### Abstract

In this article, we deal with existence and multiplicity of solutions to the *p*-Laplacian system of the type

where Ω ⊂ ℝ^{N }is a bounded domain with smooth boundary ∂Ω, Δ_{p}*u *= div(|∇*u*|^{p-2}∇*u*) is the *p*-Laplacian operator,
*p**. By using the variational method and Ljusternik-Schnirelmann theory, we prove that
the system has at least cat_{Ω}(Ω) distinct nonnegative solutions.

**AMS 2010 Mathematics Subject Classifications**: 35J50; 35B33.

##### Keywords:

*p*-Laplacian system; Ljusternik-Schnirelmann theory; critical exponent; multiple solutions

### 1 Introduction and main results

In this article, we consider the existence and multiplicity of solutions for the following
critical *p*-Laplacian system:

where Ω ⊂ ℝ^{N }is a bounded domain with smooth boundary ∂Ω, Δ_{p}*u *= div(|∇*u*|^{p-2}∇*u*) is the *p*-Laplacian operator,
*λ, δ *are positive parameters.

The starting point on the study of the system (1.1) is its scalar version:

with 2 ≤ *p *≤ *q *< *p**. In a pioneer work Brezis and Nirenberg [1] showed that, if *p *= *q *= 2, the equation (1.2) has at least one positive solution provided *N *≥ 4 and 0 < *λ *< *λ*_{1}, where *λ*_{1 }is the first eigenvalue of the operator
*N *≥ 5, *p *= *q *= 2, for *λ *small enough equation (1.2) has at least cat_{Ω}(Ω) solutions, where cat_{Ω}(Ω) denotes the Ljusternik-Schnirelmann category of Ω in itself. Furthermore, Alves
and Ding [3] obtained the existence of cat_{Ω}(Ω) positive solutions to equation (1.2) with *p *≥ 2, *p *≤ *q *< *p**.

In recent years, more and more attention have been paid to the elliptic systems. In
particular, Ding and Xiao [4] concerned the case *F*(*x, u, v*) = 2|*u*|^{α}|*v*|^{β},*α *> 1, *β *>1 satisfying *α *+ *β *= *p**, i.e., the following elliptic system

Using standard tools of the variational theory and the Ljusternik-Schnirelmann category
theory, Ding and Xiao [4] have proved that system (1.3) has at least cat_{Ω}(Ω) positive solutions if *λ, δ *satisfied a certain condition. Hsu [5] obtained the existence of two positive solutions of system (1.3) with the sublinear
perturbation of 1 < *q *< *p *< *N*. Recently, Shen and Zhang [6] extended the results in [5] to the case (1.1) with 1 < *q *< *p *< *N *and obtained similar results. In this article, we study (1.1) and complement the results
of [5,6] to the case 2 ≤ *p *≤ *q *< *p**, also extend the results of [4,7]. To the best of our knowledge, problem (1.1) has not been considered before. Thus
it is necessary for us to investigate the critical *p*-Laplacian systems (1.1) deeply. For more similar problems, we refer to [8-17], and references therein.

Before stating our results, we need the following assumptions:

(*F*_{0})

(*F*_{1})
*u, v *∈ ℝ^{+};

(*F*_{2})
*u *and *v *for all *u, v *> 0.

The main results we get are the following:

**Theorem 1.1**. *Suppose N *≥ *p*^{2 }*and F satisfies *(*F*_{0})-(*F*_{2}), *then the problem (1.1) has at least one nonnegative solution for *2 ≤ *p *< *q *< *p** *and λ, δ *> 0, *or q *= *p and λ, δ *∈ (0, Λ_{1}), *where *Λ_{1 }*is the first eigenvalue of *

**Theorem 1.2**. *Suppose N *≥ *p*^{2}, 2 ≤ *p *≤ *q *< *p* and F satisfies *(*F*_{0})-(*F*_{2}), *then there exists *Λ > 0 *such that the problem (1.1) has at least *cat_{Ω}(Ω) *distinct nonnegative solutions for λ, δ *∈ (0,Λ).

**Remark 1.1**. *Theorem 1 in *[4]*is the special case of our Theorem 1.2 corresponding to F*(*x*,*u*,*v*) = 2|*u*|^{α}|*v*|^{β},*α *> 1,*β *> 1,*α *+ *β *= *p*. There are functions F*(*x*,*u*,*v*) *satisfying the conditions of our Theorems 1.1 and 1.2. Some typical examples are:*

*(i) *

*(ii) *

*where *
*Obviously, F*(*x, u, v*) *satisfies *(*F*_{0})-(*F*_{2}).

