Research

# Multiple solutions for p-Laplacian systems with critical homogeneous nonlinearity

Dengfeng Lü

### Author affiliations

School of Mathematics and Statistics, Hubei Engineering University, Hubei 432000, P. R. China

Boundary Value Problems 2012, 2012:27  doi:10.1186/1687-2770-2012-27

 Received: 17 November 2011 Accepted: 28 February 2012 Published: 28 February 2012

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

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

where Ω ⊂ ℝN is a bounded domain with smooth boundary ∂Ω, Δpu = div(|∇u|p-2u) is the p-Laplacian operator, denotes the Sobolev critical exponent, is a homogeneous function of degree p*. By using the variational method and Ljusternik-Schnirelmann theory, we prove that the system has at least cat(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:

(1.1)

where Ω ⊂ ℝN is a bounded domain with smooth boundary ∂Ω, Δpu = div(|∇u|p-2u) is the p-Laplacian operator, denotes the Sobolev critical exponent, is a homogeneous function of degree and λ, δ are positive parameters.

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

(1.2)

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 < λ1, where λ1 is the first eigenvalue of the operator . In particular, the first multiplicity result for (1.2) has been achieved by Rey [2] in the semilinear case. Precisely Rey proved that if N ≥ 5, p = q = 2, for λ small enough equation (1.2) has at least cat(Ω) solutions, where cat(Ω) denotes the Ljusternik-Schnirelmann category of Ω in itself. Furthermore, Alves and Ding [3] obtained the existence of cat(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

(1.3)

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) 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:

(F0) and holds for all ;

(F1) , where u, v ∈ ℝ+;

(F2) are strictly increasing functions about u and v for all u, v > 0.

The main results we get are the following:

Theorem 1.1. Suppose N p2 and F satisfies (F0)-(F2), 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, Λ1), where Λ1 is the first eigenvalue of .

Theorem 1.2. Suppose N p2, 2 ≤ p q < p* and F satisfies (F0)-(F2), then there exists Λ > 0 such that the problem (1.1) has at least cat(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 (F0)-(F2).

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, Ci will denote various positive constants whose exact values are not important, → (respectively ⇀) denotes strong (respectively weak) convergence. O(εt) denotes |O(εt)|/εt C, om(1) denotes om(1) → 0 as m → ∞. Ls(Ω)(1 ≤ s < +∞) denotes Lebesgue spaces, the norm Ls is denoted by | · |s for 1 ≤ s < + ∞. Let Br(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 endowed with the norm , and the norm .

Using assumption of (F1), we have the so-called Euler identity

(2.1)

In addition, we can extend the function F(x,u,v) to the whole by considering , where u+ = max{u,0}. It is easy to check that is of class C1 and its restriction to coincides with 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 (F0)-(F2), we can verify Iλ, δ(u, v) ∈ C1(E, ℝ) (see [6]). It is well known that the weak solutions of problem (1.1) are the critical points of the energy functional Iλ, δ(u, v).

The functional I C1(E, ℝ) is said to satisfy the (PS)c condition if any sequence {um} ⊂ E such that as m → ∞, I(um) → c, I'(um) → 0 strongly in E-1 contains a subsequence converging in E to a critical point of I. In this article, we will take I = Iλ, δ(u, v) and .

As the energy functional Iλ,δ is not bounded below on E, we need to study Iλ,δ on the Nehari manifold

Note that contains every nonzero solution of problem (1.1), and define the minimax cλ,δ as

Next, we present some properties of cλ,δ and . Its proofs can be done as [18, Theorem 4.2]. First of all, we note that there exists ρ > 0, such that

It is standard to check that Iλ,δ satisfies Mountain-Pass geometry, so we can use the homogeneity of F to prove that cλ,δ can be alternatively characterized by

(2.2)

where Γ = {γ C([0, 1],E) : γ(0) = 0,Iλ,δ(γ(1)) < 0}. Moreover, for each (u, v) ∈ E\{(0,0)}, there exists a unique t* > 0 such that . The maximum of the function t Iλ,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 Iλ,δ. First let us define

(2.3)

Lemma 2.1. If N p2 and F satisfies (F0)-(F2), then the functional Iλ,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, Λ1), where Λ1 > 0 denotes the first eigenvalue of .

Proof. Let {(um, vm)} ⊂ E such that and . Now, we first prove that {(um, vm)} is bounded in E. If the above item (i) is true it suffices to use the definition of Iλ,δ to obtain C1 > 0 such that

The above expression implies that {(um, vm)} ⊂ E is bounded. When (ii) occurs, in this case, it follows that

and therefore we get

Since λ, δ ∈ (0,Λ1) the boundedness of {(um, vm)} follows as the first case.

