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Existence of solutions for weighted p(r)-Laplacian impulsive system mixed type boundary value problems

Li Yin1, Yunrui Guo2, Guizhen Zhi1 and Qihu Zhang1*

Author affiliations

1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China

2 Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China

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Citation and License

Boundary Value Problems 2011, 2011:42  doi:10.1186/1687-2770-2011-42


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/42


Received:1 July 2011
Accepted:1 November 2011
Published:1 November 2011

© 2011 Yin et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the existence of solutions for weighted p(r)-Laplacian impulsive system mixed type boundary value problems. The proof of our main result is based upon Gaines and Mawhin's coincidence degree theory. Moreover, we obtain the existence of nonnegative solutions.

Keywords:
Weighted p(r)-Laplacian; impulsive system; coincidence degree

1 Introduction

In this paper, we mainly consider the existence of solutions for the weighted p(r)-Laplacian system

- ( w ( r ) u p ( r ) - 2 u ( r ) ) + f ( r , u ( r ) , ( w ( r ) ) 1 p ( r ) - 1 u ( r ) ) = 0 , r ( 0 , T ) , r r i , (1)

where u: [0, T] → ℝN, with the following impulsive boundary conditions

lim r r i + u ( r ) - lim r r i - u ( r ) = A i ( lim r r i - u ( r ) , lim r r i - ( w ( r ) ) 1 p ( r ) - 1 u ( r ) ) , i = 1 , , k , (2)

lim r r i + w ( r ) u p ( r ) - 2 u ( r ) - lim r r i - w ( r ) u p ( r ) - 2 u ( r ) = B i ( lim r r i - u ( r ) , lim r r i - ( w ( r ) ) 1 p ( r ) - 1 u ( r ) ) , i = 1 , , k , (3)

a u ( 0 ) - b lim r 0 + ( w ( r ) ) 1 p ( r ) - 1 u ( r ) = 0 , and c u ( T ) + d lim r T - w ( r ) u p ( r ) - 2 u ( r ) = 0 , (4)

where p C ([0, T], ℝ) and p(r) > 1, -Δp(r) u:= -(w(r) |u'|p(r)-2 u'(r))' is called weighted p(r)-Laplacian; 0 < r1 < r2 < ⋯ < rk < T; Ai, Bi C(ℝN × ℝN, ℝN); a, b, c, d ∈ [0, +∞), ad + bc > 0.

Throughout the paper, o(1) means functions which uniformly convergent to 0 (as n → +∞); for any v ∈ ℝN, vj will denote the j-th component of v; the inner product in ℝN will be denoted by 〈·,·〉; |·| will denote the absolute value and the Euclidean norm on ℝN. Denote J = [0, T], J' = [0, T]\{r0, r1,..., rk+1}, J0 = [r0, r1], Ji = (ri, ri+1], i = 1, ..., k, where r0 = 0, rk+1 = T. Denote J i o the interior of Ji, i = 0, 1,..., k. Let PC(J, ℝN) = {x: J → ℝN | x C(Ji, ℝN), i = 0, 1,..., k, and lim r r i + x ( r ) exists for i = 1,..., k}; w PC(J, ℝ) satisfies 0 < w(r), ∀r J', and ( w ( ) ) - 1 p ( ) - 1 L 1 ( 0 , T ) ; P C 1 ( J , N ) = { x P C ( J , N ) x C ( J i o , N ) , lim r r i + w ( r ) x p ( r ) - 2 x ( r ) and lim r r i + 1 - w ( r ) x p ( r ) - 2 x ( r ) exists for i = 0, 1,..., k}. For any x = (x1,..., xN) ∈ PC(J, ℝN), denote |xi|0 = suprJ' |xi(r)|. Obviously, PC(J, ℝN) is a Banach space with the norm x 0 = ( i = 1 N x i 0 2 ) 1 2 , PC1(J, ℝN) is a Banach space with the norm x 1 = x 0 + ( w ( r ) ) 1 p ( r ) - 1 x 0 . In the following, PC(J, ℝN) and PC1(J, ℝN) will be simply denoted by PC and PC1, respectively. Let L1 = L1(J, ℝN) with the norm x L 1 = ( i = 1 N x i L 1 2 ) 1 2 , ∀x L1, where x i L 1 = 0 T x i ( r ) d r . We will denote

u ( r i + ) = lim r r i + u ( r ) , u ( r i - ) = lim r r i - u ( r ) ,

w ( 0 ) u p ( 0 ) - 2 u ( 0 ) = lim r 0 + w ( r ) u p ( r ) - 2 u ( r ) , w ( T ) u p ( T ) - 2 u ( T ) = lim r T - w ( r ) u p ( r ) - 2 u ( r ) .

The study of differential equations and variational problems with nonstandard p(r)-growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electro-rheological fluids, image processing, etc. (see [1-4]). Many results have been obtained on this problems, for example [1-25]. If p(r) ≡ p (a constant), (1) is the well-known p-Laplacian system. If p(r) is a general function, -Δp(r) represents a nonhomogeneity and possesses more nonlinearity, thus -Δp(r) is more complicated than -Δp; for example, if Ω ⊂ ℝN is a bounded domain, the Rayleigh quotient

λ p ( ) = inf u W 0 1 , p ( x ) ( Ω ) \ { 0 } Ω 1 p ( x ) u p ( x ) d x Ω 1 p ( x ) u p ( x ) d x

is zero in general, and only under some special conditions λp(·) > 0 (see [8,17-19]), but the property of λp > 0 is very important in the study of p-Laplacian problems.

