Dirichlet and Neumann Problems Related to Nonlinear Elliptic Systems: Solvability, Multiple Solutions, Solutions with Positive Components

Abstract

We study the solvability of Dirichlet and Neumann problems for different classes of nonlinear elliptic systems depending on parameters and with nonmonotone operators, using existence theorems related to a general system of variational equations in a reflexive Banach space. We also point out some regularity properties and the sign of the found solutions components. We often prove the existence of at least two different solutions with positive components.

1. Introduction

In this paper, we present some significant applications of the results got in [1] to Dirichlet problems (Section 2) of the type: −di=v𝜆𝐴_𝑖𝑖𝑥,𝑢₁,…,𝑢_𝑛,∇𝑢₁,…,∇𝑢_𝑛𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝑑_𝑖𝑥,𝑢₁,𝑢…,𝑢_𝑛,∇𝑢₁,…,∇𝑢_𝑛+𝑓_𝑖inΩ,_𝑖=0on𝜕Ωas𝑖=1,…,𝑛,(1.1) and to Neumann problems (Section 3) of the type: −di=v𝜆𝐴_𝑖𝑖𝑥,𝑢₁,…,𝑢_𝑛,∇𝑢₁,…,∇𝑢_𝑛𝐴𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝑑_𝑖𝑥,𝑢₁,…,𝑢_𝑛,∇𝑢₁,…,∇𝑢_𝑛+𝑓_𝑖inΩ,_𝑖𝑥=,𝑢𝜇_1𝑖,…,𝑢_𝑛,∇𝑢₁,…,∇𝑢_𝑛⋅𝜈𝑏̂_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝑑_𝑖𝑥,𝑢₁,…,𝑢_𝑛,∇𝑢₁,…,∇𝑢_𝑛+𝑓_𝑖on𝜕Ωas𝑖=1,…,𝑛,(1.2) where 𝑛≥1,𝜆_𝑖,𝜇_𝑖 are real parameters, Ω is a bounded connected open set of 𝑅^𝑁 with regular boundary 𝜕Ω, and 𝜈 is the outward orthogonal unitary vector to 𝜕Ω.

The study deals with the solvability of the problems, the existence of multiple solutions with all the components not identically equal to zero and, in the homogeneous case, the existence of solutions with positive components, bounded and locally Hölderian with their first derivatives. It is suitable to recall the problem studied in [1] with some notations and hypotheses.

Let 𝑊₁,…,𝑊_𝑛 real reflexive Banach spaces (𝑛≥1). Let 𝑊 be the product space 𝑋^𝑛ℓ=1𝑊_ℓ. Let ‖⋅‖ be the norm on 𝑊, ‖⋅‖_∗ the norm on 𝑊^∗ (dual space of 𝑊), and ⟨⋅,⋅⟩_ℓ (resp.⟨⟨⋅,⋅⟩⟩) the duality between 𝑊^∗ℓ (dual space of 𝑊_ℓ) and 𝑊_ℓ (resp. 𝑊^∗ and 𝑊). Let us denote by “𝜕” Fréchet differential operator and by “𝜕_𝑢ℓ” Fréchet differential operator with respect to 𝑢_ℓ. Let 𝐴≢0 and 𝐷_𝑗≢0(𝑗=1,…,𝑚;𝑚≥1) be real functionals defined in 𝑊,𝐵_ℓ and 𝐵_ℓ(ℓ=1,…,𝑛) real functionals defined in 𝑊_ℓ satisfying the conditions:(𝑖₁₁)𝐴 is lower weakly semicontinuous in 𝑊 and 𝐶¹(𝑊⧵{0}),𝐵_ℓand𝐵_ℓ are weakly continuous in 𝑊_ℓ and𝐶¹(𝑊_ℓ),∃𝑝>1∶𝐴(𝑡𝑣)=𝑡^𝑝𝐴(𝑣)forall𝑡≥0and forall𝑣∈𝑊,𝐵_ℓ(𝑡𝑣_ℓ)=𝑡^𝑝𝐵_ℓ(𝑣_ℓ) and 𝐵_ℓ(𝑡𝑣_ℓ)=𝑡^𝑝𝐵_ℓ(𝑣_ℓ)forall𝑡≥0 andforall𝑣_ℓ∈𝑊_ℓ;(𝑖₁₂)𝐷_𝑗 is weakly continuous in 𝑊 and 𝐶¹(𝑊⧵{0}),∃𝑞_𝑗>1∶𝐷_𝑗(𝑡𝑣)=𝑡^𝑞𝑗𝐷_𝑗(𝑣)forall𝑡≥0 and forall𝑣∈𝑊,1<𝑞₁<⋯<𝑞_𝑚if𝑚>1.

Let𝐹=(𝐹₁,…,𝐹_𝑛)with 𝐹_ℓ∈𝑊_ℓ∗,𝜆_ℓand𝜇_ℓ∈𝑅; let us consider the following problem.

Problem (𝑃). Find 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑊⧵{0}such that 𝜕_𝑢𝑖𝐴(𝑢),𝑣_𝑖_𝑖=𝜆_𝑖𝜕𝐵_𝑖𝑢_𝑖,𝑣_𝑖_𝑖+𝜇_𝑖𝜕𝐵^𝑖𝑢_𝑖,𝑣_𝑖_𝑖+^𝑚_𝑗=1𝜕_𝑢𝑖𝐷_𝑗(𝑢),𝑣_𝑖_𝑖+⟨𝐹_𝑖,𝑣_𝑖⟩_𝑖∀𝑖∈{1,…,𝑛},∀𝑣_𝑖∈𝑊_𝑖.(1.3) Obviously Problem (P) means to find the critical points 𝑢∈𝑊⧵{0} of the Euler functional: 𝐸(𝑣)=𝐴(𝑣)−^𝑛_ℓ=1𝜆^ℓ𝐵_ℓ𝑣_ℓ+𝜇_ℓ𝐵_ℓ𝑣_ℓ−^𝑚_𝑗=1𝐷_𝑗(𝑣)−⟨⟨𝐹,𝑣⟩⟩∀𝑣=𝑣₁,…,𝑣_𝑛∈𝑊,(1.4) where ⟨⟨𝐹,𝑣⟩⟩=∑^𝑛ℓ=1⟨𝐹_ℓ,𝑣_ℓ⟩_ℓ.

