Multiplicity of positive solutions to a singular ( p 1 , p 2 ) -Laplacian system with coupled integral

Multiplicity of positive solutions to a singular ( p ₁ , p ₂ ) -Laplacian system with coupled integral

boundary conditions

Jeongmi Jeong

, Chan-Gyun Kim

and Eun Kyoung Lee

1Department of mathematics, Pusan National University, Busan, 609-735, Korea

2Department of Mathematics Education, Pusan National University, Busan, 609-735, Korea

Received 14 July 2015, appeared 1 June 2016 Communicated by Gennaro Infante

Abstract. In this work, we investigate the existence and multiplicity results for positive solutions to a singular(p₁,p2)-Laplacian system with coupled integral boundary condi- tions and a parameter(µ,λ)∈_R³₊. Using sub-super solutions method and fixed point index theorems, it is shown that there exists a continuous surface C which separates R²₊×(0,∞)into two regionsO₁andO₂such that the problem under consideration has two positive solutions for(µ,λ)∈ O₁, at least one positive solution for(µ,λ)∈ C, and no positive solutions for(µ,λ)∈ O₂.

Keywords: nonlocal boundary condition, multiplicity, positive solution, singular weight function.

2010 Mathematics Subject Classification: 34B10, 35J57, 35J92.

1 Introduction

Nonlocal boundary conditions appear when the information on the boundary are connected to values inside the domain. Various types of boundary value problems involving nonlocal conditions have been extensively studied by various methods such as fixed point theorems on cones and the Leray–Schauder alternative, etc. We refer the reader to [11,14,15,24,30–33,37]

and the references therein.

For example, Ma [24] considered







u⁰⁰(t) =g₁(t,u(t),u⁰(t)) +e(t), a.e.t ∈(0, 1), u⁰(0) =0, u(1) =

∑

α_iu(ξ_i),

where g₁ : [0, 1]×_R²→ _Ris continuous,e ∈ C[0, 1], m∈_N, ξ_i ∈ (0,∞)with 0 < ξ₁ <· · · <

ξ_m < 1, and α_i ∈ (0,∞). Using the Leray–Schauder alternative, the existence of at least one

BCorresponding author. Email: cgkim75@gmail.com

solution is obtained for two cases:

∑

α_i 6=1 (Nonresonance) and

∑

α_i =1 (Resonance).

Webb and Infante [31,32] studied the existence of multiple positive solutions of nonlinear differential equations of the form

−u⁰⁰(t) =w₁(t)g₂(t,u(t)), a.e.t ∈(0, 1)

with various nonlocal boundary conditions involving linear functionals on C[0, 1] including the following conditions: either

u(0) =α₁[u], u(1) =α₂[u] or u(0) =α₁[u], u⁰(1) =α₂[u].

Here,w₁ ∈ L¹((0, 1),R+),R+:= [0,∞), g₂ :[0, 1]×_R→_R+ is continuous, and fori∈ {1, 2}, α_i is bounded linear functionals onC[0, 1]involving Stieltjes integrals with signed measures.

Recently, Zhang and Feng [37] studied the following one-dimensional singular p-Laplacian problems of the form

(

λ(ϕ_p(u⁰(t)))⁰+w₂(t)g₃(t,u(t)) =0, t ∈(0, 1), au(0)−bu⁰(0) =R1

0 w3(t)u(t)dt, u⁰(1) =0,

whereλ is a positive parameter, ϕ_p(s) =|s|^p−2s, p> 1, a,b> 0, andw₂,w₃ ∈ L¹((0, 1),R+). Using fixed point index theory on cones of Banach spaces, they obtained several results about the existence, multiplicity, and nonexistence of positive solutions under various assumptions on the nonlinearityg₃(t,s)which satisfiesL¹-Carathéodory condition.

The systems of differential equations equipped with a variety of boundary conditions have been extensively studied by many authors, see, e.g., [2–7,10,19,21,23,25,26,34]. For example, in [25], do Ó et al. considered a class of system of second-order differential equations of the

form 









−u⁰⁰= g₄(t,u,v,a,b), in(0, 1),

−v⁰⁰ =g₅(t,u,v,a,b)_{, in}(_{0, 1})_, u(0) =u(1) =v(0) =v(1) =0,

(1.1) where the nonlinearities g₄ and g₅ are superlinear at the origin as well as at infinity, and a,b∈_R+. Using fixed point theorems of cone expansion/compression type, the upper-lower solutions method and degree argument, it was shown that there exists a continuous curve Γ which splits the positive quadrant of the(a-b)-plane into disjoint setsS₁andS₂such that (1.1) has at least two positive solutions in S₁, has at least one positive solution on the boundary of S₁, and has no positive solutions in S₂. The result was applied to establish the existence and multiplicity of positive radial solutions for a certain class of semilinear elliptic systems in annular domains.

