EXISTENCE AND MULTIPLICITY OF SOLUTIONS OF A SYSTEM OF QUASILINEAR EQUATIONS

EXISTENCE AND MULTIPLICITY OF SOLUTIONS OF A SYSTEM OF QUASILINEAR EQUATIONS

PETRE SORIN ILIAS¸

Investigate the existence and multiplicity of weak solutions of the system of quasi- linear elliptic equations

−∆pu=Fu(x, u, v) in Ω

−∆_qv=Fv(x, u, v) in Ω

u=v= 0 on∂Ω.

Here 1 < p, q < ∞, Ω is a bounded domain in R^N and F : Ω×R² → R is a C¹ Carath´eodory function. We show the existence and multiplicity of weak solutions inW₀^1,p(Ω)×W₀^1,q(Ω) of the system whenF and its partial derivates with respect touandvsatisfies some growth conditions. A key role in our proof is played by Fountain Theorem which is applied for C¹ functional defined on W₀^1,p(Ω)×W₀^1,q(Ω).

AMS 2000 Subject Classification: 34A34, 35B38, 47J30.

Key words: quasilinear system, critical points, Fountain Theorem.

1. INTRODUCTION

The aim of this paper is to prove the existence and multiplicity of weak solutions inW₀^1,p(Ω)×W₀^1,q(Ω) of the system of quasilinear equations

(S)





−∆_pu=Fu(x, u, v) in Ω

−∆_qu=F_v(x, u, v) in Ω

u=v= 0 on ∂Ω,

where the real numbersp, q belongs to (1,+∞) such that the setI = (p, p^∗)∩ (q, q^∗) is nonvoid, p^∗ is the Sobolev conjugate of p, Ω is a bounded domain inR^N,−∆_p and−∆_qstand for p-laplacian, respectivelyq-laplacian operators and the given functionF : Ω×R²→Ris a continuously Fr´echet differentiable Carath´eodory function which satisfies the following conditions:

F(x,−t₁,−t₂) =−F(x, t₁, t₂), x∈Ω, t₁, t₂ ∈R, (1.1)

|F(x, t1, t2)| ≤c(1 +|t1|^r+|t2|^s), x∈Ω, t1, t2 ∈R, (1.2)

REV. ROUMAINE MATH. PURES APPL.,52(2007),2, 201–211

wherec >0 is a constant andr, s∈I;

(1.3) F(x, t1, t2)≥c1(|t₁|^r+|t₂|^s)−c2, x∈Ω, t1, t2∈R, with

(t₁, t₂)_∞≥M, wherec_1,c₂, M >0 are constants andα∈I; (1.4) θF(x, t1, t2)≤ t1∂F

∂t1(x, t1, t2) + t2∂F

∂t2(x, t1, t2), x∈Ω, t1, t2 ∈R, with

(t₁, t₂)_∞≥R, whereR >0 is a constant and θ∈I;

(1.5) ∂F

∂t1(x, t₁, t₂)

≤d₁(1 +|t₁|^m1−1+|t₂|ⁿ¹⁻¹), x∈Ω, t₁, t₂ ∈R, ∂F

∂t₂(x, t1, t2)

≤d2(1 +|t1|^m2−1+|t2|ⁿ²⁻¹) x∈Ω, t1, t2∈R, whered₁, d₂ >0 are constants andm₁, m₂, n₁, n₂ ∈I.

Here Fu and Fv are the partial derivates of F with respect to uand v.

This kind of system has been studied by many authors in the last years (see [1], [2], [3], [6]) and all of them only proved the existence of a weak solution inW₀^1,p(Ω)×W₀^1,q(Ω). In all these articles the idea of the proof is the same and is very simple. Namely, the weak solutions of system (S) correspond to the critical points of theC¹functionalH :W₀^1,p(Ω)×W₀^1,q(Ω)→Rdefined by

H(u, v) = 1 p

Ω|∇u|^pdx+ 1 q

Ω|∇v|^qdx−

ΩF(x, u, v)dx.

Here we use this equivalent characterization of weak solutions of system (S) and prove the existence of critical points for the C¹ functional H, using Fountain Theorem. This variational method has the advantage of proving the existence of a sequence of critical points (un, vn)_n∈N ⊆ W₀^1,p(Ω)×W₀^1,q(Ω) such that lim

n→∞H(u_n, v_n) = +∞. So, we obtain a sequence of weak solutions of system (S) inW₀^1,p(Ω)×W₀^1,q(Ω).

