Multiple solutions for p-Laplacian systems with critical homogeneous nonlinearity

Abstract

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

$$\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{1}{p^{*}}\frac{\partial F\left(x,u,v\right)}{\partial u}+\lambda {\left|u\right|}^{q-2}u,\hfill & \hfill x\in \Omega ,\hfill \\ -{\Delta }_{p}v=\frac{1}{p^{*}}\frac{\partial F\left(x,u,v\right)}{\partial v}+\delta {\left|v\right|}^{q-2}v,\hfill & \hfill x\in \Omega ,\hfill \\ u=v=0,\hfill & \hfill x\in \partial \Omega ,\hfill \end{array}\right.$$

where Ω ⊂ ℝ^Nis a bounded domain with smooth boundary ∂Ω, Δ _p u = div(|∇u|^p-2∇u) is the p-Laplacian operator, $N\ge p^2,2\le p\le q<p^{*},p^{*}=\frac{Np}{N-p}$ denotes the Sobolev critical exponent, $F\in C^1\left(\bar{\Omega }\times {ℝ}^{+}\times {ℝ}^{+},{ℝ}^{+}\right)$ 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.

1 Introduction and main results

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

$$\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{1}{p^{*}}\frac{\partial F\left(x,u,v\right)}{\partial u}+\lambda {\left|u\right|}^{q-2}u,\hfill & \hfill x\in \Omega ,\hfill \\ -{\Delta }_{p}v=\frac{1}{p^{*}}\frac{\partial F\left(x,u,v\right)}{\partial v}+\delta {\left|v\right|}^{q-2}v,\hfill & \hfill x\in \Omega ,\hfill \\ u=v=0,\hfill & \hfill x\in \partial \Omega ,\hfill \end{array}\right.$$

(1.1)

where Ω ⊂ ℝ^Nis a bounded domain with smooth boundary ∂Ω, Δ _p u = div(|∇u|^p-2∇u) is the p-Laplacian operator, $N\ge p^2,2\le p\le q<p^{*},p^{*}=\frac{Np}{N-p}$ denotes the Sobolev critical exponent, $F\in C^1\left(\bar{\Omega }\times {ℝ}^{+}\times {ℝ}^{+},{ℝ}^{+}\right)$ is a homogeneous function of degree $p^{*},\left(\frac{\partial F\left(x,u,v\right)}{\partial u},\frac{\partial F\left(x,u,v\right)}{\partial v}\right)=\nabla F$ and λ, δ are positive parameters.

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

$$\left\{\begin{array}{cc}-{\Delta }_{p}u={\left|u\right|}^{p^{*}-2}u+\lambda {\left|u\right|}^{q-2}u,\hfill & \hfill x\in \Omega \hfill \\ u=0,\hfill & \hfill x\in \partial \Omega ,\hfill \end{array}\right.$$

(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 $\left(-\Delta ,H_0^1\left(\Omega \right)\right)$. 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

$$\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{2\alpha }{\alpha +\beta }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }+\lambda {\left|u\right|}^{q-2}u,\hfill & \hfill x\in \Omega ,\hfill \\ -{\Delta }_{p}v=\frac{2\beta }{\alpha +\beta }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v+\delta {\left|v\right|}^{q-2}v,\hfill & \hfill x\in \Omega ,\hfill \\ u=v=0,\hfill & \hfill x\in \partial \Omega .\hfill \end{array}\right.$$

(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(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) $F\in C^1\left(\bar{\Omega }\times {ℝ}^{+}\times {ℝ}^{+},{ℝ}^{+}\right)$ and $F\left(x,tu,tv\right)=t^{p^{*}}F\left(x,u,v\right)\left(t>0\right)$ holds for all $\left(x,u,v\right)\in \bar{\Omega }\times {ℝ}^{+}\times {ℝ}^{+}$;

(F₁) $F\left(x,u,0\right)=F\left(x,0,v\right)=\frac{\partial F\left(x,u,0\right)}{\partial u}=\frac{\partial F\left(x,0,v\right)}{\partial v}=0$, where u, v ∈ ℝ⁺;

(F₂) $\frac{\partial F\left(x,u,v\right)}{\partial u},\frac{\partial F\left(x,u,v\right)}{\partial v}$ 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 ≥ p²and 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 Λ₁is the first eigenvalue of$\left(-{\Delta }_p,W_0^{1,p}\left(\Omega \right)\right)$.

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)$F\left(x,u,v\right)={\sum }_{i=1}^k{f}_i\left(x\right){\left|u\right|}^{{\alpha }_i}{\left|v\right|}^{{\beta }_i};$;

(ii) $F\left(x,u,v\right)=\left\{\begin{array}{c}{f}_1\left(x\right){\left|u\right|}^{\frac{3}{2}}{\left|v\right|}^{\frac{5}{2}}+f_2\left(x\right)\frac{u^3{v}^3}{u^2+v^2},\left(u,v\right)\ne \left(0,0\right),\hfill \\ 0,\left(u,v\right)=\left(0,0\right),\hfill \end{array}\right.$

where$f_i\left(x\right)\ge 0,f_i\left(x\right)\not\equiv 0,f_i\left(x\right)\in C\left(\bar{\Omega }\right)\cap L^{\infty }\left(\Omega \right),{\alpha }_i,{\beta }_i>1,{\alpha }_i+{\beta }_i=p^{*}$. 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, 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^sis 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 $E:=W_0^{1,p}\left(\Omega \right)\times W_0^{1,p}\left(\Omega \right)$ endowed with the norm ${∥\left(u,v\right)∥}_E={\left({∥u∥}_{W_0^{1,p}\left(\Omega \right)}^p+{∥v∥}_{W_0^{1,p}\left(\Omega \right)}^p\right)}^{\frac{1}{p}}$, and the norm ${∥u∥}_{W_0^{1,p}\left(\Omega \right)}={\left({\int }_{\Omega }{\left|\nabla u\right|}^{p}dx\right)}^{\frac{1}{p}}$.

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

$$\frac{\partial F\left(x,u,v\right)}{\partial u}u+\frac{\partial F\left(x,u,v\right)}{\partial v}v=p^{*}F\left(x,u,v\right).$$

(2.1)

In addition, we can extend the function F(x,u,v) to the whole $\bar{\Omega }\times {ℝ}^2$ by considering $\tilde{F}\left(x,u,v\right)=F\left(x,u^{+},v^{+}\right)$, where u⁺ = max{u,0}. It is easy to check that $\tilde{F}\left(x,u,v\right)$ is of class C¹ and its restriction to $\bar{\Omega }\times {ℝ}^{+}\times {ℝ}^{+}$ 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

$$\begin{array}{l}\underset{\Omega }{\int }\left({\left|\nabla u\right|}^{p-2}\nabla u\nabla {\phi }_1+{\left|\nabla v\right|}^{p-2}\nabla v\nabla {\phi }_2\right)dx-\frac{1}{p^{*}}\underset{\Omega }{\int }\left(\frac{\partial F\left(x,u,v\right)}{\partial u}{\phi }_1+\frac{\partial F\left(x,u,v\right)}{\partial v}{\phi }_2\right)dx\\ -\underset{\Omega }{\int }\left(\lambda {\left|u\right|}^{q-2}u{\phi }_1+\delta {\left|v\right|}^{q-2}v{\phi }_2\right)dx=0,\forall \left({\phi }_1,{\phi }_2\right)\in E.\end{array}$$

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

$$I_{\lambda ,\delta }\left(u,v\right)=\frac{1}{p}\underset{\Omega }{\int }\left({\left|\nabla u\right|}^p+{\left|\nabla v\right|}^p\right)dx-\frac{1}{p^{*}}\underset{\Omega }{\int }F\left(x,u,v\right)dx-\frac{1}{q}\underset{\Omega }{\int }\left(\lambda {\left|u\right|}^q+\delta {\left|v\right|}^q\right)dx.$$

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 $E:=W_0^{1,p}\left(\Omega \right)\times W_0^{1,p}\left(\Omega \right)$.