We are concerned with the existence of positive solutions to the following singular(p₁,p₂)- Laplacian system with coupled integral boundary conditions

((_Φ(u⁰(t)))⁰+λh(t)•f(t,u(t)) =_θ, t∈(_{0, 1})_,

u(0) =α[u], u(1) =µ, (P_λ,µ)

where Φ(s₁,s₂) = (ϕ_p1(s₁),ϕ_p2(s₂)), ϕ_pi(s) := |s|^pi−2s with p_i > 1 for i ∈ {1, 2}, θ is the origin of R², α : C[0, 1]×C[0, 1] → _R² is a linear transformation which is defined by, for u∈C[0, 1]×C[0, 1],

α[_u]_:=

Z ₁

0 u(_s)_k(_s)_ds

and k = (k_ij)_2×2 with k_ij ∈ L¹((0, 1),R+) for i,j ∈ {1, 2}, (µ,λ) = (µ₁,µ₂,λ) ∈ _R³₊ is a parameter, and•denotes the entrywise product, i.e.,(a₁,a2)•(b₁,b2):= (a₁b₁,a2b2).

Throughout this paper, we assume the following hypotheses are satisfied unless otherwise stated:

(H₁) f = (f₁,f2) : [0, 1]×_R² → (0,∞)² with f_i ∈ C([0, 1]×_R²,(0,∞)) and h = (h₁,h2) : (0, 1)→(0,∞)² withh_i ∈C((0, 1),(0,∞))satisfying h_i ∈ A_i fori∈ {1, 2}, where

A_i = (

hˆ : Z ¹₂

0 ϕ⁻_pi¹ Z ¹₂

s

hˆ(τ)dτ

! ds+

Z ₁

1 2

ϕ⁻_pi¹ _{Z s}

1 2

hˆ(τ)dτ

ds< _∞ )

;

(H2) fori∈ {1, 2},R1

0(1−s)k_ii(s)ds∈[0, 1); (H₃) detK>0, where

K:= I− _{Z 1}

0

(1−s)k(s)ds

and I is the identity matrix of size 2.

For convenience, we identify (a,b) ∈ _R² with the 1-by-2 matrix a b

if necessary. Con- sequently,α[u]is well defined for u∈C[0, 1]×C[0, 1].

The main purpose of this paper is to study the existence and multiplicity results for posi- tive solutions to problem (P_λ,µ) using sub-super solutions method and fixed point index theo- rems. For sub-super solutions method concerning semilinear problems with nonlocal bound- ary conditions, we refer to [27–29]. It seems not obvious that sub-super solutions method can be applicable to our problem with (_p1,p2)-Laplacian due to the coupled integral boundary condition in (P_λ,µ). Thus we prove a theorem for sub-super solutions (see Theorem 2.12), and it is shown that there exists a continuous surface C which separates R²+×(0,∞) into two regions O₁ and O₂ such that (P_λ,µ) has two positive solutions for (µ,λ) ∈ O₁, at least one positive solution for (µ,λ)∈ C, and no positive solutions for(µ,λ)∈ O₂ (see Theorem3.10).

Deng and Li [9] considered a semilinear elliptic problem of the form











∆u+A(x)u^q=0 in Ω1,

u>0 inΩ1, u∈ H_loc¹ (_Ω1)∩C(_Ω^¯₁), u|_∂Ω1 =0, u→µ₁>0 as|x| →_∞,

(1.2)

where Ω1 =_R^N\ω is an exterior domain inR^N, ω ⊂_R^N is a bounded domain with smooth boundary, N > 2, andq> 1. Among other results, whenq = (N+2)/(N−2)and 0 ≤ A∈ L¹(_Ω1)satisfies certain additional conditions, it was shown that there existsµ^∗ >0 such that (1.2) has at least two positive solutions for µ₁ ∈ (0,µ^∗), exactly one positive solution forµ₁ = µ^∗, and no positive solutions forµ₁∈ (µ^∗,∞). The existence, multiplicity and nonexistence of positive radial solutions to p-Laplacian problems similar to (1.2) were studied in [16–18].

As applications, we study existence results for positive radial solutions to p-Laplacian systems defined in an exterior domain as follows:

(div |∇z₁|^p−2∇z₁

+λK₁(|x|)f^ˆ₁(|x|,z₁,z₂) =0 inΩ, div |∇z2|^p−2∇z2

+λK2(|x|)f^ˆ2(|x|,z₁,z2) =0 inΩ, (1.3) subject to coupled integral boundary conditions

(z₁(x) =R

Ω[l₁₁(|y|)z₁(y) +l₂₁(|y|)z₂(y)]dy on|x|=r₀, z₂(x) =R

Ω[l₁₂(|y|)z₁(y) +l₂₂(|y|)z₂(y)]dy on|x|=r₀, (1.4) or

(z₁(x) =R

Ω[l₁₁(|y|)z₁(y) +l₂₁(|y|)z₂(y)]dy as|x| →_∞, z₂(x) =R

Ω[l₁₂(|y|)z₁(y) +l₂₂(|y|)z₂(y)]dy as|x| →_∞, ^(1.5) where Ω = {x ∈ _R^N : |x| > r0}, r0 > 0, N > p > 1, K_i ∈ C((r0,∞),(0,∞)), l_ij ∈ L¹((r₀,∞)_,R+)_{, and ˆ}f_i ∈C([r₀,∞)×_R²_+,(_0,∞))_fori,j∈ {_{1, 2}}_.