This article is organized as follows. In Section 2, some notations and the Mountain-Pass levels are established and the Theorem 1.1 is proved. We present some technical lemmas which are crucial in the proof of the Theorem 1.2 in Section 3. Theorem 1.2 is proved in Section 4.

### 2 Notations and proof of Theorem 1.1

Throughout this article, *C, C*_{i }will denote various positive constants whose exact values are not important, → (respectively
⇀) denotes strong (respectively weak) convergence. *O*(*ε*^{t}) denotes |*O*(*ε*^{t})|/*ε*^{t }≤ *C, o*_{m}(1) denotes *o*_{m}(1) → 0 as *m *→ ∞. *L*^{s}(Ω)(1 ≤ *s *< +∞) denotes Lebesgue spaces, the norm *L*^{s }is denoted by | · |_{s }for 1 ≤ *s *< + ∞. Let *B*_{r}(*x*) denotes a ball centered at *x *with radius *r*, the dual space of a Banach space *E *will be denoted by *E*^{-1}. We define the product space

Using assumption of (*F*_{1}), we have the so-called Euler identity

In addition, we can extend the function *F*(*x*,*u*,*v*) to the whole
*u*^{+ }= max{*u*,0}. It is easy to check that
*C*^{1 }and its restriction to
*F*(*x*,*u*,*v*). In order to simplify the notation we shall write, from now on, only *F*(*x*,*u*,*v*) to denote the above extension.

A pair of functions (*u, v*) ∈ *E *is said to be a weak solution of problem (1.1) if

Thus, by (2.1) the corresponding energy functional of problem (1.1) is defined on
*E *by

Using (*F*_{0})-(*F*_{2}), we can verify *I*_{λ, δ}(*u, v*) ∈ *C*^{1}(*E*, ℝ) (see [6]). It is well known that the weak solutions of problem (1.1) are the critical points
of the energy functional *I*_{λ, δ}(*u, v*).

The functional *I *∈ *C*^{1}(*E*, ℝ) is said to satisfy the (*PS*)_{c }condition if any sequence {*u*_{m}} ⊂ *E *such that as *m *→ ∞, *I*(*u*_{m}) → *c, I*'(*u*_{m}) → 0 strongly in *E*^{-1 }contains a subsequence converging in *E *to a critical point of *I*. In this article, we will take *I *= *I*_{λ, δ}(*u, v*) and

As the energy functional *I*_{λ,δ }is not bounded below on *E*, we need to study *I*_{λ,δ }on the Nehari manifold

Note that
*c*_{λ,δ }as

Next, we present some properties of *c*_{λ,δ }and
*ρ *> 0, such that

It is standard to check that *I*_{λ,δ }satisfies Mountain-Pass geometry, so we can use the homogeneity of *F *to prove that *c*_{λ,δ }can be alternatively characterized by

where Γ = {*γ *∈ *C*([0, 1],*E*) : *γ*(0) = 0,*I*_{λ,δ}(*γ*(1)) < 0}. Moreover, for each (*u, v*) ∈ *E*\{(0,0)}, there exists a unique *t* *> 0 such that
*t *↦ *I*_{λ,δ}(*t*(*u, v*)), for *t *≥ 0, is achieved at *t *= *t*.*

In this section, we will find the range of *c *where the (*PS*)_{c }condition holds for the functional *I*_{λ,δ}. First let us define

**Lemma 2.1**. *If N *≥ *p*^{2 }*and F satisfies *(*F*_{0})-(*F*_{2}), *then the functional I*_{λ,δ }*satisfies the *(*PS*)_{c }*condition for all *
*provide one of the following conditions holds*

*(i) *2 ≤ *p *< *q *< *p* and λ, δ *> 0;

*(ii) q *= *p, and λ, δ *∈ (0, Λ_{1}), *where *Λ_{1 }> 0 *denotes the first eigenvalue of *

*Proof*. Let {(*u*_{m}, *v*_{m})} ⊂ *E *such that
*u*_{m}, *v*_{m})} is bounded in *E*. If the above item (i) is true it suffices to use the definition of *I*_{λ,δ }to obtain *C*_{1 }> 0 such that

The above expression implies that {(*u*_{m}, *v*_{m})} ⊂ *E *is bounded. When (ii) occurs, in this case, it follows that

and therefore we get

Since *λ, δ *∈ (0,Λ_{1}) the boundedness of {(*u*_{m}, *v*_{m})} follows as the first case.

So, {(*u*_{m}*, v*_{m})} is bounded in *E*. Going if necessary to a subsequence, we can assume that

as *m *→ ∞. Clearly, we have

Moreover, a standard argument shows that

Let

By the same method of [8, Lemma 5] (or [6, Lemma 3.4]), we obtain

By (2.4)-(2.7) and the weak convergence of (*u*_{m}, *v*_{m}), we have

By using

Recalling that

where *k *is a nonnegative number.