Let us set 𝐻_𝜆𝜇(𝑣)=𝐴(𝑣)−^𝑛_ℓ=1𝜆^ℓ∀𝑣=𝑣₁𝐵_ℓ𝑣_ℓ+𝜇_ℓ𝐵_ℓ𝑣_ℓ𝑆_𝜆𝜇,…,𝑣_𝑛∈𝑊,∀𝜆=𝜆₁,…,𝜆_𝑛,𝜇=𝜇₁,…,𝜇_𝑛∈𝑅^𝑛,𝑉=𝑣∈𝑊∶𝐻_𝜆𝜇(𝑣)=1,𝑉^−𝜆𝜇=𝑣∈𝑊∶𝐻_𝜆𝜇(𝑣)<0,as𝑚₁=1,…,𝑚₊𝑆𝐷_𝑚1,…,𝐷_𝑚=𝑣∈𝑊∶^𝑚_𝑗=𝑚1𝐷_𝑗(𝑣)>0₊,𝑆𝐷𝐷₁,…,𝐷_𝑚=𝑣∈𝑊∶^𝑚_𝑗=1𝐷_𝑗(𝑣)=1_𝑗,=𝑣∈𝑊∶𝐷_𝑗(𝑣)=−1,𝑉⁺(𝐹)={𝑣∈𝑊∶⟨⟨𝐹,𝑣⟩⟩>0}.(1.5) About Problem (P), using Lagrange multipliers and the “fibering method,” different existence theorems have been proved in [1]. They base on one of the following hypotheses: (𝑖₁₃)∃𝑐(𝜆,𝜇)>0∶‖𝑣‖^𝑝≤𝑐(𝜆,𝜇)𝐻_𝜆𝜇(𝑣)forall𝑣∈𝑊; (𝑖₁₄)∃𝑐(𝜆,𝜇)>0∶‖𝑣‖^𝑝≤𝑐(𝜆,𝜇)𝐻_𝜆𝜇(𝑣)forall𝑣∈𝑉⁺(𝐷_𝑚)(if𝑉⁺(𝐷_𝑚)≠∅); (𝑖₁₅)∃𝑚₁∈{1,…,𝑚}∶𝑉^−𝜆𝜇∩𝑆(𝐷_𝑚1) is not empty and bounded in W.

Remark 1.1. In this paper, we use some existence theorems ([1], Theorems 2.1, 2.2, 3.1, and 3.2), in which as 𝑛>1, in relation to a set 𝔉⊆𝑆_𝜆𝜇, we suppose(𝑖^ℎ₁₆) for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝔉 with 𝑣_ℎ=0, there exist 𝑣_ℎ∈𝑊_ℎ⧵{0} and the real functions 𝜙₁,…,𝜙_𝑛 such that 𝜙_ℎ∈𝐶⁰([0,1])∩𝐶¹([0,1[) and 𝜙_ℎ(1)=0, 𝜙_ℓ∈𝐶¹([0,1]) and 𝜙_ℓ(1)=1 as ℓ≠ℎ, 𝑣(𝑠)=(𝜙₁(𝑠)𝑣₁,…,𝜙_ℎ(𝑠)𝑣_ℎ,…,𝜙_𝑛(𝑠)𝑣_𝑛)∈𝔉 for all 𝑠∈[𝑠₀,1](0≤𝑠₀<1), lim^{𝑠→1}₋(𝑑/𝑑𝑠)𝐷_𝑗(𝑣(𝑠))<+∞ for all 𝑗∈{1,…,𝑚}, lim_𝑠→1⁻(𝑑/𝑑𝑠)𝐷_𝑗(𝑣(𝑠))=−∞ for some 𝑗∈{1,…,𝑚}.
The condition (𝑖^ℎ₁₆) assures that for the solutions 𝑢=(𝑢₁,…,𝑢_𝑛) of Problem (P), found with the method used in the recalled theorems, we have 𝑢_ℎ≠0 if 𝐹_ℎ≡0.

Before showing Dirichlet problems (including the problem studied in [2] by Drábek and Pohozaev when 𝑛=1 and 𝑚=1) we give Propositions 2.2–2.6 which show some cases in which hypotheses (𝑖₁₃)−(𝑖₁₅) hold. These propositions are based on the comparison between the parameters 𝜆_𝑖 with suitable eigenvalues connected to 𝑝-Laplacian. About Neumann problems (including the one studied in [3] by Pohozaev and Véron when 𝑛=1) the same question is solved by Propositions 3.1–3.5 in which the parameters 𝜆_𝑖 and 𝜇_𝑖 have compared with zero. Finally, the results in Appendix are very useful: Propositions A.1 and A.2 in order to get condition (𝑖^ℎ₁₆), Propositions A.3 and A.4 to get qualitative properties of the solutions and the positive sign of the components of the found solutions.

2. Dirichlet Problems

Let Ω⊆𝑅^𝑁 be an open, bounded, connected and 𝐶^2,𝛽 set with 0<𝛽≤1. Let|⋅|_𝑁 the Lebesgue measure on 𝑅^𝑁,1<𝑝<∞,𝑝̃=𝑁𝑝/(𝑁−𝑝)if𝑁>𝑝,𝑝̃=∞otherwise.

Let us assume𝑊=𝑊^01,𝑝(Ω)^𝑛(𝑛≥1)with‖𝑣‖=^𝑛_ℓ=1_Ω𝐵||∇𝑣_ℓ||^𝑝𝑑𝑥^1/𝑝∀𝑣=𝑣₁,…,𝑣_𝑛∈𝑊,_ℓ𝑣_ℓ=𝑝⁻¹_Ω𝑏_ℓ||𝑣_ℓ||^𝑝𝑑𝑥∀𝑣_ℓ∈𝑊^01,𝑝(Ω)where𝑏_ℓ∈𝐿^∞(Ω)⧵{0},𝑏_ℓ≥0,𝐵_ℓ≡0.(2.1) Moreover we consider the functionals 𝐴 (as in (𝑖₁₁)) such that∃𝑐̃>0∶𝐴(𝑣)≥𝑝⁻¹𝑐̃‖𝑣‖^𝑝∀𝑣∈𝑊.(2.2) Let us use the notation 𝐻_𝜆(𝑆_𝜆and𝑉^−𝜆,resp.) instead of 𝐻_𝜆𝜇(𝑆_𝜆𝜇and𝑉^−𝜆𝜇,resp.).