Using the main result (Theorem3.10), we investigate the existence, multiplicity and nonex- istence of positive solutionsz = (z₁,z₂)to (1.3)+(1.4) (resp. (1.3)+(1.5)) satisfyingz(x)→µas

|x| →_∞(resp.z(x) =µon |x|=r₀) for givenµ∈_R²_{+ and}λ∈(_0,∞)(see Corollary4.1).

For u,v ∈ _R², u ≤ v (resp. u < v) meansu_i ≤ v_i (resp. u_i < v_i)for all i ∈ {1, 2}, where u_i andv_i arei-th coordinates ofu andv, respectively. For functions w¹,w² : [0, 1]→ _Rⁿ with n ∈ {_{1, 2}}_, w¹ ≤ w² (_resp. w¹ < w²) _{also means} w¹(t) ≤ w²(t) (_resp. w¹(t) < w²(t)) _for t∈ [0, 1]. We also denoteθ the zero function from[0, 1]toR² as well as the origin ofR².

This paper is organized as follows. In Section 2, well-known theorems such as generalized Picone identity and a fixed point index theorem are recalled, and a solution operator and a theorem for sub-super solutions related to problem (P_λ,µ) are also introduced. In Section 3, the main result in this paper is given (see Theorem3.10). Finally, in Section 4, applications for problem (1.3)+(1.4) or (1.3)+(1.5) are given (see Corollary4.1).

2 Preliminaries

For semilinear problems, we usually use integration by parts twice in order to obtain the useful information for solutions such as a block for parameters λ and a priori estimates for solutions. However, it is not effective for thep-Laplacian problem. The following generalized Picone identity can be used to overcome the difficulty (see Lemma3.1and Lemma 3.3). The identity can be verified by straightforward differentiation, but for completeness, we give the proof of it.

Theorem 2.1(generalized Picone identity, see, e.g., [13,20]). Let us define l_p[y] = (ϕ_p(y⁰))⁰+b₁(t)ϕ_p(y),

Lp[z] = (ϕp(z⁰))⁰+b2(t)ϕp(z),

whereϕ_p(s) =|s|^p−2s, s ∈_{R, p}> 1and b₁,b₂are continuous functions on an interval I. Let y and z be functions such that y, z, ϕ_p(y⁰)_, ϕ_p(z⁰)are differentiable on I and z(t)6= ₀for t ∈ I.Then the

generalized Picone identity can be written as d

dt

|y|^pϕ_p(z⁰)

ϕ_p(z) −yϕ_p(y⁰)

= (b₁−b₂)|y|^p

−

|y⁰|^p+ (p−1)

yz⁰ z

p

−py⁰ϕ_p yz⁰

z

−ylp[y] + |y|^p

ϕ_p(z)^Lp[z]. (2.1) Proof. By straightforward differentiation,

d dt

|y|^pϕ_p(z⁰) ϕ_p(z)

= (|y|^pϕ_p(z⁰))⁰ϕ_p(z)− |y|^pϕ_p(z⁰)(ϕ_p(z))⁰ (ϕ_p(z))²

= ^py

0ϕ_p(y)ϕ_p(z⁰) +|y|^p(L^p[z]−b₂(t)ϕ_p(z))

ϕ_p(z) −(p−1)

yz⁰ z

p

= py⁰ϕ_p yz⁰

z

+ |y|^p

ϕp(z)^Lp[z]−b₂(t)|y|^p−(p−1)

yz⁰ z

p

(2.2)

and d

dt(yϕ_p(y⁰)) =|y⁰|^p+y(l_p[y]−b₁(t)ϕ_p(y)). (2.3) Then subtracting (2.3) from (2.2) yields the identity (2.1).