So, {(um, vm)} is bounded in E. Going if necessary to a subsequence, we can assume that

as m → ∞. Clearly, we have

(2.4)

Moreover, a standard argument shows that . Thus we get

(2.5)

Let , then by Brezis-Lieb Lemma in [19] implies

(2.6)

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

(2.7)

By (2.4)-(2.7) and the weak convergence of (um, vm), we have

(2.8)

By using and (2.4), (2.6), and (2.7), we get

Recalling that , we can use the above equality and (2.8) to obtain

where k is a nonnegative number.

In view of the definition of SF, we have that

Taking the limit we get . So, if k > 0, we conclude that and therefore

which is a contradiction. Hence k = 0 and therefore (um, vm) → (u, v) strongly in E.

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

(2.9)

satisfies

(2.10)

where S is the best constant of the Sobolev embedding . Thus, using [8, Lemma 3] and the homogeneity of F, we obtain A, B > 0 such that

from which and (2.10) it follows that

(2.11)

We define a cut-off function such that ϕ(x) = 1 if |x| ≤ R; ϕ(x) = 0 if |x| ≥ 2R and 0 ≤ ϕ(x) ≤ 1, where B2R(0) ⊂ Ω, set , where Uε was defined in (2.9). So that . Then, we can get the following results from [[20], Lemma 11.1]:

(2.12)

(2.13)

where A B means C1B A C2B.

Lemma 2.2. Suppose that F satisfies (F0)-(F2), 2 ≤ p < q < p* and λ > 0, δ > 0, then . The same result holds if q = p and λ, δ ∈ (0,Λ1), where Λ1 > 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 and h(t) > 0 when t is close to 0, there exists tε > 0 such that

(2.14)

Let

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

So, for each t ≥ 0,

and therefore

(2.15)

We claim that, for some C2 > 0, there holds

Indeed, if this is not the case, we have that for some sequence εm → 0+, then,

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

(2.16)

where . We know if N p2. By N p2 and 2 ≤ p < q < p* we obtain . Thus from the above inequality we conclude that, for each ε > 0 small, there holds

Case 2. q = p.

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

Since we suppose λ, δ ∈ (0,Λ1), we can use Poincaré's inequality to obtain

Thus, there exists tε > 0 satisfying (2.14).

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

Because if N > p2 and if N = p2, then εp-1 = o(εp-1| ln ε|). If N > p2, then , so . Choosing 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 Iλ,δ satisfies the geometric conditions of the Mountain-Pass theorem, there exists {(um, vm)} ⊂ E such that . It follows from Lemmas 2.1 and 2.2 that {(um, vm)} converges, along a subsequence, to a nonzero critical point (u,v) ∈ E of Iλ,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 cλ,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, uε, gε, and tε are the same as those in the proof of Lemma 2.2, we have that . Thus

Taking the limit as ε →0+ and using (2.11), we conclude that .

In order to obtain the reverse inequality we consider {(um, vm)} ⊂ E such that I0,0 (um, vm) → c0,0 and . It is easy to show that the sequence {(um, vm)} is bounded in E and therefore . It follows that

Taking the limit in the inequality we conclude, as in the proof of Lemma 2.1, that . Hence,

and therefore .

We proceed now with the calculation of . Let {λm},{δm} ⊂ ℝ+ such that λm, δm → 0+. Since λm, δm are positive, we have that whenever (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 c0,0 which can be attained at a nonnegative solution. The above inequality implies that

(2.17)

On the other hand, it follows from Theorem 1.1 that there exists {(um, vm)} ⊂ E such that

Since is bounded, the same argument performed in the proof of Lemma 2.1 implies that {(um, vm)} is bounded in E. Since

(2.18)

Let tm > 0 be such that . Since , we have that

If {tm} 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 {tm} is bounded. A straightforward calculation shows that

(2.19)

Since , we obtain

Hence , and therefore from the above expression it follows that ∫Ω F(x, um, vm)dx C5 > 0. Thus, the boundedness of {(um, vm)} and (2.19) imply that {tm} is bounded. This completes the proof.

### 3 Some technical lemmas

In this section, we denote by the Banach space of finite Radon measures over Ω equipped with the norm . A sequence is said to converge weakly to provided σm(φ) → σ(φ) for all φ C0(Ω). By [18, Theorem 1.39], every bounded sequence contains a weakly convergent subsequence.