As the energy functional I_λ,δis not bounded below on E, we need to study I_λ,δon the Nehari manifold

$${\mathcal{N}}_{\lambda ,\delta }=\left\{\left(u,v\right)\in E\backslash \left\{\left(0,0\right)\right\}:⟨I_{\lambda ,\delta }^{\prime }\left(u,v\right),\left(u,v\right)⟩=0\right\}.$$

Note that ${\mathcal{N}}_{\lambda ,\delta }$ contains every nonzero solution of problem (1.1), and define the minimax c_λ,δas

$$c_{\lambda ,\delta }=\underset{\left(u,v\right)\in {\mathcal{N}}_{\lambda ,\delta }}{\text{inf}}{I}_{\lambda ,\delta }\left(u,v\right).$$

Next, we present some properties of c_λ,δand ${\mathcal{N}}_{\lambda ,\delta }$. Its proofs can be done as [18, Theorem 4.2]. First of all, we note that there exists ρ > 0, such that

$${∥\left(u,v\right)∥}_E\ge \rho >0,\forall \left(u,v\right)\in {\mathcal{N}}_{\lambda ,\delta }.$$

It is standard to check that I_λ,δsatisfies Mountain-Pass geometry, so we can use the homogeneity of F to prove that c_λ,δcan be alternatively characterized by

$$c_{\lambda ,\delta }=\underset{\gamma \in \Gamma }{\text{inf}}\underset{t\in \left[0,1\right]}{\text{max}}{I}_{\lambda ,\delta }\left(\gamma \left(t\right)\right)=\underset{\left(u,v\right)\in E\backslash \left\{\left(0,0\right)\right\}}{\text{inf}}\underset{t\ge 0}{\text{max}}{I}_{\lambda ,\delta }\left(t\left(u,v\right)\right)>0,$$

(2.2)

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 $t^{*}\left(u,v\right)\in {\mathcal{N}}_{\lambda ,\delta }$. 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

$$S_F=\underset{\left(u,v\right)\in E\backslash \left\{\left(0,0\right)\right\}}{\text{inf}}\left\{\frac{{\int }_{\Omega }{\left|\nabla u\right|}^p+{\left|\nabla v\right|}^{p}dx}{{\left({\int }_{\Omega }F\left(x,u,v\right)dx\right)}^{\frac{p}{p^{*}}}}:\underset{\Omega }{\int }F\left(x,u,v\right)dx>0\right\}.$$

(2.3)

Lemma 2.1. If N ≥ p²and F satisfies (F₀)-(F₂), then the functional I_λ,δsatisfies the (PS) _c condition for all$c<\frac{1}{N}{S}_F^{\frac{N}{p}}$, 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$\left(-{\Delta }_p,W_0^{1,p}\left(\Omega \right)\right)$.

Proof. Let {(u _m , v _m )} ⊂ E such that $I_{\lambda ,\delta }^{\prime }\left(u_m,v_m\right)\to 0$ and $I_{\lambda ,\delta }\left(u_m,v_m\right)\to c<\frac{1}{N}{S}_F^{\frac{N}{p}}$. 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,δto obtain C₁ > 0 such that

$$\begin{array}{ll}\hfill c+C_1{∥\left(u_m,v_m\right)∥}_E+o_m\left(1\right)& \ge I_{\lambda ,\delta }\left(u_m,v_m\right)-\frac{1}{q}⟨I_{\lambda ,\delta }^{\prime }\left(u_m,v_m\right),\left(u_m,v_m\right)⟩\\ =\left(\frac{1}{p}-\frac{1}{q}\right){∥\left(u_m,v_m\right)∥}_E^p+\left(\frac{1}{q}-\frac{1}{p^{*}}\right)\underset{\Omega }{\int }F\left(x,u_m,v_m\right)dx\\ \ge \frac{q-p}{pq}{∥\left(u_m,v_m\right)∥}_E^p.\end{array}$$

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

$$\underset{\Omega }{\int }\left(\lambda {\left|u_m\right|}^p+\delta {\left|v_m\right|}^p\right)dx\le \text{max}\left\{\lambda ,\delta \right\}\underset{\Omega }{\int }\left({\left|u_m\right|}^p+{\left|v_m\right|}^p\right)dx\le \frac{\text{max}\left\{\lambda ,\delta \right\}}{{\Lambda }_1}\underset{E}{\overset{p}{∥\left(u_m,v_m\right)∥}},$$

and therefore we get

$$\begin{array}{ll}\hfill c+C_1{∥\left(u_m,v_m\right)∥}_E+o_m\left(1\right)& \ge I_{\lambda ,\delta }\left(u_m,v_m\right)-\frac{1}{p^{*}}⟨I_{\lambda ,\delta }^{\prime }\left(u_m,v_m\right),\left(u_m,v_m\right)⟩\\ =\left(\frac{1}{p}-\frac{1}{p^{*}}\right){∥\left(u_m,v_m\right)∥}_E^p+\left(\frac{1}{p^{*}}-\frac{1}{p}\right)\underset{\Omega }{\int }\left(\lambda {\left|u_m\right|}^p+\delta {\left|v_m\right|}^p\right)dx\\ \ge \frac{1}{N}\left(1-\frac{\text{max}\left\{\lambda ,\delta \right\}}{{\Lambda }_1}\right){∥\left(u_m,v_m\right)∥}_E^p.\end{array}$$

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

$$\left\{\begin{array}{cc}\left(u_m,v_m\right)\rightharpoonup \left(u,v\right),\hfill & \hfill \mathsf{\text{in}}E,\hfill \\ \left(u_m,v_m\right)\to \left(u,v\right),\hfill & \hfill \mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}\mathsf{\text{in}}\Omega \hfill \\ \left(u_m,v_m\right)\to \left(u,v\right),\hfill & \hfill \mathsf{\text{in}}{L}^s\left(\Omega \right)\times L^s\left(\Omega \right),1\le s<p^{*},\hfill \end{array}\right.$$

as m → ∞. Clearly, we have

$$\underset{\Omega }{\int }\left(\lambda {\left|u_m\right|}^q+\delta {\left|v_m\right|}^q\right)dx=\underset{\Omega }{\int }\left(\lambda {\left|u\right|}^q+\delta {\left|v\right|}^q\right)dx+o_m\left(1\right).$$

(2.4)