Remark 2.2. By Young’s inequality, y⁰ϕ_p

yz⁰ z

≤ |y⁰|^p

p +

1− ¹

p

yz⁰ z

p

,

and the equality holds if and only if sgny⁰ =sgn(yz⁰/z)and|y⁰|^p =|yz⁰/z|^p. Thus,

|y⁰|^p+ (p−1)

yz⁰ z

p

−py⁰ϕ_p yz⁰

z

≥0, which implies, by (2.1),

d dt

|y|^pϕ_p(z⁰)

ϕ_p(z) −yϕ_p(y⁰)

≤(b₁−b₂)|y|^p−yl_p[y] + |y|^p

ϕ_p(z)^Lp[z]. (2.4) Now we recall a well-known theorem for the existence of a global continuum of solutions by Leray and Schauder [22] and a fixed point index theorem:

Theorem 2.3(see, e.g., [35, Corollary 14.12]). Let X be a Banach space with X 6={0}and letP be an order cone in X.Consider

x= H(λ,x), (2.5)

where λ ∈ _R+ and x ∈ P.If H : R+× P → P is completely continuous and H(0,x) = 0 for all x∈ P,thenC₊(P),the component of the solution set of (2.5)containing(0, 0), is unbounded.

Theorem 2.4(see, e.g., [12]). Let X be a Banach space,Pbe a cone in X andObe a bounded open set containingθ in X,whereθ is the origin of X. Let A:P ∩ O → P be completely continuous. Suppose that Ax6= νx for all x∈ P ∩∂Oand allν ≥1.Then i(A,O ∩ P,P) =1.

2.1 Solution operator

In this subsection, we define an operator related to problem (P_λ,µ) and prove the complete continuity of it.

Denote X := C[0, 1]×C[0, 1] with norm k(u₁,u2)k_X := ku₁k_∞+ku2k_∞, and P := {u = (u₁,u₂)∈X:u₁,u₂∈ K}. Then(X,k · k_X)is a Banach space andPis a cone inX. Here,C[0, 1] denotes the Banach space of continuous functions u defined on [0, 1] with usual maximum normkuk_∞:=max_t∈[0,1]|u(t)|andK:={u∈C[0, 1]: uis a nonnegative concave function}.

By(H₃), detK>0 and

K⁻¹ = ¹ detK







 1−

Z ₁

0

(1−s)k22(s)ds

Z ₁

0

(1−s)k₁₂(s)ds Z ₁

0

(1−s)k₂₁(s)ds 1−

Z ₁

0

(1−s)k₁₁(s)ds







 .

Then all entries ofK⁻¹are nonnegative by(H₂)and nonnegativity ofk_ij fori,j∈ {1, 2}. Defineβ:R²+×X→_R² by, for(µ,v)∈_R²₊×X,

β[µ,v]:= (β₁[µ,v],β2[µ,v]):=

Z ₁

0

(v(s) +µs)k(s)K⁻¹ds.

Then β[µ_n,v_n] → β[µ₀,v₀] _{in R}² _as (µ_n,v_n) → (µ₀,v₀) _{in R}²₊×X, and β[_µ,v] ∈ _R²_{+ for all} (µ,v)∈_R²₊× P.

Consider the following problem

((_Φ(v⁰(t)−β[µ,v] +µ))⁰+λh(t)•F(µ,v)(t) =θ, t ∈(0, 1),

v(0) =v(1) =θ, ( ˆP_λ,µ)

whereF:= (F₁,F₂):R²+×X→Xis defined by, for(µ,v)∈_R²₊×X,

F(µ,v)(t):= f(t,v(t) + (1−t)β[µ,v] +tµ), t∈[0, 1]. (2.6) ThenF(µ,v)>θ, since f(t,s)>θ for all(t,s)∈ [0, 1]×_R².

For i = 1, 2, define continuous transformations Lⁱ : R²+×X → X by, for t ∈ [_{0, 1}] _and (µ,u),(µ,v)∈_R²₊×X,

L¹(µ,u)(t):= (L¹₁(µ,u),L¹₂(µ,u))(t):=u(t)−((1−t)α[u] +tµ) (2.7) and

L²(_µ,v)(t)_:= (L²₁(_µ,v)_,L²₂(_µ,v))(t)_:=v(t) + ((₁−t)β[_µ,v] +tµ)_{. (2.8)} With the above transformations (2.7) and (2.8), we have the following lemma.

Lemma 2.5. Assume that(H₃)holds. Thenα[u] =β[µ,L¹(µ,u)]for(µ,u)∈ _R²₊×X andβ[µ,v] = α[L²(µ,v)]for(µ,v)∈ _R²₊×X.

Proof. We only show thatα[u] = β[µ,L¹(µ,u)], since the other case can be proved in a similar manner. For (µ,u)∈_R²₊×X,

β[µ,L¹(µ,u)] =

Z ₁

0

(L¹(µ,u)(s) +µs)k(s)K⁻¹ds

=

Z ₁

0

(u(s)−(1−s)α[u])k(s)K⁻¹ds

=

Z ₁

0

u(s)k(s)K⁻¹ds−α[u]

Z ₁

0

(1−s)k(s)K⁻¹ds

=α[u]

I−

Z ₁

0

(1−s)k(s)ds

K⁻¹=α[u], and the proof is complete.

