Multiplicity of Positive Solutions for Semilinear Elliptic Systems

Abstract

We study the effect of the coefficient ℎ(𝑥) of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems. Under suitable assumptions for 𝑓(𝑥), 𝑔(𝑥), and ℎ(𝑥), we should prove that for sufficiently small 𝜆,𝜇>0, there are at least 𝑘+1 positive solutions of the semilinear elliptic systems −Δ𝑢=𝜆𝑓(𝑥)|𝑢|^𝑞−2𝑢+(𝛼/(𝛼+𝛽))ℎ(𝑥)|𝑢|^𝛼−2𝑢|𝑣|^𝛽, −Δ𝑣=𝜇𝑔(𝑥)|𝑣|^𝑞−2𝑣+(𝛽/(𝛼+𝛽))ℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽−2𝑣, where 0∈Ω⊂ℝ^𝑁 is a bounded domain, 𝛼>1, 𝛽>1, and 𝑁/(𝑁−2)<𝑞<2<𝛼+𝛽=2^∗ for 𝑁>4.

1. Introduction and Main Results

For 𝑁≥3, 𝛼>1, 𝛽>1, and 1≤𝑞<2<𝛼+𝛽=2^∗=2𝑁/(𝑁−2), consider the semilinear elliptic systems −Δ𝑢=𝜆𝑓(𝑥)|𝑢|^𝑞−2𝛼𝑢+ℎ(𝑥𝑢|𝛼+𝛽)|^𝛼−2𝑢|𝑣|^𝛽(𝑥𝑣|inΩ,−Δ𝑣=𝜇𝑔)|^𝑞−2𝛽𝑣+ℎ(𝑥𝑢|𝛼+𝛽)|^𝛼|𝑣|^𝛽−2𝑣inΩ,𝑢=𝑣=0on𝜕Ω, 𝑃_𝜆,𝜇 where 𝜆,𝜇>0, Ω⊂ℝ^𝑁 is a bounded domain with smooth boundary 𝜕Ω.

Let 𝑓, 𝑔, and ℎ satisfy the following conditions.(𝐻1)𝑓, 𝑔, and ℎ are positive continuous functions in Ω and max_𝑥∈Ωℎ(𝑥)=1. (𝐻2)There exist 𝑘 points 𝑎₁,𝑎₂,…,𝑎_𝑘∈Ω and some 𝜎≥𝑁−2 such that ℎ(𝑎_𝑖) are strict maxima and satisfy ℎ𝑎_𝑖=max_𝑥∈Ωℎ(𝑥)=1∀1≤𝑖≤𝑘 (1)and ℎ(𝑥)=ℎ(𝑎_𝑖)+𝑂(|𝑥−𝑎_𝑖|^𝜎) as 𝑥→𝑎_𝑖 uniformly in 𝑖.

Recent studies [1–10] have investigated the elliptic systems with subcritical or critical exponents and have proved the existence of a ground state solution or the existence of at least two positive solutions for these problems. For the case of 𝑁>4, 𝛼>1, 𝛽>1, and 2<𝑞<𝛼+𝛽=2^∗=2𝑁/(𝑁−2), Lin [11] constructs the 𝑘 compact Palais-Smale sequences that are suitably localized in correspondence of 𝑘 maximum points of ℎ. Under assumptions (𝐻1)-(𝐻2), she has showed that there are at least 𝑘 positive solutions of the problem 𝑃_𝜆,𝜇 for sufficiently small 𝜆,𝜇>0. In this paper, we study the problem 𝑃_𝜆,𝜇 and complement the results of [11] to the case 1≤𝑞<2. Under assumptions (𝐻1)-(𝐻2), we should prove that there exist at least 𝑘+1 positive solutions of the problem 𝑃_𝜆,𝜇 for sufficiently small 𝜆,𝜇>0.

Let 𝐸=𝐻¹₀(Ω)×𝐻¹₀(Ω) be the space with the standard norm ‖(𝑢,𝑣)‖_𝐸=_Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥^1/2. (2) Associated with the problem 𝑃_𝜆,𝜇, we consider the 𝐶¹-functional𝐼_𝜆,𝜇, for (𝑢,𝑣)∈𝐸, 𝐼_𝜆,𝜇()=1𝑢,𝑣2‖(𝑢,𝑣)‖²_𝐸−1𝑞_Ω𝜆𝑓(𝑥)|𝑢|^𝑞+𝜇𝑔(𝑥)|𝑣|^𝑞−1𝑑𝑥2^∗_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥. (3) The weak solution (𝑢,𝑣)∈𝐸 of the problem 𝑃_𝜆,𝜇 is the critical point of the functional 𝐼_𝜆,𝜇; that is, (𝑢,𝑣)∈𝐸 satisfies _Ω∇𝑢∇𝜑₁+∇𝑣∇𝜑₂𝑑𝑥−𝜆_Ω𝑓(𝑥)|𝑢|^𝑞−2𝑢𝜑₁𝑑𝑥−𝜇_Ω𝑔(𝑥)|𝑣|^𝑞−2𝑣𝜑₂𝛼𝑑𝑥−2^∗_Ωℎ(𝑥)|𝑢|^𝛼−2𝑢|𝑣|^𝛽𝜑₁−𝛽𝑑𝑥2^∗_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽−2𝑣𝜑₂𝑑𝑥=0 (4) for any (𝜑₁,𝜑₂)∈𝐸.

Let 𝐷^1,2(ℝ^𝑁)={𝑢∈𝐿^2∗(ℝ^𝑁)∣∇𝑢∈(𝐿²(ℝ^𝑁))^𝑁} with the norm ‖𝑢‖²=∫_ℝ𝑁|∇𝑢|²𝑑𝑥, and let 𝑆 be the best Sobolev constant defined by 𝑆=inf_{𝑢∈𝐷1,2ℝ𝑁⧵{0}}∫_ℝ𝑁|∇𝑢|²𝑑𝑥∫_ℝ𝑁|𝑢|^2∗𝑑𝑥^2/2∗=inf_{𝑢∈𝐻10(Ω)⧵{0}}∫_Ω|∇𝑢|²𝑑𝑥∫_Ω|𝑢|^2∗𝑑𝑥^2/2∗>0, (5) and let 𝑆_𝛼,𝛽=inf_{𝑢,𝑣∈𝐻10(Ω)⧵{0}}∫_Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥∫_Ω|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥^{2/(𝛼+𝛽)}; (6) then, by [1, Theorem 5], we have 𝑆_𝛼,𝛽=𝛼𝛽^{𝛽/(𝛼+𝛽)}+𝛽𝛼^{𝛼/(𝛼+𝛽)}𝑆, (7) where 𝛼+𝛽=2^∗.

Set Λ₁=2−𝑞2^∗−𝑞^2/(2∗−2)2^∗−𝑞𝛾_∞2^∗|Ω|−2^{(2∗−𝑞)/2∗−2/(2−𝑞)}×𝑆^{𝑁/2+𝑞/(2−𝑞)}>0, (8) where 𝛾_∞=max{|𝑓|_𝐿^∞_(Ω),|𝑔|_𝐿^∞_(Ω)}.

The main results of this paper are given as follows.

Theorem 1. Assume that (𝐻1) holds. If 𝜆,𝜇>0 satisfy 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₁, then there exists at least one positive ground state solution of the problem 𝑃_𝜆,𝜇.

Theorem 2. Under the assumptions (𝐻1)-(𝐻2), and 𝑁/(𝑁−2)<𝑞<2 and 𝑁>4, there exists a positive number Λ^∗∈(0,Λ₁) such that for 𝜆,𝜇>0 and 𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ^∗, the problem 𝑃_𝜆,𝜇 has 𝑘+1 positive solutions.

This paper is organized as follows. In Section 2, we consider the Nehari manifold 𝒩_𝜆,𝜇=(𝑢,𝑣)∈𝐸⧵{0}∣𝐼^′_𝜆,𝜇(𝑢,𝑣),(𝑢,𝑣)=0, (9) where 𝐼^′_𝜆,𝜇(),(=𝑢,𝑣𝑢,𝑣)‖(𝑢,𝑣)‖²_𝐸−_Ω𝜆𝑓(𝑥)|𝑢|^𝑞+𝜇𝑔(𝑥)|𝑣|^𝑞−𝑑𝑥_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥. (10) Note that 𝒩_𝜆,𝜇 contains all nontrivial weak solution of the problem 𝑃_𝜆,𝜇. Using the argument of Tarantello [12, 13], we split 𝒩_𝜆,𝜇 into two parts 𝒩⁺_𝜆,𝜇 and 𝒩⁻_𝜆,𝜇 for 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₁. In Section 3, we prove Theorem 1. In Section 4, since 𝐼_𝜆,𝜇 satisfies the (PS)_𝛾-condition for 𝛾∈(−∞,(1/𝑁)(𝑆_𝛼,𝛽)^𝑁/2−𝐶₀(𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞))), for sufficiently small 𝜆, 𝜇, and some restriction on 𝑞 and 𝑁, we construct the 𝑘 compact Palais-Smale sequences which are suitably localized in correspondence with the 𝑘 maximum points of ℎ and which converge to distinct solutions of the problem 𝑃_𝜆,𝜇 belonging to 𝒩⁻_𝜆,𝜇. Hence, we prove Theorem 2 (one is the ground state solution belonging to 𝒩⁺_𝜆,𝜇 and the others are in 𝒩⁻_𝜆,𝜇).

2. Nehari Manifold

Throughout this paper, (𝐻1) will be assumed. First, we give some notations.