Impulsive differential equations have been studied extensively in recent years. Such equations arise in many applications such as spacecraft control, impact mechanics, chemical engineering and inspection process in operations research (see [26-28] and the references therein). It is interesting to note that p(r)-Laplacian impulsive boundary problems are about comparatively new applications like ecological competition, respiratory dynamics and vaccination strategies. On the Laplacian impulsive differential equation boundary value problems, there are many results (see [29-37]). There are many methods to deal with this problem, e.g., subsupersolution method, fixed point theorem, monotone iterative method and coincidence degree. Because of the nonlinearity of -Δp, results on the existence of solutions for p-Laplacian impulsive differential equation boundary value problems are rare (see [38,39]). On the Laplacian (p(x) ≡ 2) impulsive differential equations mixed type boundary value problems, we refer to [30,32,34].

In [39], Tian and Ge have studied nonlinear IBVP

- ( ρ ( t ) Φ p ( x ( t ) ) ) + s ( t ) Φ p ( x ( t ) ) = f ( t , x ( t ) ) , t t i , a . e . t [ a , b ] , lim t t i + ρ ( t ) Φ p ( x ( t ) ) - lim t t i - ρ ( t ) Φ p ( x ( t ) ) = I i ( x ( t i ) ) , i = 1 , , l , α x ( a ) - β x ( a ) = σ 1 , γ x ( b ) + σ x ( b ) = σ 2 , (5)

where Φp(x) = |x|p-2 x, p > 1, ρ, s L[a, b] with essin f[a, b] ρ > 0, and essin f[a,b] s > 0, 0 < ρ(a), p(b) <∞, σ1 ≤ 0, σ2 ≥ 0, α, β, γ, σ > 0, a = t0 < t1 < ⋯ < tl < tl+1 = b, Ii C([0, +∞), [0, ∞)), i = 1,..., l, f C ([a, b] × [0, +∞), [0, ∞)), f(·, 0) is nontrivial. By using variational methods, the existence of at least two positive solutions was obtained.

In [24,25], the present author investigates the existence of solutions of p(r)-Laplacian impulsive differential equation (1-3) with periodic-like or multi-point boundary value conditions.

In this paper, we consider the existence of solutions for the weighted p(r)-Laplacian impulsive differential system mixed type boundary value condition problems, when p(r) is a general function. The proof of our main result is based upon Gaines and Mawhin's coincidence degree theory. Since the nonlinear term f in (5) is independent on the first-order derivative, and the impulsive conditions are simpler than (2), our main results partly generalized the results of [30,32,34,39]. Since the mixed type boundary value problems are different from periodic-like or multi-point boundary value conditions, and this paper gives two kinds of mixed type boundary value conditions (linear and nonlinear), our discussions are different from [24,25] and have more difficulties. Moreover, we obtain the existence of nonnegative solutions. This paper was motivated by [24-26,38,40].

Let N ≥ 1, the function f: J × ℝN × ℝN → ℝN is assumed to be Caratheodory; by this, we mean:

(i) for almost every t J, the function f(t, ·, ·) is continuous;

(ii) for each (x, y) ∈ ℝN × ℝN, the function f(·, x, y) is measurable on J;

(iii) for each R > 0, there is a αR L1 (J, ℝ), such that, for almost every t J and every (x, y) ∈ ℝN × ℝN with |x| ≤ R, |y| ≤ R, one has

f ( t , x , y ) α R ( t ) .

We say a function u: J → ℝN is a solution of (1) if u PC1 with w(·) |u'|p(·)-2 u'(·) absolutely continuous on every J i o , i = 0, 1,..., k, which satisfies (1) a.e. on J.

This paper is divided into three sections; in the second section, we present some preliminary. Finally, in the third section, we give the existence of solutions and nonnegative solutions of system (1)-(4).

2 Preliminary

Let X and Y be two Banach spaces and L: D(L) ⊂ X Y be a linear operator, where D(L) denotes the domain of L. L will be a Fredholm operator of index 0, i.e., ImL is closed in Y and the linear spaces KerL and coImL have the same dimension which is finite. We define X1 = KerL and Y1 = coImL, so we have the decompositions X = X1 coKerL and Y = Y1 ImL. Now, we have the linear isomorphism Λ: X1 Y1 and the continuous linear projectors P: X X1 and Q: Y Y1 with KerQ = ImL and ImP = X1.

Let Ω be an open bounded subset of X with Ω ∩ D(L) ≠ ∅. Operator S : Ω ¯ Y be a continuous operator. In order to define the coincidence degree of (L, S) in Ω, as in [40,41], denoted by d(L - S, Ω), we assume that

L x S x for all  x Ω .

It is easy to see that the operator M : Ω ¯ X , M = (L + ΛP)-1 (S + ΛP) is well defined, and

L x * = S x *  if and only if  x * = M x * .