Moreover, a standard argument shows that $I_{\lambda ,\delta }^{\prime }\left(u,v\right)=0$. Thus we get

$$\begin{array}{ll}\hfill I_{\lambda ,\delta }\left(u,v\right)& =\frac{1}{p}{∥\left(u,v\right)∥}_E^p-\frac{1}{p^{*}}\underset{\Omega }{\int }F\left(x,u,v\right)dx-\frac{1}{q}\underset{\Omega }{\int }\left(\lambda {\left|u\right|}^q+\delta {\left|v\right|}^q\right)dx\\ =\left(\frac{1}{p}-\frac{1}{q}\right){∥\left(u,v\right)∥}_E^p+\left(\frac{1}{q}-\frac{1}{p^{*}}\right)\underset{\Omega }{\int }F\left(x,u,v\right)dx\\ \ge 0.\end{array}$$

(2.5)

Let $\left({ũ}_m,{ṽ}_m\right)=\left(u_m-u,v_m-v\right)$, then by Brezis-Lieb Lemma in [19] implies

$${∥\left(u_m,v_m\right)∥}_E^p={∥\left(u,v\right)∥}_E^p+{∥\left({ũ}_m,{ṽ}_m\right)∥}_E^p+o_m\left(1\right).$$

(2.6)

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

$$\underset{\Omega }{\int }F\left(x,u_m,v_m\right)dx=\underset{\Omega }{\int }F\left(x,u,v\right)dx+\underset{\Omega }{\int }F\left(x,{ũ}_m,{ṽ}_m\right)dx+o_m\left(1\right).$$

(2.7)

By (2.4)-(2.7) and the weak convergence of (u _m , v _m ), we have

$$\begin{array}{ll}\hfill c+o_m\left(1\right)& =I_{\lambda ,\delta }\left(u,v\right)+\frac{1}{p}{∥\left({ũ}_m,{ṽ}_m\right)∥}_E^p-\frac{1}{p^{*}}\underset{\Omega }{\int }F\left(x,{ũ}_m,{ṽ}_m\right)dx\\ \ge \frac{1}{p}{∥\left({ũ}_m,{ṽ}_m\right)∥}_E^p-\frac{1}{p^{*}}\underset{\Omega }{\int }F\left(x,{ũ}_m,{ṽ}_m\right)dx.\end{array}$$

(2.8)

By using $I_{\lambda ,\delta }^{\prime }\left(u_m,v_m\right)\to 0$ and (2.4), (2.6), and (2.7), we get

$$\begin{array}{ll}\hfill o_m\left(1\right)& =⟨I_{\lambda ,\delta }^{\prime }\left(u_m,v_m\right),\left(u_m,v_m\right)⟩\\ ={∥\left(u_m,v_m\right)∥}_E^p-\underset{\Omega }{\int }F\left(x,u_m,v_m\right)dx-\underset{\Omega }{\int }\left(\lambda {\left|u_m\right|}^q+\delta {\left|v_m\right|}^q\right)dx\\ =⟨I_{\lambda ,\delta }^{\prime }\left(u,v\right),\left(u,v\right)⟩+{∥\left({ũ}_m,{ṽ}_m\right)∥}_E^p-\underset{\Omega }{\int }F\left(x,{ũ}_m,{ṽ}_m\right)dx.\end{array}$$

Recalling that $I_{\lambda ,\delta }^{\prime }\left(u,v\right)=0$, we can use the above equality and (2.8) to obtain

$$\underset{m\to \infty }{\text{lim}}{∥\left({ũ}_m,{ṽ}_m\right)∥}_E^p=k=\underset{m\to \infty }{\text{lim}}\underset{\Omega }{\int }F\left(x,{ũ}_m,{ṽ}_m\right)dx,c\ge \left(\frac{1}{p}-\frac{1}{p^{*}}\right)k=\frac{1}{N}k,$$

where k is a nonnegative number.

In view of the definition of S _F , we have that

$${∥\left({ũ}_m,{ṽ}_m\right)∥}_E^p\ge S_F{\left(\underset{\Omega }{\int }F\left(x,{ũ}_m,{ṽ}_m\right)dx\right)}^{\frac{p}{p^{*}}}.$$

Taking the limit we get $k\ge S_F{k}^{\frac{p}{p*}}$. So, if k > 0, we conclude that $k\ge S_F^{\frac{N}{p}}$ and therefore

$$\frac{1}{N}{S}_F^{\frac{N}{p}}\le \frac{1}{N}k\le c<\frac{1}{N}{S}_F^{\frac{N}{p}},$$

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

$$U_{\epsilon }\left(x\right)=\frac{C_N\cdot {\epsilon }^{\frac{N-p}{p^2}}}{{\left(\epsilon +{\left|x\right|}^{\frac{p}{p-1}}\right)}^{\frac{N-p}{p}}},C_N={\left(N{\left(\frac{N-p}{p-1}\right)}^{p-1}\right)}^{\frac{N-p}{p^2}},x\in {ℝ}^N$$

(2.9)

satisfies

$${\left|\nabla U_{\epsilon }\left(x\right)\right|}_p^p={\left|U_{\epsilon }\left(x\right)\right|}_{p^{*}}^{p^{*}}=S^{\frac{N}{p}},$$

(2.10)

where S is the best constant of the Sobolev embedding $D^{1,p}\left({ℝ}^N\right)\hookrightarrow L^{p^{*}}\left({ℝ}^N\right)$. Thus, using [8, Lemma 3] and the homogeneity of F, we obtain A, B > 0 such that

$$S_F=\frac{{∥\left(A{U}_{\epsilon },B{U}_{\epsilon }\right)∥}_E^p}{{\left({\int }_{{ℝ}^N}F\left(x,A{U}_{\epsilon }B{U}_e\right)dx\right)}^{\frac{p}{p^{*}}}}=\frac{A^p+B^p}{{\left(F\left(x,A,B\right)\right)}^{\frac{p}{p^{*}}}}\cdot \frac{S^{\frac{N}{p}}}{{\left|U_{\epsilon }\right|}_{p^{*}}^p},$$

from which and (2.10) it follows that

$$S_F=\frac{A^p+B^p}{{\left(F\left(x,A,B\right)\right)}^{\frac{p}{p^{*}}}}S.$$

(2.11)

We define a cut-off function $\varphi \left(x\right)\in C_0^{\infty }\left({ℝ}^N\right)$ 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 $u_{\epsilon }=\frac{\varphi \left(x\right)U_{\epsilon }}{{\left|\varphi U_{\epsilon }\right|}_{p^{*}}}$, where U _ε was defined in (2.9). So that ${\left|u_{\epsilon }\right|}_{p^{*}}=1$. Then, we can get the following results from [[20], Lemma 11.1]:

$${∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p=S+O\left({\epsilon }^{\frac{N-p}{p}}\right),$$

(2.12)

$$\underset{\Omega }{\int }{\left|u_{\epsilon }\right|}^{\xi }dx\approx \left\{\begin{array}{cc}{\epsilon }^{\frac{N-p}{p^2}\xi },\hfill & \hfill \mathsf{\text{if}}1<\xi <p^{*}\left(1-\frac{1}{p}\right),\hfill \\ {\epsilon }^{\frac{N-p}{p^2}\xi }\left|\text{ln}\epsilon \right|,\hfill & \hfill \mathsf{\text{if}}\xi =p^{*}\left(1-\frac{1}{p}\right),\hfill \\ {\epsilon }^{\frac{\left(p-1\right)\left(Np-\xi \left(N-p\right)\right)}{p^2}},\hfill & \hfill \mathsf{\text{if}}{p}^{*}\left(1-\frac{1}{p}\right)<\xi <p^{*},\hfill \end{array}\right.$$

(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$c_{\lambda ,\delta }<\frac{1}{N}{S}_F^{\frac{N}{p}}$. The same result holds if q = p and λ, δ ∈ (0,Λ₁), where Λ₁ > 0 denotes the first eigenvalue of$\left(-{\Delta }_p,W_0^{1,p}\left(\Omega \right)\right)$.