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 {(um,vm)} ⊂ D1,p(ℝN) × D1,p(ℝN) satisfies

and define

(3.1)

then it follows that

(3.2)

(3.3)

(3.4)

(3.5)

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 SF, for each nonnegative function we have

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 , the same argument of step 3 of the proof of Lemma 1.40 in [18] implies that the measures 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 {(um, vm)} ⊂ E such that Ω F(x, um, vm)dx = 1 and . Then there exist {rm} ⊂ (0, +∞) and {ym} ⊂ ℝN such that

(3.6)

contains a convergent subsequence denoted again by such that in D1,p(ℝN) × D1,p(ℝN). Moreover, as m → ∞, we have rm → 0 and .

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

Since for every m,

there exist rm > 0 and a sequence satisfying

Recalling that , we conclude that is bounded. Hence, up to a subsequence, and we obtain

We shall prove that the above sequences {rm} and {ym} satisfy the statements of the lemma. First notice that

(3.7)

By (3.6), a straightforward calculation provides

Hence, we can apply Lemma 3.1 to obtain (ω1,ω2) ∈ D1,p(ℝN) × D1,p (ℝN) satisfying

(3.8)

(3.9)

The second equality in (3.8) implies that . If one of these values belongs to the open interval (0,1), we can use (3.8), and (3.9) to get

which is a contradiction. Thus . Actually, it follows from (3.7) that for any 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 SF ≥ ||μ|| + ||σ||. On the other hand, the first inequality in (3.9) provides ||μ|| + ||σ|| ≥ SF. Hence, we conclude that ||μ|| + ||σ|| = SF. Since we suppose that ||ν|| = 1 we obtain . It follows from Remark 3.1 that for some x0 ∈ ℝN. Thus, from (3.7), we get

This contradiction proves that ∥ν∥ = 0.

Since ∥ν∥ = ν= 0, we have that . This and (3.8) provide

So, and therefore strongly in D1,p(ℝN) × D1,p(ℝN) and for a.e. x ∈ ℝN. In order to conclude the proof we notice that

Since {(um, vm)} is bounded and (ω1, ω2) ≢ (0,0), we infer from the above equality that, up to a subsequence, rm r0 ≥ 0. If |ym| → ∞, for each fixed x ∈ ℝN, we have that there exists mx N such that rmx + ym ∉ Ω for m mx. For such values of m we have that . Taking the limit and recalling that x ∈ ℝ is arbitrary, we conclude that (ω1, ω2) = (0,0), which is a contradiction. So, along a subsequence, ym y ∈ ℝN.

We claim that r0 = 0. Indeed, suppose by contradiction that r0 > 0. Then, as m becomes large, the set Ωm = (Ω-ym)/rm approaches Ω0 = (Ω -y)/r0 ≠ ℝN. This implies that ω1,ω2 has compact support in ℝN. On the other hand, since (ω1,ω2) achieves the infimum in (2.3) and F is homogeneous, we can use the Lagrange Multiplier Theorem to conclude that ( ω1, ω2) satisfies

for . It follows from (F2) and the maximum principle that at least one of the functions ω1,ω2 is positive in ℝN. But this contradicts supp (ω1,ω2) ⊂ Ω0. Hence, we conclude that r0 = 0. Finally, if we obtain rmx + ym ∉ Ω for large values of m, and therefore we should have (ω1, ω2) ≡ (0, 0) again. Thus, and the proof is completed.

Up to translations, we may assume that 0 ∈ Ω, since Ω is a smooth bounded domain of ℝN, we can choose r > 0 small enough such that Br = Br(0) = {x ∈ ℝN : d(x, 0) < r} ⊂ Ω and the sets

are homotopically equivalent to Ω. Let

and

We define the functional

and set

where

Clearly, mλ,δ is nonincreasing in λ, δ. Note that mλ,δ > 0 for all λ, δ > 0.

Arguing as in the proof of Lemma 2.3 and Theorem 1.1, we obtain the following result.

Lemma 3.3. Suppose F satisfies (F0)-(F2), then the infimum mλ,δ is attained by a nonneg-ative radial function (uλ,δ, vλ,δ) ∈ Erad whenever 2 ≤ p < q < p* and λ,δ > 0, or q = p and λ,δ ∈ (0,Λ1,rad), where Λ1,rad > 0 is the first eigenvalue of the operator . Moreover,

We introduce the barycenter map as follows

This map has the following property.

Lemma 3.4. If N p2,2 ≤ p q < p* and F satisfies (F0)-(F2), then there exists λ* > 0 such that whenever and Iλ,δ(u, v) ≤ mλ,δ.

Proof. By way of contradiction, we suppose that there exist {λm}, {δm} ⊂ ℝ+ and such that λm, δm → 0+ as but .

