We show the existence of the negative solution for the problem

http://dx.doi.org/10.11568/kjm.2015.23.2.259

DIRICHLET BOUNDARY VALUE PROBLEM FOR A CLASS OF THE NONCOOPERATIVE ELLIPTIC

SYSTEM

Tacksun Jung^∗ and Q-Heung Choi^†

Abstract. This paper is devoted to investigate the existence of the solutions for a class of the noncooperative elliptic system involving critical Sobolev exponents. We show the existence of the negative solution for the problem. We show the existence of the unique neg- ative solution for the system of the linear part of the problem under some conditions, which is also the negative solution of the nonlinear problem. We also consider the eigenvalue problem of the matrix.

1. Introduction

Let Ω be a bounded domain of Rⁿ with smooth boundary ∂Ω, n ≥ 3, α, β, γ be real constants and φi, i ≥ 1, be the normalized eigenfunctions associated to eigenvalues λi of the eigenvalue problem −∆u = λu in Ω, u|_∂Ω = 0. We note that φ₁ is the positive normalized eigenfunction associated to λ₁. In this paper we investigate the existence of the solu- tions for the following class of the systems of the noncooperative elliptic

Received January 14, 2015. Revised June 2, 2015. Accepted June 2, 2015.

2010 Mathematics Subject Classification: 35J50, 35J55.

Key words and phrases: Noncooperative elliptic system, critical Sobolev expo- nents nonlinear term, eigenvalue problem of the matrix.

∗Corresponding author.

†This work was supported by Basic Science Research Program through the Na- tional Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (KRF-2013010343).

The Kangwon-Kyungki Mathematical Society, 2015.c

This is an Open Access article distributed under the terms of the Creative com- mons Attribution Non-Commercial License (http://creativecommons.org/licenses/by -nc/3.0/) which permits unrestricted non-commercial use, distribution and reproduc- tion in any medium, provided the original work is properly cited.

equations with Dirichlet boundary condition







−∆u = αu + βv + _p+q^2p u^p−1₊ v₊^q + ¯f in Ω,

∆v = βu + γv + _p+q^2q u^p₊v₊^q−1+ ¯g in Ω,

u = v = 0 on ∂Ω,

(1.1)

where u₊ = max{u, 0}, p, q > 1 are real constants, p + q = 2^∗, 2^∗ = _n−2²ⁿ , n ≥ 3 and we may write ¯f , ¯g as

f = τ φ¯ ₁+ f, g = tφ¯ ₁+ g, where τ, t are real constants and f , g ∈ L²(Ω) with

Z

Ω

f φ₁ = Z

Ω

gφ₁ = 0.

Our problems are characterized as Ambrosetti-Prodi type problems.

Since the pioneering work on the subject in [1], these problem have been investigated in many ways. For a survey on the scalar case we recommend the paper [3] and the references therein. In the last decades, some works on the matter were published focusing some other obsta- cles added to this kind of nonlinearities problems having critical growth and the case involving systems. Ambrosetti-Prodi type problems for the critical growth case were studied in [4] and in [5]. For systems, these problems were firstly investigated in [7], and lately, in [2].

Let H₀¹(Ω) be the Sobolev space and H = H₀¹(Ω) × H₀¹(Ω). Then H is a Hilbert space endowed with the norm

k(u, v)k²_H = kuk²_H1

0(Ω)+ kvk²_H1 0(Ω).

We are looking for the weak solutions of (1.1) in H. The weak solutions in H satisfy

Z

Ω

[(−∆u, ∆v) · (z, w) − (αu + βv, βu + γv) · (z, w)

−(τ φ1+ f, tφ1+ g) · (z, w)]dx = 0 ∀(z, w) ∈ H.

(1.2)

We observe that the weak solutions of (1.1) coincide with the critical points of the the associated functional

J : H → R ∈ C^1.1, J (u, v) = Q_αβγ(u, v) −

Z

Ω

[ 2

p + qu^p₊v^q₊+ τ φ₁u + f u + tφ₁v + gv]dx, (1.3)

where

Q_αβγ(u, v) = 1 2

Z

Ω

[|∇u|²− |∇v|²− αu²− 2βuv − γv²]dx.