As ℓ=1,…,𝑛 let 𝜆^∗ℓand𝑢^∗ℓ, respectively, the first eigenvalue and the first eigenfunction of the problem: 𝑢_ℓ∈𝑊^01,𝑝(Ω)∶−𝑐̃div||∇𝑢_ℓ||^𝑝−2∇𝑢_ℓ=𝜃𝑏^ℓ||𝑢_ℓ||^𝑝−2𝑢_ℓinΩ.(2.3) Let us remember that [4]𝑢^∗ℓ∈𝐶^1,𝛼ℓ(Ω) with 0<𝛼_ℓ<1,𝑢^∗ℓ>0 in Ω;𝜆^∗ℓ=𝑐̃∫_Ω|∇𝑢^∗ℓ|^𝑝𝑑𝑥/∫_Ω𝑏_ℓ|𝑢^∗ℓ|^𝑝𝑑𝑥=min{𝑐̃∫_Ω|∇𝑣_ℓ|^𝑝𝑑𝑥/∫_Ω𝑏_ℓ|𝑣_ℓ|^𝑝𝑑𝑥∶∫_Ω𝑏_ℓ|𝑣_ℓ|^𝑝𝑑𝑥>0};𝜆^∗ℓ is simple, that is, each eigenfunction of (2.3) related to 𝜆^∗ℓ is of the type 𝑐_ℓ𝑢^∗ℓ with 𝑐_ℓ∈𝑅⧵{0};𝜆^∗ℓ is isolate, that is, there exists 𝑎>0 such that 𝜆^∗ℓ is the only eigenvalue of (2.3) belonging to ]0,𝑎[.

Remark 2.1. About the results related to problem (2.3), it is sufficient to suppose 𝑏_ℓ∈𝐿^∞(Ω) and 𝑏^+ℓ=max{𝑏_ℓ,0}≢0 as ℓ=1,…,𝑛. This holds also for the results of this section if we limit to consider only the parameters 𝜆₁,…,𝜆_𝑛 nonnegative.

Let us start by presenting some sufficient conditions such that (𝑖₁₃),(𝑖₁₄), and (𝑖₁₅) hold.

Using the variational characterization of 𝜆^∗ℓ it is easy to verify the following proposition.

Proposition 2.2. If 𝜆_ℓ<𝜆^∗ℓforallℓ∈{1,…,𝑛}, then (𝑖₁₃) holds. Consequently, (𝑖₁₄) holds when 𝑉⁺(𝐷_𝑚)≠∅.

When 𝜆_ℓ≥𝜆^∗ℓ  for some ℓ∈{1,…,𝑛}, it is possible to fulfil (𝑖₁₄) with an additional condition on 𝐷_𝑚. Let 𝐼={1,…,𝑛}. For any 𝐼^∗⊆𝐼 let 𝑉^∗=𝑣𝑣_ℓ=𝑣₁,…,𝑣_𝑛∈𝑊∶𝑣_ℓ≡0ifℓ∈𝐼⧵𝐼^∗,=𝑐_ℓ𝑢^∗ℓifℓ∈𝐼^∗with𝑐_ℓ∈𝑅and𝑐_ℓ≠0forsomeℓ,(2.4) and let us suppose(𝑖₂₁) There exists 𝐼^∗⊆𝐼∶𝐷_𝑚(𝑣)<0forall𝑣∈𝑉^∗.

Proposition 2.3. Let (𝑖₂₁) holds with 𝐼^∗≠𝐼. Let 𝑉⁺(𝐷_𝑚)≠∅. If we fix the parameters set (𝜆_ℓ)_{ℓ∈𝐼⧵𝐼∗} with 𝜆_ℓ<𝜆^∗ℓ, then there exists 𝛿^∗>0 such that (𝑖₁₄) also holds for any (𝜆_ℓ)_{ℓ∈𝐼∗}∈𝑋_ℓ∈𝐼^∗[𝜆^∗ℓ,𝜆^∗ℓ+𝛿^∗[.

Proof. Arguing by contradiction, for any 𝑘∈ℕ there exist (𝜆^𝑘ℓ)_{ℓ∈𝐼∗}∈𝑋_ℓ∈𝐼^∗[𝜆^∗ℓ,𝜆^∗ℓ+𝑘⁻¹[ and 𝑣^𝑘=(𝑣^𝑘1,…,𝑣^𝑘𝑛)∈𝑉⁺(𝐷_𝑚) such that 𝐴𝑣^𝑘−𝑝⁻¹_ℓ∈_{𝐼⧵𝐼∗}𝜆_ℓ_Ω𝑏_ℓ||𝑣_ℓ||^𝑝𝑑𝑥−𝑝⁻¹_ℓ_∈𝐼∗𝜆^𝑘ℓ_Ω𝑏_ℓ||𝑣^𝑘ℓ||^𝑝𝑑𝑥<𝑘⁻¹‖‖𝑣^𝑘‖‖^𝑝.(2.5) Set 𝑤^𝑘=‖𝑣^𝑘‖⁻¹𝑣^𝑘, we have 𝑐̃_{ℓ∈𝐼⧵𝐼∗}𝐷_𝑚𝑤^𝑘>0,_Ω||∇𝑤^𝑘ℓ||^𝑝𝑑𝑥−_{ℓ∈𝐼⧵𝐼∗}𝜆_ℓ_Ω𝑏_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝑥+𝑐̃_{ℓ∈𝐼∗}_Ω||∇𝑤^𝑘ℓ||^𝑝𝑑𝑥−_{ℓ∈𝐼∗}𝜆^𝑘ℓ_Ω𝑏_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝑥<𝑝𝑘⁻¹,(2.6) moreover, since ‖𝑤^𝑘‖=1, there exists 𝑤∈𝑊 such that (within a subsequence) 𝑤^𝑘⟶𝑤weaklyin𝑊,𝑤^𝑘⟶𝑤stronglyin(𝐿^𝑝(Ω))^𝑛.(2.7) Taking into account that 𝐷_𝑚 is weakly continuous in 𝑊, from (2.6) as 𝑘→+∞ we get 𝐷_𝑚(𝑤)≥0,(2.8)_{ℓ∈𝐼⧵𝐼∗}𝑐̃_Ω||∇𝑤_ℓ||^𝑝𝑑𝑥−𝜆_ℓ_Ω𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥+_{ℓ∈𝐼∗}𝑐̃_Ω||∇𝑤_ℓ||^𝑝𝑑𝑥−𝜆^∗ℓ_Ω𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥≤0.(2.9) Since 𝑤_ℓ≢0⟹𝑐̃_Ω||∇𝑤_ℓ||^𝑝𝑑𝑥−𝜆_ℓ_Ω𝑐̃_Ω𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥>0,||∇𝑤_ℓ||^𝑝𝑑𝑥−𝜆^∗ℓ_Ω𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥≥0,(2.10) from (2.9), we deduce that 𝑤_ℓ≡0∀ℓ∈𝐼⧵𝐼^∗,∀ℓ∈𝐼^∗∃𝑐_ℓ∈𝑅∶𝑤_ℓ=𝑐_ℓ𝑢^∗ℓ.(2.11) Let us add that𝑐_ℓ≠0 for some ℓ∈𝐼^∗, since if 𝑐_ℓ=0forallℓ∈𝐼^∗ we have the contradiction 𝑐̃=𝑐̃lim_𝑘→+∞‖𝑤^𝑘‖^𝑝=0. Then 𝑤∈𝑉^∗, and consequently 𝐷_𝑚(𝑤)<0 from (𝑖₂₁). This last inequality contradicts (2.8).