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

$$I_{\lambda ,\delta }\left(tA{u}_{\epsilon },tB{u}_{\epsilon }\right)=\frac{t^p}{p}\left(A^p+B^p\right){∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p-\frac{t^{p^{*}}}{p^{*}}F\left(x,A,B\right)-\frac{t^q}{q}\left(\lambda A^q+\delta B^q\right){\left|u_{\epsilon }\right|}_q^q.$$

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 $\underset{t\to +\infty }{\text{lim}}h\left(t\right)=-\infty$ and h(t) > 0 when t is close to 0, there exists t _ε > 0 such that

$$h\left(t_{\epsilon }\right)=\underset{t\ge 0}{\text{max}}h\left(t\right).$$

(2.14)

Let

$$g\left(t\right)=\frac{t^p}{p}\left(A^p+B^p\right){∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p-\frac{t^{p^{*}}}{p^{*}}F\left(x,A,B\right),t\ge 0,$$

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

$${\tilde{t}}_{\epsilon }={\left(\frac{\left(A^p+B^p\right){∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p}{F\left(x,A,B\right)}\right)}^{\frac{1}{p^{*}-p}}.$$

So, for each t ≥ 0,

$$g\left(t\right)\le g\left({\tilde{t}}_{\epsilon }\right)=\frac{1}{N}{\left(\frac{\left(A^p+B^p\right){∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p}{{\left(F\left(x,A,B\right)\right)}^{\frac{p}{p^{*}}}}\right)}^{\frac{N}{p}},$$

and therefore

$$h\left(t_{\epsilon }\right)\le \frac{1}{N}{\left(\frac{\left(A^p+B^p\right){∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p}{{\left(F\left(x,A,B\right)\right)}^{\frac{p}{p^{*}}}}\right)}^{\frac{N}{p}}-\frac{t_{\epsilon }^q}{q}\left(\lambda A^q+\delta B^q\right){\left|u_{\epsilon }\right|}_q^q.$$

(2.15)

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

$$t_{\epsilon }^q\left(\lambda A^q+\delta B^q\right)\ge C_2.$$

Indeed, if this is not the case, we have that $t_{{\epsilon }_m}\to 0$ for some sequence ε _m → 0⁺, then,

$$0<c_{\lambda ,\delta }\le \underset{t\ge 0}{\text{sup}}{I}_{\lambda ,\delta }\left(tA{u}_{{\epsilon }_m},tB{u}_{{\epsilon }_m}\right)=I_{\lambda ,\delta }\left(t_{{\epsilon }_m}A{u}_{{\epsilon }_m},t_{{\epsilon }_m}B{u}_{{\epsilon }_m}\right)\to 0,$$

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

$$\begin{array}{ll}\hfill h\left(t_{\epsilon }\right)& \le \frac{1}{N}{\left(\frac{A^p+B^p}{{\left(F\left(x,A,B\right)\right)}^{\frac{p}{p^{*}}}}\left(S+O\left({\epsilon }^{\frac{N-p}{p}}\right)\right)\right)}^{\frac{N}{p}}-C_3{\left|u_{\epsilon }\right|}_q^q\\ \le \frac{1}{N}{S}_F^{\frac{N}{p}}+O\left({\epsilon }^{\frac{N-p}{p}}\right)-C_3{\left|u_{\epsilon }\right|}_q^q\\ \le \frac{1}{N}{S}_F^{\frac{N}{p}}+O\left({\epsilon }^{\frac{N-p}{p}}\right)-O\left({\epsilon }^{\frac{\left(p-1\right)\left(Np-q\left(N-p\right)\right)}{p^2}}\right),\end{array}$$

(2.16)

where $C_3=\frac{C_2}{q}$. We know $p^{*}\left(1-\frac{1}{p}\right)\le p<q<p^{*}$ if N ≥ p². By N ≥ p² and 2 ≤ p < q < p* we obtain $\frac{N-p}{p}>\frac{\left(p-1\right)\left(Np-q\left(N-p\right)\right)}{p^2}$. Thus from the above inequality we conclude that, for each ε > 0 small, there holds

$$c_{\lambda ,\delta }\le \underset{t\ge 0}{\text{sup}}{I}_{\lambda ,\delta }\left(tA{u}_{{\epsilon }_m},tB{u}_{{\epsilon }_m}\right)=h\left(t_{\epsilon }\right)<\frac{1}{N}{S}_F^{\frac{N}{p}}.$$

Case 2. q = p.

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

$$\left(A^p+B^p\right){∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p-\left(\lambda A^p-\delta B^p\right){\left|u_{\epsilon }\right|}_p^p=t^{p*-p}F\left(x,A,B\right).$$

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

$$\begin{array}{ll}\hfill \left(\lambda A^p+\delta B^p\right){\left|u_{\epsilon }\right|}_p^p& \le \text{max}\left\{\lambda ,\delta \right\}\left(A^p+B^p\right)\underset{p}{\overset{p}{\left|u_{\epsilon }\right|}}\\ <{\Lambda }_1\left(A^p+B^p\right){\left|u_{\epsilon }\right|}_p^p\\ \le \left(A^p+B^p\right){∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p.\end{array}$$

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

$$\begin{array}{ll}\hfill h\left(t_{\epsilon }\right)& \le \frac{1}{N}{S}_F^{\frac{N}{p}}+O\left({\epsilon }^{\frac{N-p}{p}}\right)-C_3{\left|u_{\epsilon }\right|}_p^p\\ =\left\{\begin{array}{c}\frac{1}{N}{S}_F^{\frac{N}{p}}+O\left({\epsilon }^{p-1}\right)-O\left({\epsilon }^{p-1}\left|\text{ln}\epsilon \right|\right),N=p^2,\hfill \\ \frac{1}{N}{S}_F^{\frac{N}{p}}+O\left({\epsilon }^{\frac{N-p}{p}}\right)-O\left({\epsilon }^{p-1}\right),N>p^2.\hfill \end{array}\right.\end{array}$$

