Under some suitable assumptions on the nonlinearities, we show the existence result by using a fixed point theorem

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ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3(2011), Pages 220-226.

ON A CLASS OF QUASILINEAR ELLIPTIC SYSTEMS IN R^N INVOLVING CRITICAL SOBOLEV EXPONENTS

(COMMUNICATED BY VICENTIU RADULESCU)

G.A.AFROUZI, T.A.ROUSHAN

Abstract. We study here a class of quasilinear elliptic systems involving the p-Laplacian operator. Under some suitable assumptions on the nonlinearities, we show the existence result by using a fixed point theorem.

1. Introduction and Preliminaries

This paper is concerned with the existence of nontrivial solution to the quasilinear elliptic system of the form







−∆_pu=f(x)|u|^p^∗⁻²u+λ^∂F_∂u(x, u, v), inR^N,

−∆qv=g(x)|v|^q^∗⁻²v+µ^∂F_∂v(x, u, v), inR^N, u(x), v(x)→0, as|x|→+∞

(1.1)

where ∆pis the so calledp-Laplacian operator, i.e. ∆pu=div(| ∇u|^p−2∇u). f, g andF are real-valued functions satisfying some assumptions;uandv are unknown real valued functions defined inR^N and belonging to appropriate function spaces;λ andµare positive parameters, which can be taken equal to 1, and the parameters p and q are real numbers satisfying 2 ≤ p, q < N. The real number p^∗ = _N^{N p}_−p designates the critical Sobolev exponent ofp.

In recent years, several authors use different methods to solve quasilinear equations or systems defined in bounded or unbounded domains. Djellit and Tas [6] investi- gated a system such as (1.1) by employing variational approach.

In this work, motivated by [A. Djellit, S. Tas. On some nonlinear elliptic systems.

Nonl. Anal. 59 (2004), 695-706], we show an existence result by using a fixed point theorem due to Bohnenblust-Karlin.

This paper is divided into three sections, organized as follows: in Section 2,we give some notation and hypotheses; Section 3 is devoted to establish an existence theorem.

2000Mathematics Subject Classification. 35P65, 35P30.

Key words and phrases. p-Laplacian operator, Critical Sobolev exponent, Fixed point theorem.

c

2011 Universiteti i Prishtinës, Prishtinë, Kosovë.

Submitted June 2, 2011. Published August 6, 2011.

220

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2. Notation and hypotheses

We denote byD^1,m(R^N) the completion ofC₀^∞(R^N) in the norm kuk_1,m≡k ∇uk_m= (

Z

R^N

| ∇u|^mdx)^m¹; 1< m < N.

It is well known that D^1,m(R^N) is a uniformly convex Banach space and may be written as

D^1,m(R^N) ={u∈L^m^∗(R^N); ∇u∈(L^m(R^N))^N}.

Moreover, we have the following Sobolev constant defined by

Sm≡C^−m(N, m) = inf

kuk^m_1,m kuk^m_m∗

, u∈D^1,m(R^N)\ {0}

.

We denote Z by the product space Z ≡ D^1,p(R^N)×D^1,q(R^N) with the norm k(u, v)k_Z=kuk_1,p+kvk_1,q; Z^∗ is the dual space ofZ equipped with the dual normk.k_∗.

In addition, letT andN be two operators defined fromZ intoZ^∗ by T(u, v)(w, z) =

Z

R^N

| ∇u|^p−2∇u∇wdx+ Z

R^N

| ∇v|^q−2∇v∇zdx,

and

N(u, v)(w, z) = Z

R^N

[(f(x)|u|^p^∗⁻²u+λ∂F

∂u(x, u, v))w + (g(x)|v|^q^∗⁻²v+µ∂F

∂v(x, u, v))z]dx, ∀(u, v),(w, z)∈Z.

Now, we recall the fixed point theorem due to Bohnenblust-Karlin (see [11]).

Theorem 2.1.([11]) Let Z be a Banach space, let B ⊂Z be a nonempty, closed, convex set and letS:B →2^B be a set-valued mapping satisfying

(a) for eachU ∈Z, the setSU is nonempty, closed and convex, (b)S is closed,

(c) the setS(B) =S

U∈BSU is relatively compact.

ThenS has a fixed point inB i.e. there isU ∈B such thatU ∈SU.

Our aim is to find the condition of the above theorem. The fixed points of the set- valued mappingSare precisely the weak solutions of system (1.1). In other words, we state the existence of a pair (u, v)∈Z such that T(u, v)(w, z) =N(u, v)(w, z),

∀(w, z)∈Z, under the following assumptions.

(H1)f andg are positive and bounded functions.

(H2)F ∈C¹(R^N,R,R) andF(x,0,0) = 0.