If M is continuous and compact, then S is called L-compact, and the Leray-Schauder degree of IX - M (where IX is the identity mapping of X) is well defined in Ω, and we will denote it by dLS (IX - M, Ω, 0). This number is independent of the choice of P, Q and Λ (up to a sign) and we can define

d ( L - S , Ω ) : = d L S ( I X - M , Ω , 0 ) .

Definition 2.1. (see [40,41]) The coincidence degree of (L, S) in Ω, denoted by d(L - S, Ω), is defined as d(L - S, Ω) = dLS (IX - M, Ω, 0).

There are many papers about coincidence degree and its applications (see [40-43]).

Proposition 2.2. (see [40]) (i) (Existence property). If d(L - S, Ω) ≠ 0, then there exists x ∈ Ω such that Lx = Sx.

(ii) (Homotopy invariant property). If H : Ω ¯ × [ 0 , 1 ] Y is continuous, L-compact and Lx H(x, λ) for all x ∈ ∂Ω and λ ∈ [0, 1], then d(L - H (·, λ), Ω) is independent of λ.

The effect of small perturbations is negligible, as is proved in the next Proposition (see [41] Theorem III.3, page 24).

Proposition 2.3. Assume that Lx Sx for each x ∈ ∂Ω. If Sε is such that supx∈∂Ω||Sεx||Y is sufficiently small, then Lx Sx + Sεx for all x ∈ ∂Ω and d(L - S - Sε, Ω) = d(L - S, Ω).

For any (r, x) ∈ (J × ℝN), denote φp(r)(x) = |x|p(r)-2x. Obviously, φ has the following properties

Proposition 2.4 (see [41]) φ is a continuous function and satisfies

(i) For any r ∈ [0, T], φp(r)(·) is strictly monotone, i.e.,

φ p ( r ) ( x 1 ) - φ p ( r ) ( x 2 ) , x 1 - x 2 > 0 , x 1 , x 2 N , x 1 x 2 ;

(ii) There exists a function η: [0, +∞) → [0, +∞), η(s) → +∞ as s → +∞, such that

φ p ( r ) ( x ) , x η ( x ) x , for all  x N .

It is well known that φp(r)(·) is a homeomorphism from ℝN to ℝN for any fixed r J. Denote

φ p ( r ) - 1 ( x ) = x 2 - p ( r ) p ( r ) - 1 x , for x N \ { 0 } , φ p ( r ) - 1 ( 0 ) = 0 .

It is clear that φ p ( r ) - 1 ( ) is continuous and sends bounded sets to bounded sets, and φ p ( r ) - 1 ( ) = φ q ( r ) ( ) where 1 p ( r ) + 1 q ( r ) 1 . Let X = {(x1, x2) | x1 PC, x2 PC} with the norm ||(x1, x2)||X = || x1||0 + ||x2||0, Y = L1 × L1 × ℝ2(k + 1)N, and we define the norm on Y as

( y 1 , y 2 , z 1 , , z 2 ( k + 1 ) ) Y = y 1 L 1 + y 2 L 1 + m = 1 2 ( k + 1 ) z m , ( y 1 , y 2 , z 1 , , z 2 ( k + 1 ) ) Y ,

where y1, y2 L1, zm ∈ ℝN, m = 1,..., 2(k + 1), then X and Y are Banach spaces.

Define L: D(L) ⊂ X Y and S: X Y as the following

L x = ( x 1 , x 2 , Δ x 1 ( r i ) , Δ x 2 ( r i ) , 0 , 0 ) ,

S x = φ q ( r ) x 2 w ( r ) , f ( r , x 1 , φ q ( r ) ( x 2 ) ) , A i , B i , a x 1 ( 0 ) - b φ q ( 0 ) ( x 2 ( 0 ) ) , c x 1 ( T ) + d x 2 ( T ) ,

where

Δ x j ( r i ) = x j ( r i + ) - x j ( r i - ) , j = 1 , 2 , i = 1 , , k ; A i = A i ( x 1 ( r i - ) , φ q ( r i ) ( x 2 ( r i - ) ) ) , B i = B i ( x 1 ( r i - ) , φ q ( r i ) ( x 2 ( r i - ) ) ) , i = 1 , , k . (6)

Obviously, the problem (1)-(4) can be written as Lx = Sx, where L: X Y is a linear operator, S: X Y is a nonlinear operator, and X and Y are Banach spaces.

Since

I m L = { ( y 1 , y 2 , a i , b i , 0 , 0 ) y 1 , y 2 L 1 , a i , b i N , i = 1 , , k } ,

we have dimKerL = dim(Y/ImL) = 2N < +∞ is even and ImL is closed in Y, then L is a Fredholm operator of index zero. Define

P : X X , ( x 1 , x 2 ) ( x 1 ( 0 ) , x 2 ( 0 ) ) , Q : Y Y , ( y 1 , y 2 , a i , b i , h 1 , h 2 ) ( 0 , 0 , 0 , 0 , h 1 , h 2 ) , y 1 , y 2 L 1 , a i , b i , h 1 , h 2 N , i = 1 , , k ,

at the same time the projectors P: X X and Q: Y Y satisfy

d i m ( I m P ) = d i m ( K e r L ) = d i m ( Y I m L ) = d i m ( I m Q ) .