Because $p^{*}\left(1-\frac{1}{p}\right)<p$ if N > p² and $p^{*}\left(1-\frac{1}{p}\right)=p$ if N = p², then ε^p-1= o(ε^p-1| ln ε|). If N > p², then $\frac{N-p}{p}>p-1$, so ${\epsilon }^{\frac{N-p}{p}}=o\left({\epsilon }^{p-1}\right)$. Choosing ε > 0 small enough, we have

$$c_{\lambda ,\delta }\le \underset{t\ge 0}{\text{sup}}{I}_{\lambda ,\delta }\left(tA{u}_{\epsilon },tB{u}_{\epsilon }\right)=h\left(t_{\epsilon }\right)<\frac{1}{N}{S}_F^{\frac{N}{p}}.$$

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,δsatisfies the geometric conditions of the Mountain-Pass theorem, there exists {(u _m , v _m )} ⊂ E such that $I_{\lambda ,\delta }\left(u_m,v_m\right)\to c_{\lambda ,\delta },I_{\lambda ,\delta }^{\prime }\left(u_m,v_m\right)\to 0$. 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

$$\begin{array}{ll}\hfill 0=⟨I_{\lambda ,\delta }^{\prime }\left(u,v\right),\left(u^{-},v^{-}\right)⟩& =-{∥\left(u^{-},v^{-}\right)∥}_E^p-\frac{1}{p^{*}}\underset{\Omega }{\int }\left(\frac{\partial F\left(x,u,v\right)}{\partial u}{u}^{-}+\frac{\partial F\left(x,u,v\right)}{\partial v}{v}^{-}\right)dx\\ -\underset{\Omega }{\int }\left(\lambda {\left|u\right|}^{q-2}u{u}^{-}+\delta {\left|v\right|}^{q-2}v{v}^{-}\right)dx\\ \le -{∥\left(u^{-},v^{-}\right)∥}_E^p,\end{array}$$

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,δas both the parameters λ, δ approach zero.

Lemma 2.3. $\underset{\lambda ,\delta \to 0^{+}}{\text{lim}}{c}_{\lambda ,\delta }=c_{0,0}=\frac{1}{N}{S}_F^{\frac{N}{p}}.$.

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 $\left(t_{\epsilon }A{u}_{\epsilon },t_{\epsilon }B{u}_{\epsilon }\right)\in {\mathcal{N}}_{0,0}$. Thus

$$\begin{array}{ll}\hfill c_{0,0}& \le I_{0,0}\left(t_{\epsilon }A{u}_{\epsilon },t_{\epsilon }B{u}_{\epsilon }\right)\\ =\frac{1}{N}{\left(\frac{A^p+B^p}{{\left(F\left(x,A,B\right)\right)}^{\frac{p}{p^{*}}}}{∥u_{\epsilon }∥}_{W_0^{1,p}\left(\Omega \right)}^p\right)}^{\frac{N}{p}}\\ =\frac{1}{N}{\left(\frac{A^p+B^p}{{\left(F\left(x,A,B\right)\right)}^{\frac{p}{p^{*}}}}\left(S+O\left({\epsilon }^{\frac{N-p}{p}}\right)\right)\right)}^{\frac{N}{p}}.\end{array}$$

Taking the limit as ε →0⁺ and using (2.11), we conclude that $c_{0,0}\le \frac{1}{N}{S}_F^{\frac{N}{p}}$.

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 $I_{0,0}^{\prime }\left(u_m,v_m\right)\to 0$. It is easy to show that the sequence {(u _m , v _m )} is bounded in E and therefore $⟨I_{0,0}^{\prime }\left(u_m,v_m\right),\left(u_m,v_m\right)⟩={∥\left(u_m,v_m\right)∥}_E^p-{\int }_{\Omega }F\left(x,u_m,v_m\right)dx=o_m\left(1\right)$. It follows that

$$\underset{m\to \infty }{\text{lim}}{∥\left(u_m,v_m\right)∥}_E^p=l=\underset{m\to \infty }{\text{lim}}\underset{\Omega }{\int }F\left(x,u_m,v_m\right)dx.$$

Taking the limit in the inequality $S_F{\left({\int }_{\Omega }F\left(x,u_m,v_m\right)dx\right)}^{\frac{p}{p^{*}}}\le {∥\left(u_m,v_m\right)∥}_E^p$ we conclude, as in the proof of Lemma 2.1, that $N{c}_{0,0}=l\ge S_F^{\frac{N}{p}}$. Hence,

$$c_{0,0}=\underset{m\to \infty }{\text{lim}}{I}_{0,0}\left(u_m,v_m\right)=\underset{m\to \infty }{\text{lim}}\left(\frac{1}{p}{∥\left(u_m,v_m\right)∥}_E^p-\frac{1}{p^{*}}\underset{\Omega }{\int }F\left(x,u_m,v_m\right)dx\right)=\frac{1}{N}l\ge \frac{1}{N}{S}_F^{\frac{N}{p}},$$

and therefore $c_{0,0}=\frac{1}{N}{S}_F^{\frac{N}{p}}$.

We proceed now with the calculation of $\underset{\lambda ,\delta \to 0^{+}}{\text{lim}}{c}_{\lambda ,\delta }$. Let {λ _m },{δ _m } ⊂ ℝ⁺ such that λ _m , δ _m → 0⁺. Since λ _m , δ _m are positive, we have that ${\int }_{\Omega }\left({\lambda }_m{\left|u\right|}^q+{\delta }_m{\left|v\right|}^q\right)dx\ge 0$ whenever (u, v) is nonnegative. Thus, for this kind of function, we have that $I_{{\lambda }_m,{\delta }_m}\left(u,v\right)\le I_{0,0}\left(u,v\right)$.

It follows that

$$\begin{array}{ll}\hfill c_{{\lambda }_m,{\delta }_m}& =\underset{\left(u,v\right)\ne \left(0,0\right)}{\text{inf}}\underset{t\ge 0}{\text{max}}{I}_{{\lambda }_m,{\delta }_m}\left(t\left(u,v\right)\right)\\ \le \underset{\underset{\left(u,v\right)\ge 0}{(u,v)\ne (0,0),}}{\text{inf}}\underset{t\ge 0}{\text{max}}{I}_{{\lambda }_m,{\delta }_m}\left(t\left(u,v\right)\right)\\ \le \underset{\underset{\left(u,v\right)\ge 0}{(u,v)\ne (0,0),}}{\text{inf}}\underset{t\ge 0}{\text{max}}{I}_{0,0}\left(t\left(u,v\right)\right)=c_{0,0},\end{array}$$

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

$$\underset{m\to \infty }{\text{lim}\text{sup}}{c}_{{\lambda }_{m,}{\delta }_m}\le c_{0,0}.$$

(2.17)

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

$$I_{{\lambda }_m,{\delta }_m}\left(u_m,v_m\right)=c_{{\lambda }_m,{\delta }_m},I_{{\lambda }_m,{\delta }_m}^{\prime }\left(u_m,v_m\right)\to 0.$$

Since $c_{{\lambda }_m,{\delta }_m}$ is bounded, the same argument performed in the proof of Lemma 2.1 implies that {(u _m , v _m )} is bounded in E. Since

$$\underset{m\to \infty }{\text{lim}}\underset{\Omega }{\int }\left({\lambda }_m{\left|u_m\right|}^q+{\delta }_m{\left|v_m\right|}^q\right)dx=0.$$

(2.18)