Let B be  α β β γ



∈ M_2×2(R) and µ⁺_λ

i and µ⁻_λ

i be the eigenvalues of the matrix

 λi− α −β

−β −λi− γ



∈ M_2×2(R), i. e.,

µ⁺_λ

i = 1

2{−γ − α +p

(−γ − α)²− 4{(λ_i− α)(−λ_i− γ) − β²}}, µ⁻_λ

i = 1

2{−γ − α −p

((−γ − α))²− 4{(λ_i− α)(−λ_i− γ) − β²}}.

We note that if −γ = α, then µ⁺_λ

i = µ⁻_λ

i = 0,

if − γ > α and 4{(λ₁− α)(−λ₁− γ) − β²} > 0, then 0 < µ⁻_λ

1 < µ⁺_λ

1, if − γ < α and 4{(λ1− α)(−λ1− γ) − β²} > 0, then µ⁻_λ1 < µ⁺_λ1 < 0, and

if 4{(λi− α)(−λi− γ) − β²} < 0, i ≥ 2, then µ⁻_λ

i < 0 < µ⁺_λ

i. In this paper we show the existence of the unique negative solution for the system of the linear part of the problem under some conditions, which is also the negative solution of (1.1). We also consider some property of the eigenvalue problem of the matrix.

The outline of the proofs of main results are as follows: In section 2, we show the existence of the unique negative solution of the linear part of (1.1), which is also a negative solution under the conditions (B1)-(B3) and some condition for τ and tso. In section 3, we obtain some result for the eigenvalue problem of the matrix.

2. A negative solution

Lemma 2.1. Assume that the conditions (B1), (B2) and (B3) hold.

Let N_αβγ : H → H be the operator defined by N_αβγ(u, v) = (−∆u − αu − βv, ∆v − βu − γv). Then the operator

N_αβγ⁻¹ : H → H

is well defined and continuous, and the system







−∆u = αu + βv + f in Ω,

∆v = βu + γv + g, in Ω,

u = v = 0 on ∂Ω

(2.1) has a unique solution (u_{f g}, v_{f g}), which is of the form

u_{f g} =X

m

( (−λ_m− γ)f_m+ βg_m

(λ_m− α)(−λ_m− γ) − β²)φ_m, v_{f g} =X

m

( (λ_m− α)g_m+ βf_m

(λ_m− α)(−λ_m− gamma) − β²)φ_m, where f = P

mf_mφ_m with P

mf_m² < +∞ and g = P

mg_mφ_m with P

mg_m² < +∞.

Proof. let us take (f, g) in H. We can write f = P

mf_mφ_m with P

mf_m² < +∞ and g = P

mg_mφ_m with P

mg_m² < +∞. We define, for m integers,

u_m = (−λ_m− γ)f_m+ βg_m

(λm− α)(−λ_m− γ) − β², v_m = (λ_m− α)g_m+ βf_m (λm− α)(−λ_m− γ) − β², which make sense since (λ_m − α)(−λ_m − γ) − β² 6= 0 for every m. We have

|u_m| ≤ C

|λ_m|(|f_m| + |g_m|), from which it follows that

λ²_mu²_m ≤ C₁(f_m² + g_m²)

for suitable constants C, C₁ not depending on m. We apply the same in- equality for vm. So if uf g =P

mumφm, vf g =P

mvmφm, then (uf g, vf g) ∈ H. We can check easily that

N_αβγ(u_{f g}, v_{f g}) = (f, g).

So N⁻¹ : H → H is well defined, so the lemma is proved.

Using Lemma 2.1 we can derive the following Lemma 2.2.

Lemma 2.2. Assume that the conditions (B1), (B2) and (B3) hold.

Then the system







−∆u = αu + βv in Ω,

∆v = βu + γv, in Ω, u = v = 0 on ∂Ω

(2.2)

has only the trivial solution (0, 0).

Lemma 2.3. Assume that the conditions (B1), (B2) and (B3) hold.