2. PRELIMINARIES

To simplify notation, we consider the reflexive and separable Banach spaceX =W₀^1,p(Ω)×W₀^1,q(Ω) equipped with the norm

(u, v)_∞ = max

u_1,p,v_1,q .

First, we are interested in stating some basic results on the Nemytskii operator (see de Figueiredo [5]). LetM be the set of all measurable function u: Ω→R.

Proposition2.1. If F : Ω×R² →R is a Carath´eodory function, then for each u, v∈M the function NF(u, v) : Ω→Rdefined by

NF(u, v)(x) =F(x, u(x), v(x)), x∈Ω, is measurable in Ω.

By Proposition 2.1, we can associate with the Carath´eodory function F : Ω ×R² → R the operator NF : M ×M → M, called the Nemytskii operator.

The conditions imposed on the Carath´eodory function F ensures that

∂t∂F1, ^∂F_∂t

2 are Carath´eodory functions. Denote by NFu and NFv the Nemytskii operators associated Carath´eodory functions ^∂F_∂t

1, respectively _∂t^∂F

2.

The propositions with the below state sufficient conditions for a Nemyt- skii operator on theL^p1×L^p2 space to map it into anotherL^p3 space.

Proposition 2.2. Suppose F : Ω×R² → Ris a Carath´eodory function and the growth condition

|F(x, t₁, t₂)| ≤c(1 +|t₁|^r+|t₂|^s) x∈Ω, t₁, t₂∈R,

is satisfied, wherec >0is a constant and r, s∈I. Then N_F(L^r(Ω)×L^s(Ω))⊆ L¹(Ω).Moreover, NF is continuous from L^r(Ω)×L^s(Ω)into L¹(Ω)and takes bounded sets into bounded sets.

Proposition2.3. Suppose F : Ω×R² →Ris a C¹-Carath´eodory func- tion such that ^∂F_∂t

1, _∂t^∂F

2, the partial derivates of F with respect of t1 and t2, satisfy the growth conditions

∂F

∂t1(x, t₁, t₂) ≤d₁

1 +|t₁|^m1−1+|t₂|ⁿ¹⁻¹ , x∈Ω, t₁, t₂∈R, ∂F

∂t2(x, t1, t2) ≤d2

1 +|t1|^m2−1+|t2|ⁿ²⁻¹ , x∈Ω, t1, t2∈R, where d₁, d₂ >0 are constants and m₁, m₂, n₁, n₂ ∈I. Then N_Fu(L^m1(Ω)× Lⁿ¹(Ω))⊆L^β(Ω)and N_Fv(L^m2(Ω)×Lⁿ²(Ω))⊆L^β(Ω)for any real number max{m₁, m₂, n₁, n₂}< β <min{p^∗, q^∗}. Moreover, N_Fuand N_Fv are contin- uous from L^m1(Ω)×Lⁿ¹(Ω) into L^β(Ω), respectively from L^m2(Ω)×Lⁿ²(Ω) into L^β(Ω), and take bounded sets into bounded sets.

Remark that the restrictionr, s, m₁, m₂, n₁, n₂∈I ensures the existence of compact imbbedings X → L^r(Ω)×L^s(Ω), X → L^m1(Ω)×Lⁿ¹(Ω), X →

L^m2(Ω)×Lⁿ²(Ω)). It is natural to give results concerning the compactness of some important functionals defined onX.

Proposition 2.4. Assume (1.2). Then the functional Ψ : X → R de- fined by

Ψ(u, v) =

ΩN_F(u, v)(x)dx is well-defined and strongly continuous on X.

Proposition 2.5. Let F : Ω×R² →R be a C¹-Carath´eodory function such that ^∂F_∂t

1, ^∂F_∂t

2 satisfies (1.5). Then the functionals Ψ₁ :X→ W^−1,p(Ω), Ψ₂ :X→W^−1,q(Ω)defined by

Ψ₁(u, v), w₁=

ΩN_Fu(u, v)(x)w₁(x)dx, w₁∈W₀^1,p(Ω) Ψ₂(u, v), w₂=

ΩN_Fv(u, v)(x)w₂(x)dx, w₂ ∈W₀^1,q(Ω) are well-defined and strongly continuous on X.