In the same way the following propositions can be proved.

Proposition 2.4. Let (𝑖₂₁) holds with 𝐼^∗=𝐼. Let 𝑉⁺(𝐷_𝑚)≠∅. Then, there exists 𝛿^∗>0 such that (i₁₄) also holds for any (𝜆_ℓ)_ℓ∈𝐼∈𝑋_ℓ∈𝐼[𝜆^∗ℓ,𝜆^∗ℓ+𝛿^∗[.

Let us pass to (𝑖₁₅) and suppose(𝑖₂₂) there exist𝐼^∗⊆𝐼 and 𝑚₁∈{1,…,𝑚}such that𝐷_𝑚1(𝑣)<0and𝐴(𝑣)=𝑐̃𝑝⁻¹∑_{ℓ∈𝐼∗}∫_Ω|∇𝑣_ℓ|^𝑝𝑑𝑥 for any 𝑣∈𝑉^∗.

Proposition 2.5. If (i₂₂) holds with𝐼^∗≠𝐼, then 𝑉^−𝜆∩𝑆𝐷_𝑚1≠∅∀𝜆_ℓ_ℓ∈𝐼with𝜆_ℓ_ℓ∈𝐼^∗∈𝑋_{ℓ∈𝐼∗}𝜆^∗ℓ,+∞⧵𝜆^∗ℓ_ℓ∈𝐼^∗.(2.12) Moreover, if we fix the parameters set (𝜆_ℓ)_{ℓ∈𝐼⧵𝐼∗} with 𝜆_ℓ<𝜆^∗ℓ, then there exists 𝛿^∗>0 such that 𝑉^−𝜆∩𝑆𝐷_𝑚1isboundedin𝑊∀𝜆_ℓ_ℓ∈𝐼^∗∈𝑋_{ℓ∈𝐼∗}𝜆^∗ℓ,𝜆^∗ℓ+𝛿^∗⧵𝜆^∗ℓ_ℓ∈𝐼^∗.(2.13)

Proof. Let us prove (2.12). Let 𝑣∈𝑉^∗with 𝑣_ℓ=𝑢^∗ℓif ℓ∈𝐼^∗, then 𝐷_𝑚1(𝑣)<0. Let 𝑤=|𝐷_𝑚1(𝑣)|^{−1⧵𝑞𝑚1}𝑣, we have 𝐻𝐷_𝑚1(𝑤)=||𝐷_𝑚1(𝑣)||⁻¹𝐷_𝑚1(𝑣)=−1,_𝜆(𝑤)=𝑝⁻¹_ℓ_∈𝐼∗𝑐̃_Ω||∇𝑤_ℓ||^𝑝𝑑𝑥−𝜆_ℓ_Ω𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥<0.(2.14) Let us prove (2.13). Arguing by contradiction, for any 𝑘∈ℕ there exist (𝜆^𝑘ℓ)_{ℓ∈𝐼∗}∈𝑋_ℓ∈𝐼^∗[𝜆^∗ℓ,𝜆^∗ℓ+𝑘⁻¹[ with (𝜆^𝑘ℓ)_{ℓ∈𝐼∗}≠(𝜆^∗ℓ)_{ℓ∈𝐼∗} and (𝑣^𝑘,ℎ)_ℎ∈ℕ⊆𝑉^−𝜆_𝑘∩𝑆(𝐷_𝑚1), where 𝜆^𝑘ℓ=𝜆_ℓ if ℓ∈𝐼⧵𝐼^∗, such that sup_ℎ∈ℕ‖‖𝑣^𝑘,ℎ‖‖=+∞.(2.15) Relation (2.15) implies that there exists (ℎ_𝑘)_𝑘∈ℕ⊆ℕ strictly increasing such that 𝛿_𝑘=‖‖𝑣^𝑘,ℎ𝑘‖‖⟶+∞as𝑘⟶+∞.(2.16) Let 𝑤^𝑘=𝛿^𝑘−1𝑣^𝑘,ℎ𝑘, we have _ℓ∈_{𝐼⧵𝐼∗}𝑐̃_Ω||∇𝑤^𝑘ℓ||^𝑝𝑑𝑥−𝜆_ℓ_Ω𝑏_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝑥+_{ℓ∈𝐼∗}𝑐̃_Ω||∇𝑤^𝑘ℓ||^𝑝𝑑𝑥−𝜆^𝑘ℓ_Ω𝐷𝑏_ℓ||𝑤^𝑘||^𝑝𝑑𝑥<0,_𝑚1∃𝑤∈𝑊∶(withinasubsequence)𝑤_𝑘𝑤^𝑘=−𝛿^−𝑞𝑚1_𝑘,⟶𝑤weaklyin𝑊,𝑤^𝑘⟶𝑤stronglyin(𝐿^𝑝(Ω))^𝑛.(2.17) Then, as 𝑘→+∞ we get _ℓ∈_{𝐼⧵𝐼∗}𝑐̃_Ω||∇𝑤_ℓ||^𝑝𝑑𝑥−𝜆_ℓ_Ω𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥+_{ℓ∈𝐼∗}𝑐̃_Ω||∇𝑤_ℓ||^𝑝𝑑𝑥−𝜆^∗ℓ_Ω𝐷𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥≤0,(2.18)_𝑚1(𝑤)=0.(2.19) From (2.18), we get that 𝑤∈𝑉^∗. Then since (𝑖₂₂) inequality 𝐷_𝑚1(𝑤)<0 holds, which contradicts (2.19).

