### Abstract

This paper investigates the existence of solutions for a weighted

**MSC: **
34B37.

##### Keywords:

weighted### 1 Introduction

In this paper, we consider the existence of solutions and nonnegative solutions for
the following weighted

where

where
*T* and *S* are linear operators defined by

If

Throughout the paper,
*j*th component of *v*; the inner product in

For any

Obviously,

In the following,
*PC* and

The study of differential equations and variational problems with nonstandard
*p*-Laplacian problem. If

(a) If

is zero in general, and only under some special conditions
*p*-Laplacian problems, for example, in [20], the authors use this property to deal with the existence of solutions.

(b) If
*u* is concave, this property is used extensively in the study of one-dimensional *p*-Laplacian problems (see [21]), but it is invalid for

In recent years, many results have been devoted to the existence of solutions for
the Laplacian impulsive differential equation boundary value problems; see, for example,
[22-29]. There are some methods to deal with these problems, for example, sub-super-solution
method, fixed point theorem, monotone iterative method, coincidence degree. Because
of the nonlinear property of
*p*-Laplacian impulsive differential equation boundary value problems are rare (see [30-33]). In [34], using the coincidence degree method, the present author investigates the existence
of solutions for

In this paper, when

where
*p*-Laplacian impulsive problems have two kinds of impulsive conditions, including LI
and NLI; but Laplacian impulsive problems only have LI in general. It is another difference
between *p*-Laplacian impulsive problems and Laplacian impulsive problems. Moreover, since the
Rayleigh quotient
*p*-Laplacian impulsive problems, the nonlinear term only needs to satisfy the sub-

Let

(i) For almost every

(ii) For each
*J*;

(iii) For each

We say a function
*J*.

In this paper, we always use

We say *f* satisfies the sub-
*f* satisfies

where

We will discuss the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) in the following three cases:

Case (i):

Case (ii):

Case (iii):

This paper is organized as five sections. In Section 2, we present some preliminaries
and give the operator equation which has the same solutions of (1)-(4) in the three
cases, respectively. In Section 3, we give the existence of solutions for system (1)-(4)
or (1) with (2), (4) and (5) when

### 2 Preliminary

For any
*φ* has the following properties.

**Lemma 2.1** (see [34])

*φ**is a continuous function and satisfies*:

(i) *For any*
*is strictly monotone*, *i*.*e*.,

(ii) *There exists a function*
*as*
*such that*

It is well known that

It is clear that

In this section, we will do some preparation and give the operator equation which has the same solutions of (1)-(4) in three cases, respectively. At first, let us now consider the following simple impulsive problem with boundary value condition (4):

where

Denote

We will discuss it in three cases, respectively.

#### 2.1 Case (i)

Suppose that
*u* is a solution of (6) with (4), we have

Denote
*a*, *b* and

By solving for

which together with boundary value condition (4) implies

and

Denote

then *W* is a Banach space.

For any

Denote

Throughout the paper, we denote

**Lemma 2.2***Suppose that*
*on*
*and*
*on*
*Then the function*
*has the following properties*:

(i) *For any fixed*
*the equation*

*has a unique solution*

(ii) *The function*
*defined in* (i), *is continuous and sends bounded sets to bounded sets*. *Moreover*, *for any*
*we have*

*where the notation*
*means*

*Proof* (i) From Lemma 2.1, it is immediate that

and hence, if (8) has a solution, then it is unique.

Set

Suppose that

Thus the
*J*, namely, for any

Obviously, we have

then it is easy to see that the

Let us consider the equation

According to the preceding discussion, all the solutions of (9) belong to

it means the existence of solutions of

In this way, we define a function

(ii) By the proof of (i), we also obtain

It only remains to prove the continuity of
*W* and

Now we denote by
*f* defined by

We define

where

It is clear that

If *u* is a solution of (6) with (4), we have

For fixed

Define

**Lemma 2.3** (i) *The operator*
*is continuous and sends equi*-*integrable sets in*
*to relatively compact sets in*

(ii) *The operator*
*is continuous and sends bounded sets in*
*to relatively compact sets in*

*Proof* (i) It is easy to check that

it is easy to check that

Let now *U* be an equi-integrable set in

We want to show that

Let

Hence the sequence
*PC*. According to the bounded continuity of the operator
*PC*, then is convergent in *PC*.