By a non-negative solution (resp. a positive solution)uto problem (P_λ,µ) or (Pˆ_λ,µ), we mean u= (u₁,u₂)is a solution to problem (P_λ,µ) or (Pˆ_λ,µ) which satisfies u_i(t) ≥_{0 (resp.}u_i(t)> ₀₎ for all t∈(0, 1)and alli∈ {1, 2}.

Remark 2.6. (1) Let v = (v₁,v2) be a solution to problem (Pˆ_λ,µ). Then for i ∈ {1, 2}, v⁰_i is decreasing in (0, 1), and v_i is concave on(0, 1). Since v(0) = v(1) = θ, v ∈ P. Moreover, v is a positive solution to problem (Pˆ_λ,µ) if λ > 0, and v = θ only if λ = 0. Similarly, let u be a non-negative solution to problem (P_λ,µ). Then u∈ P, andu is a positive solution to problem (P_λ,µ) if λ>0.

(2)By Lemma2.5, for fixed(λ,µ)∈_R³₊, ifvis a (non-negative) solution to problem (Pˆ_λ,µ), thenu= L²(µ,v)is a non-negative solution to problem (P_λ,µ), and conversely ifuis a solution to problem (P_λ,µ), thenv= L¹(µ,u)is a (non-negative) solution to problem (Pˆ_λ,µ).

Lemma 2.7. Assume that(H₁)–(H₃)hold. For fixed(λ,µ,v)∈(0,∞)×_R²₊×X withµ= (µ₁,µ₂) and i∈ {1, 2}, there exists a unique constant M_i =M_i(λ,µ,v)∈(0, 1)satisfying

(β_i[µ,v]−µ_i)M_i+

Z _Mi

0 ϕ⁻_pi¹

ϕ_pi(µ_i−β_i[µ,v]) +λ Z _Mi

s h_i(τ)F_i(µ,v)(τ)dτ

ds

= −(β_i[µ,v]−µ_i)(1−M_i) +

Z ₁

Mi

ϕ⁻_pi¹

ϕ_pi(−µ_i+β_i[µ,v]) +λ Z _s

Mi

h_i(τ)F_i(µ,v)(τ)dτ

ds.

(2.9)

Proof. Let (λ,µ,v) ∈ (0,∞)×_R²₊×X with µ = (µ₁,µ₂) and i ∈ {1, 2} be fixed. Define a continuous function x_i = x_i(λ,µ,v):(0, 1)→_Rby

x_i(t) =β_i[µ,v]−µ_i+x¹_i(t) +x²_i(t), t ∈(0, 1), where

x¹_i(t) =

Z _t

0 ϕ⁻_pi¹

ϕ_pi(µ_i−β_i[µ,v]) +λ Z _t

s h_i(τ)F_i(µ,v)(τ)dτ

ds and

x²_i(t) =

Z ₁

t ϕ⁻_pi¹

ϕp_i(µ_i−β_i[µ,v])−λ Z _s

t h_i(τ)F_i(µ,v)(τ)dτ

ds.

We claim thatx_iis strictly increasing in(0, 1). Indeed, by(H₁), lim_t→0⁺x¹_i(t) =lim_t→1⁻x²_i(t) = 0, lim_t→1⁻ x¹_i(t)> µ_i−β_i[µ,v], and lim_t→0⁺x²_i(t)<µ_i−β_i[µ,v]. Consequently,

tlim→0⁺x_i(t)<0 and lim

t→1⁻x_i(t)>0.

For 0<t₁<t₂<1, one has x_i¹(t₂)−x¹_i(t₁)>

Z _t2

t₁ ϕ⁻_pi¹

ϕ_pi(µ_i−β_i[µ,v]) +λ Z _t2

s h_i(τ)F_i(µ,v)(τ)dτ

ds and

x²_i(t₂)−x²_i(t₁)>−

Z _t2

t1

ϕ⁻_pi¹

ϕp_i(µ_i−β_i[µ,v]) +λ Z _t1

s h_i(τ)F_i(µ,v)(τ)dτ

ds

>−

Z _t2

t1

ϕ⁻_pi¹

ϕ_pi(µ_i−β_i[µ,v]) +λ Z _t2

s h_i(τ)F_i(µ,v)(τ)dτ

ds.

Thus,x_i(t₂)−x_i(t₁)>0 for 0<t₁< t₂ <1, and there exists a uniqueM_i = M_i(λ,µ,v)∈(0, 1) such thatx_i(M_i) =0. Consequently,M_i satisfies (2.9).