Let t _m > 0 be such that $t_m\left(u_m,v_m\right)\in {\mathcal{N}}_{0,0}$. Since $\left(u_m,v_m\right)\in {\mathcal{N}}_{{\lambda }_m,{\delta }_m}$, we have that

$$\begin{array}{ll}\hfill c_{0,0}& \le I_{0,0}\left(t_m\left(u_m,v_m\right)\right)\\ =I_{{\lambda }_m,{\delta }_m}\left(t_m\left(u_m,v_m\right)\right)+\frac{t_m^q}{q}\underset{\Omega }{\int }\left({\lambda }_m{\left|u_m\right|}^q+{\delta }_m{\left|v_m\right|}^q\right)dx\\ \le I_{{\lambda }_m,{\delta }_m}\left(u_m,v_m\right)+\frac{t_m^q}{q}\underset{\Omega }{\int }\left({\lambda }_m{\left|u_m\right|}^q+{\delta }_m{\left|v_m\right|}^q\right)dx\\ =c_{{\lambda }_m,{\delta }_m}+\frac{t_m^q}{q}\underset{\Omega }{\int }\left({\lambda }_m{\left|u_m\right|}^q+{\delta }_m{\left|v_m\right|}^q\right)dx.\end{array}$$

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

$$c_{0,0}\le \underset{m\to \infty }{\text{lim}\text{inf}}{c}_{{\lambda }_m,{\delta }_m}.$$

This and (2.17) we get

$$c_{0,0}\le \underset{m\to \infty }{\text{lim}\text{inf}}{c}_{{\lambda }_m,{\delta }_m}\le \underset{m\to \infty }{\text{lim}\text{sup}}{c}_{{\lambda }_m,{\delta }_m}\le c_{0,0},$$

that is $c_{0,0}=\underset{m\to \infty }{\text{lim}}{c}_{{\lambda }_m,{\delta }_m}$.

It remains to check that {t _m } is bounded. A straightforward calculation shows that

$$t_m={\left(\frac{{∥\left(u_m,v_m\right)∥}_E^p}{{\int }_{\Omega }F\left(x,u_m,v_m\right)dx}\right)}^{\frac{1}{p^{*}-p}}.$$

(2.19)

Since $\left(u_m,v_m\right)\in {\mathcal{N}}_{{\lambda }_m,{\delta }_m}$, we obtain

$${∥\left(u_m,v_m\right)∥}_E^p=\underset{\Omega }{\int }F\left(x,u_m,v_m\right)dx+\underset{\Omega }{\int }\left({\lambda }_m{\left|u_m\right|}^q+{\delta }_m{\left|v_m\right|}^q\right)dx\le S_F^{-\frac{p^{*}}{p}}{∥\left(u_m,v_m\right)∥}_E^{p^{*}}+o_m\left(1\right).$$

Hence ${∥\left(u_{m,}{v}_m\right)∥}_E^p\ge C_4>0$, and therefore from the above expression it follows that ∫_ΩF(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.

3 Some technical lemmas

In this section, we denote by $ℳ(\Omega )$ the Banach space of finite Radon measures over Ω equipped with the norm $\Vert \sigma \Vert ={\sup }_{\underset{{\left|\phi \right|}_{\infty }=1}{\phi \in C_0(\Omega )}}\left|\sigma (\phi )\right|$. A sequence $\left\{{\sigma }_m\right\}\subset ℳ\left(\Omega \right)$ is said to converge weakly to $\sigma \in ℳ\left(\Omega \right)$ provided σ _m (φ) → σ(φ) for all φ ∈ C₀(Ω). By [18, Theorem 1.39], every bounded sequence $\left\{{\sigma }_m\right\}\subset ℳ\left(\Omega \right)$ 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

$$\begin{array}{l}\left(u_m,v_m\right)\rightharpoonup \left(u,v\right)in{D}^{1,p}\left({ℝ}^N\right)\times D^{1,p}\left({ℝ}^N\right),\\ \left(u_m,v_m\right)\to \left(u,v\right)a.e.x\in {ℝ}^N,\\ \left(\nabla u_m,\nabla v_m\right)\to \left(\nabla u,\nabla v\right)a.e.x\in {ℝ}^N,\\ {\left|\nabla \left(u_m-u\right)\right|}^p\rightharpoonup \mu ,{\left|\nabla \left(v_m-v\right)\right|}^p\rightharpoonup \sigma inℳ\left({ℝ}^N\right),\\ F\left(x,u_m-u,v_m-v\right)\rightharpoonup \nu inℳ\left({ℝ}^N\right),\end{array}$$

and define

$$\left\{\begin{array}{c}{\mu }_{\infty }=\underset{R\to \infty }{\text{lim}}\underset{m\to \infty }{\text{lim}\text{sup}}\underset{\left|x\right|>R}{\int }{\left|\nabla u_m\right|}^{p}dx,\hfill \\ {\sigma }_{\infty }=\underset{R\to \infty }{\text{lim}}\underset{m\to \infty }{\text{lim}\text{sup}}\underset{\left|x\right|>R}{\int }{\left|\nabla v_m\right|}^{p}dx,\hfill \\ {\nu }_{\infty }=\underset{R\to \infty }{\text{lim}}\underset{m\to \infty }{\text{lim}\text{sup}}\underset{\left|x\right|>R}{\int }F\left(x,u_m,v_m\right)dx,\hfill \end{array}\right.$$

(3.1)

then it follows that

$$\underset{m\to \infty }{\text{lim}\text{sup}}\underset{{ℝ}^N}{\int }{\left|\nabla u_m\right|}^{p}dx=∥\mu ∥+{\mu }_{\infty }+\underset{{ℝ}^N}{\int }{\left|\nabla u\right|}^{p}dx,$$

(3.2)

$$\underset{m\to \infty }{\text{lim}\text{sup}}\underset{{ℝ}^N}{\int }{\left|\nabla v_m\right|}^{p}dx=∥\sigma ∥+{\sigma }_{\infty }+\underset{{ℝ}^N}{\int }{\left|\nabla u\right|}^{p}dx,$$

(3.3)

$$\underset{m\to \infty }{\text{lim}\text{sup}}\underset{{ℝ}^N}{\int }F\left(x,u_m,v_m\right)dx=∥\nu ∥+{\nu }_{\infty }+\underset{{ℝ}^N}{\int }F\left(x,u,v\right)dx,$$

(3.4)

$${∥\nu ∥}^{\frac{p}{p^{*}}}\le S_F^{-1}\left(∥\mu ∥+∥\sigma ∥\right),{\nu }_{\infty }^{\frac{p}{p^{*}}}\le S_F^{-1}\left({\mu }_{\infty }+{\sigma }_{\infty }\right).$$

(3.5)

Moreover, if (u,v) = (0,0) and${∥\nu ∥}^{\frac{p}{p^{*}}}=S_F^{-1}\left(∥\mu ∥+∥\sigma ∥\right)$, 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 $\phi \in C_0^{\infty }\left({ℝ}^N\right)$ we have

$${\left(\underset{{ℝ}^N}{\int }{\left|\phi \right|}^{p^{*}}F\left(x,u_m,v_m\right)dx\right)}^{\frac{p}{p^{*}}}={\left(\underset{{ℝ}^N}{\int }F\left(x,\phi u_m,\phi v_m\right)dx\right)}^{\frac{p}{p^{*}}}\le S_F^{-1}{∥(\phi u_m,\phi v_m)∥}_E^p.$$