Second, we state results concerning differentials of someC¹-functionals.

Define as usual the continuous linear applications

pr₁:X →W₀^1,p(Ω), pr₁(u, v) =u, pr₂:X →W₀^1,q(Ω), pr₂(u, v) =v.

Proposition 2.6. Let F : Ω×R² →R be a C¹-Carath´eodory function such that F satisfies (1.2) and (1.5). Then the functional Ψ : X → R de- fined by Ψ(u, v) =

ΩN_F(u, v)(x)dxis continuously Fr´echet differentiable and Ψ(u, v) = Ψ₁(u, v)◦pr₁+Ψ₂(u, v)◦pr₂for all (u, v)∈X,where the functionals Ψ₁ :X→W^−1,p(Ω), Ψ₂:X →W^−1,q(Ω)are defined in Proposition 2.5.

Proposition2.7. Let p, q∈(1,+∞). Then the functional L:X→R defined by L(u, v) = ¹_p

Ω|∇u|^pdx+ ¹_q

Ω|∇v|^qdx is continuously Fr´echet differentiable and L(u, v) = (−∆_pu)◦pr₁+ (−∆_qv)◦pr₂ for all (u, v) ∈X.

Concerning the differential of L, we can generalise the properties of the p-laplacian in case of L.

Proposition 2.8. The differential L :X → X has the following pro- perties:

(i) L is strictly monotone homeomorphism from X into X,that is, L(u1, v1)−L(u2, v2),(u1, v1)−(u2, v2)

>0 for all (u₁, v₁),(u₂, v₂)∈X, (u₁, v₁)= (u₂, v₂);

(ii) L satisfies a (δ+) condition: if (un, vn) (u, v) in X and lim sup

n→∞ L(u_n, v_n),(u_n, v_n)−(u, v) ≤0,then lim

n→∞(u_n, v_n) = (u, v).

3. THE MAIN RESULT

Recall that (u0, v0)∈X is a weak solution of system (S) if and only if

Ω|∇u₀|^p−2∇u₀∇udx=

ΩF_u(x, u₀, v₀)udx

Ω|∇v₀|^q−2∇v₀∇vdx=

ΩFv(x, u0, v0)vdx for all (u, v)∈X.

Now, consider the functionalH :X→R defined by H(u, v) = 1

p

Ω|∇u|^pdx+ 1 q

Ω|∇v|^qdx−

ΩF(x, u, v)dx.

It is obvious, using the notation into from Section 2, that H(u, v) = L(u, v)−Ψ(u, v) for all (u, v)∈X. Moreover, combining Proposition 2.6 and Proposition 2.7, we obtain that H is continuously Fr´echet differentiable on X and H(u, v) = (−∆_pu−Ψ₁(u, v))◦pr₁+ (−∆_qv−Ψ₂(u, v))◦pr₂ for all (u, v)∈X.

We can reformulate the concept of weak solution for system (S) as follows:

(u₀, v₀) ∈ X is a weak solution of system (S) if and only if H(u₀, v₀) = 0 inX.

The main tool in searching for the critical points of H is Fountain The- orem (see Willem [7]). We need the preliminary result below.

Lemma3.1.Let Xbe a reflexive and separable Banach space. Then there exist {e_j}_j∈N⊆X and {f_j}_j∈N ⊆X such that

X= Span{ej |j∈N}, X= Span{fj |j ∈N} and f_j, ej=δij for all i, j∈N.

For convenience, we write X_k = Span{e_k}, Y_k = Span{e₁, e₂, . . . , e_k}, Z_k=⊕^∞_j=kX_j.

Remark 3.1. For all k, p ∈ N with k > p, we have fp(x) = 0 for all x∈Zk.

Theorem 3.1 (Fountain Theorem, [7]). Let X be a reflexive and sepa- rable Banach space, H ∈C¹(X,R) an even functional and the subspaces Xk, Y_k, Z_k defined above. If for each k∈N there exist ϕ_kr_k0 such that

x∈Zkinf,x=rkH(x)→ ∞ask→ ∞, max

x∈Yk,x=ϕkH(x)≤0,

and H satisfies the (PS)condition for every c >0, then H has a sequence of critical values tending to +∞.