(H3) For allU = (u, v)∈R² and for almost everyx∈R^N

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| ∂F

∂u(x, U)|≤a1(x)|U |^p¹⁻¹+a2(x)|U |^p²⁻¹

| ∂F

∂v(x, U)|≤b1(x)|U |^q¹⁻¹+b2(x)|U |^q²⁻¹

where 1< p₁, q₁<min(p, q), max(p, q)< p₂, q₂<min(p^∗, q^∗) a_i∈L^αⁱ(R^N)∩L^βⁱ(R^N), b_i∈L^γⁱ(R^N)∩L^δⁱ(R^N), i= 1,2.

α_i= p^∗ p^∗−pi

, γ_i= q^∗ q^∗−qi

, β_i= p^∗q^∗

p^∗q^∗−p^∗(pi−1)−q^∗, δi= p^∗q^∗

p^∗q^∗−q^∗(qi−1)−p^∗.

3. Existence of solutions The goal of this section is to establish the following result.

Theorem 3.1. Under hypotheses (H₁)−(H₃), the equationT(u, v) =N(u, v) has a solution inZ.

First, two preliminary results. The first one concerns the properties of the oper- atorT while the second one describes the property of the operatorN.

Lemma 3.2. The operatorT is monotone, hemicontinuous, coercive and satisfies the following property:

[(un, vn)*(u, v), T(un, vn)→T(u, v)]⇒(un, vn)→(u, v). (3.1) Proof. Let us denote byT_pthe operator defined fromD^1,p(R^N) into (D^1,p(R^N))^∗ by

T_p(u)w= Z

R^N

| ∇u|^p−2∇u· ∇wdx, ∀u, w∈D^1,p(R^N) andT_q the corresponding one with preplaced byq.

Observe that T(u, v)(w, z) =T_p(u)w+T_q(v)z, ∀(u, v),(w, z)∈Z. T_p, T_q are du- ality mappings on D^1,p(R^N) and D^1,q(R^N) corresponding to the Guage functions Φ_p(T) = t^p−1 and Φ_q(t) = t^q−1, respectively. Hence T_p, T_q are demicontinuous(see[3,p.175]).

So, for (u_n, v_n) → (u, v) in Z, we have T_p(u_n) * T_p(u) in (D^1,p(R^N))^∗ and Tq(vn)* Tq(v) in (D^1,q(R^N))^∗.SinceD^1,p(R^N) andD^1,q(R^N) are reflexive, and the dual space of any reflexive space is reflexive. we getT(un, vn) =Tp(un) +Tq(vn)* Tp(u) +Tq(v) =T(u, v) in Z^∗, i.e. T is demicontinuous. So, it is hemicontinuous.

We note according to [2] that∀λ, µ∈R^N

|λ−µ|^p≤(|λ|^p−2λ− |µ|^p−2µ)·(λ−µ) ifp≥2.

Replacingλandµby∇u,∇v respectively and integrating overR^N, we obtain Z

R^N

| ∇u− ∇v|^p≤ Z

R^N

(| ∇u|^p−2∇u− | ∇v|^p−2∇v)·(∇u− ∇v) ifp≥2. (3.2)

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By virtue of (3.2) we show thatTp (similarlyTq) is monotone, indeed, (Tpu−Tpw)(u−w) = Tpu(u−w)−Tpw(u−w)

= Z

R^N

(| ∇u|^p−2∇u∇(u−w)dx

− Z

R^N

(| ∇w|^p−2∇w∇(u−w)dx

= Z

R^N

(| ∇u|^p−2∇u− | ∇w|^p−2∇w)(∇u− ∇w)dx

≥ Z

R^N

| ∇u− ∇w|^p=ku−wk^p_1,p≥0.

So, T is monotone.On the other hand, T is coercive since T(u, v)(u, v) =k uk^p_1,p +kvk^q_1,q. Now we show thatT satisfies property (3.1).

Let us take a sequence (u_n, v_n)∈Zsuch that (u_n, v_n)*(u, v) inZandT(u_n, v_n)→ T(u, v) inZ^∗. ThenT(u_n, v_n)(u_n, v_n)→T(u, v)(u, v).Soku_nk^p_1,p+kv_nk^q_1,q→ kuk^p_1,p+kvk^q_1,q .According to the uniform convexity of Z, (un, vn)→(u, v) in

Z.

Lemma 3.3. Under hypothesis (H₁)−(H₃), the operatorN is compact.

Proof. LetB_Rbe the ball of radiusR, centered at the origin ofR^N. We putB_R⁰ = R^N−BRand we designateNRthe operator defined fromZR≡D^1,p(BR)×D^1,q(BR) intoZ_R^∗ by

NR(u, v)(w, z) = Z

BR

[(f(x)|u|^p^∗⁻²u+λ∂F

∂u(x, u, v))w + (g(x)|v|^q^∗⁻²v+µ∂F

∂v(x, u, v))z]dx.