From and we have that {(um, vm)} is bounded in E. Moreover,

Since λm, δm → 0+, we can use the boundedness of {(um, vm)} to get

from which it follows that

Notice that

Recalling that and both converge to , we can use the above expression and ∫(λm|um|q + δm|vm|q)dx → 0 again to conclude that , that is,

(3.10)

Let and notice that tm(um, vm) satisfies the hypotheses of Lemma 3.2. Using Lemma 3.2, there exist sequences {rm} ⊂ (0,+∞) and {ym} ⊂ ℝN satisfying we have that in D1,p (ℝN) × D1,p (ℝN).

The definition of β(u, v), (3.10), the strong convergence of and Lebesgue's theorem provide

Since and ∫F(x,ω1,ω2)dx = 1, the above expression implies that

According to Lemma 3.3, for each λ, δ > 0 small the infimum mλ,δ is attained by a nonnegative radial function . We consider

and define the function by setting, for each ,

(3.11)

A change of variables and straightforward calculations show that the map γ is well defined. Since σλ,δ is radial, we have that . Hence, for each , we obtain

where .

Along the way of proving Lemma 3.4 we can check easily the following

Lemma 3.5. If λ,δ → 0+, αλ,δ → 1.

Proof. By Lemma 3.3, we have

As before . Thus, , the above expression and the same arguments used in the proof of Lemma 3.3 imply that

The above equality and the definition of αλ,δ imply that αλ,δ → 1. The lemma is proved.

Next we define by

We have the following

Lemma 3.6. if F satisfies (F0)-(F2), then there exists λ** > 0 such that

(3.12)

for all λ, δ ∈ (0, λ**).

Proof. Arguing by contradiction, we suppose that there exist sequences {λm},{δm} ⊂ ℝ+ and such that λm, δm → 0+, as m → ∞, and for all m. Up to a subsequence tm t0 ∈ [0, 1]. Moreover, the compactness of and Lemma 3.4 imply that, up to a subsequence, . From Lemma 3.5 . So, we can use the definition of Hλ,δ to conclude that , which is a contradiction. The lemma is proved.

### 4 Proof of Theorem 1.2

We begin with the following lemma.

Lemma 4.1. If (u, v) is a critical point of Iλ,δ on , then it is a critical point of Iλ,δ in E.

Proof. The proof is almost the same as that [4, Lemma 4.1] and is omitted here.

Lemma 4.2. Suppose F satisfies (F0)-(F2), then any sequence such that and contains a convergent subsequence for λ,δ > 0 if q > p and λ,δ ∈ (0, λ*) if q = p for some small λ* > 0.

Proof. By hypothesis there exists a sequence θm ∈ ℝ such that as m → ∞, where . Thus

Recall that

If , we have

Consequently ∥(um,vm)∥E → 0.

On the other hand, if it follows that

for some C > 0. Hence we arrive at a contradiction if λ, δ > 0 and q > p or λ, δ ∈ (0, λ*) for small λ* > 0 when q = p. Thus we may assume that . Since , we conclude that θm = 0, consequently, . Using this information we have

so by Lemma 2.1 the proof is completed.

Below we denote by the restriction of Iλ,δ on .

Lemma 4.3. Suppose N p2,2 ≤ p q < p* and F satisfies (F0)-(F2), let Λ = min{λ*,λ**} > 0, λ, δ ∈ (0,Λ), then , where λ*, λ** given by Lemmas 34 and 3.6, respectively.

Proof. Assume that , where Aj,j = 1,2,...,m, are closed and contractible sets in , i.e., there exists such that

where ϑ Aj is fixed. Consider Bj = γ-1(Aj), 1 ≤ j m. The sets Bj are closed and

We define the deformation gj : [0, 1] × Bj by setting

for λ,δ ∈ (0,Λ). Note that

implies

and gj(1,y) = Hλ,δ(1, hj(1,γ (y))) = β(hj(1,γ(y))) implies

Thus the sets Bj are contractible in . It follows that .

Proof of Theorem 1.2.

Using Lemmas 2.1, 2.2, and 3.3 we know that for λ,δ ∈ (0,Λ). Moreover, by Lemma 4.2, satisfies the (PS)c condition for all . Therefore, by Lemma 4.3, a standard deformation argument implies that, for contains at least catΩ(Ω) critical points of the restriction of Iλ,δ on . Now Lemma 4.1 implies that Iλ,δ has at least catΩ(Ω) critical points, and therefore at least catΩ(Ω) nontrivial solutions of (1.1). As Theorem 1.1, the obtained solutions are nonnegative in Ω. The proof is completed.

### Competing interests

The author declares that he has no competing interests.

### Acknowledgements

The author would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This study was supported by the Youth Foundation of Hubei Engineering University (No. Z2012003).

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