Fori=1, 2, defineT_i :R³+×X→C[_{0, 1}]_{by, for}(_λ,µ,v)∈_R³₊×X,

T_i(λ,µ,v)(t):=



















(β_i[µ,v]−µ_i)t+Rt 0 ϕ⁻_pi¹

ϕp_i(µ_i−β_i[µ,v]) +λRMi

s h_i(τ)F_i(_µ,v)(τ)dτ

ds, 0≤t ≤M_i,

−(β_i[µ,v]−µ_i)(1−t) +R1 t ϕ⁻_pi¹

ϕ_pi(−µ_i+β_i[µ,v]) +λRs

Mih_i(τ)F_i(µ,v)(τ)dτ

ds, M_i ≤t ≤1,

where M_i = M_i(λ,µ,v) ∈ (0, 1)is a constant satisfying (2.9) forλ > 0 and M_i may be taken arbitrary number in(0, 1)forλ=0, since T_i(0,µ,v) =0 for all (µ,v)∈ _R²₊×X.

For i ∈ {1, 2}, by Lemma 2.7, we can see that T_i is well-defined, kT_i(λ,µ,v)k_∞ = T_i(λ,µ,v)(M_i), and (T_i(λ,µ,v))⁰(M_i) = 0. Moreover, since h_i(t)f_i(t,s) ≥ 0 for all (t,s) ∈ [0, 1]×_R²,(T_i(λ,µ,v))⁰ is decreasing in(0, 1)for all(λ,µ,v)∈_R³₊×X. SinceT_i(λ,µ,v)(0) = T_i(λ,µ,v)(1) =0,T_i(_R³₊×X)⊆ K.

Define T : R³+×X → P by T(λ,µ,v) := (T₁(λ,µ,v),T2(λ,µ,v)) for (λ,µ,v) ∈ _R³₊×X.

ThenTis well defined, and by standard argument we have the following lemma.

Lemma 2.8. Assume that (H₁)–(H₃) hold. Let (λ,µ) ∈ _R³₊ be fixed. Then problem (Pˆ_λ,µ) has a (non-negative) solution v if and only if T(_λ,µ,·)has a fixed point v inP. Moreover, ifθ is a solution to problem(Pˆ_λ,µ), thenλ=0,and ifλ=0,thenθ is a unique solution to problem(Pˆ_λ,µ).

To show the complete continuity ofT, we first prove the following lemma.

Lemma 2.9. Assume that(H₁)–(H₃)hold, let{(λⁿ,µⁿ,vⁿ)}be a bounded sequence inR³+×X and let i∈ {1, 2}be fixed. If M_iⁿ = M_i(λⁿ,µⁿ,vⁿ) →0 (or1)as n → _∞,thenkT_i(λⁿ,µⁿ,vⁿ)k_∞ → 0 as n→_∞.

Proof. We only prove the caseMⁿ_i →0, since the other case can be proved in a similar manner.

Since {(λⁿ,µⁿ,vⁿ)} is a bounded sequence in R³+×X, there exists C > 0 such that µⁿ_i + λⁿkF_i(µⁿ,vⁿ)k_∞+|β_i[µⁿ,vⁿ]| ≤Cfor alln∈_{N. Here,}µⁿ_i is thei-th component ofµⁿ. Then

kT_i(λⁿ,µⁿ,vⁿ)k_∞ = T_i(λⁿ,µⁿ,vⁿ)(Mⁿ_i)

= (β_i[µⁿ,vⁿ]−µⁿ_i)Mⁿ_i +

Z _Mⁿ

i

0 ϕ⁻_pi¹

ϕ_pi(µⁿ_i −β_i[µⁿ,vⁿ]) +λⁿ Z _Mⁿ

i

s h_i(τ)F_i(µⁿ,vⁿ)(τ)dτ

ds

≤CMⁿ_i +γ 1 pi−1

Z _Mⁿ

i

0

C+ϕ⁻_pi¹(C)ϕ⁻_pi¹ _{Z M}n

i

s h_i(τ)dτ

ds,

whereγ_q=max{1, 2^q−1}forq>0. It follows fromh_i ∈ Athat, asn→_∞,kT_i(λ_n,µ_n,v_n)k_∞ →0, and thus the proof is complete.

Combining Lemma 2.9 with the standard arguments (see, e.g., [1, Lemma 3] and [14, Lemma 2.4]), we have the following lemma. We omit the proof of it.

Lemma 2.10. Assume that(H₁)–(H₃)hold. Then T:R³+×X→ P is completely continuous.

2.2 Sub-super solutions theorem

In this subsection, we give a theorem for sub-super solutions to problem (P_λ,µ).

Definition 2.11. Letλ >0 andµ∈_R²₊be given. We say that ζ = (ζ₁,ζ2)is asupersolutionto problem (P_λ,µ) if ζ_i ∈C¹(0, 1)with ϕ_pi(ζ_i⁰)absolutely continuous fori=1, 2, and

((_Φ(ζ⁰(t)))⁰+λh(t)•f(t,ζ(t))≤θ, t ∈(0, 1), ζ(₀)≥α[ζ]_, ζ(₁)≥ _µ.

