand,

n

X

i=1

Z

Ω

|∇(v^(k))_i|²≥S_FZ

Ω

F(v^(k))2/2^∗

and, lettingk→ ∞,b≥S_Fb^2/2∗. It follows thatb= 0 orb≥S_F^N/2. In this last case,

c^∗= 1 2 − 1

2^∗ S_F^N/2

N ≤ 1

2 − 1 2^∗

b N ≤c, a contradiction. Hence,b= 0 and v^(k)→0 strongly.

Proof of Theorem 1.1 part 2) In the same way of [12, Theorem 1.45], it suffices to apply the Mountain Pass Theorem with a valuec < c^∗. We shall find the maximum of the functionh: [0,1]→Rgiven by

h(t) =ϕ(tv) = t²

2 kvk²−t^2∗ 2^∗

Z

Ω

F(v)

=At² 2 −Bt^2∗ Since 2^∗−2 = _N⁴

−2, we obtain the critical point t0= A

2^∗B

^(N−2)/4

with

h(t0) = A 2^∗B

N/2 2^∗ 2 −1

B >0 Then, it is easy to conclude thatc < c^∗.

Now we consider the special case [u] =|u|q=Xⁿ

i=1

|ui|^q1/q

and Fq(u) =Xⁿ

i=1

|ui|^q2∗/q

for proving Corollary 1.2.

Lemma 3.4 SF(Ω) =S forq≥2 whereS is the best constant for the Sobolev inequality withn= 1.

Proof Suppose first thatq≥2^∗, then we have the estimate hZ

Ω

Xⁿ

i=1

|ui|^q2^∗/qi2/2^∗

≤hZ

Ω n

X

i=1

|ui|^2∗i2/2^∗

=hXⁿ

i=1

Z

Ω

|ui|^2∗i2/2^∗

≤hXⁿ

i=1

(S⁻¹ Z

Ω

|∇ui|²2)^2∗/2i^2/2∗

≤

n

X

i=1

S⁻¹ Z

Ω

|∇ui|²

It follows thatSF ≥S. For 2≤q≤2^∗we use Minkowski inequality:

hZ

Ω

Xⁿ

i=1

|ui|^q2∗/qi^2/2∗

= (

hZ

Ω

Xⁿ

i=1

|ui|^q2∗/qi^q/2∗ )^2/q

≤hXⁿ

i=1

Z

Ω

|u_i|^2∗q/2∗i^2/q

=

n

X

i=1

Z

Ω

|u_i|^2∗2/2^∗

≤

n

X

i=1

S⁻¹ Z

Ω

|∇u_i|² The inequality S_F ≤ S is verified easily taking functions of the form u = (u₁,0, ...,0).

Proof of Corollary 1.2 First we note that by the 2^∗-homogeneity of F,SF

does not depend on Ω. Taking u(x) =U(x)ei, where U(x) = [N(N−2)]^(N−2)/4 (1 +|x|²)^(N−2)/2

is the function that attains Sobolev’s best constant in one dimension [12, The- orem 1.42], it follows that S_F is achieved when Ω = R^N (where N ≥4). By translation invariance of the problem,S_Fis also achieved withu_ε(x) =U_ε(x)·e_i, for

U_ε(x) =ε^(2−N)/2U(x/ε)

We shall see thatSF,A< SF. Indeed, we may assume that 0∈Ω and choose isuch thataii >0. Then, if we defineu(x) =vε(x)ei, withvε(x) =ψ(x)Uε(x), andψa smooth function with compact support in Ω such thatψ≡1 inB(0, ρ), we obtain as in [12, Lemma 1.46]:

R

Ω

Pn

i=1|∇ui|²−R

ΩhAu, ui R

ΩF(u)^2/p = R

Ω|∇uε|²−aiiR

Ωu²_ε R

Ω|uε|^p < S

forεsmall enough.