Then for any (τ, t) with τ < 0 and t < 0, the linear system







−∆u = αu + βv + τ φ₁ in Ω,

∆v = βbu + γv + tφ₁ in Ω, u = v = 0 on ∂Ω

(2.3)

has a unique positive solution (u_{τ t}, v_{τ t}) ∈ H, which is of the form uτ t= [ β²τ + βt(λ₁− α)

(λ₁− α)((λ₁− α)(−λ₁− γ) − β²)+ τ

λ₁− α]φ1 < 0, v_{τ t}= [ βτ + t(λ₁− α)

(λ1− α)(−λ1− γ) − β²]φ₁ < 0.

Since for any (τ, t) with τ < 0 and t < 0, u_{τ t} < 0 and v_{τ t} < 0, we can choose (τ₀, t₀) such that for any (τ, t) with τ < τ₀ and t < t₀, u₀ = u_{f g}+ u_{τ t}< 0 and v₀ = v_{f g} + v_{τ t}< 0.

Proof. We note that (u_{τ t}, v_{τ t}) is a solution of system (2.3) with u_{τ t}< 0 and v_{τ t} < 0 for any (τ, t) with τ < 0 and t < 0, and the uniqueness is the consequence of Lemma 2.1.

Combing Lemma 2.1 with Lemma 2.3, we obtain the following lemma.

Lemma 2.4. Assume that the conditions (B1), (B2) and (B3) hold.

there exist constants τ₀ < 0 and t₀ < 0 such that for any (τ, t) with τ < τ₀ and t < t₀, the linear system







−∆u = αu + βv + τ φ₁+ f in Ω,

∆v = βu + γv + tφ₁+ g in Ω, u = v = 0 on ∂Ω

(2.4) has a unique negative solution (u₀, v₀) ∈ H with u₀ < 0 and v₀ < 0, which is of the form

u0 = X

m

( (−λ_m− γ)f_m+ βg_m (λ_m− α)(−λ_m− γ) − β²)φm

+[ β²τ + βt(λ₁− α)

(λ₁− α)((λ₁− α)(−λ₁− γ) − β²) + τ

λ₁− α]φ₁, v₀ = X

m

( (λ_m− α)g_m+ βf_m

(λ_m− α)(−λ_m− γ) − β²)φ_m+ [ βτ + t(λ₁ − α)

(λ₁− α)(−λ₁− γ) − β²]φ₁,

where f = P

mfmφm with P

mf_m² < +∞ and g = P

mgmφm with P

mg_m² < +∞.

Proof. Since for any (τ, t) with τ < 0 and t < 0, u_{τ t} > 0 and v_{τ t}< 0, we can choose (τ₀, t₀) such that for any (τ, t) with τ > τ₀ and t < t₀, u₀ = u_{f g}+ u_{τ t}< 0 and v₀ = v_{f g} + v_{τ t}< 0.

Theorem 2.1. Assume that (B1) det λ₁− α −β

−β −λ₁− γ



> 0, (B2) det λi− α −β

−β −λi− γ



< 0 for i ≥ 2, (B3) α > 0, β > 0, γ < 0, λ₁− α > 0.

Then there exists (τ₀, t₀) with τ₀ < 0 and t₀ < 0 such that for any (τ, t) with τ < τ0 and t < t0, (1.1) has at least one negative solution (u, v).

Proof. We note that the negative solution (u₀, v₀) with u₀ < 0 and v₀ < 0 of (2.4) is also a negative solution of (1.1).

3. Eigenvalue problem of the matrix Let us set

Eλi = span{φj| λj = λi}.

Let us denote by (ξ_λ⁺

i, η_λ⁺

i) and (ξ_λ⁻

i, η_λ⁻

i) the eigenvectors of  α β β γ



∈ M_2×2(R) corresponding to µ⁺_λ

i and µ⁻_λ

i respectively. Let us also set D_λ1 = {(α, β, γ) ∈ R³| (λ₁− α)(−λ₁− γ) − β² ≥ 0},

D_λi = {(α, β, γ) ∈ R³| (λ_i− α)(−λ_i− γ) − β² ≤ 0} for i ≥ 2, D⁰_λi = D_λi∩ {−γ ≤ α},