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

$$\underset{{ℝ}^N}{\int }\phi F\left(x,u_m-u,v_m-v\right)dx=\underset{{ℝ}^N}{\int }\phi F\left(x,u_m,v_m\right)dx-\underset{{ℝ}^N}{\int }\phi F\left(x,u,v\right)dx+o_m\left(1\right).$$

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 ${∥\nu ∥}^{\frac{p}{p^{*}}}=S_F^{-1}\left(∥\mu ∥+∥\sigma ∥\right)$, 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$\left({ũ}_m,{ṽ}_m\right)=\left(u_m-u,v_m-v\right)$and notice that

$$\begin{array}{c}\left({ũ}_m,{ṽ}_m\right)\rightharpoonup \left(ũ,ṽ\right)in{D}^{1,p}\left({ℝ}^N\right)\times D^{1,p}\left({ℝ}^N\right),\\ \left({ũ}_m,{ṽ}_m\right)\to \left(0,0\right)a.e.x\in {ℝ}^N,\\ \left(\nabla {ũ}_m,\nabla {ṽ}_m\right)\to \left(\nabla ũ,\nabla ṽ\right)a.e.x\in {ℝ}^N,\\ {\left|\nabla \left({ũ}_m-ũ\right)\right|}^p\rightharpoonup \tilde{\mu },{\left|\nabla \left({ṽ}_m-ṽ\right)\right|}^p\rightharpoonup \tilde{\sigma }inℳ\left({ℝ}^N\right),\\ F\left(x,{ũ}_m-ũ,{ṽ}_m-ṽ\right)\rightharpoonup \tilde{\nu }inℳ\left({ℝ}^N\right),\end{array}$$

Since$\left({ũ}_m-ũ,{ṽ}_m-ṽ\right)=\left(u_m-u,v_m-v\right)$and therefore$\tilde{\mu }=\mu ,\tilde{\sigma }=\sigma$, and$ṽ=v$, where μ,σ, and ν are the same as those in Lemma 3.1. Thus, if${∥\nu ∥}^{\frac{p}{p^{*}}}=S_F^{-1}\left(∥\mu ∥+∥\sigma ∥\right)$we also have that${∥\tilde{\nu }∥}^{\frac{p}{p^{*}}}=S_F^{-1}\left(∥\tilde{\mu }∥+∥\tilde{\sigma }∥\right)$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 ∫_ΩF(x, u _m , v _m )dx = 1 and$\underset{m\to \infty }{\text{lim}}{∥\left(u_m,v_m\right)∥}_E^p=S_F$. Then there exist {r _m } ⊂ (0, +∞) and {y _m } ⊂ ℝ^Nsuch that

$$\left({\omega }_m^1\left(x\right),{\omega }_m^2\left(x\right)\right)=r_m^{\frac{N-p}{p}}\left(u_m\left(r_{m}x+y_m\right),v_m\left(r_{m}x+y_m\right)\right)$$

(3.6)

contains a convergent subsequence denoted again by$\left\{\left({\omega }_m^1\left(x\right),{\omega }_m^2\left(x\right)\right)\right\}$such that$\left({\omega }_m^1\left(x\right),{\omega }_m^2\left(x\right)\right)\to \left({\omega }_1,{\omega }_2\right)$in D^1,p(ℝ^N) × D^1,p(ℝ^N). Moreover, as m → ∞, we have r _m → 0 and$y_m\to y\in \bar{\Omega }$.

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

$$H_m\left(r\right)=\underset{y\in {ℝ}^N}{\text{sup}}\underset{B_r\left(y\right)}{\int }F\left(x,u_m,v_m\right)dx.$$

Since for every m,

$$\underset{r\to 0^{+}}{\text{lim}}{H}_m\left(r\right)=0,\underset{r\to \infty }{\text{lim}}{H}_m\left(r\right)=1,$$

there exist r _m > 0 and a sequence $\left\{y_m^k\right\}\subset {ℝ}^N$ satisfying

$$\frac{1}{2}=H_m\left(r_m\right)=\underset{k\to \infty }{\text{lim}}\underset{B_{r_m}\left(y_m^k\right)}{\int }F\left(x,u_m,v_m\right)dx.$$

Recalling that $\underset{\left|y\right|\to \infty }{\text{lim}}{\int }_{B_{r_m}\left(y\right)}F\left(x,u_m,v_m\right)dx=0$, we conclude that $\left\{y_m^k\right\}$ is bounded. Hence, up to a subsequence, $\underset{k\to \infty }{\text{lim}}{y}_m^k=y_m\in {ℝ}^N$ and we obtain

$$\frac{1}{2}=\underset{B_{r_m}\left(y_m\right)}{\int }F\left(x,u_m,v_m\right)dx.$$

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

$$\frac{1}{2}=\underset{B_{r_m}\left(y_m\right)}{\int }F\left(x,u_m,v_m\right)dx=\underset{B_1\left(0\right)}{\int }F\left(x,{\omega }_m^1,{\omega }_m^2\right)dx=\underset{y\in {ℝ}^N}{\text{sup}}\underset{B_1\left(y\right)}{\int }F\left(x,{\omega }_m^1,{\omega }_m^2\right)dx.$$

(3.7)

By (3.6), a straightforward calculation provides

$$\underset{m\to \infty }{\text{lim}}{∥\left({\omega }_m^1,{\omega }_m^2\right)∥}_E^p=\underset{m\to \infty }{\text{lim}}{∥\left(u_m,v_m\right)∥}_E^p=S_F,\underset{{ℝ}^N}{\int }F\left(x,{\omega }_m^1,{\omega }_m^2\right)dx=1.$$

Hence, we can apply Lemma 3.1 to obtain (ω₁,ω₂) ∈ D^1,p(ℝ^N) × D^1,p(ℝ^N) satisfying

$$S_F=∥\mu ∥+{\mu }_{\infty }+∥\sigma ∥+{\sigma }_{\infty }+{∥\left({\omega }_1,{\omega }_2\right)∥}_E^p,1=∥\nu ∥+{\nu }_{\infty }+\underset{{ℝ}^N}{\int }F\left(x,{\omega }_1,{\omega }_2\right)dx,$$

(3.8)

$${∥\nu ∥}^{\frac{p}{p^{*}}}\le S_F^{-1}\left(∥\mu ∥+∥\sigma ∥\right),{\nu }_{\infty }^{\frac{p}{p^{*}}}\le S_F^{-1}\left({\mu }_{\infty }+{\sigma }_{\infty }\right).$$

(3.9)