Notations. We make use of the following notations. 𝐿^𝑝(Ω), 1≤𝑝≤∞, denote Lebesgue spaces; the norm 𝐿^𝑝 is denoted by |⋅|_𝐿^𝑝_(Ω) for 1≤𝑝≤∞.  𝐸=[𝐻¹₀(Ω)]², endowed with norm ‖𝑧‖²_𝐸=‖(𝑢,𝑣)‖²_𝐸=|∇𝑢|²₂+|∇𝑣|²₂.  The dual space of a Banach space 𝐸 will be denoted by 𝐸⁻¹.  |𝑧|=|(𝑢,𝑣)|=(|𝑢|,|𝑣|) and 𝑡𝑧=𝑡(𝑢,𝑣)=(𝑡𝑢,𝑡𝑣) for all 𝑧∈𝐸 and 𝑡∈ℝ.  𝑧=(𝑢,𝑣) is said to be nonnegative in Ω if 𝑢≥0 and 𝑣≥0 in Ω.  𝑧=(𝑢,𝑣) is said to be positive in Ω if 𝑢>0 and 𝑣>0 in Ω.  |Ω| is the Lebesgue measure of Ω.  𝐵_𝑟(𝑎)={𝑥∈ℝ^𝑁∣|𝑥−𝑎|<𝑟} is a ball in ℝ^𝑁.  𝑂(𝜀^𝑡) denotes |𝑂(𝜀^𝑡)|/𝜀^𝑡≤𝐶 as 𝜀→0 for 𝑡≥0.  𝑂₁(𝜀^𝑡) means that there exist the constants 𝐶₁,𝐶₂>0 such that 𝐶₁𝜀^𝑡≤𝑂₁(𝜀^𝑡)≤𝐶₂𝜀^𝑡 as 𝜀 is small.  𝑜_𝑛(1) denotes 𝑜_𝑛(1)→0 as 𝑛→∞.  𝛾_∞=max{|𝑓|_𝐿^∞_(Ω),|𝑔|_𝐿^∞_(Ω)}.  𝐶, 𝐶_𝑖 will denote various positive constants, the exact values of which are not important.

Let 𝐾_𝜆,𝜇∶𝐸→ℝ be the functional defined by 𝐾_𝜆,𝜇(𝑧)=_Ω𝜆𝑓(𝑥)|𝑢|^𝑞+𝜇𝑔(𝑥)|𝑣|^𝑞𝑑𝑥∀𝑧=(𝑢,𝑣)∈𝐸. (11)

We know that 𝐼_𝜆,𝜇 is not bounded below on 𝐸. From the following lemma, we have that 𝐼_𝜆,𝜇 is bounded from below on the Nehari manifold 𝒩_𝜆,𝜇 defined in (9).

Lemma 3. The energy functional 𝐼_𝜆,𝜇 is coercive and bounded below on 𝒩_𝜆,𝜇.

Proof. If 𝑧=(𝑢,𝑣)∈𝒩_𝜆,𝜇, then by (10), the Hölder inequality, and the Sobolev embedding theorem, we get 𝐼_𝜆,𝜇(𝑧)=2^∗−22^∗2‖𝑧‖²_𝐸−2^∗−𝑞2^∗𝑞𝐾_𝜆,𝜇(𝑧)≥1𝑁‖𝑧‖²_𝐸−2^∗−𝑞2^∗𝑞𝛾_∞𝑆^−𝑞/2|Ω|^{(2∗−𝑞)/2∗}×𝜆^2/(2−𝑞)+𝜇^{2/(2−𝑞)(2−𝑞)/2}‖𝑧‖^𝑞_𝐸. (12)(13) Hence, we have that 𝐼_𝜆,𝜇 is coercive and bounded below on 𝒩_𝜆,𝜇.

Define Φ_𝜆,𝜇(𝑧)=𝐼^′_𝜆,𝜇(𝑧),𝑧. (14) Then, for 𝑧∈𝒩_𝜆,𝜇, Φ^′_𝜆,𝜇(𝑧)‖𝑧‖,𝑧=2²_𝐸−𝑞𝐾_𝜆,𝜇(𝑧)−2^∗_Ωℎ(𝑥)|𝑢|^𝛼𝑣^𝛽𝑑𝑥=2−𝑞‖𝑧‖²_𝐸−2^∗−𝑞_Ωℎ(𝑥)|𝑢|^𝛼𝑣^𝛽𝑑𝑥=2^∗−𝑞𝐾_𝜆,𝜇(𝑧)−2^∗−2‖𝑧‖²_𝐸. (15)(16)(17) We apply the method in [12]; let 𝒩⁺_𝜆,𝜇=𝑧∈𝒩_𝜆,𝜇∣Φ^′_𝜆,𝜇(𝑧),𝒩,𝑧>0_0𝜆,𝜇=𝑧∈𝒩_𝜆,𝜇∣Φ^′_𝜆,𝜇(𝑧),𝒩,𝑧=0_−𝜆,𝜇=𝑧∈𝒩_𝜆,𝜇∣Φ^′_𝜆,𝜇(𝑧),𝑧<0. (18)

By using equality (17), we get that 𝐾_𝜆,𝜇(𝑧)>0 for 𝑧∈𝒩⁺_𝜆,𝜇. Moreover, we have the following results.

Lemma 4. Let Λ₁ be a constant defined as in (8). If 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₁, then 𝒩⁰_𝜆,𝜇=∅.

Proof. Assuming the contrary, there exist 𝜆,𝜇>0 with 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₁ such that 𝒩⁰_𝜆,𝜇≠∅. Then, by (16) and (17), for 𝑢∈𝒩⁰_𝜆,𝜇, we have ‖𝑧‖²_𝐸=2^∗−𝑞2−𝑞_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽2𝑑𝑥=^∗−𝑞2^∗𝐾−2^𝜆,𝜇(𝑧). (19) Using (𝐻1) and both the Hölder and the Sobolev inequalities, we get ‖𝑧‖_𝐸≥2−𝑞2^∗𝑆−𝑞^{2∗/21/(2∗−2)},‖𝑧‖_𝐸≤2^∗−𝑞2^∗𝑆−2^−𝑞/2|Ω|^{(2∗−𝑞)/2∗}𝛾_∞^1/(2−𝑞)×𝜆^2/(2−𝑞)+𝜇^{2/(2−𝑞)1/2}. (20)(21) This implies 𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)≥2−𝑞2^∗−𝑞^2/(2∗−2)2^∗−𝑞𝛾_∞2^∗|Ω|−2^{(2∗−𝑞)/2∗−2/(2−𝑞)}×𝑆^{𝑁/2+𝑞/(2−𝑞)}=Λ₁, (22) which is a contradiction.

For each 𝑧∈𝐸 with ∫_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥>0, we write 𝑡_max=2−𝑞‖𝑧‖²_𝐸2^∗−𝑞∫_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥^1/(2∗−2)>0. (23) Then, the following lemma holds.

Lemma 5. Suppose that 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₁, and 𝑧∈𝐸 with ∫_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥>0. Then, there exist unique 0<𝑡⁺<𝑡_max<𝑡⁻ such that 𝑡⁺𝑧∈𝒩⁺_𝜆,𝜇, 𝑡⁻𝑧∈𝒩⁻_𝜆,𝜇 and 𝐼_𝜆,𝜇𝑡⁺𝑧=inf_{0≤𝑡≤𝑡max}𝐼_𝜆,𝜇();𝐼𝑡𝑧_𝜆,𝜇(𝑡⁻𝑧)=sup_𝑡≥0𝐼_𝜆,𝜇(𝑡𝑧). (24)

Proof. This is similar to the proof of Hsu [14, Lemma 2.7].

Applying Lemma 4 (𝒩⁰_𝜆,𝜇=∅ for 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₁), we write 𝒩_𝜆,𝜇=𝒩⁺_𝜆,𝜇∪𝒩⁻_𝜆,𝜇 and define 𝜃_𝜆,𝜇=inf_{𝑧∈𝒩𝜆,𝜇}𝐼_𝜆,𝜇(𝑧);𝜃⁺_𝜆,𝜇=inf_{𝑧∈𝒩+𝜆,𝜇}𝐼_𝜆,𝜇(𝑧);𝜃⁻_𝜆,𝜇=inf_{𝑧∈𝒩−𝜆,𝜇}𝐼_𝜆,𝜇(𝑧). (25)

The following lemma shows that the minimizers on 𝒩_𝜆,𝜇 are usual critical points for 𝐼_𝜆,𝜇.

Lemma 6. For the case when 𝜆∈(0,Λ₁), if 𝑧₀ is a local minimizer for 𝐼_𝜆,𝜇 on 𝒩_𝜆,𝜇, then 𝐼^′_𝜆,𝜇(𝑧₀)=0 in 𝐸⁻¹.

Proof. See Brown and Zhang [15, theorem 2.3].

Lemma 7. (i) If 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₁ and 𝑧=(𝑢,𝑣)∈𝒩⁺_𝜆,𝜇, then one has 𝐾_𝜆,𝜇(𝑧)>0,𝐼_𝜆,𝜇(𝑧)<0. (26) In particular, 𝜃_𝜆,𝜇≤𝜃⁺_𝜆,𝜇<0.
(ii) If 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<(𝑞/2)^2/(2−𝑞)Λ₁ and 𝑧=(𝑢,𝑣)∈𝒩⁻_𝜆,𝜇, then one has 𝑢≢0, 𝑣≢0 in Ω, ‖𝑧‖_𝐸>2−𝑞2^∗−𝑞^1/(2∗−2)𝑆^𝑁/4, (27) and 𝜃⁻_𝜆,𝜇>𝑑₀ for some positive constant 𝑑₀=𝑑₀(𝜆,𝜇,𝑞,𝑁,𝑆,𝛾_∞,|Ω|).