Proposition 2.6. If (𝑖₂₂) holds with 𝐼^∗=𝐼, then ∃𝛿_∗𝑉^−𝜆∩𝑆𝐷_𝑚1≠∅∀𝜆=𝜆_ℓ_ℓ∈𝐼∈𝑋_ℓ∈𝐼𝜆^∗ℓ,+∞⧵𝜆^∗ℓ_ℓ∈𝐼,>0∶𝑉^−𝜆∩𝑆𝐷_𝑚1isboundedin𝑊∀𝜆=𝜆_ℓ_ℓ∈𝐼∈𝑋_ℓ∈𝐼𝜆^∗ℓ,𝜆^∗ℓ+𝛿^∗⧵𝜆^∗ℓ_ℓ∈𝐼.(2.20)
The proof as in Proposition 2.5.

Remark 2.7. The applications we now show, except the first one, deal with systems with 𝑛>1 equations. We consider the functionals 𝐴 with 𝑐̃=1, and we suppose 𝑏_ℓ∈𝐿^∞(Ω)⧵{0},𝑏_ℓ≥0.

Application 2.8. Let 𝑛=1. Let us consider the problem −div||∇𝑢||^𝑝−2∇𝑢=𝜆₁𝑏₁|𝑢|^𝑝−2𝑢+^𝑚_𝑗=1𝑑_𝑗|𝑢|^𝑞𝑗−2𝑢inΩ,𝑢=0on𝜕Ω,(2.21) where 𝑝<𝑞₁𝑝<𝑞₁<𝑝,̃𝑑₁∈𝐿^∞(Ω)⧵{0}if𝑚=1,<⋯𝑑<𝑞_𝑚<𝑝,̃𝑑_𝑗∈𝐿^∞(Ω)⧵{0}as𝑗=1,…,𝑚,_𝑗≤0as𝑗=1,…,𝑚−1if𝑚>1.(2.22) Evidently 𝐴(𝑣)=𝑝⁻¹_Ω||∇𝑣||^𝑝𝑑𝑥,𝐷_𝑗(𝑣)=𝑞^𝑗−1_Ω𝑑_𝑗|𝑣|^𝑞𝑗𝑑𝑥∀𝑣∈𝑊.(2.23) Let us advance the conditions: 𝑑^+𝑚≢0⟹𝑉⁺𝐷_𝑚≠∅,(2.24)_Ω𝑑_𝑚𝑢^∗1^𝑞𝑚𝑑𝑥<0⟹𝐷_𝑚𝑐₁𝑢^∗1<0∀𝑐₁∈𝑅⧵{0}.(2.25) Let us note that (Propositions 2.2, 2.4, and 2.6) (2.24)and(2(.22.52)4⟹)⟹∃𝛿_∗1𝑖₁₄holdsif𝜆₁<𝜆^∗1,(2.25)⟹∃𝛿_∗2>0∶𝑖₁₄holdsif𝜆₁<𝜆^∗1+𝛿^∗1,>0∶𝑖₁₅holdsif𝜆₁∈𝜆^∗1,𝜆^∗1+𝛿^∗2.(2.26)

Proposition 2.9 (see [1], Theorems 2.1, 2.2, 4.1, and 4.2; Remarks 2.1, 2.3, 4.1, and 4.4; Proposition A.3; [5, 6]). Under assumptions (2.22) we have:(i)When (2.24) holds, with 𝜆₁<𝜆^∗1 [resp. (2.24) and (2.25) hold, with 𝜆₁<𝜆^∗1+𝛿^∗1] problem (2.21) has at least two weak solutions 𝑢⁰ and−𝑢⁰(𝑢⁰=𝜏⁰𝑣⁰,𝜏⁰=const.>0,𝑣⁰∈𝑆_𝜆1∩𝑉⁺(𝐷_𝑚)), and it results in 𝑢⁰∈𝐿^∞(Ω)∩𝐶^1,𝛼0_ℓ𝑜𝑐(Ω),𝑢⁰>0;(ii)When (2.25) holds, with 𝜆₁∈]𝜆^∗1,𝜆^∗1+𝛿^∗2[ problem (2.21) has at least two weak solutions 𝑢𝑎𝑛𝑑−𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆₁∩𝑆(𝐷_𝑚)), and it results in 𝑢∈𝐿^∞(Ω)∩𝐶^{1,𝛼ℓ𝑜𝑐}(Ω),𝑢>0.
Consequently, when (2.24) and (2.25) hold, with 𝜆₁∈]𝜆^∗1,𝜆^∗1+min{𝛿^∗1,𝛿^∗2}[ problem (2.21) has at least four different weak solutions.

Remark 2.10. Our results include the ones of Drábek and Pohozaev [2] when 𝑚=1.

Application 2.11. Let us consider the system: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+|||||^𝑛_ℓ=1𝑑_ℓ𝑢_ℓ|||||^𝑞1−2^𝑛_ℓ=1𝑑_ℓ𝑢_ℓ𝑑^𝑖𝑢−𝑑_𝑖||𝑢_𝑖||^𝑞1−2𝑢_𝑖inΩ,_𝑖=0on𝜕Ωas𝑖=1,…,𝑛,(2.27) where 1<𝑞₁<𝑝,̃𝑞₁≠𝑝,𝑑_ℓ,𝑑_ℓ∈𝐿^∞(Ω),𝑑_ℓ,𝑑_ℓ>0.(2.28) System (2.27) is included among Problem (P) with: 𝐴(𝑣)=𝑝^𝑛−1_ℓ=1_Ω𝐷||∇𝑣_ℓ||^𝑝𝑑𝑥,₁(𝑣)=𝑞¹⁻¹⎡⎢⎢⎣_Ω|||||_𝑛_ℓ=1𝑑_ℓ𝑣_ℓ|||||^𝑞1𝑑𝑥−^𝑛_ℓ=1_Ω𝑑_ℓ||𝑣_ℓ||^𝑞1𝑑𝑥⎤⎥⎥⎦∀𝑣=𝑣₁,…,𝑣_𝑛∈𝑊.(2.29) Let us advance the conditions (compatible): 𝑑^𝑞1_ℓ<𝑑_ℓ∀ℓ∈{1,…,𝑛}⟹𝐷₁0,…,𝑐_𝑖𝑢^∗𝑖,…,0<0as𝑖=1,…,𝑛,𝑐_𝑖∈𝑅⧵{0},(2.30) there exist Ω⁺⊆Ω and a constant 𝑐̃_𝑗>0 such that |Ω⁺|_𝑁>0 and _ℓ≠𝑗𝑑_ℓ+𝑐̃_𝑗𝑑_𝑗^𝑞1>_ℓ≠𝑗𝑑_ℓ+𝑐̃^𝑞1_𝑗𝑑_𝑗inΩ⁺⟹𝑉⁺𝐷₁≠∅(PropositionA.1).(2.31) Then (Propositions 2.2, 2.3, and 2.5) (2.31)⟹𝑖₁₄holdsif𝜆_ℓ<𝜆^∗ℓ∀ℓ∈{1,…,𝑛},(2.32) and set 𝑖∈{1,…,𝑛}(2.3(02).30an)d⟹(2.31w)it⟹h𝜆_ℓwith𝜆_ℓ<𝜆^∗ℓ∀ℓ≠𝑖∃𝛿^∗1>0∶𝑖₁₄holdsif𝜆_𝑖<𝜆^∗𝑖+𝛿^∗1,(2.33)<𝜆^∗ℓ∀ℓ≠𝑖∃𝛿^∗2>0∶𝑖₁₅holdsif𝜆_𝑖∈𝜆^∗𝑖,𝜆^∗𝑖+𝛿^∗2.(2.34)