Recall thatHis said to satisfy the (PS) condition if any sequence (xn)⊆ X such that H(x_n) ⊆R is bounded and H(x_n) → 0 as n → ∞ , possesses a convergent subsequence. Also, H is said to satisfy the (PS)_c condition for c∈Rif any sequence (x_n)⊆X such that lim

n→∞H(x_n) =cand H(x_n)→0 as n→ ∞, possesses a convergent subsequence. It is clear that if H satisfies the (PS) condition, thenH satisfies the (PS)_c condition for all c∈R.

Proposition 3.1. Assume (1.4) and (1.5). Then the functional H sa- tisfies the (PS) condition.

Proof. Let a sequence (u_n, v_n)⊆X for which H(u_n, v_n)⊆R is bounded andH(un, vn)→0 asn→ ∞. There is d∈Rsuch thatH(un, vn)≥dfor all n∈N. Define the sets

Ω₁ ={x∈Ω| |u_n(x)| ≥R or |v_n(x)| ≥R for all n∈N}, Ω₂ ={x∈Ω| |un(x)|R and |vn(x)|< R for alln∈N}, whereR >0 is the constant from (1.4).

Combining (1.4) and (1.5), it is easy to establish the existence of two constantsk1, k2 >0 such that

Ω2

F(x, un, vn)dx ≤k1,

(1)

Ω2

(unFu(x, un, vn) +vnFv(x, un, vn))dx ≤k2. From (1.4) and (1), we obtain the inequality

(2) d≥ 1

pun^p_1,p+1

q vn^q_1,q−

−1 θ

Ω(unFu(x, un, vn) +vnFv(x, un, vn))dx−k1−k2.

Since H(un, vn) → 0 as n → ∞, there is n0 ∈ N such that

|H(un, vn),(un, vn)| ≤ (un, vn)_∞ forn≥n0. Consequently, for all n≥n0

we have

un^p_1,p+vn^q_1,q−

Ω(unFu(x, un, vn) +vnFv(x, un, vn))dx ≤

≤ (un, vn)_∞≤ un_1,p+vn_1,q, which implies

(3) −1

θ

Ω(unFu(x, un, vn) +vnFv(x, un, vn))dx≥

≥ −1

θu_n1,p−1

θv_n1,q− 1

θu_n^p_1,p−1

θv_n^q_1,q

for alln≥n0. Now, it follows from (2) and (3) that d≥

1 p− 1

θ

u_n^p_1,p+ 1

q −1 θ

v_n^q_1,q−1

θu_n1,p−1

θv_n1,q−k₁−k₂ for all n ≥n₀, and, since θ > p, q, we deduce that the sequences (u_n)_n∈N ⊆ W₀^1,p(Ω) and (v_n)_n∈N⊆W₀^1,q(Ω) are bounded.

Using Smuljian Theorem, we can extract a subsequence (unk, vnk)_k∈N weakly convergent to some (u, v)∈X. AsH(u_nk, v_nk)→0, we get

(4) lim

n→∞

H(u_nk, v_nk),(u_nk, v_nk)−(u, v)

= 0.

But Ψ(unk, vnk),(unk, vnk)−(u, v)≤

≤

Ω|u_nk−u| |F_u(x, unk, vnk)|+|v_nk−v| |F_v(x, unk, vnk)|dx≤

≤ unk−u_βNFu(unk, vnk)_β+vnk−v_βNFv(unk, vnk)_β, so by the properties of the Nemytskii operators NFu and NFv from Proposi- tion 2.3 and the compact imbeddingsX⊆L^m1(Ω)×Lⁿ¹(Ω), X ⊆L^m2(Ω)× Lⁿ²(Ω), we get

(5) lim

n→∞

Ψ(u_nk, v_nk),(u_nk, v_nk)−(u, v)

= 0 From (4) and (5) we obtain that

n→∞lim

L(unk, vnk),(unk, vnk)−(u, v)

= 0.

By Proposition 2.8 the differential ofLsatisfies the (δ+)-condition. Hence we conclude that lim

k→∞(unk, vnk) = (u, v) in X.

We now state some technical lemmas needel in proof of the main result.

Lemma3.1.Suppose that(1.2)is satisfied. Then for all k∈Nthe upper bound α_k = sup

(u, v)_r,s | (u, v) ∈ Z_k, (u, v)_∞ = 1

is infinite, where (u, v)_r,s= max

ur,vs

. Moreover, lim

k→∞α_k= 0.