Proof. (i) Let 𝑧=(𝑢,𝑣)∈𝒩⁺_𝜆,𝜇. By (16) and (17), we have 𝐾_𝜆,𝜇(𝑧)>2^∗−22^∗‖𝑧‖−𝑞^2𝐸>0,2−𝑞2^∗‖𝑧‖−𝑞^2𝐸>_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥. (28) Then, 𝐼_𝜆,𝜇(𝑧)=12−1𝑞‖𝑧‖²_𝐸+1𝑞−12^∗_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽<1𝑑𝑥2−1𝑞+1𝑞−12^∗2−𝑞2^∗𝑧‖−𝑞‖²_𝐸=−2−𝑞‖𝑧‖𝑞𝑁^2𝐸<0. (29) By the definition of 𝜃_𝜆,𝜇, 𝜃⁺_𝜆,𝜇, we deduce that 𝜃_𝜆,𝜇≤𝜃⁺_𝜆,𝜇<0.
(ii) Let 𝑧∈𝒩⁻_𝜆,𝜇; by (16) and the Hölder and the Sobolev inequalities, we get 2−𝑞2^∗‖𝑧‖−𝑞^2𝐸<_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥≤𝑆^−2∗/2‖𝑧‖^2∗_𝐸. (30) This implies _Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽2𝑑𝑥>^∗−𝑞‖𝑧‖2−𝑞^𝐸>2−𝑞2^∗1−𝑞2^∗𝑆𝑁−24∀𝑧∈𝒩⁻_𝜆,𝜇. (31) By (13) and (31), we obtain that 𝑢≢0, 𝑣≢0 in Ω, and 𝐼_𝜆,𝜇(𝑧)≥‖𝑧‖^𝑞_𝐸1𝑁‖𝑧‖_𝐸^2−𝑞−2^∗−𝑞2^∗𝑞𝛾_∞𝑆^−𝑞/2|Ω|^{(2∗−𝑞)/2∗}×𝜆^2/(2−𝑞)+𝜇^{2/(2−𝑞)(2−𝑞)/2}>2−𝑞2^∗−𝑞^{𝑞/(2∗−2)}𝑆^𝑞𝑁/4×1𝑁2−𝑞2^∗−𝑞^{(2−𝑞)/(2∗−2)}𝑆^{(2−𝑞)𝑁/4}−2^∗−𝑞2^∗𝑞𝛾_∞𝑆^−𝑞/2|Ω|^{(2∗−𝑞)/2∗}×𝜆^2/(2−𝑞)+𝜇^{2/(2−𝑞)(2−𝑞)/2}. (32) Thus, if 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<(𝑞/2)^2/(2−𝑞)Λ₁, for all 𝑧∈𝒩⁻_𝜆,𝜇, then 𝐼_𝜆,𝜇(𝑧)≥𝑑₀𝜆,𝜇,𝑞,𝑁,𝑆,𝛾_∞,|Ω|>0. (33)

3. Existence of a Ground State Solution

First of all, we define the Palais-Smale (denote by (PS)) sequences and (PS)-condition in 𝐸 for 𝐼_𝜆,𝜇 as follows.

Definition 8. (i) For 𝛾∈ℝ, a sequence {𝑧_𝑛} is a (PS)_𝛾-sequence in 𝐸 for 𝐼_𝜆,𝜇 if 𝐼_𝜆,𝜇(𝑧_𝑛)=𝛾+𝑜_𝑛(1) and 𝐼^′_𝜆,𝜇(𝑧_𝑛)=𝑜_𝑛(1) strongly in 𝐸⁻¹ as 𝑛→∞.
(ii) 𝐼_𝜆,𝜇 satisfies the (PS)_𝛾-condition in 𝐸 if any (PS)_𝛾-sequence {𝑧_𝑛} in 𝐸 for 𝐼_𝜆,𝜇 contains a convergent subsequence.

Proof of Theorem 1. Using the same argument as in Wu [16, Proposition 9] or Hsu [14, Proposition 3.3], there exists a minimizing sequence {𝑧_𝑛} for 𝐼_𝜆,𝜇 on 𝒩_𝜆,𝜇 such that 𝐼_𝜆,𝜇𝑧_𝑛=𝜃_𝜆,𝜇+𝑜_𝑛(1),𝐼^′_𝜆,𝜇𝑧_𝑛=𝑜_𝑛(1)in𝐸⁻¹. (34) Since 𝐼_𝜆,𝜇 is coercive on 𝒩_𝜆,𝜇 (see Lemma 3), we get that {𝑧_𝑛} is bounded in 𝐸. Then, there exist a subsequence {𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛)} and 𝑧¹_𝜆,𝜇=(𝑢¹_𝜆,𝜇,𝑣¹_𝜆,𝜇)∈𝐸 such that 𝑢_𝑛⇀𝑢¹_𝜆,𝜇,𝑣_𝑛⇀𝑣¹_𝜆,𝜇weaklyin𝐻¹₀(Ω),𝑢_𝑛⟶𝑢¹_𝜆,𝜇,𝑣_𝑛⟶𝑣¹_𝜆,𝜇𝑢almosteverywhereinΩ,_𝑛⟶𝑢¹_𝜆,𝜇,𝑣_𝑛⟶𝑣¹_𝜆,𝜇stronglyin𝐿^𝑠(Ω)∀1≤𝑠<2^∗. (35) This implies 𝐾_𝜆,𝜇𝑧_𝑛=𝐾_𝜆,𝜇𝑧¹_𝜆,𝜇+𝑜_𝑛(1)as𝑛⟶∞. (36) First, we claim that 𝑧¹_𝜆,𝜇 is a nontrivial solution of 𝑃_𝜆,𝜇. By (34) and (35), it is easy to verify that 𝑧¹_𝜆,𝜇 is a weak solution of 𝑃_𝜆,𝜇. From 𝑧_𝑛∈𝒩_𝜆,𝜇 and (12), we deduce that 𝐾_𝜆,𝜇𝑧_𝑛=𝑞2^∗−222^∗𝑧−𝑞^𝑛2𝐸−2^∗𝑞2^∗𝐼−𝑞^𝜆,𝜇𝑧_𝑛. (37) Let 𝑛→∞ in (37); by (34), (36), and 𝜃_𝜆,𝜇<0, we get 𝐾_𝜆,𝜇𝑧¹_𝜆,𝜇2≥−^∗𝑞2^∗𝜃−𝑞^𝜆,𝜇>0. (38) Thus, 𝑧¹_𝜆,𝜇∈𝒩_𝜆,𝜇 is a nontrivial solution of 𝑃_𝜆,𝜇. Now, we prove that 𝑧_𝑛→𝑧¹_𝜆,𝜇 strongly in 𝐸 and 𝐼_𝜆,𝜇(𝑧¹_𝜆,𝜇)=𝜃_𝜆,𝜇. By (37), if 𝑧∈𝒩_𝜆,𝜇, then 𝐼_𝜆,𝜇(𝑧)=1𝑁‖𝑧‖²_𝐸−2^∗−𝑞2^∗𝑞𝐾_𝜆,𝜇(𝑧). (39) In order to prove that 𝐼_𝜆,𝜇(𝑧¹_𝜆,𝜇)=𝜃_𝜆,𝜇, it suffices to recall that 𝑧¹_𝜆,𝜇∈𝒩_𝜆,𝜇, by (39) and applying Fatou's lemma to get 𝜃_𝜆,𝜇≤𝐼_𝜆,𝜇𝑧¹_𝜆,𝜇=1𝑁𝑧¹_𝜆,𝜇²_𝐸−2^∗−𝑞2^∗𝑞𝐾_𝜆,𝜇𝑧¹_𝜆,𝜇≤liminf_𝑛→∞1𝑁𝑧_𝑛²_𝐸−2^∗−𝑞2^∗𝑞𝐾_𝜆,𝜇𝑧_𝑛≤liminf_𝑛→∞𝐼_𝜆,𝜇𝑧_𝑛=𝜃_𝜆,𝜇. (40) This implies that 𝐼_𝜆,𝜇(𝑧¹_𝜆,𝜇)=𝜃_𝜆,𝜇 and lim_𝑛→∞‖𝑧_𝑛‖²_𝐸=‖𝑧¹_𝜆,𝜇‖²_𝐸. Let 𝑧_𝑛=𝑧_𝑛−𝑧¹_𝜆,𝜇; then Brézis-Lieb lemma [17] implies 𝑧_𝑛²_𝐸=𝑧_𝑛²_𝐸−𝑧¹_𝜆,𝜇²_𝐸. (41) Therefore, 𝑧_𝑛→𝑧¹_𝜆,𝜇 strongly in 𝐸. Since 𝐼_𝜆,𝜇(𝑧¹_𝜆,𝜇)=𝐼_𝜆,𝜇(|𝑧¹_𝜆,𝜇|)=𝜃_𝜆,𝜇 and |𝑧¹_𝜆,𝜇|∈𝒩⁺_𝜆,𝜇, by Lemma 6 we may assume that 𝑧¹_𝜆,𝜇 is a nontrivial nonnegative solution of 𝑃_𝜆,𝜇. By an argument of Hsu [18, Lemma 4.2], we can deduce that 𝑢¹_𝜆,𝜇≢0 and 𝑣¹_𝜆,𝜇≢0 in Ω. Finally, from the maximum principle [19], we deduce that 𝑧¹_𝜆,𝜇 is positive in Ω.

Remark 9. 𝑧¹_𝜆,𝜇∈𝒩⁺_𝜆,𝜇 and 𝐼_𝜆,𝜇(𝑧¹_𝜆,𝜇)=𝜃_𝜆,𝜇=𝜃⁺_𝜆,𝜇.