D⁰⁰_λi = D_λi∩ {−γ ≥ α},

H_λi = {(ξφ, ηφ) ∈ H| (ξ, η) ∈ R², φ ∈ E_λi}, H_λ⁺

i = {(ξ_λ⁺

iφ, η_λ⁺

iφ) ∈ H| φ ∈ E_λi}, H_λ⁻

i = {(ξ_λ⁻

iφ, η_λ⁻

iφ) ∈ H| φ ∈ E_λi},

X⁺(α, β, γ) = (⊕_µ⁺

λi>0H_λ⁺

i) ⊕ (⊕_µ⁻

λi>0H_λ⁻

i), X⁻(α, β, γ) = (⊕_µ⁺

λi<0H_λ⁺

i) ⊕ (⊕_µ⁻

λi<0H_λ⁻

i), X⁰(α, β, γ) = (⊕_µ⁺

λi=0H_λ⁺

i) ⊕ (⊕_µ⁻

λi=0H_λ⁻

i).

Then X⁺(α, β, γ), X⁻(α, β, γ) and X⁰(α, β, γ) are the positive, negative and null space relative to the quadratic form Q_α,β,γ in H. Because (λ_i− α)(−λ_i− γ) − β² 6= 0, X⁰(α, β, γ) = {0}.

The following theorem can be obtained by easy computations.

Theorem 3.1. Assume that the conditions (B1), (B2) and (B3) hold.

Let (α, β, γ) ∈ R³ and i ∈ N . Then (i) H_λ⁺

i and H_λ⁻

iare the eigenspaces for the operator N_α,β,γ, N_α,β,γ(u, v) = (−∆u − αu − βv, ∆v − βu − γv) associated with Q_α,β,γ with eigenvalues

µ⁺_λi

λi and ^µ

− λi

λi respectively.

(ii) H_λ⁺

i and H_λ⁻

i generate H.

(iii)

If (α, β, γ) ∈ D⁰_λ

1, µ⁻_λ

1 < µ⁺_λ

1 < 0, if (α, β, γ) ∈ D⁰⁰_λ1, 0 < µ⁻_λ

1 < µ⁺_λ

1, if i ≥ 2, µ⁻_λ

i < 0 < µ⁺_λ

i,

i→∞lim µ⁻_λ

i

λ_i = −1, lim

i→∞

µ⁺_λ

i

λ_i = 1 for i ≥ 2, lim

(α,β,γ)→(α0,β0,γ0)µ⁻_λ

i(α, β, γ) = µ⁻_λ

i(α₀, β₀, γ₀) and

lim

(α,β,γ)→(α0,β0,γ0)µ⁺_λ

i(α, β, γ) = µ⁺_λ

i(α₀, β₀, γ₀).

uniformly with respect to i ∈ N .

Assume that the conditions (B1), (B2) and (B3) hold. Let us choose (α0, β0, γ0) ∈ ∂D_λ⁰_i and U be an any neighborhood of (α0, β0, γ0). Then U = (U ∩ D_i⁰) ⊕ (U \D_λ⁰

i),

if (α, β, γ) ∈ U ∩ D⁰_λ1, then µ⁻_λ1 < µ⁺_λ1 ≤ 0, so X⁺(α, β, γ) = ∅.

if (α, β, γ) ∈ U \D_λ⁰₁, then 0 ≤ µ⁻_λ

1 < µ⁺_λ

1, so X⁻(α, β, γ) = ∅, if i ≥ 2 and (α, β, γ) ∈ U, then µ⁻_λ

i ≤ 0 ≤ µ⁺_λ

i, so X⁺(α, β, γ) 6= ∅, X⁻(α, β, γ) 6= ∅.

Thus for all i ∈ N and (α0, β0, γ0) ∈ ∂Dλi, there exist a neighborhood U of (α0, β0, γ0) such that for all (α, β, γ) ∈ U , X⁺(α, β, γ) can be split as an orthogonal sum

X⁺(α, β, γ) = Y2(α, β, γ) ⊕ Y3(α, β, γ) with dim(Y2(α, β, γ) < +∞.

Competing Interests

The authors declare that they have no competing interests.

Authors’s contributions

All authors contributed equally to the manuscript and read and ap- proved the final manuscript.

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Tacksun Jung

Department of Mathematics Kunsan National University Kunsan 573-701, Korea

E-mail : tsjung@kunsan.ac.kr Q-Heung Choi

Department of Mathematics Education Inha University

Incheon 402-751, Korea E-mail : qheung@inha.ac.kr