### Abstract

In this paper, we consider *p*-Laplacian systems with singular weights. Exploiting Amann type three solutions theorem
for a singular system, we prove the existence, nonexistence, and multiplicity of positive
solutions when nonlinear terms have a combined sublinear effect at ∞.

**MSC: **
35J55, 34B18.

##### Keywords:

*p*-Laplacian system; singular weight; upper solution; lower solution; three solutions theorem

### 1 Introduction

In this paper, we study one-dimensional *p*-Laplacian system with singular weights of the form where
*λ* is a nonnegative parameter,
*u* for corresponding scalar problem of (

For more precise description, let us introduce the following two classes of weights;

We note that *h* given in the above example satisfies
*p*-Laplacian problems with
*p*-Laplacian with the derivative-dependent nonlinearity respectively; and De Coster
and Nicaise [11] - for semilinear elliptic problems in nonsmooth domains. For noncooperative elliptic
systems (

The three solutions theorem for our system (
*p*-Laplacian problems is not obviously extended from previous works mainly by the possibility
*f* and *g*. Therefore, the first step is to investigate certain conditions on *f* and *g* to guarantee
*p*-Laplacian systems with weights of

To cover a larger class of differential system, we consider the systems of the form
where
*F* and *G* as follows: (
*u*.; (*H*) = There exist

and

for all

**Theorem 1.1***Assume* (*H*), (
*and* (
*Let*
*be a lower solution and an upper solution and*
*be a strict lower solution and a strict upper solution of problem* (*P*) *respectively*. *Also*, *assume that all of them are contained in*
*and satisfy*
*Then problem* (*P*) *has at least three solutions*
*and*
*such that*
*and*

As an application of Theorem 1.1, we study the existence, nonexistence, and multiplicity
of positive radial solutions for the following quasilinear system on an exterior domain:
where

In recent years, the existence of positive solutions for such systems has been widely
studied, for example, in [1] and [27] for second order ODE systems, in [3,7,9,10,13,14,16] and [8] for semilinear elliptic systems on a bounded domain and in [5,15,17] and [2] for *p*-Laplacian systems on a bounded domain.

For a precise description, let us give the list of assumptions that we consider. (*k*) =

; (

*f*and

*g*are nondecreasing..

Condition (

where

where

Among the reference works mentioned above, Hai and Shivaji [17] and Ali and Shivaji [2] (with more general nonlinearities) considered problem (
*f* and *g* with

We first transform (
*p*-Laplacian systems (

It is not hard to see that if
*k*), then

**Theorem 1.2***Assume*
*and* (
*Then there exists*
*such that* (
*has no positive solution for*
*at least one positive solution at*
*and at least two positive solutions for*

As a corollary, we obtain our second main result as follows.

**Corollary 1.3***Assume* (*k*), (
*and* (
*Then there exists*
*such that* (
*has no positive radial solution for*
*at least one positive radial solution at*
*and at least two positive radial solutions for*

We finally notice that the first eigenfunctions of make an important role to construct
upper solutions in the proofs of Theorem 1.2 and Theorem 1.1. This is possible due
to a recent work of Kajikiya, Lee, and Sim [19] which exploits the existence of discrete eigenvalues and the properties of corresponding
eigenfunctions for problem (*E*) with

This paper is organized as follows. In Section 2, we state a
*p*-Laplacian systems. In addition, we introduce definitions of (strict) upper and lower
solutions, a related theorem and a fixed point theorem for later use. In Section 3,
we prove Theorem 1.2.

### 2 Three solutions theorem

In this section, we give definitions of upper and lower solutions and prove three
solutions theorem for the following singular system where

We call
*P*) if
*P*).

**Definition 2.1** We say that
*lower solution* of problem (*P*) if

We also say that
*upper solution* of problem (*P*) if
*strict* lower solution and *strict* upper solution of problem (*P*), respectively, if
*P*), respectively and satisfying

We note that the inequality on
*P*) is given as follows. The proof can be done by obvious combination from Lee [20], Lee and Lee [21] and Lü and O’Regan [22].