Since ImQ is isomorphic to KerL, there exists an isomorphism Λ: KerL ImQ. It is easy to see that L |D(L)∩KerP : D(L) ∩ KerP ImL is invertible. We denote the inverse of that mapping by Kp, then Kp : ImL D(L) ∩ KerP as

K p z = 0 t y 1 ( r ) d r + r i < t a i , 0 t y 2 ( r ) d r + r i < t b i , z = ( y 1 , y 2 , a i , b i , 0 , 0 ) I m L ,

then

K p ( I Q ) S x = ( 0 t φ q ( r ) ( ( w ( r ) ) 1 x 2 ) d r + r i < t A i ( x 1 ( r i ) , φ q ( r i ) ( x 2 ( r i ) ) ) , 0 t f ( r , x 1 , φ q ( r ) ( x 2 ) ) d r + r i < t B i ( x 1 ( r i ) , φ q ( r i ) ( x 2 ( r i ) ) ) ) .

Proposition 2.5 (i) Kp(·) is continuous;

(ii) Kp (I - Q)S is continuous and compact.

Proof. (i) It is easy to see that Kp(·) is continuous. Moreover, the operator Ψ ( y ) = 0 t y ( r ) d r sends equi-integrable set of L1 to relatively compact set of PC.

(ii) It is easy to see that Kp(I - Q)Sx X, ∀x X. Since ( w ( r ) ) - 1 p ( r ) - 1 L 1 and f is Caratheodory, it is easy to check that S is a continuous operator from X to Y, and the operators (x1, x2) → φq(r) ((w(r))-1 x2) and (x1, x2) → f (r, x1, φq(r) ((w(r))-1 x2)) both send bounded sets of X to equi-integrable set of L1. Obviously, Ai, Bi and QS are compact continuous. Since f is Caratheodory, by using the Ascoli-Arzela theorem, we can show that the operator K p ( I - Q ) S : Ω ¯ X is continuous and compact. This completes the proof.

Denote

S ( x , λ ) = λ φ q ( r ) x 2 w ( r ) , λ p ( r ) f ( r , x 1 , φ q ( r ) ( x 2 ) ) , λ 2 A i , λ p ( r i ) B i , a x 1 ( 0 ) - b φ q ( 0 ) ( x 2 ( 0 ) ) , c x 1 ( T ) + d x 2 ( T ) ,

where Ai, Bi are defined in (6), i = 1,..., k.

Consider

L x = S ( x , λ ) .

Define M ( , ) : Ω ¯ × [ 0 , 1 ] X as M(·, ·) = (L + ΛP)-1 (S(·, ·) + ΛP), then

M ( , λ ) = ( L + Λ P ) - 1 ( S ( , λ ) + Λ P ) = ( K p + Λ - 1 ) ( ( I - Q ) S ( , λ ) + Q S ( , λ ) + Λ P ) = K p ( I - Q ) S ( , λ ) + Λ - 1 ( Q S ( , λ ) + Λ P ) = K p ( I - Q ) S ( , λ ) + Λ - 1 Q S ( , λ ) + P .

Since (I - Q)S(·, 0) = 0 and Kp (0) = 0, we have

d ( L - S ( , 0 ) , Ω ) = d L S ( I X - M ( , 0 ) , Ω , 0 ) = d L S ( I X - Λ - 1 Q S ( , 0 ) - P , Ω , 0 ) .

It is easy to see that all the solutions of Lx = S(x, 0) belong to KerL, then

d L S ( I X - Λ - 1 Q S ( , 0 ) - P , Ω , 0 ) = d B ( I K e r L - Λ - 1 Q S ( , 0 ) - P K e r L , Ω K e r L , 0 ) .

Notice that P |KerL = IKerL, then

d ( L - S ( , 0 ) , Ω ) = d L S ( I X - M ( , 0 ) , Ω , 0 ) = d B ( Λ - 1 Q S ( , 0 ) , Ω K e r L , 0 ) .

Proposition 2.6 (continuation theorem) (see [40]). Suppose that L is a Fredholm operator of index zero and S is L-compact on Ω ¯ , where Ω is an open bounded subset of X. If the following conditions are satisfied,

(i) for each λ ∈ (0, 1), every solution x of

L x = S ( x , λ )

is such that x ∉ ∂Ω;

(ii) QS(x, 0) ≠ 0 for x ∈ ∂Ω ∩ KerL and dB-1 QS(·,0), Ω ∩ KerL, 0) ≠ 0, then the operator equation Lx = S(x, 1) has one solution lying in Ω ¯ .

The importance of the above result is that it gives sufficient conditions for being able to calculate the coincidence degree as the Brouwer degree (denoted with dB) of a related finite dimensional mapping. It is known that the degree of finite dimensional mappings is easier to calculate. The idea of the proof is the use of the homotopy of the problem Lx = S(x, 1) with the finite dimensional one Lx = S(x, 0).