We also say thatψ= (ψ₁,ψ₂)is asubsolutionto problem (P_λ,µ) ifψ_i ∈C¹(0, 1)with ϕ_pi(ψ⁰_i) absolutely continuous fori=1, 2, and it satisfies the reverse of the above inequalities.

To get a theorem for sub-super solutions to problem (P_λ,µ), we make the following hy- potheses:

(H₂⁰) max

i∈{1,2}

_{Z 1}

0

(1−s)[k_1i(s) +k_2i(s)]ds

=:C_k ∈ [0, 1);

(F₁) For fixed (t,u) ∈ [0, 1]×_R+, f₁ = f₁(t,u,v) is quasi-monotone nondecreasing with respect to v, i.e., f₁(t,u,v₁) ≤ f₁(t,u,v₂) whenever 0 ≤ v₁ ≤ v₂. For fixed (t,v) ∈ [0, 1]×_R+, f₂= f₂(t,u,v)is quasi-monotone nondecreasing with respect tou.

Note that(H₂⁰)implies(H₂).

Now, a theorem for sub-super solutions for the problem (P_λ,µ) is given as follows.

Theorem 2.12. Letλ > 0 andµ∈ _R²₊ be fixed. Assume that(H₁),(H₂⁰),(H₃)and(F₁)hold, and that there exist ψ andζ, respectively, a subsolution and a supersolution to problem (P_λ,µ) such that ψ≤ζ. Then problem(P_λ,µ)has at least one solution u such thatψ≤u≤ ζ.

Proof. Defineγ:[0, 1]×_R²→_R²₊by, fort ∈[0, 1]andu= (u₁,u₂)∈_R², γ(t,u):= (γ₁(t,u₁),γ₂(t,u₂)),

where, fori∈ {1, 2},γ_i :[0, 1]×_R→_R+by

γ_i(t,s):=











ζ_i(t), s≥ζ_i(t),

s, ψ_i(t)≤s ≤ζ_i(t), ψ_i(t), s≤ψ_i(t).

Letλ>0 andµ∈_R²₊ be fixed, and consider the following modified problem ((_Φ(u⁰(t)))⁰+λh(t)•f(t,γ(t,u(t))) =_θ, t∈ (_{0, 1})_,

u(0) =α^γ[u], u(1) =µ, (2.10)

whereα^γ[u]:=R1

0 γ(s,u(s))k(s)ds.

For givenv∈ X, defineg_v:R² →_R²by gv(x):=

Z ₁

0 γ(s,v(s) + (1−s)x+sµ)k(s)ds.

Then gv is a contraction mapping on (_R²,| · |_R2). Here, for (x₁,x₂) ∈ _R², |(x₁,x₂)|_R2 := max(|x₁|,|x₂|). Indeed, since |γ_i(t,s)−γ_i(t,τ)| ≤ |s−τ|for any s,τ ∈ _R and t ∈ [0, 1], for v¯ = (v¯₁, ¯v2), ˜v = (v˜₁, ˜v2)∈_R²,

|γ(t, ¯v)−γ(t, ˜v)|_R2 =max{|γ₁(t, ¯v₁)−γ₁(t, ˜v₁)|,|γ₂(t, ¯v₂)−γ₂(t, ˜v₂)|}

≤max{|v¯₁−v˜₁|,|v¯₂−v˜₂|}=|v¯−v˜|_R2. Then, fors ∈[0, 1]andx,y∈ _R²,

|γ(s,v(s) + (1−s)x+sµ)−γ(s,v(s) + (1−s)y+sµ)|_R2 ≤(1−s)|x−y|_R2,

and|g_v(x)−g_v(y)|_R2 ≤C_k|x−y|_{R2. Then}g_v is a contraction mapping on(_R²_,| · |_R2)_by(H₂⁰)_. Thus, for given v ∈ X, there is a unique solution β^γ[v] ∈ _R² of the equation x = gv(x), in other words, it is the unique element ofR²which satisfies that

β^γ[v] =

Z ₁

0 γ(s,v(s) + (1−s)β^γ[v] +sµ)k(s)ds.

From this fact, it follows that

α^γ[u] =β^γ[v] under the transformations

u(t):=v(t) + (1−t)β^γ[v] +tµ, v(t):= u(t)−((1−t)α^γ[u] +tµ). (2.11) Thus (2.10) can be equivalently rewritten as follows:

((_Φ(v⁰(t)−β^γ[v] +µ))⁰+λh(t)•F^γ(v)(t) =θ, t∈(0, 1),

v(0) =v(1) =θ, (2.12)

whereF^γ := (F₁^γ,F₂^γ):X→X is defined by, forv ∈X,

F^γ(v)(t):= f(t,γ(t,v(t) + (1−t)β^γ[v] +tµ)), t∈[0, 1].

Consequently,vis a solution to problem (2.12) if and only ifu is a solution to problem (2.10) under the transformations (2.11), respectively.