**Theorem 2.2***Let*
*and*
*be a lower solution and an upper solution of problem* (*P*) *respectively such that*(
*for all*
*Assume* (
*Also assume that there exist*
*such that*(
*for all*
*Then problem* (*P*) *has at least one solution*
*such that*

**Remark 2.3** It is not hard to see that condition (*H*) implies the following condition;

For each

for

**Lemma 2.4***Assume* (*H*) *and* (
*Let*
*be a nontrivial solution of* (*P*). *Then there exists*
*such that both**u**and**v**have no interior zeros in*

*Proof* Let
*P*). Suppose, on the contrary, that there exist sequences
*u* and *v* respectively with
*v* is concave on the interval. Thus *v* should have at most one interior zero in
*u* and *v* on

and similarly,

Therefore, it follows from plugging (2.2) into (2.1) that

Since
*n*, we obtain

This contradicts (2.3) and the proof is done. □

**Theorem 2.5***Assume* (*H*) *and* (
*If*
*is a solution of* (*P*), *then*

*Proof* Let
*P*). Then

We will show

Let
*P*) over

where we use the fact that
*v* is concave. From

Thus we have

Since *ε* is arbitrary, we have

Using the fact

On the other hand, we observe the inequality

where

Since

Integrating (*P*) over

here we use the fact that

Integrating (2.9) over
*s* and using (2.8), we have

Similarly, we have

Adding (2.10) and (2.11), we have

on

Now, we consider the three solutions theorem for singular *p*-Laplacian system (*P*). For

then the zero of
*ν*. Define

It is known that *A* is completely continuous [24]. Define

If *F* and *G* satisfy condition (*H*), then for

This implies
*P*) and the regularity of solutions (Theorem 2.5) is crucial in this argument. Now,
define an operator *T* by

then we see that

**Lemma 2.6***Assume* (*H*), (
*and* (
*Let*
*and*
*be a strict lower solution and a strict upper solution of problem* (*P*) *respectively such that*
*and*
*Then problem* (*P*) *has at least one solution*
*such that*

*Moreover*, *for*
*large enough*,

*where*

*Proof* Define

and also define

Let us consider the following modified problem We first show that there exists a
constant
*H*), (

Similarly, we see that

Assume, on the contrary, that there exists

Then choosing

Now, assume

Then it is obvious that

from
*t*, we have

Similarly, we see that

Since every solution of (

Since

This completes the proof. □

We now prove three solutions theorem for (*P*).

#### Proof of Theorem 1.1

Define

and let us consider Then noting that every solution
*H*) such that

Let

We show that

Similarly, we get

Moreover,

Similarly, we also get

For

Then by Theorem 2.2, there exist two solutions
*P*) satisfying

and by the excision property, we have

This completes the proof.

### 3 Application

In this section, we prove the existence, nonexistence, and multiplicity of positive
solutions for (

and define

where

respectively. And define

Then it is known that

**Remark 3.1** If

For later use, we introduce the following well-known result. See [12] for proof and details.

**Proposition 3.2***Let**X**be a Banach space*,
*an order cone in**X*. *Assume that*
*and*
*are bounded open subsets in**X**with*
*and*
*Let*
*be a completely continuous operator such that either*

(i)
*and*

*or*

(ii)
*and*

*Then**A**has a fixed point in*

**Lemma 3.3***Assume*
*and* (
*Let*
*be a compact subset of*
*Then there exists a constant*
*such that for all*
*and all possible positive solutions*
*of* (
*one has*

*Proof* If it is not true, then there exist

where

This implies both

Thus we get

as

**Lemma 3.4***Assume*
*and* (
*If* (
*has a lower solution*
*for some*
*then* (
*has a solution*
*such that*

*Proof* It suffices to show the existence of an upper solution

(Case I) Both *f* and *g* are bounded.

Since

Similarly,

(Case II)

Using (

Let

And

Thus

(Case III) *g* is bounded and

Choose

Then

And

Consequently, by Theorem 2.2, (

□

**Lemma 3.5***Assume*
*and* (
*Then there exists*
*such that if* (
*has a positive solution*
*then*

*Proof* Let

From (3.1) and (

where

Thus we have

□

**Lemma 3.6***Assume*
*and* (
*Then for each*
*there exists*
*such that for*
*has a positive solution*
*with*
*and*

*Proof* We know that if

where

If

If

By the concavity of

By similar argument as the above, with (3.3), we may show that

Let

Let

By Proposition 3.2, (

We now prove one of the main results for this paper.

#### Proof of Theorem 1.2

From Lemma 3.6 and Lemma 3.5, we know that the set
*T*, there exist sequences

But from (