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_pu = div(|∇u|^p-2∇u) 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(Ω) distinct nonnegative solutions.

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

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_pu = div(|∇u|^p-2∇u) 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:

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(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:

(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(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.

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 endowed with the norm , and the norm .

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

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 I_{λ, 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 I_{λ, δ}(u, v).

The functional I ∈ C¹(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 contains every nonzero solution of problem (1.1), and define the minimax c_λ,δ as

Next, we present some properties of c_λ,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 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 . The maximum of the function 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 ≥ p² and F satisfies (F₀)-(F₂), 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, Λ₁), where Λ₁ > 0 denotes the first eigenvalue of .

Proof. Let {(u_m, v_m)} ⊂ E such that and . Now, we first prove 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 C₁ > 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,Λ₁) 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 . Thus we get

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

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 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 S_F, 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 (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 . 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

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 B_2R(0) ⊂ Ω, set , where U_ε was defined in (2.9). So that . Then, we can get the following results from [[20], Lemma 11.1]:

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 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 C₂ > 0, there holds

Indeed, if this is not the case, we have that for some sequence 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 . 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 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 if N > p² and if N = p², then ε^p-1 = o(ε^p-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 I_λ,δ satisfies the geometric conditions of the Mountain-Pass theorem, there exists {(u_m, v_m)} ⊂ E such that . It follows from Lemmas 2.1 and 2.2 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 . Thus

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 . It is easy to show that the sequence {(u_m, v_m)} 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 {λ_m},{δ_m} ⊂ ℝ⁺ such that λ_m, δ_m → 0⁺. Since λ_m, δ_m 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 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 is bounded, the same argument performed in the proof of Lemma 2.1 implies that {(u_m, v_m)} is bounded in E. Since

Let t_m > 0 be such that . Since , we have 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 , we obtain

Hence , and therefore from the above expression it follows that ∫_Ω F(x, u_m, v_m)dx ≥ C₅ > 0. Thus, the boundedness of {(u_m, v_m)} and (2.19) imply that {t_m} is bounded. This completes the proof.

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 σ_m(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 {(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 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 {(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 satisfying

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

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 (ω₁,ω₂) ∈ D^1,p(ℝ^N) × D^1,p (ℝ^N) satisfying

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 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 . It follows from Remark 3.1 that for some x₀ ∈ ℝ^N. 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 D^1,p(ℝ^N) × D^1,p(ℝ^N) and for a.e. x ∈ ℝ^N. In order to conclude the proof we notice that

Since {(u_m, v_m)} is bounded and (ω₁, ω₂) ≢ (0,0), we infer from the above equality that, up to a subsequence, r_m → r₀ ≥ 0. If |y_m| → ∞, for each fixed x ∈ ℝ^N, we have that there exists m_x ∈ N such that r_mx + y_m ∉ Ω for m ≥ m_x. 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, y_m →y ∈ ℝ^N.

We claim that r₀ = 0. Indeed, suppose by contradiction that r₀ > 0. Then, as m becomes large, the set Ω_m = (Ω-y_m)/r_m approaches Ω₀ = (Ω -y)/r₀ ≠ ℝ^N. This implies that ω₁,ω₂ has compact support in ℝ^N. 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 ℝ^N. But this contradicts supp (ω₁,ω₂) ⊂ Ω₀. Hence, we conclude that r₀ = 0. Finally, if we obtain r_mx + y_m ∉ 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 ℝ^N, we can choose r > 0 small enough such that B_r = B_r(0) = {x ∈ ℝ^N : d(x, 0) < r} ⊂ Ω and the sets

are homotopically equivalent to Ω. Let

and

We define the functional

and set

where

Clearly, m_λ,_δ is nonincreasing in λ, δ. Note that m_λ,_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 m_λ,δ is attained by a nonneg-ative radial function (u_λ,δ, v_λ,δ) ∈ E_rad whenever 2 ≤ p < q < p* and λ,δ > 0, or q = p and λ,δ ∈ (0,Λ_1,rad), where Λ_1,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 I_λ,δ(u, v) ≤ m_λ,δ.

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

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

Since λ_m, δ_m → 0⁺, we can use the boundedness of {(u_m, v_m)} to get

from which it follows that

Notice that

Recalling that and both converge to , we can use the above expression and ∫_Ω(λ_m|u_m|^q + δ_m|v_m|^q)dx → 0 again to conclude that , that is,

Let and notice that t_m(u_m, v_m) satisfies the hypotheses of Lemma 3.2. Using Lemma 3.2, there exist sequences {r_m} ⊂ (0,+∞) and {y_m} ⊂ ℝ^N satisfying we have that in D^1,p (ℝ^N) × D^1,p (ℝ^N).

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

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

which contradicts .

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

and define the function by setting, for each ,

A change of variables and straightforward calculations show that the map γ is well defined. Since σ_λ,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⁺, α_λ,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 α_λ,δ imply that α_λ,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

for all λ, δ ∈ (0, λ**).

Proof. Arguing by contradiction, we suppose that there exist sequences {λ_m},{δ_m} ⊂ ℝ⁺ and such that λ_m, δ_m → 0⁺, as m → ∞, and for all m. Up to a subsequence t_m → 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 H_λ,δ to conclude that , which is a contradiction. The lemma is proved.

We begin with the following lemma.

Lemma 4.1. If (u, v) is a critical point of I_λ,δ on , then it is a critical point of I_λ,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 θ_m ∈ ℝ such that as m → ∞, where . Thus

Recall that

If , we have

Consequently ∥(u_m,v_m)∥_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 θ_m = 0, consequently, . Using this information we have

so by Lemma 2.1 the proof is completed.

Below we denote by the restriction of I_λ,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 A_j,j = 1,2,...,m, are closed and contractible sets in , i.e., there exists such that

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

We define the deformation g_j : [0, 1] × B_j by setting

for λ,δ ∈ (0,Λ). Note that

implies

and g_j(1,y) = H_λ,δ(1, h_j(1,γ (y))) = β(h_j(1,γ(y))) implies

Thus the sets B_j 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 cat_Ω(Ω) critical points of the restriction of I_λ,δ on . Now Lemma 4.1 implies that I_λ,δ has at least cat_Ω(Ω) critical points, and therefore at least cat_Ω(Ω) nontrivial solutions of (1.1). As Theorem 1.1, the obtained solutions are nonnegative in Ω. The proof is completed.

The author declares that he has no competing interests.

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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