Taking into account that 𝐷₁(𝑣₁,…,𝑣_𝑛)≤𝐷₁(|𝑣₁|,…,|𝑣_𝑛|) and 𝐷₁(−𝑣)=𝐷₁(𝑣), from ([1], Theorem 2.1, Remark 2.1, and Theorem 4.1) we get the following proposition.

Proposition 2.12. Under assumptions (2.28) we have:(i)When (2.31) holds, ((2.30) and (2.31) hold resp.), choosing 𝜆₁,…,𝜆_𝑛 as in (2.32) (resp. (2.33)) system (2.27) has at least two weak solutions 𝑢⁰ and −𝑢⁰ with 𝑢^0ℓ≥0 as ℓ=1,…,𝑛(𝑢⁰=𝜏⁰𝑣⁰,𝜏₀=const.>0,𝑣⁰∈𝑆_𝜆∩𝑉⁺(𝐷₁)); (ii)When (2.30) holds, choosing𝜆₁,…,𝜆_𝑛 as in (2.34) system (2.27) has at least two weak solutions 𝑢 and −𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆∩𝑆(𝐷₁)).
Consequently, when (2.30) and (2.31) hold, with 𝜆_ℓ<𝜆^∗ℓforallℓ≠𝑖 and 𝜆_𝑖∈]𝜆^∗𝑖,𝜆^∗𝑖+min{𝛿^∗1,𝛿^∗2}[ system (2.27) has at least four different weak solutions.

The following proposition is obvious.

Proposition 2.13. The following relations hold: ∃ℎ,𝑘∈𝑢^0𝑖{≢1,0…a,s𝑛}𝑖=∶1,…,𝑛,𝑢_ℎ≢0,𝑢_𝑘≢0.(2.35)

Proposition 2.14. If 𝑝<𝑞₁, then as 𝑖=1,…,𝑛: 𝑢^0𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼0𝑖_ℓ𝑜𝑐(Ω),𝑢^0𝑖>0.(2.36)

Proof. It is easy to prove that ^𝑛_𝑖=1_Ω||∇𝑢^0𝑖||^𝑝−2∇𝑢^0𝑖⋅∇𝑣_𝑖𝑑𝑥≤_Ω𝑔^𝑛_𝑖=1𝑢^0𝑖^𝑝−1^𝑛_𝑖=1𝑣_𝑖∀𝑣=𝑣₁𝑑𝑥,…,𝑣_𝑛∈𝑊^01,𝑝(Ω)∩𝐿^∞(Ω)^𝑛with𝑣_𝑖≥0,(2.37) where 𝑔∈𝐿^{𝑞1/(𝑞1−𝑝)}(Ω). Then (Proposition A.3) 𝑢^0𝑖∈𝐿^∞(Ω) and consequently [5] 𝑢^0𝑖∈𝐶^1,𝛼0𝑖_ℓoc(Ω).
Let us note that 𝑢^0𝑖 is a weak supersolution to the equation: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖−𝑑_𝑖||𝑢_𝑖||^𝑞1−2𝑢_𝑖inΩ.(2.38) Then, since (2.35), it must be [6] 𝑢^0𝑖>0.

Let us continue the analysis of system (2.27) under the condition: _ℓ≠𝑖𝑑_ℓ^𝑞1<min𝑑₁,…,𝑑_𝑛∀𝑖∈{1,…,𝑛},(2.39) then 𝐷₁𝑐₁𝑢^∗1,…,𝑐_𝑛𝑢^∗𝑛<0∀𝑐₁,…,𝑐_𝑛∈𝑅^𝑛⧵{0}with𝑐_𝑖=0foratleastone𝑖∈{1,…,𝑛}.(2.40) Hence (Proposition 2.5) if 𝐼^∗⊆𝐼 and 𝐼^∗≠𝐼: (2.39)⟹as𝜆_ℓ<𝜆^∗ℓ∀ℓ∈𝐼⧵𝐼^∗∃𝛿^∗>0∶𝑖₁₅holdsif𝜆_ℓ_ℓ∈𝐼^∗∈𝑋_{ℓ∈𝐼∗}𝜆^∗ℓ,𝜆^∗ℓ+𝛿^∗⧵𝜆^∗ℓ_ℓ∈𝐼^∗(2.4.1)

Proposition 2.15. Under assumptions (2.28) and (2.39), choosing 𝜆₁,…,𝜆_𝑛 as in (2.41) system (2.27) has at least two weak solutions 𝑢and−𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆∩𝑆(𝐷₁)) with 𝑢_𝑖≢0 as 𝑖=1,…,𝑛.