Let us now consider the following simple impulsive problem

( w ( r ) φ p ( r ) ( u ( r ) ) ) = g ( r ) , r J , lim r r i + u ( r ) - lim r r i - u ( r ) = a i , i = 1 , , k , lim r r i + w ( r ) φ p ( r ) ( u ( r ) ) - lim r r i - w ( r ) φ p ( r ) ( u ( r ) ) = b i , i = 1 , , k , a u ( 0 ) - b lim r 0 + ( w ( r ) ) 1 p ( r ) - 1 u ( r ) = 0 , and c u ( T ) + d lim r T - w ( r ) u p ( r ) - 2 u ( r ) = 0 , (7)

where J' = [0, T]\{r0, r1, ..., rk+1}, ai, bi ∈ ℝN; g L1.

If u is a solution of (7), then we have

w ( r ) φ p ( r ) ( u ( r ) ) = w ( 0 ) φ p ( 0 ) ( u ( 0 ) ) + r i < r b i + 0 r g ( t ) d t , r J . (8)

Denote ρ0 = w(0)φp(0) (u'(0)). Obviously, ρ0 is dependent on g, ai, bi. Define F: L1 PC as

F ( g ) ( r ) = 0 r g ( t ) d t , r J , g L 1 .

By (8), we have

u ( r ) = u ( 0 ) + r i < r a i + F φ q ( r ) ( w ( r ) ) - 1 ρ 0 + r i < r b i + F ( g ) ( r ) ( r ) , r J . (9)

If a ≠ 0, then the boundary condition a u ( 0 ) - b lim r 0 + ( w ( r ) ) 1 p ( r ) - 1 u ( r ) = 0 implies that

u ( r ) = b a φ q ( 0 ) ( ρ 0 ) + r i < r a i + F φ q ( r ) ( w ( r ) ) - 1 ρ 0 + r i < r b i + F ( g ) ( r ) ( r ) , r J .

The boundary condition c u ( T ) + d lim r T - w ( r ) u p ( r ) - 2 u ( r ) = 0 implies that

c b a φ q ( 0 ) ( ρ 0 ) + c i = 1 k a i + c F φ q ( r ) ( w ( r ) ) - 1 ρ 0 + r i < r b i + F ( g ) ( r ) ( T ) + d ρ 0 + i = 1 k b i + F ( g ) ( T ) = 0 .

Denote H = L1 × ℝ2kN with the norm

h H = g L 1 + i = 1 k a i + i = 1 k b i , h = ( g , a i , b i ) H ,

then H is a Banach space. For fixed h H, we denote

Θ h ( ρ ) = c b a φ q ( 0 ) ( ρ ) + c i = 1 k a i + c F φ q ( r ) ( w ( r ) ) - 1 ρ + r i < r b i + F ( g ) ( r ) ( T ) + d ρ + i = 1 k b i + F ( g ) ( T ) .

Lemma 2.7 The mapping Θh(·) has the following properties

(i) For any fixed h H, the equation

Θ h ( ρ ) = 0 (10)

has a unique solution ρ(h) ∈ ℝN.

(ii) The mapping ρ: H → ℝN, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, ρ ( h ) 3 N [ ( 2 N E + 1 E i = 1 k a i ) p # - 1 + i = 1 k b i + g L 1 ] , where h = (g, ai, bi) ∈ H, E = 0 T ( w ( r ) ) - 1 p ( r ) - 1 d r , the notation p# means C p # - 1 = C p + - 1 , C > 1 C p - - 1 , C 1 .

Proof. (i) From Proposition 2.4, it is immediate that

Θ h ( ρ 1 ) - Θ h ( ρ 2 ) , ρ 1 - ρ 2 > 0 , f o r ρ 1 ρ 2 ,

and hence, if (10) has a solution, then it is unique.

Let R 0 = 3 N [ ( 2 N E + 1 E i = 1 k a i ) p # - 1 + i = 1 k b i + g L 1 ] . Since ( w ( r ) ) - 1 p ( r ) - 1 L 1 ( 0 , T ) and F(g) ∈ PC, if |ρ| > R0, it is easy to see that there exists a j0 such that, the j0-th component ρ j 0 of ρ satisfies

ρ j 0 1 N ρ . (11)

Obviously,

| r i < r b i + F ( g ) ( r ) | r i < r b i + | F ( g ) ( r ) | i = 1 k | b i | + g L 1 , r 0, T ] ,

then

r i < r b i + F ( g ) ( r ) i = 1 k b i + g L 1 R 0 3 N < ρ 3 N , r [ 0 , T ] , (12)

and

ρ + r i < r b i + F ( g ) ( r ) ρ + r i < r b i + F ( g ) ( r ) < 4 ρ 3 , r [ 0 , T ] . (13)

By (11) and (12), the j0-th component of ρ + r i < r b i + F ( g ) ( r ) keeps the same sign of ρ j 0 on J and

ρ j 0 + r i < r b i j 0 + F ( g ) j 0 ( r ) ρ j 0 - r i < r b i j 0 + F ( g ) j 0 ( r ) > 2 ρ 3 N , r J . (14)

Combining (13) and (14), the j0-th component φ q ( r ) j 0 ( w ( r ) ) - 1 ρ + r i < r b i + F ( g ) ( r ) of φ q ( r ) ( w ( r ) ) - 1 ρ + r i < r b i + F ( g ) ( r ) satisfies

φ q ( r ) j 0 ( ρ + r i < r b i + F ( g ) ( r ) ) = ρ + r i < r b i + F ( g ) ( r ) q ( r ) - 2 ρ j 0 + r i < r b i j 0 + F ( g ) j 0 ( r ) > 2 3 N ρ + r i < r b i + F ( g ) ( r ) q ( r ) - 2 ρ > 1 2 N ρ + r i < r b i + F ( g ) ( r ) q ( r ) - 1 .