Proof of Theorem 1.3: Necessary conditions for the existence of non- negative solutions We recall the following theorem in [8].

Lemma 3.5 (Perron-Frobenius Theorem) Let A≥0 be irreducible. Then A has a positive simple eigenvalue µmax such that |µ| ≤µmax for any µ eigen- value ofA. Furthermore, there exists an eigenvector ofµmax with positive coor- dinates.

Now we are able to prove Theorem 1.3.

Suppose that the system has a nonnegative nontrivial solution. Ife₁ is the first eigenfunction of −∆ in H₀¹(Ω), thene₁ ∈C^∞(Ω) and e₁(x)> 0∀x∈ Ω.

Then

λ₁ Z

Ω

u_ie₁= Z

Ω

−∆u_ie₁= Z

Ω

f_i(u)e₁+

n

X

j=1

a_ij Z

Ω

u_je₁

Ifz_i=R

Ωu_ie₁, then

λ₁z≥Az,

and the inequality between the i-th components is strict if u_i 6= 0 for some i.

Sincez≥0, andz_i>0 for somei, we obtain λ1|z|²>hAz, zi. SinceAis symmetric and positive definite,

λ₁|z|²> µ_min|z|² and λ₁> µ_min. This proves the first claim of the theorem.

For A≥0 and irreducible, let v be the eigenvector of A^t corresponding to µmax, then from the Perron-Frobenius Theorem [8],vi>0 for anyi and

λ1hz, vi>hAz, vi= z, A^tv

=µmaxhz, vi

and sincehz, vi>0, it follows thatλ₁> µ_maxand the second claim is proved.

Finally when A ≥0 is symmetric, we have µ_max = kAk, and the proof is

complete.

Proof of Theorem 1.4 The proof of Theorem 1.4 consists of the next lemma and the next corollary.

Lemma 3.6 Suppose that u∈C²(Ω,Rⁿ)is a classical solution of the gradient elliptic system

−∆u_i=g_i(u) in Ω u= 0on ∂Ω whereg_i= _∂u^∂G

i,G∈C¹(Rⁿ),G(0) = 0andΩ⊂R^N is a bounded open set with smooth boundary. Then for a fixedy,

n

X

k=1

Z

∂Ω

|∇u_k|²(x−y)·n(x)dS= 2N Z

Ω

G(u)dx−(N−2)

n

X

k=1

Z

Ω

g_k(u)u_kdx

Proof Multiply thek-th equation by (x−y)· ∇uk =PN

i=1(xi−yi)^∂u_∂x^k

i and integrate by parts, then we have

Z

Ω N

X

i=1

(xi−yi)∂u_k

∂xi

gk(u)

= Z

Ω

|∇u_k|²+ Z

Ω n

X

i,j=1

(x_i−y_i)∂u_k

∂xj

∂²u_k

∂xixj

− Z

∂Ω

|∇u_k|²(x−y)·n(x)dS Hence,

Z

Ω N

X

i=1

(xi−yi)∂uk

∂x_igk(u) = 1−N 2

Z

Ω

|∇uk|²−1 2

Z

∂Ω

|∇uk|²(x−y)·n(x)dS

Adding this identities fork= 1,2, . . . n, Z

Ω N

X

i=1

(xi−yi)

n

X

k=1

∂u_k

∂xi

gk(u)

= 1−N 2

n

X

k=1

Z

Ω

|∇uk|²−1 2

n

X

k=1

Z

∂Ω

|∇u|²(x−y)·n(x)dS By the chain rule we have

Z

Ω N

X

i=1

(xi−yi)

N

X

k=1

∂uk

∂x_igk(u) = Z

Ω N

X

i=1

(xi−yi)∂G(u)

∂x_i

=−N Z

Ω

G(u) +

n

X

i=1

Z

∂Ω

G(u)(xi−yi)·ni(x)dS . SinceG(u) = 0 on∂Ω,

−N Z

Ω

G(u) = (1−N 2)

N

X

k=1

Z

Ω

|∇uj|²−1 2

N

X

k=1

Z

∂Ω

|∇u|²(x−y)·n(x)dS Finally

Z

Ω

|∇u_k|²= Z

Ω

g_k(u)u_k and

N

X

k=1

Z

∂Ω

|∇uk|²(x−y)·n(x) = 2N Z

Ω

G(u)−(N−2)

N

X

k=1

Z

Ω

gk(u)uk

With the following corollary, we complete the proof of Theorem 1.4.