Proof. We claim that 𝑧¹_𝜆,𝜇∈𝒩⁺_𝜆,𝜇. On the contrary, assume that 𝑧¹_𝜆,𝜇∈𝒩⁻_𝜆,𝜇 (𝒩⁰_𝜆,𝜇=∅ for 𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)∈(0,Λ₁)); then by Lemma 5, there exist unique 𝑡⁺₁ and 𝑡⁻₁ such that 𝑡⁺₁𝑧¹_𝜆,𝜇∈𝒩⁺_𝜆,𝜇 and 𝑡⁻₁𝑧¹_𝜆,𝜇∈𝒩⁻_𝜆,𝜇. In particular, we have 𝑡⁺₁<𝑡⁻₁=1. Since 𝑑𝐼𝑑𝑡^𝜆,𝜇𝑡⁺₁𝑧¹_𝜆,𝜇𝑑=0,²𝑑𝑡²𝐼_𝜆,𝜇𝑡⁺₁𝑧¹_𝜆,𝜇>0, (42) there exists 𝑡⁺₁<𝑡≤𝑡⁻₁ such that 𝐼_𝜆,𝜇(𝑡⁺₁𝑧¹_𝜆,𝜇)<𝐼_𝜆,𝜇(𝑡𝑧¹_𝜆,𝜇). By Lemma 5, 𝐼_𝜆,𝜇𝑡⁺₁𝑧¹_𝜆,𝜇<𝐼_𝜆,𝜇𝑡𝑧¹_𝜆,𝜇≤𝐼_𝜆,𝜇𝑡⁻₁𝑧¹_𝜆,𝜇=𝐼_𝜆,𝜇𝑧¹_𝜆,𝜇, (43) which is a contradiction. Hence, 𝑧¹_𝜆,𝜇∈𝒩⁺_𝜆,𝜇 and 𝐼_𝜆,𝜇(𝑧¹_𝜆,𝜇)=𝜃_𝜆,𝜇=𝜃⁺_𝜆,𝜇.

4. Existence of 𝑘+1 Solutions

Throughout this section, (𝐻1)-(𝐻2) will be assumed. First of all, we want to show that 𝐼_𝜆,𝜇 satisfies the (PS)_𝛾-condition in 𝐸 for 𝛾∈(−∞,(1/𝑁)(𝑆_𝛼,𝛽)^𝑁/2−𝐶₀(𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞))), where 𝐶₀ is defined in the following lemma.

Lemma 10. If {𝑧_𝑛}⊂𝐸 is a (PS)_𝛾-sequence for 𝐼_𝜆,𝜇 with 𝑧_𝑛⇀𝑧 weakly in 𝐸, then 𝐼^′_𝜆,𝜇(𝑧)=0 and there exists a constant 𝐶₀=𝐶₀(𝑞,𝑁,𝑆,𝛾_∞,|Ω|)>0 such that 𝐼_𝜆,𝜇(𝑧)≥−𝐶₀(𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)).

Proof. Let 𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛) and 𝑧=(𝑢,𝑣). If {𝑧_𝑛} is a (PS)_𝛾-sequence for 𝐼_𝜆,𝜇 with 𝑧_𝑛⇀𝑧 weakly in 𝐸, it is easy to check that 𝐼^′_𝜆,𝜇(𝑧)=0 in 𝐸⁻¹. Then, we get ⟨𝐼^′_𝜆,𝜇(𝑧),𝑧⟩=0; that is, ∫_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥=‖𝑧‖²_𝐸−𝐾_𝜆,𝜇(𝑧). Thus, by (13), the Hölder, the Young, and the Sobolev inequalities, we have 𝐼_𝜆,𝜇(𝑧)≥1𝑁‖𝑧‖²_𝐸−2^∗−𝑞2^∗𝑞𝛾_∞𝑆^−𝑞/2|Ω|^{(2∗−𝑞)/2∗}×𝜆^2/(2−𝑞)+𝜇^{2/(2−𝑞)(2−𝑞)/2}‖𝑧‖^𝑞_𝐸≥1𝑁‖𝑧‖²_𝐸−1𝑁‖𝑧‖²_𝐸−𝐶₀𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)=−𝐶₀𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞), (44) where 𝐶₀=𝐶₀(𝑞,𝑁,𝑆,𝛾_∞,|Ω|)>0.

Lemma 11. If {𝑧_𝑛}⊂𝐸 is a (PS)_𝛾-sequence for 𝐼_𝜆,𝜇, then {𝑧_𝑛} is bounded in 𝐸.

Proof. See Hsu and Lin [8, Lemma 2.3].

Recall that 𝑆_𝛼,𝛽=inf_{𝑢,𝑣∈𝐻10(Ω)⧵{0}}‖(𝑢,𝑣)‖²_𝐸∫_Ω|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥^{2/(𝛼+𝛽)}, (45) and let 𝑐^∗=1𝑁𝑆_𝛼,𝛽^𝑁/2−𝐶₀𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞), (46) where 𝐶₀>0 is given in Lemma 10.

Lemma 12. 𝐼_𝜆,𝜇 satisfies the (PS)_𝛾-condition in 𝐸 for 𝛾∈(−∞,𝑐^∗).

Proof. Let {𝑧_𝑛}⊂𝐸 be a (PS)_𝛾-sequence for 𝐼_𝜆,𝜇 with 𝛾∈(−∞,𝑐^∗). Write 𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛). We know from Lemma 11 that {𝑧_𝑛} is bounded in 𝐸, and then 𝑧_𝑛⇀𝑧=(𝑢,𝑣) weakly up to a subsequence; 𝑧 is a critical point of 𝐼_𝜆,𝜇. Furthermore, we may assume that 𝑢_𝑛⇀𝑢, 𝑣_𝑛⇀𝑣 weakly in 𝐻¹₀(Ω) and 𝑢_𝑛→𝑢, 𝑣_𝑛→𝑣 strongly in 𝐿^𝑠(Ω) for all 1≤𝑠<2^∗, and 𝑢_𝑛→𝑢, 𝑣_𝑛→𝑣 a.e. on Ω. Hence, we have that 𝐼^′_𝜆,𝜇(𝑧)=0 and 𝐾_𝜆,𝜇𝑧_𝑛=𝐾_𝜆,𝜇(𝑧)+𝑜_𝑛(1). (47) Let 𝑢_𝑛=𝑢_𝑛−𝑢,𝑣_𝑛=𝑣_𝑛−𝑣 and 𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛). Then, we obtain 𝑧_𝑛²_𝐸=𝑧_𝑛²_𝐸−‖𝑧‖²_𝐸+𝑜_𝑛(1), (48) and by an argument of Han [20, Lemma 2.1], _Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽=𝑑𝑥_Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥−_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥+𝑜_𝑛(1). (49) Since 𝐼_𝜆,𝜇(𝑧_𝑛)=𝛾+𝑜_𝑛(1), 𝐼^′_𝜆,𝜇(𝑧_𝑛)=𝑜_𝑛(1) in 𝐸⁻¹ and (47)–(49), we deduce that 12𝑧_𝑛²_𝐸−12^∗_Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=𝛾−𝐼_𝜆,𝜇(𝑧)+𝑜_𝑛(1),𝑧_𝑛²_𝐸−_Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=𝑜_𝑛(1). (50)(51) Hence, we may assume that 𝑧_𝑛²_𝐸⟶𝑙,_Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥⟶𝑙. (52) Assume that 𝑙≠0; by the definition of 𝑆_𝛼,𝛽, |ℎ|_𝐿^∞_(Ω)=1 and (52), we obtain 𝑆_𝛼,𝛽𝑙^2/2∗=𝑆_𝛼,𝛽lim_𝑛→∞Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥^2/2∗≤|ℎ|^2/2∗_𝐿∞(Ω)lim_𝑛→∞𝑧_𝑛²=𝑙, (53) which implies that 𝑙≥(𝑆_𝛼,𝛽)^𝑁/2. In addition, from Lemma 10, (50), and (52), we get 1𝛾=2−12^∗𝑙+𝐼_𝜆,𝜇(𝑧)≥1𝑁𝑆_𝛼,𝛽^𝑁/2−𝐶₀𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞), (54) which is a contradiction. Hence, 𝑙=0; that is, 𝑧_𝑛→𝑧 strongly in 𝐸.

From assumption (𝐻2), we can choose 𝑟₀∈(0,1) such that 𝐵_𝑟0𝑎_𝑖∩𝐵_𝑟0𝑎_𝑗=∅for𝑖≠𝑗,1≤𝑖,𝑗≤𝑘, (55) and ∪^𝑘_𝑖=1𝐵_𝑟0(𝑎_𝑖)⊂Ω, where 𝐵_𝑟0(𝑎_𝑖)={𝑥∈ℝ^𝑁∣|𝑥−𝑎_𝑖|≤𝑟₀} and ℎ(𝑎_𝑖)=|ℎ|_∞=1 for 1≤𝑖≤𝑘.

Define 𝑄_𝑖(𝑧)=∫_Ω𝜓_𝑖(𝑥)|∇𝑢|²+|∇𝑣|²𝑑𝑥∫_Ω|∇𝑢|²+|∇𝑣|²(){0},𝑑𝑥,𝑧=𝑢,𝑣∈𝐸⧵ (56) where 𝜓_𝑖(𝑥)=min{1,|𝑥−𝑎_𝑖|}, 𝑖=1,2…,𝑘.

Then, we have the following separation result.

Lemma 13. If 𝑄_𝑖(𝑧)≤𝑟₀/3 and 𝑄_𝑗(𝑧)≤𝑟₀/3 for 𝑧∈𝐸⧵{0}, then 𝑖=𝑗.