From the definition φ q ( r ) ( ) = φ p ( r ) - 1 ( ) , we have 1 p ( r ) + 1 q ( r ) 1 , then q ( r ) - 1 = 1 p ( r ) - 1 , and

φ q ( r ) j 0 ρ + r i < r b i + F ( g ) ( r ) > 1 2 N ρ + r i < r b i + F ( g ) ( r ) 1 p ( r ) - 1 1 2 N ρ - r i < r b i + F ( g ) ( r ) 1 p ( r ) - 1 1 2 N 2 N E + 1 E i = 1 k a i p # - 1 1 p ( r ) - 1 1 2 N 2 N E + 1 E i = 1 k a i = E + 1 E i = 1 k a i .

Without loss of generality, we may assume that ρ j 0 > 0 , then we have

F j 0 φ q ( r ) ( w ( r ) ) - 1 ρ + r i < r b i + F ( g ) ( r ) ( T ) > ( E + 1 ) i = 1 k a i i = 1 k a i j 0 .

Therefore, the j0-th component of i = 1 k a i + F { φ q ( r ) [ ( w ( r ) ) - 1 ( ρ + r i < r b i + F ( g ) ( r ) ) ] } ( T ) keeps the same sign of ρ j 0 . Since the j0-th component of ρ + i = 1 k b i + F ( g ) ( T ) keeps the same sign of ρ j 0 , a, b, c, d ∈ [0, +∞) and ad + bc > 0, we can easily see that the j0-th component of Θh(ρ) keeps the same sign of ρ j 0 , and thus

Θ h ( ρ ) 0 .

Let us consider the equation

λ Θ h ( ρ ) + ( 1 - λ ) ρ = 0 , λ [ 0 , 1 ] . (15)

According to the above discussion, all the solutions of (15) belong to b(R0 + 1) = {x ∈ ℝN| |x| < R0 + 1}. So, we have

d B [ Θ h ( ρ ) , b ( R 0 + 1 ) , 0 ] = d B [ I , b ( R 0 + 1 ) , 0 ] 0 .

It means the existence of solutions of Θh(ρ) = 0.

In this way, we define a mapping ρ(h): H → ℝN, which satisfies

Θ h ( ρ ( h ) ) = 0 .

(ii) By the proof of (i), we also obtain ρ sends bounded set to bounded set, and

ρ ( h ) 3 N 2 N E + 1 E i = 1 k a i p # - 1 + i = 1 k b i + g L 1 .

It only remains to prove the continuity of ρ. Let {un} is a convergent sequence in H and un u, as n → +∞. Since {ρ(un)} is a bounded sequence, it contains a convergent subsequence { ρ ( u n j ) } satisfies ρ ( u n j ) ρ * as j → +∞. Since Θh(ρ) consists of continuous functions, and

Θ u n j ( ρ ( u n j ) ) = 0 ,

Letting j → +∞, we have

Θ u ( ρ * ) = 0 ,

from (i) we get ρ* = ρ(u), it means that ρ is continuous.

This completes the proof.

If a = 0, the boundary condition a u ( 0 ) - b lim r 0 + ( w ( r ) ) 1 p ( r ) - 1 u ( r ) = 0 implies that

lim r 0 + ( w ( r ) ) 1 p ( r ) - 1 u ( r ) = 0 .

Since ad + bc > 0, we have c > 0. Thus,

u ( r ) = u ( 0 ) + r i < r a i + F φ q ( r ) ( w ( r ) ) - 1 r i < r b i + F ( g ) ( r ) ( r ) , r J ,

the boundary condition c u ( T ) + d lim r T - w ( r ) u p ( r ) - 2 u ( r ) = 0 implies that

u ( 0 ) + i = 1 k a i + F φ q ( r ) ( w ( r ) ) - 1 r i < r b i + F ( g ) ( r ) ( T ) + d c i = 1 k b i + F ( g ) ( T ) = 0 .

Denote G: H → ℝN as

G ( h ) = - i = 1 k a i - F φ q ( r ) ( w ( r ) ) - 1 r i < r b i + F ( g ) ( r ) ( T ) - d c i = 1 k b i + F ( g ) ( T ) .

It is easy to see that

Lemma 2.8 The function G(·) is continuous and sends bounded sets to bounded sets. Moreover, G ( h ) 3 N ( c + d ) c [ i = 1 k a i + E ( i = 1 k b i + g L 1 ) 1 p * - 1 + i = 1 k b i + g L 1 ] , where E = 0 T ( w ( r ) ) - 1 p ( r ) - 1 d r , the notation p* means C 1 p * - 1 = C 1 p + - 1 , C 1 C 1 p - - 1 , C > 1 .