Corollary 3.7 Assume that F ∈ C¹(Rⁿ) is homogeneous of degree p = 2^∗ = 2N/(N−2), withF(0) = 0. Further, assume thatAis symmetric and negative definite, and thatΩis star shaped. Then the system

−∆u_j =f_k(u) +

k

X

j=1

a_jku_j inΩ u= 0 on∂Ω

withfk= _∂u^∂F

k admits only the trivial solution.

Proof LetG(u) =F(u) +¹₂hAu, ui. SinceF is homogeneous of degreep,

N

X

k=1

fk(u)uk =pF(u) and

N

X

k=1

Z

∂Ω

|∇uk|²(x−y)·n(x) = [2N−p(N−2)]

Z

Ω

F(u) + 2

N

X

k=1

Z

Ω

hAu, ui

Sincep= 2N/(N−2),

N

X

k=1

Z

∂Ω

|∇u_k|²(x−y)·n(x) = 2

N

X

k=1

Z

Ω

hAu, ui

Now, becauseA is negative definite,hAu, ui ≤M|u|² whereM <0 and then

N

X

k=1

Z

∂Ω

|∇uk|²(x−y)·n(x)≤2M

n

X

k=1

Z

Ω

|u|²

Since Ω is star shaped, (x−y)·n(x)>0 on ∂Ω, and we conclude thatu= 0.

References

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[2] L. Boccardo and D. de Figueiredo,Some Remarks on a system of quasilin- ear elliptic equations, to appear.

[3] H. Brezis and E. Lieb, A relation between point wise convergence of func- tions and convergence of functionals, Proc. A.M.S. vol. 48, No 3 (1993), pp. 486-499

[4] P. Cl´ement, E. Mitidieri and R., Man´asevich,Positive solutions for a quasi- linear system via blow up,Comm. in Part. Diff. Eq., Vol. 18, No. 12, (1993).

[5] D. de Figueiredo,Semilinear Elliptic Systems, Notes of the course on semi- linear elliptic systems of the EDP Chile-CIMPA Summer School (Univer- sidad de la Frontera, Temuco, Chile, January 1999).

[6] D. de Figueiredo,Semilinear Elliptic Systems: A survey of superlinear prob- lems.Resenhas IME-USP 1996, vol. 2, No. 4, pp. 373-391.

[7] D. Gilbarg, and N. S. Trudinger, Elliptic partial differential equations of second order, Springer- Verlag (1983).

[8] F. R. Grantmacher,The Theory of matrices, Chelsea (1959).

[9] P. L. Lions, The concentration compactness principle in the calculus of variations. The limit case.Rev. Mat. Iberoamericana (1985), pp. 145-201 and 45-121.

[10] S. Pohozaev,Eigenfunctions of the equation∆u+λf(u) = 0.Nonlinearity Doklady Akad. Nauk SSRR 165, (1965), pp. 36-9.

[11] P. Pucci and J. Serrin, A general variational identity, Indiana University Journal, Vol. 35, No. 3, (1986), 681-703.

[12] M. Willem,Minimax Theorems, Birkhauser (1986) Pablo Amster(e-mail: [email protected])

Pablo De N´apoli(e-mail: [email protected])

Maria Cristina Mariani(e-mail: [email protected]) Departamento. de Matem´atica

Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires.

Pabell´on I, Ciudad Universitaria (1428) Buenos Aires, Argentina