Proof. For any 𝑧∈𝐸⧵{0} satisfying 𝑄_𝑖(𝑧)≤𝑟₀/3(1≤𝑖≤𝑘), we get 𝑟₀3‖𝑧‖²_𝐸≥_Ω𝜓_𝑖(𝑥)|∇𝑢|²+|∇𝑣|²≥𝑑𝑥_{Ω⧵𝐵𝑟0(𝑎𝑖)}𝜓_𝑖(𝑥)|∇𝑢|²+|∇𝑣|²𝑑𝑥≥𝑟_{0Ω⧵𝐵𝑟0(𝑎𝑖)}|∇𝑢|²+|∇𝑣|²𝑑𝑥, (57) which implies that ‖𝑧‖²_𝐸≥3_{Ω⧵𝐵𝑟0(𝑎𝑖)}|∇𝑢|²+|∇𝑣|²𝑑𝑥,1≤𝑖≤𝑘. (58) Hence, from (58), we obtain 2‖𝑧‖²_𝐸≥3_{Ω⧵𝐵𝑟0(𝑎𝑖)}|∇𝑢|²+|∇𝑣|²𝑑𝑥+_{Ω⧵𝐵𝑟0(𝑎𝑗)}|∇𝑢|²+|∇𝑣|²‖𝑧‖𝑑𝑥≥3²_𝐸if𝑖≠𝑗, (59) which is a contradiction.

For 𝑖=1,2,…,𝑘, we set 𝒩^𝑖_𝜆,𝜇=𝑢∈𝒩⁻_𝜆,𝜇∣𝑄_𝑖(𝑧)<𝑟₀3,𝜕𝒩^𝑖_𝜆,𝜇=𝑢∈𝒩⁻_𝜆,𝜇∣𝑄_𝑖(𝑧)=𝑟₀3, (60) and define 𝜃^𝑖_𝜆,𝜇=inf_{𝒩𝑖𝜆,𝜇}𝐼_𝜆,𝜇(𝑧),𝜃^𝑖_𝜆,𝜇=inf_{𝜕𝒩𝑖𝜆,𝜇}𝐼_𝜆,𝜇(𝑧). (61)

Recall that the best Sobolev constant 𝑆 is defined as 𝑆=inf_{𝑢∈𝐷1,2ℝ𝑁⧵{0}}∫_ℝ𝑁|∇𝑢|²𝑑𝑥∫_ℝ𝑁|𝑢|^2∗𝑑𝑥^2/2∗. (62) It is well known that 𝑈(𝑥)=[𝑁(𝑁−2)]^(𝑁−2)/41+|𝑥|^{2(𝑁−2)/2} (63) is a minimizer of 𝑆, and |∇𝑈|²_{𝐿2(ℝ𝑁)}=|𝑈|^2∗_𝐿2^∗_(ℝ𝑁)=𝑆^𝑁/2. Fix a maximum point 𝑎_𝑖 of ℎ (1≤𝑖≤𝑘). Let 𝜂_𝑖∈𝐶^∞₀(Ω) be a cut-off function such that 0≤𝜂_𝑖≤1, |∇𝜂_𝑖|≤𝐶, and 𝜂_𝑖(𝑥)=1 for |𝑥−𝑎_𝑖|<𝑟₀/2, 𝜂_𝑖(𝑥)=0 for |𝑥−𝑎_𝑖|>𝑟₀. We define 𝑢^𝑖_𝜀(𝑥)=𝜀^(2−𝑁)/2𝜂_𝑖(𝑥)𝑈𝑥−𝑎_𝑖𝜀=𝑐₁𝜀^(𝑁−2)/2𝜂_𝑖(𝑥)𝜀²+𝑥−𝑎_𝑖^{2(𝑁−2)/2}, (64) where 𝑐₁=[𝑁(𝑁−2)]^(𝑁−2)/4 and 𝜀>0.

From now on, we assume that 𝑁/(𝑁−2)<𝑞<2 and 𝑁>4.

Lemma 14. There exist 𝜀₀>0, Λ₂∈(0,(𝑞/2)^2/(2−𝑞)Λ₁), such that for 𝜀∈(0,𝜀₀) and 𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)∈(0,Λ₂), one has sup_𝑡≥0𝐼_𝜆,𝜇𝑡√𝛼𝑢^𝑖_𝜀,𝑡𝛽𝑢^𝑖_𝜀<𝑐^∗𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑖, (65) where 𝑐^∗ is the positive constant given in Lemma 12.
In particular, 0<𝜃⁻_𝜆,𝜇≤𝜃^𝑖_𝜆,𝜇<𝑐^∗ for all 1≤𝑖≤𝑘.

Proof. It is well known that (or see Brézis and Nirenberg [21], Cheng and Ma [22, Lemma 3.2], Struwe [23], and Willem [24, Lemma 1.46]) as 𝜀→0⁺, 𝑢^𝑖_𝜀²_𝐿2∗(Ω)=|𝑈|²_𝐿2^∗_ℝ𝑁+𝑂𝜀^𝑁−2,∇𝑢^𝑖_𝜀²_𝐿2(Ω)|∇𝑈|²_𝐿2ℝ𝑁+𝑂𝜀^𝑁−2. (66)(67) For 𝑁/(𝑁−2)<𝑞<2, 𝑁>4 and 𝜀<𝑟₀/2, 𝑢^𝑖_𝜀^𝑞_𝐿𝑞(Ω)=_{𝐵𝑟0/2(𝑎𝑖)}𝜀^(2−𝑁)/2𝑈𝑥−𝑎_𝑖𝜀^𝑞𝑑𝑥+𝑂𝜀^𝑁−2≥𝐶𝜀^𝜃+𝑂𝜀^𝑁−2()𝑞,where𝜃=𝑁−𝑁−22. (68)
Set 𝑧^𝑖_𝜀=(√𝛼𝑢^𝑖_𝜀,𝛽𝑢^𝑖_𝜀). By Lemma 5, there exists 𝑡^𝑖_𝜀>0 such that 𝑧^𝑖_𝜀=𝑡^𝑖_𝜀𝑧^𝑖_𝜀∈𝒩⁻_𝜆,𝜇 for 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₁. Furthermore, 𝑄_𝑖𝑧^𝑖_𝜀=∫_Ω𝜓_𝑖(𝑥)∇𝑢^𝑖_𝜀²𝑑𝑥∫_Ω∇𝑢^𝑖_𝜀²=∫𝑑𝑥_{(Ω−𝑎𝑖)/𝜀}𝜓_𝑖𝑎_𝑖+𝜀𝑦∇𝜂_𝑖𝑎_𝑖+𝜀𝑦𝑈𝑦²𝑑𝑦∫_{(Ω−𝑎𝑖)/𝜀}∇𝜂_𝑖𝑎_𝑖+𝜀𝑦𝑈𝑦²𝑑𝑦⟶𝜓_𝑖𝑎_𝑖=0as𝜀⟶0. (69) Hence, there exists 𝜀₀>0 for any 𝜀∈0,𝜀₀,𝑄_𝑖𝑧^𝑖_𝜀<𝑟₀3, (70) which implies 𝑧^𝑖_𝜀=𝑡^𝑖_𝜀𝑧^𝑖_𝜀∈𝒩^𝑖_𝜆,𝜇for𝜀∈0,𝜀₀, (71) and then 𝜃⁻_𝜆,𝜇≤𝜃^𝑖_𝜆,𝜇≤𝐼_𝜆,𝜇𝑧^𝑖_𝜀≤sup_𝑡≥0𝐼_𝜆,𝜇𝑡𝑡^𝑖_𝜀𝑧^𝑖_𝜀=sup_𝑡≥0𝐼_𝜆,𝜇𝑡𝑧^𝑖_𝜀. (72) First, we consider the functional 𝐼_0,0∶𝐸→ℝ defined by 𝐼_0,0()=1𝑢,𝑣2‖(𝑢,𝑣)‖²_𝐸−12^∗_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥. (73)
Step I. Show that sup_𝑡≥0𝐼_0,0(𝑧^𝑖_𝜀)≤(1/𝑁)(𝑆_𝛼,𝛽)^𝑁/2+𝑂(𝜀^𝑁−2).
According to condition (𝐻2), we conclude that _Ωℎ(𝑥)𝑢^𝑖_𝜀(𝑥)^2∗𝑑𝑥−_Ωℎ𝑎_𝑖𝑢^𝑖_𝜀(𝑥)^2∗≤𝑑𝑥_Ωℎ(𝑥)−ℎ𝑎_𝑖𝑢^𝑖_𝜀(𝑥)^2∗𝑑𝑥=𝑂_{𝐵𝑟0(𝑎𝑖)}𝑥−𝑎_𝑖^𝜎𝑢^𝑖_𝜀(𝑥)^2∗𝜀𝑑𝑥=𝑂^𝜎. (74) From (66), (74), ℎ(𝑎_𝑖)=1, and 𝜎≥𝑁−2, we can deduce that _Ωℎ(𝑥)𝑢^𝑖_𝜀(𝑥)^2∗𝑑𝑥^2/2∗=𝑢^𝑖_𝜀^2∗_𝐿2∗(Ω)+𝑂𝜀^𝜎2/2∗=𝑢^𝑖_𝜀²_𝐿2∗(Ω)+𝑂𝜀^𝜎=|𝑈|²_𝐿2^∗_ℝ𝑁+𝑂𝜀^𝑁−2. (75) Using (67) and (75), then ∇𝑢^𝑖_𝜀²_𝐿2(Ω)∫_Ωℎ(𝑥)𝑢^𝑖_𝜀(𝑥)^2∗𝑑𝑥^2/2∗=|∇𝑈|²_𝐿2ℝ𝑁+𝑂𝜀^𝑁−2|𝑈|²_𝐿2^∗_ℝ𝑁+𝑂𝜀^𝑁−2=𝑆+𝑂𝜀^𝑁−2. (76) Since sup_𝑡≥0𝐴2𝑡²−𝐵2^∗𝑡^2∗=1𝑁𝐴𝐵^2/2∗𝑁/2,forany𝐴>0,𝐵>0, (77) by (7) and (76), we conclude that sup_𝑡≥0𝐼_0,0𝑡𝑧^𝑖_𝜀=1𝑁𝛼+𝛽∇𝑢^𝑖_𝜀²_𝐿2(Ω)𝛼^𝛼/2𝛽^𝛽/2∫_Ωℎ(𝑥)𝑢^𝑖_𝜀(𝑥)^2∗𝑑𝑥^2/2∗𝑁/2≤1𝑁𝑆_𝛼,𝛽^𝑁/2+𝑂𝜀^𝑁−2. (78)
Step II. Let 𝐶₀ be the positive constant given in Lemma 10. We can choose 𝛿₁>0 such that for all 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<𝛿₁, we have 𝑐^∗=1𝑁𝑆_𝛼,𝛽^𝑁/2−𝐶₀𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)>0. (79) Since 𝐼_𝜆,𝜇 is continuous in 𝐸, 𝐼_𝜆,𝜇(0)=0, and {𝑧^𝑖_𝜀} is uniformly bounded in 𝐸 for any 0<𝜀<min{𝜀₀,𝑟₀/2} (see (67)), then there exists 𝑡₀>0 (independent of 𝜀) such that for any 0<𝜀<min{𝜀₀,𝑟₀/2}, sup_{0≤𝑡≤𝑡0}𝐼_𝜆,𝜇𝑡𝑧^𝑖_𝜀<𝑐^∗,uniformlyin𝑖,∀0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<𝛿₁. (80) According to condition (𝐻1), 𝑓_min=min_𝑥∈Ω𝑓(𝑥)>0 and 𝑔_min=min_𝑥∈Ω𝑔(𝑥)>0. Applying the results of Step I and (68), we have that for 𝑁/(𝑁−2)<𝑞<2 and 𝑁>4, sup_𝑡≥𝑡0𝐼_𝜆,𝜇𝑡𝑧^𝑖_𝜀=sup_𝑡≥𝑡0𝐼_0,0𝑡𝑧^𝑖_𝜀−𝑡^𝑞𝑞𝐾_𝜆,𝜇𝑡𝑧^𝑖_𝜀≤1𝑁𝑆_𝛼,𝛽^𝑁/2+𝑂𝜀^𝑁−2−𝑡^𝑞₀𝑞𝑚×𝜆+𝜇_{𝐵𝑟0/2(𝑎𝑖)}𝑢^𝑖_𝜀^𝑞≤1𝑑𝑥𝑁𝑆_𝛼,𝛽^𝑁/2+𝑂𝜀^𝑁−2−𝜆+𝜇𝑂₁𝜀^𝜃, (81) where 𝑚=min{𝛼^𝑞/2𝑓_min,𝛽^𝑞/2𝑔_min} and 𝜃=𝑁−((𝑁−2)𝑞)/2.
Therefore, we can choose 𝜆=𝑂₁(𝜀^𝜏1) and 𝜇=𝑂₁(𝜀^𝜏2) such that 2−𝑞𝑞𝜃<𝜏₁,𝜏₂<(𝑁−2)−𝜃. (82)
This implies that min𝜏₁,𝜏₂2+𝜃<𝜏2−𝑞min₁,𝜏₂,min𝜏₁,𝜏₂𝑂+𝜃<𝑁−2,𝜆+𝜇₁𝜀^𝜃=𝑂₁𝜀^{min{𝜏1,𝜏2}+𝜃},𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)=𝑂₁𝜀^{2/(2−𝑞)min{𝜏1,𝜏2}}. (83) There exist 𝛿₂>0, 𝜀₀∈(0,min{𝜀₀,𝑟₀/2}) such that for all 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<𝛿₂ and 0<𝜀<𝜀₀, we have 𝑂𝜀^𝑁−2−𝜆+𝜇𝑂₁𝜀^𝜃<−𝐶₀𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞). (84) Thus, we can choose Λ₂=min{(𝑞/2)^2/(2−𝑞)Λ₁,𝛿₁,𝛿₂}>0. Then, for all 𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)∈(0,Λ₂), there holds sup_𝑡≥0𝐼_𝜆,𝜇𝑡𝑧^𝑖_𝜀<𝑐^∗uniformlyin𝑖. (85)
Step III. For 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ₂ and 0<𝜀<𝜀₀, by Lemma 7, (72), and (85), we get 0<𝜃⁻_𝜆,𝜇≤𝜃^𝑖_𝜆,𝜇≤𝐼_𝜆,𝜇𝑡𝑧^𝑖_𝜀<𝑐^∗∀1≤𝑖≤𝑘. (86)