3 Main results and proofs

In this section, we will apply coincidence degree to deal with the existence of solutions for (1)-(4). In the following, we always use C and Ci to denote positive constants, if it cannot lead to confusion.

Theorem 3.1 Assume that Ω is an open bounded set in X such that the following conditions hold.

(10) For each λ ∈ (0, 1) the problem

L x = S ( x , λ ) (16)

has no solution on ∂Ω.

(20) (0, 0) ∈ Ω.

Then, problem (1)-(4) has a solution u satisfies ( u , v ) Ω ¯ , where v = w(r)φp(r)(u'(r)), ∀r J'.

Proof. Let us consider the following operator equation

L x = S ( x , λ ) . (17)

It is easy to see that x = (x1, x2) is a solution of Lx = S(x, 1) if and only if x1(r) is a solution of (1)-(4) and x 2 ( r ) = w ( r ) φ p ( r ) ( x 1 ( r ) ) , ∀r J'.

According to Proposition 2.5, we can conclude that S(·, ·) is L-compact from X × [0, 1] to Y. We assume that for λ = 1, (16) does not have a solution on ∂Ω, otherwise we complete the proof. Now from hypothesis (10), it follows that (16) has no solutions for (x, λ) ∈ ∂Ω × (0, 1]. For λ = 0, (17) is equivalent to Lx = S(x, 0), namely the following usual problem

x 1 = 0 , r ( 0 , T ) , x 2 = 0 , r ( 0 , T ) , a x 1 ( 0 ) - b φ q ( 0 ) ( x 2 ( 0 ) ) = 0 , c x 1 ( T ) + d x 2 ( T ) = 0 .

The problem (??) is a usual differential equation. Hence,

x 1 c 1 , x 2 c 2 ,

where c1, c2 ∈ ℝN are constants. The boundary value condition of (??) holds,

a c 1 - b φ q ( 0 ) ( c 2 ) = 0 , c c 1 + d c 2 = 0 .

Since (ad + bc) > 0, we have

c 1 = 0 , c 2 = 0 ,

which together with hypothesis (20), implies that (0, 0)∈ Ω. Thus, we have proved that (16) has no solution on ∂Ω × [0, 1]. It means that the coincidence degree d[L - S(·, λ), Ω] is well defined for each λ ∈ [0, 1]. From the homotopy invariant property of that degree, we have

d [ L - S ( , 1 ) , Ω ] = d [ L - S ( , 0 ) , Ω ] . (18)

Now, it is clear that the following problem

L x = S ( x , 1 ) (19)

is equivalent to problem (1)-(4), and (18) tells us that problem (19) will have a solution if we can show that

d [ L - S ( , 0 ) , Ω ] 0 .

Since by hypothesis (20), this last degree

d [ L - S ( , 0 ) , Ω ] = d B [ ω * , Ω 2 N , 0 ] 0 ,

where ω*(c1, c2) = (ac1 - q(0)(c2), cc1 + dc2). This completes the proof.

Our next theorem is a consequence of Theorem 3.1. Denote

z - = min r J z ( r ) , z + = max r J z ( r ) , for z C ( J , ) .

Theorem 3.2 Assume that the following conditions hold

(10) a > 0;

(20) lim|u| + |v| → +∞ (f(r, u, v)/(|u| + |v|)β(r) -1) = 0, for r J uniformly, where β(r) ∈ C(J, ℝ), and 1<β - β + < p -;

(30) i = 1 k A i ( u , v ) C 1 ( u + v ) θ when |u| + |v| is large enough, where 0 < θ < p - - 1 p + - 1 ;

(40) i = 1 k B i ( u , v ) C 2 ( u + v ) ε when |u| + |v| is large enough, where 0 ≤ ε < β + - 1.

Then, problem (1)-(4) has at least one solution.

Proof. Now, we consider the following operator equation

L x = S ( x , λ ) . (20)

For any λ ∈ (0, 1], x = (x1, x2) = (u, v) is a solution of (20) if and only if v ( r ) = 1 λ p ( r ) - 1 w ( r ) φ p ( r ) ( u ( r ) ) ( r J ) and u(r) is a solution of the following

1 λ p ( r ) - 1 w ( r ) φ p ( r ) ( u ( r ) ) ) = λ p ( r ) f ( r , u , 1 λ ( w ( r ) ) 1 p ( r ) - 1 u ) r ( 0 , T ) , r r i , lim r r i + u ( r ) - lim r r i - u ( r ) = λ 2 A i ( lim r r i - u ( r ) , 1 λ lim r r i - ( w ( r ) ) 1 p ( r ) - 1 u ( r ) ) , i = 1 , , k , lim r r i + 1 λ p ( r ) - 1 w ( r ) φ p ( r ) ( u ( r ) ) - lim r r i - 1 λ p ( r ) - 1 w ( r ) φ p ( r ) ( u ( r ) ) = λ p ( r i ) B i ( lim r r i - u ( r ) , 1 λ lim r r i - ( w ( r ) ) 1 p ( r ) - 1 u ( r ) ) , i = 1 , , k , a u ( 0 ) - b 1 λ ( w ( 0 ) ) 1 p ( 0 ) - 1 u ( 0 ) = 0 , and c u ( T ) + d 1 λ p ( T ) - 1 w ( T ) u p ( T ) - 2 u ( T ) = 0 . (21)