Proof. Thanks to ([1], Theorem 4.1), there exists 𝑣∈𝑉^−𝜆∩𝑆(𝐷₁) such that 𝐻_𝜆𝑣=inf𝐻_𝜆(𝑣)∶𝑣∈𝑉^−𝜆∩𝑆𝐷₁=𝑒,𝑢=𝜏𝑣isaweaksolutionofsystem(2.27),(2.42) where 𝜏=(−𝑝𝑞¹⁻¹𝑒)^{1/(𝑞1−𝑝)}.
Reasoning by contradiction, let, for example, 𝑢₁≡0. Since −1=𝐷₁(𝑣)≤𝐷₁(0,|𝑣₂|,…,|𝑣_𝑛|) and from (2.39) 𝐷₁(0,|𝑣₂|,…,|𝑣_𝑛|)<0, setting 𝛿=|𝐷₁(0,|𝑣₂|,…,|𝑣_𝑛|)|^−1/𝑞1 we have 𝐷₁0,𝛿||𝑣₂||,…,𝛿||𝑣_𝑛||=−1,𝐻_𝜆0,𝛿||𝑣₂||,…,𝛿||𝑣_𝑛||=𝛿^𝑝𝐻_𝜆𝑣≤𝐻_𝜆𝑣,(2.43) then 𝐻_𝜆(0,𝛿|𝑣₂|,…,𝛿|𝑣_𝑛|)=𝐻_𝜆(𝑣). This implies that ([1], see the proof of Theorem 4.1) (0,𝜏𝛿|𝑣₂|,…,𝜏𝛿|𝑣_𝑛|) is a weak solution of system (2.27). Then (∑^𝑛ℓ=2𝑑_ℓ|𝑣_ℓ|)^𝑞1−1≡0from which 𝑢_ℓ≡0too as ℓ=2,…,𝑛.
Condition (2.39) holds in particular when ^𝑛_ℓ=1𝑑_ℓ^𝑞1<min𝑑₁,…,𝑑_𝑛.(2.44)

Proposition 2.16. Replacing in Proposition 2.15 (2.39) with (2.44), it is right to say that 𝑢_𝑖≥0 and 𝑢_𝑖≢0 as 𝑖=1,…,𝑛. Consequently, if 𝑝<𝑞₁𝑢_𝑖∈𝐿^∞(Ω)∩𝐶^1,𝑎𝑖_ℓ𝑜𝑐(Ω),𝑢_𝑖>0as𝑖=1,…,𝑛.(2.45)

Proof. Set 𝛿=|𝐷₁(|𝑣₁|,…,|𝑣_𝑛|)|^−1/𝑞1, as in Proposition 2.15(𝜏𝛿|𝑣₁|,…,𝜏𝛿|𝑣_𝑛|) is a weak solution to system (2.27).
Let us add that since (2.44)⇒𝐷₁(𝑐₁𝑢^∗1,…,𝑐_𝑛𝑢^∗𝑛)<0forall(𝑐₁,…,𝑐_𝑛)∈𝑅^𝑛⧵{0}, there exists (Proposition 2.6) 𝛿^∗∗>0 such that 𝑖₁₅holdsif𝜆_ℓ_ℓ∈𝐼∈𝑋^𝑛_ℓ=1𝜆^∗ℓ,𝜆^∗ℓ+𝛿^∗∗⧵𝜆^∗ℓ_ℓ∈𝐼.(2.46) Then the existence of 𝑢 is assured also choosing 𝜆₁,…,𝜆_𝑛 as in (2.46), and the conclusions of Proposition 2.16 hold.

Application 2.17. Let us set 𝐴𝜆(𝑣₁)==⋯𝑝_𝑛−1=𝜆_𝑛=𝜆,𝑏₁=⋯=𝑏_𝑛=𝑏then𝜆^∗1=⋯=𝜆^∗𝑛=𝜆^∗,𝑢^∗1=⋯=𝑢^∗𝑛=𝑢^∗,_ℓ=1_Ω||∇𝑣_ℓ||^𝑝𝑑𝑥,𝐷₁(𝑣)=𝑞¹⁻¹_Ω𝑑₁^𝑛_ℓ=1||𝑣_ℓ||^𝛾^𝑞1/𝛾𝑑𝑥,∀𝑣=𝑣₁,…,𝑣_𝑛∈𝑊,(2.47) where 1<𝛾<𝑞₁<𝑝,̃𝑞₁≠𝑝,𝑑₁∈𝐿^∞(Ω).(2.48) Let us consider the system: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆𝑏||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝑑₁^𝑛_ℓ=1||𝑢_ℓ||^𝛾^{(𝑞1/𝛾)−1}𝑢||𝑢_𝑖||^𝛾−2𝑢_𝑖inΩ,_𝑖=0on𝜕Ωas𝑖=1,…,𝑛.(2.49) We advance the conditions 𝑑⁺¹≢0⟹𝑉⁺𝐷₁≠∅,(2.50)_Ω𝑑₁𝑢^∗^𝑞1𝑑𝑥<0⟹𝐷₁𝑐₁𝑢^∗,…,𝑐_𝑛𝑢^∗<0∀𝑐₁,…,𝑐_𝑛∈𝑅^𝑛⧵{0}.(2.51) Therefore, (2.50)and(2(.25.15)0⟹)⟹∃𝛿_∗1𝑖₁₄holdsif𝜆<𝜆^∗(Proposition2.2),(2.51)⟹∃𝛿_∗2>0∶𝑖₁₄holdsif𝜆<𝜆^∗+𝛿^∗1(Proposition2.4),>0∶𝑖₁₅holdsif𝜆∈𝜆^∗,𝜆^∗+𝛿^∗2(Proposition2.6).(2.52) Then ([1], Theorems 2.1 and 4.1, and Remarks 2.1 and 4.1).

Proposition 2.18. Under assumption (2.48), we have:(i)When (2.50) holds, ((2.50) and (2.51) hold resp.), if 𝜆<𝜆^∗(resp.𝜆<𝜆^∗+𝛿^∗1) system (2.49) has at least two weak solutions 𝑢⁰and −𝑢⁰ with 𝑢^0ℓ≥0 as ℓ=1,…,𝑛(𝑢⁰=𝜏⁰𝑣⁰,𝜏₀=const.>0,𝑣⁰∈𝑆_𝜆∩𝑉⁺(𝐷₁));(ii)When (2.51) holds,if𝜆∈]𝜆^∗,𝜆^∗+𝛿^∗2[system (2.49) has at least two weak solutions 𝑢 and −𝑢 with 𝑢_ℓ≥0 as ℓ=1,…,𝑛(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆∩𝑆(𝐷₁)).
Consequently, when (2.50) and (2.51) hold, with𝜆∈]𝜆^∗,𝜆^∗+min{𝛿^∗1,𝛿^∗2}[ system (2.49) has at least four different weak solutions.