To proceed, we need to quote the concentration-compactness principle (see [24, 25]) about the case of systems.

Lemma 15. Let {𝑢_𝑛,𝑣_𝑛}⊂𝐻¹₀(Ω)×𝐻¹₀(Ω) be a sequence such that 𝑢_𝑛⇀𝑢,𝑣_𝑛⇀𝑣𝑤𝑒𝑎𝑘𝑙𝑦𝑖𝑛𝐻¹₀(Ω);𝑢_𝑛⟶𝑢,𝑣_𝑛∇𝑢⟶𝑣𝑎.𝑒.𝑜𝑛Ω,_𝑛−𝑢²+∇𝑣_𝑛−𝑣²⇀𝜇𝑢𝑤𝑒𝑎𝑘𝑙𝑦𝑖𝑛𝑡ℎ𝑒𝑠𝑒𝑛𝑠𝑒𝑜𝑓𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠,_𝑛−𝑢^𝛼𝑣_𝑛−𝑣^𝛽⇀𝜈𝑤𝑒𝑎𝑘𝑙𝑦𝑖𝑛𝑡ℎ𝑒𝑠𝑒𝑛𝑠𝑒𝑜𝑓𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠. (87) Then, it follows that limsup_{𝑛→∞ℝ𝑁}∇𝑢_𝑛²+|∇𝑣|_𝑛²=𝑑𝑥_ℝ𝑁|∇𝑢|²+|∇𝑣|²𝑑𝑥+𝜇,limsup_{𝑛→∞ℝ𝑁}𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=_ℝ𝑁|𝑢|^𝛼|𝑣|^𝛽‖𝜈‖,‖𝜈‖𝑑𝑥+_{2/(𝛼+𝛽)}≤𝑆⁻¹_𝛼,𝛽𝜇. (88) Moreover, if 𝑢≡𝑣≡0 and ‖𝜈‖^{2/(𝛼+𝛽)}=𝑆⁻¹_𝛼,𝛽‖𝜇‖, then 𝜇 and 𝜈 concentrate at a single point.

Proof. See Han [20, Lemma 2.2].

Lemma 16. For any 𝑖∈{1,2,…,𝑘}, there exist Λ_𝑖>0 such that 𝜃^𝑖_𝜆,𝜇>1𝑁𝑆_𝛼,𝛽^𝑁/2∀0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ_𝑖. (89)