Let{(un, vn)} be a bounded sequence in Z.There is a subsequence denoted again as{(un, vn)}, weakly convergent to (u, v) inZ. For (w, z)∈Z, we have

|N(un, vn)(w, z)−N(u, v)(w, z)|

=|NR(un, vn)(w, z)−NR(u, v)(w, z)| +|

Z

B_R⁰

f(x)(|un|^p^∗⁻²un− |u|^p^∗⁻²u)wdx| +|

Z

B_R⁰

g(x)(|vn|^q^∗⁻²vn− |v|^q^∗⁻²v)zdx| +|

Z

B_R⁰

λ(∂F

∂u(x, un, vn)−∂F

∂u(x, u, v))wdx| +|

Z

B_R⁰

µ(∂F

∂v(x, u_n, v_n)−∂F

∂v(x, u, v))zdx|. (3.3) Since the restriction operator (u, v) → (u, v)|B_R is continuous from D^1,p(R^N)× D^1,q(R^N) into D^1,p(BR)×D^1,q(BR), we have (un, vn) * (u, v) in D^1,p(BR)× D^1,q(BR).We have also that the embeddingsD^1,p(BR),→L^p(BR) andD^1,q(BR),→ L^q(BR) are compact, so

un→u a.e. inBR, v_n→v a.e. inB_R.

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Hypothesis (H3) gives

| ∂F

∂u(x, un, vn)w| ≤ [a1(x)(|un|^p¹⁻¹+|vn|^p¹⁻¹)

+a2(x)(|un|^p²⁻¹+|vn|^p²⁻¹)]|w|, (3.4) and

| ∂F

∂v(x, un, vn)z| ≤ [b1(x)(|un|^q¹⁻¹+|vn|^q¹⁻¹)

+b2(x)(|un|^q²⁻¹+|vn |^q²⁻¹)]|z|. (3.5) Using Holder’s inequality and Sobolev’s imbedding, and the fact that

a_i∈L^αⁱ(R^N)TL^βⁱ(R^N), b_i∈L^γⁱ(R^N)TL^δⁱ(R^N), we get that the right hand side of inequalities (3.4), (3.5) belong toL¹(BR). Hence under hypotheses (H1)−(H3) and by using Holder’s inequality and Sobolev’s imbedding, according to Dominated convergence theorem, we obtain,the first expression on the right hand side of the inequality (3.3) tends to 0 as n →+∞; Taking (H1) and (H3) into account, and the fact that fori= 1,2,

ka_ik_Lαi(B⁰_R)+ka_ik_Lβi(B_R⁰)→0, kb_ikLγi(B_R⁰)+kb_ik_Lδi(B⁰_R)→0,

asR→+∞; we obtain, the other expressions tend also to 0 asRsufficiently large.

So, the compactness ofN follows.

Lemma 3.4. Suppose that (H1) and (H3) hold. There is a constant k >0 such thatT(u, v) =N(σ, ρ) andk(σ, ρ)kZ≤kimpliesk(u, v)kZ≤k.

Proof. Let (u, v),(σ, ρ)∈Z be such that T(u, v) =N(σ, ρ), then T(u, v)(w, z) =N(σ, ρ)(w, z), ∀(w, z)∈Z.

In particular, we haveT(u, v)(u,0) =N(σ, ρ)(u,0) i.e.

kuk^p_1,p= Z

R^N

| ∇u|^pdx= Z

R^N

(f(x)|σ|^p^∗⁻²σ+λ∂F

∂u(x, σ, ρ))udx. (3.6) In view of (H₁) and (H₃), by using Holder’s inequality and Sobolev’s imbedding we obtain

Z

R^N

f(x)|σ|^p^∗⁻²σudx ≤ c⁰ Z

R^N

|σ|^p^∗⁻¹|u|dx

≤ c⁰ kuk_p^∗kσk^p_p^∗∗⁻¹≤c₁kuk_1,pkσk^p_1,p^∗⁻¹, (3.7) and

Z

R^N

λ∂F

∂u(x, σ, ρ)udx ≤ c1kuk1,p(ka1kα₁kσk^p_1,p¹⁻¹+ka1kβ₁kρk^p_1,q¹⁻¹ +ka₂kα2kσk^p_1,p²⁻¹+ka₂kβ2kρk^p_1,q²⁻¹) (3.8) So, by virtue of (3.6), (3.7) and (3.8) we get

kuk^p−1_1,p ≤ c₁λ(kσk^p_1,p^∗⁻¹+ka₁kα1kσk^p_1,p¹⁻¹+ka₁kβ1kρk^p_1,q¹⁻¹ +ka₂kα2kσk^p_1,p²⁻¹+ka₂kβ2kρk^p_1,q²⁻¹). (3.9)