In view of the definition of *S*_{F}, we have that

Taking the limit we get
*k *> 0, we conclude that

which is a contradiction. Hence *k *= 0 and therefore (*u*_{m}, *v*_{m}) → (*u, v*) strongly in *E*.

Before presenting our next result we recall that, for each *ε *> 0, the function

satisfies

where *S *is the best constant of the Sobolev embedding
*F*, we obtain *A, B *> 0 such that

from which and (2.10) it follows that

We define a cut-off function
*ϕ*(*x*) = 1 if |*x*| ≤ *R*; *ϕ*(*x*) = 0 if |*x*| ≥ 2*R *and 0 ≤ *ϕ*(*x*) ≤ 1, where *B*_{2R}(0) ⊂ Ω, set
*U*_{ε }was defined in (2.9). So that

where *A *≈ *B *means *C*_{1}*B *≤ *A *≤ *C*_{2}*B*.

**Lemma 2.2**. *Suppose that F satisfies *(*F*_{0})-(*F*_{2}), 2 ≤ *p *< *q *< *p** *and λ *> 0, *δ *> 0, *then *
*The same result holds if q *= *p and λ, δ *∈ (0,Λ_{1}), *where *Λ_{1 }> 0 *denotes the first eigenvalue of *

*Proof*. We can use the homogeneity of *F *to get, for any *t *≥ 0,

We shall denote by *h*(*t*) the right-hand side of the above equality and consider two distinct cases.

**Case 1**. 2 ≤ *p *< *q *< *p**.

From the fact that
*h*(*t*) > 0 when *t *is close to 0, there exists *t*_{ε }> 0 such that

Let

and notice that the maximum value of *g*(*t*) occurs at the point

So, for each *t *≥ 0,

and therefore

We claim that, for some *C*_{2 }> 0, there holds

Indeed, if this is not the case, we have that
*ε*_{m }→ 0^{+}, then,

which is a contradiction. So, the claim holds and we infer from (2.15) and (2.11)-(2.13) that

where
*N *≥ *p*^{2}. By *N *≥ *p*^{2 }and 2 ≤ *p *< *q *< *p* *we obtain

**Case 2**. *q *= *p*.

In this case, we have that *h'*(*t*) = 0 if and only if,

Since we suppose *λ, δ *∈ (0,Λ_{1}), we can use Poincaré's inequality to obtain

Thus, there exists *t*_{ε }> 0 satisfying (2.14).

Arguing as in the first case we conclude that, from (2.16) for *ε *> 0 small, there holds

Because
*N *> *p*^{2 }and
*N *= *p*^{2}, then *ε*^{p-1 }= *o*(*ε*^{p-1}| ln *ε*|). If *N *> *p*^{2}, then
*ε *> 0 small enough, we have

This concludes the proof.

By Lemmas 2.1 and 2.2 we can prove our first result.

**Proof of Theorem 1.1**.

Since *I*_{λ,δ }satisfies the geometric conditions of the Mountain-Pass theorem, there exists {(*u*_{m}, *v*_{m})} ⊂ *E *such that
*u*_{m}, *v*_{m})} converges, along a subsequence, to a nonzero critical point (*u*,*v*) ∈ *E *of *I*_{λ,δ}. Then, if we denote by *u*^{- }= max{-*u*,0} and *v*^{- }= max{-*v*,0} the negative part of *u *and *v*, respectively, we get

it follows that (*u*^{-},*v*^{-}) = (0,0). Hence, *u*,*v *≥ 0 in Ω. The Theorem 1.1 is proved.

We finalize this section with the study of the asymptotic behavior of the minimax
level *c*_{λ,δ }as both the parameters *λ, δ *approach zero.

**Lemma 2.3**.

*Proof*. We first prove the second equality. It follows from *λ *= *δ *= 0 that *λ*|*u*|^{q }+ *δ*|*v*|^{q }≡ 0. If *A, B, u*_{ε}, *g*_{ε}, and *t*_{ε }are the same as those in the proof of Lemma 2.2, we have that

Taking the limit as *ε *→0^{+ }and using (2.11), we conclude that

In order to obtain the reverse inequality we consider {(*u*_{m}, *v*_{m})} ⊂ *E *such that *I*_{0,0 }(*u*_{m}, *v*_{m}) → *c*_{0,0 }and
*u*_{m}, *v*_{m})} is bounded in *E *and therefore