Now, defineT^γ = (T₁^γ,T₂^γ)_: X→ P by, for eachi=_{1, 2 and}v∈ X,

T_i^γ(v)(t):=



















(β^γ_i[v]−µ_i)t+Rt 0 ϕ⁻_pi¹

ϕ_pi(µ_i−β^γ_i[v]) +λRM^γ_i

s h_i(τ)F_i^γ(v)(τ)dτ

ds, 0≤t≤ M^γ_i,

−(β^γ_i[v]−µ_i)(1−t) +R1 t ϕ⁻_pi¹

ϕp_i(−µ_i+β^γ_i[v]) +λRs

M^γ_i h_i(τ)F_i^γ(v)(τ)dτ

ds, M^γ_i ≤ t≤_1,

where M^γ_i = M_i^γ(v)is the constant satisfying (β^γ_i[v]−µ_i)M_i^γ+

Z _M^γ

i

0 ϕ⁻_pi¹ ϕ_pi(µ_i−β^γ_i [v]) +λ Z _M^γ

i

s h_i(τ)F_i^γ(v)(τ)dτ

! ds

=−(β^γ_i[v]−µ_i)(1−M^γ_i ) +

Z ₁

M^γ_i

ϕ⁻_pi¹

ϕ_pi(−µ_i+β^γ_i[v]) +λ Z _s

M^γ_i h_i(τ)F_i^γ(v)(τ)dτ

ds.

Thenvis a fixed point ofT^γinXif and only ifvis a solution to problem (2.12). It follows that T^γ is completely continuous onXandT^γ(X)is bounded inX. Then, by Theorem2.4,T^γ has a fixed point v, and consequently (2.10) has a solutionuunder the first transformation (2.11).

Now if we prove thatψ≤u≤ζ, then, by the definition ofγ, (P_λ,µ) has a solutionusuch that ψ ≤ u ≤ ζ and the proof is complete. In order to show u ≤ ζ, assume on the contrary that u₁6≤ζ₁or u₂6≤ζ₂. We only consider the caseu₁ 6≤ζ₁, since the case u₂6≤ζ₂can be dealt in a similar manner. Set X₁(t) := u₁(t)−ζ₁(t)for t ∈ [0, 1]. Then, since X₁(0) =u₁(0)−ζ₁(0)≤ R1

0 ∑²i=1[γ_i(s,u_i(s))−ζ_i(s)]k_i1(s)ds ≤0 and X₁(1) =u₁(1)−ζ₁(1)≤ 0, there exists σ ∈ (0, 1) such that X₁(σ) = _maxt∈[0,1]X₁(t) > _{0. Then} X⁰₁(σ) = 0 and we may assume that there is a∈(σ, 1)such thatX₁⁰(t)<0 andX₁(t)>0 fort∈(σ,a], which imply that

u⁰₁(σ) =ζ₁⁰(σ), u⁰₁(t)<ζ₁⁰(t), and u₁(t)>ζ₁(t) fort ∈(σ,a]. (2.13) By the quasi-monotonicity of f₁ and (2.13), fort ∈(_σ,a]_,

−(ϕp₁(u⁰₁(t)))⁰ = λh₁(t)f₁(t,γ₁(t,u₁(t)),γ₂(t,u₂(t)))

= λh₁(t)f₁(t,ζ₁(t),γ2(t,u2(t)))

≤ λh₁(t)f₁(t,ζ₁(t)_,ζ₂(t))≤ −(ϕ_p1(ζ⁰₁(t)))⁰_. For t∈(σ,a], integrating this inequality fromσtot, we get

ϕ_p1(u⁰₁(t))≥ ϕ_p1(ζ₁⁰(t)), fort∈ (σ,a].

Since ϕ_p1 is monotone increasing, u⁰₁(t) ≥ ζ₁⁰(t) _fort ∈ (_σ,a], and it contradicts (2.13). Thus u₁ ≤ ζ₁, and we can show that u ≤ ζ. In a similar manner, it is shown thatψ ≤ u, and thus the proof is complete.

3 Main results

First, we give a hypothesis which will be used in this section:

(F_∞) For eachi∈ {1, 2}, there exists an interval I_i := [θ₁ⁱ,θ₂ⁱ]⊂(0, 1)withθⁱ₁<θⁱ₂such that f_i^∞:= lim

|s1|+|s2|→_∞

f_i(t,s₁,s₂) s_i^pi−1

=_∞ uniformly int ∈ I_i. Set

m_h := min

t∈I1∪I2

{h₁(t),h2(t)}>0. (3.1) Now we givea prioriestimates for solutions as follows.

Lemma 3.1. Assume that (H₁)–(H₃) and (F_∞) hold. Let α ∈ (0,∞) be given. Then there exists M = M(α) > 0 such that kuk_X < M for any non-negative solutions u to problem (P_λ,µ) for all λ∈ [_α,∞)andµ∈_R²₊.