The second equality in (3.8) implies that $∥\nu ∥,{\nu }_{\infty },{\int }_{{ℝ}^N}F\left(x,{\omega }_1,{\omega }_2\right)dx\in \left[0,1\right]$. If one of these values belongs to the open interval (0,1), we can use (3.8), $\frac{p}{p^{*}}<1,({{\int }_{{ℝ}^N}F(x,{\omega }_1,{\omega }_2)dx)}^{\frac{p}{p^{*}}}\le S_F^{-1}{\Vert ({\omega }_1,{\omega }_2)\Vert }_E^p$ and (3.9) to get

$$\begin{array}{ll}\hfill S_F& =S_F\left(∥\nu ∥+{\nu }_{\infty }+\underset{{ℝ}^N}{\int }F\left(x,{\omega }_1,{\omega }_2\right)dx\right)\\ <S_F\left({∥\nu ∥}^{\frac{p}{p^{*}}}+{\nu }_{\infty }^{\frac{p}{p^{*}}}+{\left(\underset{{ℝ}^N}{\int }F\left(x,{\omega }_1,{\omega }_2\right)dx\right)}^{\frac{p}{p^{*}}}\right)\\ \le S_F,\end{array}$$

which is a contradiction. Thus $∥\nu ∥,{\nu }_{\infty },{\int }_{{ℝ}^N}F\left(x,{\omega }_1,{\omega }_2\right)dx\in \left\{0,1\right\}$. Actually, it follows from (3.7) that ${\int }_{\left|x\right|>R}F\left(x,{\omega }_m^1,{\omega }_m^2\right)dx\le \frac{1}{2}$ 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 ${∥\nu ∥}^{\frac{p}{p^{*}}}=S_F^{-1}\left(∥\mu ∥+∥\sigma ∥\right)$. It follows from Remark 3.1 that $\nu ={\delta }_{x_0}$ for some x₀ ∈ ℝ^N. Thus, from (3.7), we get

$$\frac{1}{2}\ge \underset{m\to \infty }{\text{lim}}\underset{B_1\left(x_0\right)}{\int }F\left(x,{\omega }_m^1,{\omega }_m^2\right)dx=\underset{B_1\left(x_0\right)}{\int }d\nu =∥\nu ∥=1.$$

This contradiction proves that ∥ν∥ = 0.

Since ∥ν∥ = ν_∞ = 0, we have that ${\int }_{{ℝ}^N}F\left(x,{\omega }_1,{\omega }_2\right)dx=1$. This and (3.8) provide

$$\underset{m\to \infty }{\text{lim}}{∥\left({\omega }_m^1,{\omega }_m^2\right)∥}_E^p=S_F\ge {∥\left({\omega }_1,{\omega }_2\right)∥}_E^p\ge S_F{\left(\underset{{ℝ}^N}{\int }F\left(x,{\omega }_1,{\omega }_2\right)dx\right)}^{\frac{p}{p^{*}}}=S_F.$$

So, ${∥\left({\omega }_1,{\omega }_2\right)∥}_E^p=S_F$ and therefore $\left({\omega }_m^1,{\omega }_m^2\right)\to \left({\omega }_1,{\omega }_2\right)\not\equiv \left(0,0\right)$ strongly in D^1,p(ℝ^N) × D^1,p(ℝ^N) and $\left({\omega }_m^1\left(x\right),{\omega }_m^2\left(x\right)\right)\to \left({\omega }_1\left(x\right),{\omega }_2\left(x\right)\right)$ for a.e. x ∈ ℝ^N. In order to conclude the proof we notice that

$${∥\left({\omega }_m^1,{\omega }_m^2\right)∥}_{L^p\left({ℝ}^N\right)\times L^p\left({ℝ}^N\right)}=\frac{1}{r_m^p}{∥\left(u_m,v_m\right)∥}_{L^p\left(\Omega \right)\times L^p\left(\Omega \right).}$$

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 _m x + y _m ∉ Ω for m ≥ m _x . For such values of m we have that $\left({\omega }_m^1\left(x\right),{\omega }_m^2\left(x\right)\right)=\left(0,0\right)$. 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

$$-{\Delta }_{p}u=\theta \frac{\partial F\left(x,u,v\right)}{\partial u},-{\Delta }_{p}v=\theta \frac{\partial F\left(x,u,v\right)}{\partial u},x\in {ℝ}^N,$$

for $\theta =\frac{S_F}{p^{*}}>0$. 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 $y\notin \bar{\Omega }$ we obtain r _m x + y _m ∉ Ω for large values of m, and therefore we should have (ω₁, ω₂) ≡ (0, 0) again. Thus, $y\in \bar{\Omega }$ 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

$${\Omega }_r^{+}=\left\{x\in {ℝ}^N:\mathsf{\text{dist}}\left(x,\Omega \right)<r\right\},{\Omega }_r^{-}=\left\{x\in {ℝ}^N:\mathsf{\text{dist}}\left(x,\partial \Omega \right)>r\right\},$$

are homotopically equivalent to Ω. Let

$$W_{0,\mathsf{\text{rad}}}^{1,p}\left(B_r\right)=\left\{u\in W_0^{1,p}\left(B_r\right):u\mathsf{\text{is}}\mathsf{\text{radial}}\right\}$$

and

$$E_{\mathsf{\text{rad}}}\left(B_r\right)=W_{0,\mathsf{\text{rad}}}^{1,p}\left(B_r\right)\times W_{0,\mathsf{\text{rad}}}^{1,p}\left(B_r\right).$$

We define the functional

$$I_{B_r}\left(u,v\right)=\frac{1}{p}\underset{B_r}{\int }\left({\left|\nabla u\right|}^p+{\left|\nabla v\right|}^p\right)dx-\frac{1}{p^{*}}\underset{B_r}{\int }F\left(x,u,v\right)dx-\frac{1}{q}\underset{B_r}{\int }\left(\lambda {\left|u\right|}^q+\delta {\left|v\right|}^q\right)dx,\left(u,v\right)\in E_{\mathsf{\text{rad}}}\left(B_r\right),$$

and set

$$m_{\lambda ,\delta }=\underset{\left(u,v\right)\in {\mathcal{N}}_{\lambda ,\delta }^{B_r}}{\text{inf}}{I}_{B_r}\left(u,v\right),$$

where

$${\mathcal{N}}_{\lambda ,\delta }^{B_r}:=\left\{\left(u,v\right)\in E_{\mathsf{\text{rad}}}\left(B_r\right)\backslash \left\{\left(0,0\right)\right\}:⟨I_{B_r}^{\prime }\left(u,v\right),\left(u,v\right)⟩=0\right\}.$$

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,δis attained by a nonneg-ative radial function (u_λ,δ, v_λ,δ) ∈ E_radwhenever 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$\left(-{\Delta }_p,W_{0,\mathsf{\text{rad}}}^{1,p}\left(B_r\right)\right)$. Moreover,

$$m_{\lambda ,\delta }<\frac{1}{N}{S}_F^{\frac{N}{p}},\underset{\lambda ,\delta \to 0^{+}}{\text{lim}}{m}_{\lambda ,\delta }=\frac{1}{N}{S}_F^{\frac{N}{p}}.$$

We introduce the barycenter map $\beta :{\mathcal{N}}_{\lambda ,\delta }\to {ℝ}^N$ as follows

$$\beta \left(u,v\right)=S_F^{-\frac{N}{p}}\underset{\Omega }{\int }F\left(x,u,v\right)xdx.$$

This map has the follo