Proof.By the compact imbeddingX⊆L^r(Ω)×L^s(Ω), the set

(u, v)_r,s | (u, v)∈Z_k,(u, v)_∞= 1

⊆[0,+∞) is bounded. So, α_k is finite.

From the definitions of subspaces Xk, Yk, Zk, it is obvious that αk ≥ α_k+1 ≥0, so that α_k→α ask→ ∞. For eachk∈N we can take (u_k, v_k)∈ Z_k,(u_k, v_k)_∞= 1 such that

(6) 0≤α_k− (u_k, v_k)_r,s 1 k.

As X is reflexive and the sequence {(u_k, v_k)}_k∈N ⊆ X is bounded, we can extract a weakly convergent subsequence. Without loss of generality, we su- posse that (uk, vk) (u0, v0) in X. Recall that X = Span{fj |j ∈N}

and fj(u, v) = 0 for all j > k and (u, v) ∈ Z_k. The weak convergence (u_k, v_k) (u₀, v₀) in X and the previous assertion show that f_j(u₀, v₀) = 0 for allj∈N. Consequently,f(u0, v0) = 0 for allf ∈X and this fact implies (u₀, v₀) = 0.

Now, the compact imbeddingX ⊆L^r(Ω)×L^s(Ω) implies that

(7) lim

k→∞(uk, vk)_r,s= 0.

From (6) and (7) we conclude that lim

k→∞αk = 0.

Lemma3.2. Suppose that (1.2) and (1.3) are satisfied. Then there exist real numbers α1, α2 >0 such that

F(x, t1, t2)≥α1(|t1|^α+|t2|^α)−α2

for allx∈Ω, t1, t2 ∈R.

Proof. We already know that the inequality (8) F(x, t₁, t₂)≥c₁(|t₁|^α+|t₂|^α)−c₂ holds for all (x, t₁, t₂)∈Ω×R² with(t₁, t₂)_∞≥M.

If(t1, t2)_∞< M, by (1.2), we have the inequality |F(x, t1, t2)| ≤c(1 + M^r+M^s). Let σ=c(1 +M^r+M^s)0. It is obvious that

(9) F(x, t₁, t₂)≥ −σ

for all (x, t₁, t₂) ∈Ω×R² with (t₁, t₂)_∞ M. Take α₂ >max{c₂, σ} and 0 < α1 < min

c1,α2−σ 2M^α

. Combining (8) and (9), it is easy to show that F(x, t1, t2)≥α1(|t₁|^α+|t₂|^α)−α2 for all x∈Ω, t1, t2 ∈R.

We shall now prove our main result. The main tool in the proof is Foun- tain Theorem, because the weak solutions of system (S) correspond to the critical points of theC¹ functionalH.

Theorem3.2.If the Carath´eodory function F satisfies(1.1), (1.2), (1.3), (1.4), (1.5) and is continuously Fr´echet differentiable, then system (S) has an infinity of weak solutions {(u_n, v_n)} ⊆ X such that H(u_n, v_n) ≥ 0 for all n∈N and lim

n→∞H(un, vn) = +∞.

Proof.Let the C¹ functional H :X→Rbe defined by H(u, v) = 1

p

Ω|∇u|^pdx+ 1 q

Ω|∇v|^qdx−

ΩF(x, u, v)dx.

From Proposition 3.1 we deduce that the functionalH satisfies the (PS) condition. It follows thatHsatisfies the (PS)_ccondition for allc >0. Because F is odd with respect tot1, t2, the functionalH is even.

Let us verify the conditions in Fountain forH Theorem. First, we prove that for eachk∈N there exists r_k >0 such that

(u,v)∈Zk,(u,v)inf ∞=rkH(u, v)→ ∞ ask→ ∞.

From the inequalities

ΩF(x, u, v)dx≤

Ωc(1+|u(x)|^r+|v(x)|^s)dxand (u, v)_r,s≤αk (u, v)_∞ for all (u, v)∈Zk, we obtain

H(u, v)≥ 1

pu^p_1,p+1

qv^q_1,q−cλ(Ω)−c(α^r_k(u, v)^r_∞+α^s_k(u, v)^s_∞) for all (u, v) ∈Zk. Without loss of generality, suppose thatp ≥q and r ≥s. The previous inequality becomes

(10) H(u, v)≥ 1

pu^p_1,p+1

q v^q_1,q−cλ(Ω)−c(u, v)^r_∞(α^r_k+α^s_k) for all (u, v)∈Z_k with (u, v)_∞≥1.