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In the same way, we have

kvk^q−1_1,q ≤ c2µ(kρk^q_1,q^∗⁻¹+kb1kδ₁kσk^q_1,p¹⁻¹+kb1kγ₁kρk^q_1,q¹⁻¹ +kb2kδ₂kσk^q_1,p²⁻¹+kb2kγ₂kρk^q_1,q²⁻¹). (3.10) Ifk (σ, ρ)kZ=kσk1,p+k ρk1,q≤k, we have kσk1,p≤k andk ρk1,q≤k. So, in view of (3.9), (3.10) we get

kuk^p−1_1,p ≤ c λ(k^p^∗⁻¹+k^p¹⁻¹+k^p²⁻¹), kvk^q−1_1,q ≤ c µ(k^q^∗⁻¹+k^q¹⁻¹+k^q²⁻¹).

Sincep₁< p₂< p^∗ andq₁< q₂< q^∗,there is ak >0 such thatc(k^p^∗⁻¹+k^p¹⁻¹+ k^p²⁻¹)≤(^k₂)^p−1 andc(k^q^∗⁻¹+k^q¹⁻¹+k^q²⁻¹)≤(^k₂)^q−1.So,kσk1,p+kρk1,q≤k implieskuk1,p+kvk1,q≤k.

We have on the following proposition, which is standard in the theory of monotone operators.

Proposition 3.5. Let X be a real normed space, T : X → X^∗ be a monotone, hemicontinuous operator and letw∈X,f ∈X^∗.

The following two assertions are equivalent (a)T w=f

(b)hT z−f, z−wi ≥0 for allz∈X.

Now, we are ready to give the following proof.

Proof of Theorem 3.1. In view of lemma 3.4, let B ⊂Z be the closed ball of radiuskcentered at the origin. We define the operatorS fromB into 2^B by

(σ, ρ)7→S(σ, ρ) ={(u, v); T(u, v) =N(σ, ρ)}.

By virtue of lemma 3.2, T is monotone, hemicontinuous and coercive, then according to Browder’s Theorem (see[13,p.557]),S(σ, ρ) is nonempty, convex, closed and bounded for every (σ, ρ) ∈ B. Furthermore, the operator S is closed, indeed, let {(σn, ρn)} ⊂ B; (σn, ρn) → (σ, ρ) ∈ Z, and {(un, vn)} ⊂ Z such that (un, vn)∈S(σn, ρn) and (un, vn)→(u, v) inZ.

SinceNis continuous, it is demicontinuous. We have also thatT is demicontinuous, so we can write

T(u_n, v_n)* T(u, v), N(σn, ρn)* N(σ, ρ).

Since (un, vn) ∈ S(σn, ρn), we have T(un, vn) = N(σn, ρn). Hence T(un, vn) * N(σ, ρ) .Since the weak limit is unique, we get

T(u, v) =N(σ, ρ).

On the other hand ,B is closed, consequently (σ, ρ)∈B and then (u, v)∈S(σ, ρ).

Now, let us show thatS(B) =S

(σ,ρ)∈BS(σ, ρ) is relatively compact.

Let (un, vn)⊂S

(σ,ρ)∈BS(σ, ρ) and (σn, ρn)⊂B be such that

T(un, vn) =N(σn, ρn). (3.11) In view of lemma 3.3, N(B) is relatively compact. So there exists H ∈ Z^∗ such that N(σ_n, ρ_n) → H, Hence by (3.11) we have T(u_n, v_n) → H. Consequently T(u_n, v_n) is bounded. Since T is coercive, (u_n, v_n) is also bounded; otherwise, if

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k(un, vn)k→ ∞,we haveT(un, vn)→ ∞, which is a contradiction. Hence, we may choose a subsequence denoted again by{(un, vn)}, weakly convergent to (u0, v0) in Z.

The monotonicity ofT leads to (T(u, v)−T(u_n, v_n))(u−un, v−vn)≥0,∀(u, v)∈Z, and passing to the limit, we obtain

(T(u, v)−H)(u−u0, v−v0)≥0, ∀(u, v)∈Z,

i.e. hT(u, v)−H,(u, v)−(u0, v0)i ≥ 0,∀(u, v) ∈ Z. So by virtue of proposition 3.5, we have T(u0, v0) = H. Taking the condition (3.1) into account, we obtain the convergence of (un, vn) to (u0, v0). Finally, by Bohnenblust-Karlin fixed point theorem,Spossesses a fixed point.i.e. there exist (σ0, ρ0)∈Bsuch thatT(σ0, ρ0) = N(σ0, ρ0).

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G.A. Afrouzi, T.A. Roushan, Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

E-mail address:[email protected]; [email protected]