Taking the limit in the inequality

and therefore

We proceed now with the calculation of
*λ*_{m}},{*δ*_{m}} ⊂ ℝ^{+ }such that *λ*_{m}, *δ*_{m }→ 0^{+}. Since *λ*_{m}, *δ*_{m }are positive, we have that
*u, v*) is nonnegative. Thus, for this kind of function, we have that

It follows that

in the last equality, we have used the infimum *c*_{0,0 }which can be attained at a nonnegative solution. The above inequality implies that

On the other hand, it follows from Theorem 1.1 that there exists {(*u*_{m}, *v*_{m})} ⊂ *E *such that

Since
*u*_{m}, *v*_{m})} is bounded in *E*. Since

Let *t*_{m }> 0 be such that

If {*t*_{m}} is bounded, we can use the above estimate and (2.18) to get

This and (2.17) we get

that is

It remains to check that {*t*_{m}} is bounded. A straightforward calculation shows that

Since

Hence
_{Ω }*F*(*x, u*_{m}, *v*_{m})*dx *≥ *C*_{5 }> 0. Thus, the boundedness of {(*u*_{m}, *v*_{m})} and (2.19) imply that {*t*_{m}} is bounded. This completes the proof.

### 3 Some technical lemmas

In this section, we denote by
*σ*_{m}(*φ*) → *σ*(*φ*) for all *φ *∈ *C*_{0}(Ω). By [18, Theorem 1.39], every bounded sequence

The next lemma is a version of the second concentration-compactness lemma of Lions [21]. It is also inspired by [18, Lemma 1.40] and [[22], Lemma 2.4].

**Lemma 3.1**. *Suppose that the sequence *{(*u*_{m},*v*_{m})} ⊂ *D*^{1,p}(ℝ^{N}) × *D*^{1,p}(ℝ^{N}) *satisfies*

*and define*

*then it follows that*

*Moreover, if *(*u*,*v*) = (0,0) *and *
*then the measures μ*,*ν, and σ are concentrated at a single point, respectively.*

*Proof*. We first recall that, in view of the definition of *S*_{F}, for each nonnegative function

Moreover, arguing as [8, Lemma 5], we have that

Since *F *is *p**-homogeneous, we can use the two above expressions and argue along the same line of
the proof of Lemma 1.40 in [18] to conclude that (3.2)-(3.5) hold. If (*u, v*) = (0,0) and
*μ, ν *and *σ *are concentrated at a single point, respectively.

**Remark 3.1**. *We notice that the last conclusion of the above result holds even if *(*u, v*) ≢ (0,0). *Indeed, in this case we can define *
*and notice that*

*Since *
*and therefore *
*and *
*where μ*,*σ, and ν are the same as those in Lemma 3.1. Thus, if *
*we also have that *
*and the result follows from the last part of Lemma 3.1.*

Now, we introduce the following Lemma.

**Lemma 3.2**. *Suppose *{(*u*_{m}, *v*_{m})} ⊂ *E such that *∫_{Ω }*F*(*x, u*_{m}, *v*_{m})*dx *= 1 *and *
*Then there exist *{*r*_{m}} ⊂ (0, +∞) *and *{*y*_{m}} ⊂ ℝ^{N }*such that*

*contains a convergent subsequence denoted again by *
*such that *
*in D*^{1,p}(ℝ^{N}) × *D*^{1,p}(ℝ^{N}). *Moreover, as m *→ ∞, *we have r*_{m }→ 0 *and *

*Proof*. For each *r *> 0, we consider the Lévy concentration functions

Since for every *m*,

there exist *r*_{m }> 0 and a sequence

Recalling that

We shall prove that the above sequences {*r*_{m}} and {*y*_{m}} satisfy the statements of the lemma. First notice that

By (3.6), a straightforward calculation provides

Hence, we can apply Lemma 3.1 to obtain (*ω*_{1},*ω*_{2}) ∈ *D*^{1,p}(ℝ^{N}) × *D*^{1,p }(ℝ^{N}) satisfying

The second equality in (3.8) implies that

which is a contradiction. Thus
*R *> 1. Thus, we conclude that *ν*_{∞ }= 0.

Let us prove that ||*ν*|| = 0. Arguing by contradiction, then ||*ν*|| = 1. It follows from the first equality in (3.8) that *S*_{F }≥ ||*μ*|| + ||*σ*||. On the other hand, the first inequality in (3.9) provides ||*μ*|| + ||*σ*|| ≥ *S*_{F}. Hence, we conclude that ||*μ*|| + ||*σ*|| = *S*_{F}. Since we suppose that ||*ν*|| = 1 we obtain
*x*_{0 }∈ ℝ^{N}. Thus, from (3.7), we get

This contradiction proves that ∥*ν*∥ = 0.

Since ∥*ν*∥ = *ν*_{∞ }= 0, we have that