We claim that all the solutions of (21) are uniformly bounded for λ ∈ (0, 1]. In fact, if it is false, we can find a sequence (un, λn) of solutions for (21), such that ||un||1 > 1 and ||un||1 → +∞ when n → +∞, λn ∈ (0, 1]. Since (un, λn) are solutions of (21), we have

w ( r ) u n p ( r ) - 2 u n ( r ) = λ n p ( r ) - 1 1 λ n p ( 0 ) - 1 ρ n + r i < r λ n p ( r i ) B i + 0 r λ n p ( t ) f t , u n , 1 λ n ( w ( t ) ) 1 p ( t ) - 1 u n d t ,

for any r J', where ρ n = w ( 0 ) φ p ( 0 ) ( u n ( 0 ) ) and

A i = A i lim r r i - u n ( r ) , 1 λ n lim r r i - ( w ( r ) ) 1 p ( r ) - 1 u n ( r ) , B i = B i lim r r i - u n ( r ) , 1 λ n lim r r i - ( w ( r ) ) 1 p ( r ) - 1 u n ( r ) .

By computation, we have

r i < r λ n 2 A i C 1 λ n 2 u n 0 + 1 λ n ( w ( r ) ) 1 p ( r ) - 1 u n 0 θ λ n C 3 u n 1 θ , r i < r λ n p ( r i ) B i C 2 λ n p ( r i ) u n 0 + 1 λ n ( w ( r ) ) 1 p ( r ) - 1 u n 0 β + - 1 C 4 u n 1 β + - 1 , 0 T λ n p ( r ) f r , u n , 1 λ n ( w ( r ) ) 1 p ( r ) - 1 u n d r C 5 λ n p - u n 0 + 1 λ n ( w ( r ) ) 1 p ( r ) - 1 u n 0 β + - 1 C 5 u n 1 β + - 1 . (22)

Denote

Γ n ( r ) = 1 λ n p ( 0 ) - 1 ρ n + r i < r λ n p ( r i ) B i + 0 r λ n p ( t ) f t , u n , 1 λ n ( w ( t ) ) 1 p ( t ) - 1 u n d t , r J .

We claim that

1 λ n p ( 0 ) - 1 ρ n 3 N C 6 u n 1 θ * ( p + - 1 ) + u n 1 β + - 1 , n = 1 , 2 , , where  θ * θ , p - - 1 p + - 1 . (23)

If it is false, without loss of generality, we may assume that

1 λ n p ( 0 ) - 1 ρ n > 3 N ( C 4 + C 5 ) u n 1 θ * ( p + - 1 ) + u n 1 β + - 1 , n = 1 , 2 , ,

then for any n = 1, 2, ..., there is a jn ∈ {1, ..., N} such that the jn-th component ρ n j n of ρn satisfies

1 λ n p ( 0 ) - 1 ρ n j n > 3 ( C 4 + C 5 ) u n 1 θ * ( p + - 1 ) + u n 1 β + - 1 , n = 1 , 2 ,

Thus, when n is large enough, the jn-th component Γ n j n ( r ) of Γn(r) keeps the same sign as ρ n j n and satisfies

Γ n j n ( r ) > ( C 4 + C 5 ) u n 1 θ * ( p + - 1 ) + u n 1 β + - 1 , r J , n = 1 , 2 ,

When n is large enough, we can conclude that the jn-th component F j n { φ q ( r ) [ Γ n ( r ) ] } ( T ) of F{φq(r)n(r)]} (T) keeps the same sign as ρ n j n and satisfies

F j n { φ q ( r ) [ Γ n ( r ) ] } > C 7 u n 1 θ * , r J . (24)

Since

u n ( r ) = u n ( 0 ) + r i < r λ n 2 A i + λ n F { φ q ( r ) ( w ( r ) ) - 1 Γ n ( r ) } = b a φ q ( 0 ) 1 λ n p ( 0 ) - 1 ρ n + r i < r λ n 2 A i + λ n F { φ q ( r ) [ ( w ( r ) ) - 1 Γ n ( r ) ] } , r J , n = 1 , 2 , ,

from (22) and (24), we can see that u n j n ( r ) ( r J ) keeps the same sign as ρ n j n , when n is large enough.

But the boundary value conditions (4) mean that

c u n ( T ) + d 1 λ n p ( T ) - 1 lim r T - w ( r ) u n p ( r ) - 2 u n ( r ) = c u n ( T ) + d Γ n ( T ) = 0 , n = 1 , 2 ,

It is a contradiction. Thus (23) is valid. Therefore,

w ( r ) u n ( r ) p ( r ) - 1 C 7 u n 1 θ * ( p + - 1 ) + u n 1 β + - 1 , r J .

It means that

( w ( r ) ) 1 p ( r ) - 1 u n 0 o ( 1 ) u n 1 ,  where  o ( 1 )  tends to  0  uniformly as  n . (25)

From (22), (23) and (25), for any r J, we have

u n ( r ) = u n ( 0 ) + 0 r u n ( t ) d t + r i < r λ n 2 A i u n ( 0 ) + 0 r