Under our assumptions, it is obvious that

(11) 1

pu^p_1,p+1

q v^q_1,q ≥ 1

p(u, v)^q_∞

for all (u, v)∈X with(u, v)_∞≥1. By (10) and (11), we have (12) H(u, v)≥ 1

p(u, v)^q_∞−cλ(Ω)−c(u, v)^r_∞(α^r_k+α^s_k)

for all (u, v) ∈Z_k with (u, v)_∞ ≥ 1. Let r_k = [c(s+p)(α^r_k+α^s_k)]^q−r1 . On account of the inequality _q−r¹ < 0 and since lim

k→∞α_k = 0, we have lim

k→∞r_k = +∞. Also, we can suppose thatr_k≥1 for allk∈N. From (12) we then obtain

(13) inf

(u,v)∈Zk,(u,v)∞=rkH(u, v)≥r_k^q 1

p− 1 s+p

−cλ(Ω) for all (u, v)∈Z_k with (u, v)_∞=r_k. It is obvious from (13) that

(u,v)∈Zk,(u,v)inf ∞=rkH(u, v)→ ∞ ask→ ∞.

Next, we prove that for each k∈N there exists ρ_kr_k 0 such that

(u,v)∈Yk,(u,v)max ∞=ϕkH(u, v)≤0. From Lemma 3.2 we have

F(x, t1, t2)≥α1(|t1|^α+|t2|^α)−α2

for all x ∈ Ω, t1, t2 ∈ R, where α1, α2 > 0 and α ∈ I. The functional

α :X→R defined by (u, v)_α =

Ω|u(x)|^α+|v(x)|^αdx _α¹

is a norm on X. The norms _α and _∞ being equivalent on the finite dimensional subspace Y_k, there is a constant σ > 0 such that (u, v)_α ≥ σ(u, v)_∞ for all (u, v) ∈X. Hence, by Lemma 3.2,

H(u, v)≤ 1

pu^p_1,p+1

q v^q_1,q−α₁(u, v)^α_α+α₂λ(Ω) for all (u, v)∈X.

On the finite dimensional subspaceY_k the last inequality becomes H(u, v)≤ 1

pu^p_1,p+1

qv^q_1,q−α₁σ^α(u, v)^α_∞+α₂λ(Ω) for all (u, v)∈Y_k.But

u_1,p ≤ (u, v)_∞,v_1,q ≤ (u, v)_∞ for all (u, v)∈X and, takingw= max{¹_p,¹_q}, we obtain

(14) H(u, v)≤w((u, v)^p_∞+(u, v)^q_∞)−α1σ^α(u, v)^α_∞+α2λ(Ω) for all (u, v)∈Yk.

It is obvious that

t→∞lim w(t^p+t^q)−α₁σ^αt^α+α₂λ(Ω) =−∞.

For each k∈N we can choose a real numberρ_k> r_k>0 such that

(15) w

ϕ^p_k+ϕ^q_k

−α₁σ^αϕ^α_k +α₂λ(Ω)≤0. From (14) and (15) we conclude that

(u,v)∈Yk,(u,v)max ∞=ϕkH(u, v)≤w

ϕ^p_k+ϕ^q_k

−α₁σ^αϕ^α_k +α₂λ(Ω)≤0. Clearly, from Fountain Theorem we deduce that the functionalH has a sequence of critical values tending to +∞. So, we obtain a sequence {(un, vn)}n∈N ⊆ X of critical points of H for which H(un, vn) ≥ 0 and

n→∞lim H(u_n, v_n) = +∞. Thus, this is a sequence of weak solutions of sys- tem (S).

REFERENCES

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[2] D.G. Costa and C.A. Magalhaes,A variational approach to noncooperative elliptic sys- tems.Nonlinear Anal.25(1995), 699–715.

[3] D.G. Costa and C.A. Magalhaes,Variational elliptic problems which are non-quadratic at infinity.Nonlinear Anal.23(1994), 1401–1412.

[4] X. Fan and X. Han,Existence and multiplicity of solutions for p(x)-Laplacian equations in R^N. Nonlinear Anal.59(2004), 173–188.

[5] D.G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detour. Tata Institute of Fundamental Research, Springer-Verlag, 1989.

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Received 1st March 2006 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania [email protected]