Proof. Fix 𝑖∈{1,2,…,𝑘}. Assume the contrary. There then exists a sequence {(𝜆_𝑛,𝜇_𝑛)} with (𝜆_𝑛,𝜇_𝑛)→(0,0) as 𝑛→∞ such that 𝜃^𝑖_{𝜆𝑛,𝜇𝑛}→𝑐≤(1/𝑁)(𝑆_𝛼,𝛽)^𝑁/2 as 𝑛→∞. Consequently, there exists a sequence {𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛)}⊂𝜕𝒩^𝑖_{𝜆𝑛,𝜇𝑛} such that as 𝑛→∞, _Ω∇𝑢_𝑛²+∇𝑣_𝑛²=𝑑𝑥_Ω𝜆_𝑛𝑓(𝑥)𝑢_𝑛^𝑞+𝜇_𝑛𝑔(𝑥)𝑣_𝑛^𝑞𝑑𝑥+_Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥,𝐼_{𝜆𝑛,𝜇𝑛}𝑧_𝑛1⟶𝑐≤𝑁𝑆_𝛼,𝛽^𝑁/2as𝑛⟶∞. (90)(91) It then follows easily that {𝑧_𝑛} is uniformly bounded in 𝐸, and since 𝑓 and 𝑔 are continuous on Ω, we obtain 𝐾_{𝜆𝑛,𝜇𝑛}𝑧_𝑛=_Ω𝜆_𝑛𝑓(𝑥)𝑢_𝑛^𝑞+𝜇_𝑛𝑔(𝑥)𝑣_𝑛^𝑞𝑑𝑥=𝑜_𝑛(1)as𝑛⟶∞. (92) From (90), and by the Hölder and the Sobolev inequalities, we can fix 𝑚₀>0 such that _Ω∇𝑢_𝑛²+∇𝑣_𝑛²𝑑𝑥≥𝑚₀,_Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥≥𝑚₀. (93) Thus, up to a subsequence, we infer that lim_𝑛→∞Ω∇𝑢_𝑛²+∇𝑣_𝑛²𝑑𝑥=lim_𝑛→∞Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=𝑙>0. (94) Furthermore, by |ℎ|_𝐿^∞_(Ω)=1, we deduce 𝑙=lim_𝑛→∞Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽≤|ℎ|𝑑𝑥_𝐿∞(Ω)lim_𝑛→∞Ω𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥≤𝑆^−2∗/2_𝛼,𝛽lim_𝑛→∞Ω∇𝑢_𝑛²+∇𝑣_𝑛²𝑑𝑥^2∗/2≤𝑆^−2∗/2_𝛼,𝛽𝑙^2∗/2, (95) which implies 𝑙≥𝑆_𝛼,𝛽^𝑁/2. (96) On the other hand, we have, as 𝑛→∞, 1𝑁1𝑙=2𝑧_𝑛²1𝑑𝑥−2^∗_Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽−1𝑑𝑥𝑞𝐾_{𝜆𝑛,𝜇𝑛}𝑧_𝑛+𝑜_𝑛(1)=𝐼_{𝜆𝑛,𝜇𝑛}𝑧_𝑛+𝑜_𝑛(1)≤1𝑁𝑆_𝛼,𝛽^𝑁/2. (97) Hence, together with (96), we get 𝑙=𝑆_𝛼,𝛽^𝑁/2, (98) and then from (95), we also have lim_𝑛→∞Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=lim_𝑛→∞Ω𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=𝑙. (99) Therefore, lim_𝑛→∞Ω𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=𝑙. (100) Set 𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛)=𝑧_𝑛/‖𝑧_𝑛‖; then, we have ‖𝑧_𝑛‖=1. Moreover, by (94),(98), and (100), we get lim_𝑛→∞Ω𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=lim_𝑛→∞∫_Ω𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥𝑧_𝑛^2∗=𝑆^{−𝑁/(𝑁−2)}_𝛼,𝛽. (101) Thus, up to a subsequence, we may assume that 𝑢_𝑛⇀𝑢,𝑣_𝑛⇀𝑣weaklyin𝐻¹₀(Ω);𝑢_𝑛⟶𝑢,𝑣_𝑛∇𝑢⟶𝑣a.e.onΩ,_𝑛−𝑢²+∇𝑣_𝑛−𝑣²⇀𝜇𝑢weaklyinthesenseofmeasures,_𝑛−𝑢^𝛼𝑣_𝑛−𝑣^𝛽⇀𝜈weaklyinthesenseofmeasures. (102) Since Ω is bounded, from (101) and Lemma 15, we deduce that 1=_Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥+𝜇,𝑆^{−𝑁/(𝑁−2)}_𝛼,𝛽=_Ω|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥+‖𝜈‖,‖𝜈‖^{2/(𝛼+𝛽)}≤𝑆⁻¹_𝛼,𝛽𝜇. (103)(104)(105) If ∫_Ω(|∇𝑢|²+|∇𝑣|²)𝑑𝑥≠0 and ‖𝜇‖≠0, we deduce that 1=_Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥+𝜇^{(𝛼+𝛽)/2}>_Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥^{(𝛼+𝛽)/2}+𝜇^{(𝛼+𝛽)/2}≥𝑆^{(𝛼+𝛽)/2}_𝛼,𝛽Ω|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥+𝑆^{(𝛼+𝛽)/2}_𝛼,𝛽‖𝜈‖=𝑆^{(𝛼+𝛽)/2}_𝛼,𝛽⋅𝑆^{−𝑁/(𝑁−2)}_𝛼,𝛽=1, (106) which is a contradiction.
Thus, ∫_Ω(|∇𝑢|²+|∇𝑣|²)𝑑𝑥=0 or ‖𝜇‖=0. If ‖𝜇‖=0, from (103)–(105), we get ∫_Ω(|∇𝑢|²+|∇𝑣|²)𝑑𝑥=1 and ∫_Ω|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥=𝑆^{−𝑁/(𝑁−2)}_𝛼,𝛽. Then, ∫_Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥∫_Ω|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥^{2/(𝛼+𝛽)}=𝑆_𝛼,𝛽, (107) which means that 𝑆_𝛼,𝛽 is achieved by (𝑢,𝑣). It is impossible since 𝑆_𝛼,𝛽 cannot be achieved on any bounded domain Ω. Hence, _Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥=0,𝜇=1. (108) Then, 𝑢≡𝑣≡0 on Ω, and from (103), (104), we easily have ‖𝜈‖^{2/(𝛼+𝛽)}=𝑆⁻¹_𝛼,𝛽=𝑆⁻¹_𝛼,𝛽‖𝜇‖. By Lemma 15, we conclude that 𝑥₀∈Ω such that ∇𝑢_𝑛²+|∇𝑣|̃²⇀𝛿_𝑥0𝑢weaklyinthesenseofmeasures,_𝑛𝛼𝑣_𝑛^𝛽⇀𝑆^{−𝑁/(𝑁−2)}_𝛼,𝛽𝛿_𝑥0weaklyinthesenseofmeasures. (109) Observe that 𝑄_𝑖(𝑧_𝑛)=𝑄_𝑖(𝑧_𝑛)=𝑟₀/3; 𝑟₀3=lim_𝑛→∞𝑄_𝑖𝑧_𝑛=lim_𝑛→∞∫_Ω𝜓_𝑖(𝑥)∇𝑢_𝑛²+∇𝑣_𝑛²𝑑𝑥∫_Ω∇𝑢_𝑛²+∇𝑣_𝑛²𝑑𝑥=𝜓_𝑖𝑥₀, (110) which implies that 𝑥₀≠𝑎_𝑖 by the definition of 𝜓_𝑖(𝑥). On the other hand, from (95) and (101), we get 𝑆^{−𝑁/(𝑁−2)}_𝛼,𝛽ℎ𝑥₀=lim_𝑛→∞Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=lim_𝑛→∞∫_Ωℎ(𝑥)𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥𝑧_𝑛^2∗=lim_𝑛→∞∫_Ω𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥𝑧_𝑛^2∗=lim_𝑛→∞Ω𝑢_𝑛^𝛼𝑣_𝑛^𝛽𝑑𝑥=𝑆^{−𝑁/(𝑁−2)}_𝛼,𝛽, (111) which is impossible, because ℎ(𝑥) is not a constant function by condition (𝐻2).

Throughout this section, take Λ^∗=min{Λ₂,min_{1≤𝑖≤𝑘}Λ_𝑖}; Λ₂ and Λ_𝑖 are as in Lemmas 14 and 16. Using the idea of Tarantello [12], we have the following results. For 𝑧=(𝑢,𝑣), 𝜑=(𝜑₁,𝜑₂)∈𝐸, we define 𝑧−𝜑=𝑢−𝜑₁,𝑣−𝜑₂,𝑧,𝜑=_Ω∇𝑢∇𝜑₁+∇𝑣∇𝜑₂𝐺𝑑𝑥,_𝜆,𝜇𝑧,𝜑=_Ω𝜆𝑓(𝑥)|𝑢|^𝑞−2𝑢𝜑₁+𝜇𝑔(𝑥)|𝑣|^𝑞−2𝑣𝜑₂𝐻=𝛼𝑑𝑥,𝑧,𝜑𝛼+𝛽_Ωℎ(𝑥)|𝑢|^𝛼−2𝑢|𝑣|^𝛽𝜑₁+𝛽𝑑𝑥𝛼+𝛽_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽−2𝑣𝜑₂𝑑𝑥. (112)

Lemma 17. For each 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ^∗ and 𝑧=(𝑢,𝑣)∈𝒩^𝑖_𝜆,𝜇(1≤𝑖≤𝑘), there exist 𝜖>0 and a differentiable function 𝜉∶𝐵_𝜖(0)⊂𝐸→ℝ⁺ such that 𝜉(0)=1, 𝜉(𝜙)(𝑧−𝜙)∈𝒩^𝑖_𝜆,𝜇 for all 𝜙∈𝐵_𝜖(0) and 𝜉^′(0)=2,𝜑𝑧,𝜑−𝑞𝐺_𝜆,𝜇𝑧,𝜑−2^∗𝐻𝑧,𝜑2−𝑞‖𝑧‖²_𝐸−2^∗𝐻(),−𝑞𝑧,𝑧 (113) for all 𝜑=(𝜑₁,𝜑₂)∈𝐸.

Proof. For 𝑧∈𝒩^𝑖_𝜆,𝜇, define a function 𝐹_𝑧∶ℝ×𝐸→ℝ by 𝐹_𝑧𝜉,𝜙=𝐼^′_𝜆,𝜇𝜉𝑧−𝜙,𝜉𝑧−𝜙=𝜉²𝑧−𝜙²−𝜉^𝑞𝐺_𝜆,𝜇𝑧−𝜙,𝑧−𝜙−𝜉^𝛼+𝛽𝐻𝑧−𝜙,𝑧−𝜙. (114) Then, 𝐹_𝑢(1,0)=⟨𝐼^′_𝜆,𝜇(𝑧),𝑧⟩=0 and 𝑑𝐹𝑑𝜉^𝑧(1,0)=2‖𝑧‖²_𝐸−𝑞𝐺_𝜆,𝜇()−𝐻()=‖𝑧‖𝑧,𝑧𝛼+𝛽𝑧,𝑧2−𝑞²_𝐸−2^∗−𝑞𝐻(𝑧,𝑧)<0. (115) According to the implicit function theorem, there exist 𝜖>0 and a differentiable function 𝜉∶𝐵_𝜖(0)⊂𝐸→ℝ such that 𝜉(0)=1; 𝜉^′(0)=2,𝜑𝑧,𝜑−𝑞𝐺_𝜆,𝜇𝑧,𝜑−2^∗𝐻𝑧,𝜑2−𝑞‖𝑧‖²_𝐸−2^∗𝐻(),𝐹−𝑞𝑧,𝑧_𝑧𝜉𝜑,𝜑=0∀𝜑∈𝐵_𝜖(0), (116) which is equivalent to 𝐼^′_𝜆,𝜇𝜉𝜑𝑧−𝜑,𝜉𝜑𝑧−𝜑=0∀𝜑∈𝐵(0;𝜖); (117) that is, 𝜉(𝜑)(𝑧−𝜑)∈𝒩_𝜆,𝜇 for all 𝜑∈𝐵_𝜖(0). Furthermore, by the continuity of the functions 𝜉 and 𝑄_𝑖, we have that 2−𝑞𝜉𝜑𝑧−𝜑²−2^∗𝐻𝜉𝜑𝜑𝑄−𝑞𝑧−𝜑,𝜉𝑧−𝜑<0,_𝑖𝜉𝜑<𝑟𝑧−𝜑⁰3 (118) still holds if 𝜖 is sufficiently small. This implies that 𝜉(𝜑)(𝑧−𝜑)∈𝒩^𝑖_𝜆,𝜇.