Since

it follows from the continuity of
*PC*. Thus

(ii) It is easy to see from (i) and Lemma 2.2.

This completes the proof. □

Let us define

It is easy to see that

**Lemma 2.4***Suppose that*
*on*
*and*
*on*
*Then**u**is a solution of* (1)-(4) *if and only if**u**is a solution of the following abstract operator equation*:

*Proof* Suppose that *u* is a solution of (1)-(4). By integrating (1) from 0 to *t*, we find that

It follows from (13) and (4) that

Combining the definition of

Conversely, if *u* is a solution of (12), then (2) is satisfied. It is easy to check that

and

By the condition of the mapping

Thus

It follows from (15) and (16) that (4) is satisfied.

From (12), we have

It follows from (17) that (3) is satisfied.

Hence *u* is a solution of (1)-(4). This completes the proof. □

#### 2.2 Case (ii)

Suppose that
*u* is a solution of (6) with (4), we have

Denote
*a*, *b* and

For any

Throughout the paper, we denote .

**Lemma 2.5***The function*
*has the following properties*:

(i) *For any fixed*
*the equation*
*has a unique solution*

(ii) *The function*
*defined in* (i), *is continuous and sends bounded sets to bounded sets*. *Moreover*, *for any*
*we have*

*where the notation*
*means*

*Proof* Similar to the proof of Lemma 2.2, we omit it here. □

We define

It is clear that

For fixed

Define

Similar to the proof of Lemma 2.3, we have the following.

**Lemma 2.6** (i) *The operator*
*is continuous and sends equi*-*integrable sets in*
*to relatively compact sets in*

(ii) *The operator*
*is continuous and sends bounded sets in*
*to relatively compact sets in*

*Let us define*
*as*

*It is easy to see that*
*is compact continuous*.

**Lemma 2.7***Suppose that*
*then**u**is a solution of* (1)-(4) *if and only if**u**is a solution of the following abstract operator equation*:

*Proof* Similar to the proof of Lemma 2.4, we omit it here. □

#### 2.3 Case (iii)

Suppose that
*u* is a solution of (6) with (4), we have

Denote
*a*, *b* and

From

From

For fixed

From (18) and (19), we have

Obviously,

Denote

**Lemma 2.8***Suppose that*
*g*, *h**satisfy one of the following*:

(1^{0})

(2^{0})
*on*
*and*
*on*

*Then the function*
*has the following properties*:

(i) *For any fixed*
*the equation*
*has a unique solution*

(ii) *The function*
*defined in* (i), *is continuous and sends bounded sets to bounded sets*. *Moreover*, *for any*
*we have*

*where the notation*
*means*

*Proof* Similar to the proof of Lemma 2.2, we omit it here. □

We define

It is clear that

For fixed

Define

Similar to the proof of Lemma 2.3, we have

**Lemma 2.9** (i) *The operator*
*is continuous and sends equi*-*integrable sets in*
*to relatively compact sets in*

(ii) *The operator*
*is continuous and sends bounded sets in*
*to relatively compact sets in*

*Let us define*
*as*

*It is easy to see that*
*is compact continuous*.

**Lemma 2.10***Suppose that*
*and*
*g*, *h**satisfy one of the following*:

(1^{0})

(2^{0})
*on*
*and*
*on*

*Then**u**is a solution of* (1)-(4) *if and only if**u**is a solution of the following abstract operator equation*:

*Proof* Similar to the proof of Lemma 2.4, we omit it here. □

### 3 Existence of solutions in Case (i)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions
for system (1)-(4) or (1) with (2), (4) and (5) when

When *f* satisfies the sub-

**Theorem 3.1***Suppose that*
*on*
*and*
*on*
*f**satisfies the sub*-
*growth condition*; *and operators**A**and**B**satisfy the following conditions*:

*then problem* (1)-(4) *has at least a solution*.

*Proof* First we consider the following problem:

Denote

where

Obviously, (

It is easy to see that the operator is compact continuous for any

We claim that all the solutions of (21) are uniformly bounded for