Proposition 18. If 0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ^∗, then there exists a (PS)_𝜃^𝑖_𝜆,𝜇-sequence {𝑧^𝑖_𝑛}⊂𝒩^𝑖_𝜆,𝜇 in 𝐸 for 𝐼_𝜆,𝜇.

Proof. If 𝒩^𝑖_𝜆,𝜇 denotes the closure of 𝒩^𝑖_𝜆,𝜇, at first we note that 𝒩^𝑖_𝜆,𝜇=𝒩^𝑖_𝜆,𝜇∪𝜕𝒩^𝑖_𝜆,𝜇 for all 𝑖=1,2,…,𝑘. It then follows from Lemmas 14 and 16, that 𝜃^𝑖_𝜆,𝜇<𝜃^𝑖_𝜆,𝜇for𝑖=1,2,…,𝑘,0<𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)<Λ^∗. (119) Hence, 𝜃^𝑖_𝜆,𝜇=inf𝐼_𝜆,𝜇(𝑧)∣𝑧∈𝒩^𝑖_𝜆,𝜇for𝑖=1,2,…,𝑘. (120) Now, we fix 𝑖∈{1,2,…,𝑘}. Applying the Ekeland variational principle [26], there exists a minimizing sequence {𝑧^𝑖_𝑛}⊂𝒩^𝑖_𝜆,𝜇 such that 𝐼_𝜆,𝜇𝑧^𝑖_𝑛<𝜃^𝑖_𝜆,𝜇+1𝑛,𝐼_𝜆,𝜇𝑧^𝑖_𝑛≤𝐼_𝜆,𝜇𝜑+1𝑛𝜑−𝑧^𝑖_𝑛𝐸foreach𝜑∈𝒩^𝑖_𝜆,𝜇. (121) Using (119), we may assume that 𝑧^𝑖_𝑛∈𝒩^𝑖_𝜆,𝜇 for 𝑛 sufficiently large. Applying Lemma 17 with 𝑧=𝑧^𝑖_𝑛, we obtain the function 𝜉_𝑛∶𝐵_𝜖𝑛(0)→ℝ for some 𝜖_𝑛>0 such that 𝜉_𝑛(𝜑)(𝑧^𝑖_𝑛−𝜑)∈𝒩^𝑖_𝜆,𝜇 for all 𝜑∈𝐵_𝜖𝑛(0). Let 0<𝛿<𝜖_𝑛 and 𝑧∈𝐸⧵{0}; we set 𝜑_𝛿=𝛿𝑧‖𝑧‖_𝐸 (122) and 𝑧_𝛿=𝜉_𝑛(𝜑_𝛿)(𝑧^𝑖_𝑛−𝜑_𝛿). Since 𝑧_𝛿∈𝒩^𝑖_𝜆,𝜇, we deduce from (121) that 𝐼_𝜆,𝜇𝑧_𝛿−𝐼_𝜆,𝜇𝑧^𝑖_𝑛1≥−𝑛𝑧_𝛿−𝑧^𝑖_𝑛𝐸. (123) By the mean-value theorem, we obtain 𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛,𝑧_𝛿−𝑧^𝑖_𝑛+𝑜𝑧_𝛿−𝑧^𝑖_𝑛𝐸1≥−𝑛𝑧_𝛿−𝑧^𝑖_𝑛𝐸. (124) Therefore, 𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛,−𝜑_𝛿+𝜉_𝑛𝜑_𝛿−1𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛,𝑧^𝑖_𝑛−𝜑_𝛿1≥−𝑛𝑧_𝛿−𝑧^𝑖_𝑛𝐸+𝑜𝑧_𝛿−𝑧^𝑖_𝑛𝐸. (125) Now, we observe that 𝜉_𝑛(𝜑_𝛿)(𝑧^𝑖_𝑛−𝜑_𝛿)∈𝒩^𝑖_𝜆,𝜇, and consequently we get from (125) that −𝛿𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛,𝑧‖𝑧‖_𝐸+𝜉_𝑛𝜑_𝛿−1𝜉_𝑛𝜑_𝛿𝐼^′_𝜆,𝜇𝑧_𝛿,𝜉_𝑛𝜑_𝛿𝑧^𝑖_𝑛−𝜑_𝛿+𝜉_𝑛𝜑_𝛿−1𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛−𝐼^′_𝜆,𝜇𝑧_𝛿,𝑧^𝑖_𝑛−𝜑_𝛿1≥−𝑛𝑧_𝛿−𝑧^𝑖_𝑛𝐸+𝑜𝑧_𝛿−𝑧^𝑖_𝑛𝐸. (126) Then, we write the pervious inequality in the following form: 𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛,𝑧‖𝑧‖_𝐸≤𝑧_𝛿−𝑧^𝑖_𝑛𝐸+𝑜𝑧𝛿𝑛^𝛿−𝑧^𝑖_𝑛𝐸𝛿+𝜉_𝑛𝜑_𝛿−1𝛿𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛−𝐼^′_𝜆,𝜇𝑧_𝛿,𝑧^𝑖_𝑛−𝜑_𝛿. (127) We can find a constant 𝐶>0 independent of 𝛿 such that 𝑧_𝛿−𝑧^𝑖_𝑛≤𝛿+𝐶𝜉_𝑛𝜑_𝛿−1,lim_𝛿→0𝜉_𝑛𝜑_𝛿−1𝛿≤𝜉^′_𝑛(0)≤𝐶. (128) For a fixed 𝑛, let 𝛿→0 in (127). Using the fact that lim_𝛿→0𝑧_𝛿−𝑧^𝑖_𝑛𝐸=0, (129) we obtain 𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛,𝑧‖𝑧‖_𝐸≤𝐶𝑛. (130) This implies 𝐼_𝜆,𝜇𝑧^𝑖_𝑛=𝜃^𝑖_𝜆,𝜇+𝑜_𝑛(1),𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛=𝑜_𝑛(1)in𝐸⁻¹. (131)

Now, we complete the proof of Theorem 2. By Lemmas 12, 14 and Proposition 18, for all 𝜆^2/(2−𝑞)+𝜇^2/(2−𝑞)∈(0,Λ^∗), there exists a sequence {𝑧^𝑖_𝑛}⊂𝒩^𝑖_𝜆,𝜇 and 𝑧^𝑖₀=(𝑢^𝑖₀,𝑣^𝑖₀)∈𝐸, 1≤𝑖≤𝑘, such that 𝐼_𝜆,𝜇𝑧^𝑖_𝑛=𝜃^𝑖_𝜆,𝜇+𝑜_𝑛(1),𝐼^′_𝜆,𝜇𝑧^𝑖_𝑛=𝑜_𝑛(1)in𝐸⁻¹,𝑧^𝑖_𝑛⟶𝑧^𝑖₀stronglyin𝐸. (132) Moreover, {𝑧^𝑖_𝑛}⊂𝒩⁻_𝜆,𝜇, and by Lemma 7 (ii), we get 𝑧^𝑖₀∈𝒩⁻_𝜆,𝜇, 𝑢^𝑖₀≢0, 𝑣^𝑖₀≢0 in Ω, 𝑧^𝑖_0𝐸>2−𝑞2^∗−𝑞^1/(2∗−2)𝑆^𝑁/4,𝜃^𝑖_𝜆,𝜇≥𝜃⁻_𝜆,𝜇>0for𝑖=1,2,…,𝑘. (133) Thus, 𝑧^𝑖₀ is a nontrivial solution of the problem 𝑃_𝜆,𝜇 and 𝐼_𝜆,𝜇(𝑧^𝑖₀)=𝜃^𝑖_𝜆,𝜇 for 𝑖=1,2…,𝑘. Set 𝑢₊=max{𝑢,0} and 𝑣₊=max{𝑣,0}. Replace the terms ∫_Ωℎ(𝑥)|𝑢|^𝛼|𝑣|^𝛽𝑑𝑥 and ∫_Ω(𝜆𝑓(𝑥)|𝑢|^𝑞+𝜇𝑔(𝑥)|𝑣|^𝑞)𝑑𝑥 of the functional 𝐼_𝜆,𝜇 by ∫_Ωℎ(𝑥)𝑢^𝛼₊𝑣^𝛽₊𝑑𝑥 and ∫_Ω(𝜆𝑓(𝑥)𝑢^𝑞₊+𝜇𝑔(𝑥)𝑣^𝑞₊)𝑑𝑥, respectively. It then follows that 𝑧^𝑖₀ is a nonnegative solution of the problem 𝑃_𝜆,𝜇. Applying the maximum principle [19], 𝑧^𝑖₀ is a positive solution of the problem 𝑃_𝜆,𝜇. Since 𝑄_𝑖(𝑧^𝑖₀)<𝑟₀/3, 𝑧¹_𝜆,𝜇∈𝒩⁺_𝜆,𝜇,𝑧^𝑖₀∈𝒩^𝑖_𝜆,𝜇⊂𝒩⁻_𝜆,𝜇for𝑖=1,2,…,𝑘, (134) where 𝑧¹_𝜆,𝜇 is a positive solution of equation 𝑃_𝜆,𝜇 as in Theorem 1. From Lemma 13, we conclude that 𝒩^𝑖_𝜆,𝜇 are disjoint for 𝑖=1,2…,𝑘. This implies that 𝑧^𝑖₀(1≤𝑖≤𝑘) and 𝑧¹_𝜆,𝜇 are distinct positive solutions of the problem 𝑃_𝜆,𝜇.