In order to establish some properties of 𝑢⁰ and 𝑢 it is useful to recall that ([1], Theorems 2.1 and 4.1)𝐷𝐻₁𝑣⁰=sup𝐷₁(𝑣)∶𝑣∈𝑆_𝜆∩𝑉⁺𝐷₁=𝑒,𝜏⁰=𝑞₁𝑝⁻¹𝑒^{1/(𝑝−𝑞1)},(2.53)_𝜆𝑣=inf𝐻_𝜆(𝑣)∶𝑣∈𝑉^−𝜆∩𝑆𝐷₁=𝑒,𝜏=−𝑝𝑞¹⁻¹𝑒^{1/(𝑞1−𝑝)}.(2.54)

Proposition 2.19. When 𝑝<𝑞₁, we have 𝑢^0𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼0𝑖_ℓ𝑜𝑐(Ω),(2.55) besides 𝑢^0𝑖≢0∀𝑖∈{1,…,𝑛}if𝛾<𝑝.(2.56)

Proof. The relation 𝑢^0𝑖∈𝐿^∞(Ω) comes from Proposition A.3. Then [5] 𝑢^0𝑖∈𝐶^1,𝛼0𝑖_ℓoc(Ω).
About (2.56), it is sufficiently (Remark 1.1) to prove that 𝑖^ℎ16holds∀ℎ∈{1,…,𝑛}with𝔉=𝑆_𝜆∩𝑉⁺𝐷₁.(2.57)
Let 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑆_𝜆∩𝑉⁺(𝐷₁) with 𝑣_ℎ≡0. Since 𝑣∈𝑉⁺𝐷₁⟹∃acompactset𝕂⊆Ω∶||𝕂||_𝑁>0,𝑑₁>0and𝜓=_ℓ≠ℎ||𝑣_ℓ||^𝛾>0in𝕂,(2.58) let (Proposition A.1) (𝜑_𝜀)_0<𝜀<𝜀0⊆𝐶^∞0(Ω) with 0≤𝜑_𝜀≤1 such that 𝜑_𝜀⟶𝜒stronglyin𝐿^𝑠(Ω),_Ω||∇𝜑_𝜀||^𝑠𝑑𝑥⟶+∞as𝜀⟶0⁺∀𝑠∈[1,+∞[,(2.59) where 𝜒 is the characteristic function of 𝕂. Set 𝜀 such that _Ω𝑑₁𝜓^{(𝑞1/𝛾)−1}𝜑^𝛾𝜀𝑑𝑥>0,𝛿=𝑝⁻¹_Ω||∇𝜑_𝜀||^𝑝𝑑𝑥−𝜆_Ω𝑏𝜑^𝑝𝜀𝑑𝑥>0,(2.60) with 𝑣(𝑠)=(𝑠^1/𝑝𝑣₁,…,(1−𝑠)^1/𝑝𝛿^−1/𝑝𝜑_𝜀,…,𝑠^1/𝑝𝑣_𝑛) it results in 𝐻_𝜆(𝑣(𝑠))=𝛿⁻¹(1−𝑠)𝑝⁻¹_Ω∃𝑠₀||∇𝜑_𝜀||^𝑝𝑑𝑥−𝜆_Ω𝑏𝜑^𝑝𝜀𝑑𝑥+𝑠𝐻_𝜆(𝑣)=1∀𝑠∈[0,1],∈[0,1[∶𝐷₁(𝑣(𝑠))>0∀𝑠∈𝑠₀,1,lim_𝑠→1−𝑑𝑑𝑠𝐷¹(𝑣(𝑠))=−∞.(2.61)

Proposition 2.20. When 𝑝<𝑞₁, we have 𝑢_𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼𝑖_ℓ𝑜𝑐(Ω),(2.62)𝑢_𝑖>0∀𝑖∈{1,…,𝑛}if𝑝<𝛾.(2.63)

Proof. We can get (2.62) from Proposition A.3 and [5].
About (2.63), it is sufficiently [6] to prove that 𝑢_𝑖≢0 as 𝑖=1,…,𝑛. Reasoning by contradiction, let, for example, 𝑣₁≡0. We note that 𝑣∈𝑉^−𝜆⟹∃ℓ∈{2,…,𝑛}∶_Ω||∇𝑣_ℓ||^𝑝𝑑𝑥−𝜆_Ω𝑏𝑣^𝑝ℓ𝑑𝑥<0.(2.64) Let us suppose ℓ=2 and set 𝑣(𝑠)=((1−𝑠)^1/𝛾𝑣₂,𝑠^1/𝛾𝑣₂,𝑣₃,…,𝑣_𝑛). Then 𝐷₁(𝑣(𝑠))=−1∀𝑠∈[0,1li]m,_𝑠→1−∃𝑠₀∈[0,1[∶𝐻_𝜆(𝑣(𝑠))<0∀𝑠∈𝑠₀,1,𝑑𝑑𝑠𝐻^𝜆(𝑣(𝑠))=+∞.(2.65) Set 𝑠₁∈[𝑠₀,1[ such that (𝑑/𝑑𝑠)𝐻_𝜆(𝑣(𝑠))>0forall𝑠∈[𝑠₁,1[ and taking into account (2.54), we get the contradiction: 𝐻_𝜆𝑣≤𝐻_𝜆(𝑣(𝑠))<𝐻_𝜆𝑣∀𝑠∈𝑠₁,1.(2.66)

Proposition 2.21. When 𝛾=𝑝<𝑞₁, we allow that as 𝑖=1,…,𝑛: 𝑢^0𝑖>0,𝑢_𝑖>0.(2.67)

Proof. The assumption 𝛾=𝑝 implies that ∃𝑣̃=𝑣₁̃∀𝑣=𝑣₁,…,𝑣_𝑛∈𝑊⧵{0}with𝑣_ℎ≡0forsomeℎ∈{1,…,𝑛},,…,𝑣_𝑛̃∈𝑊∶𝑣_ℓ̃≢0asℓ=1,…,𝑛,𝐻_𝜆(𝑣)̃=𝐻_𝜆(𝑣),𝐷₁(𝑣)̃=𝐷₁(𝑣).(2.68) Let, for example, 𝑣₁≡0 and 𝑣₂≢0. Set 𝑠∈]0,1[ and 𝑣¹¹=(1−𝑠)^1/𝑝𝑣₂,𝑣¹²=𝑠^1/𝑝𝑣₂,𝑣^1ℓ=𝑣_ℓ asℓ>2, with 𝑣¹=(𝑣¹¹,…,𝑣^1𝑛), we have 𝐻_𝜆𝑣¹=𝐻_𝜆(𝑣),𝐷₁𝑣¹=𝐷₁(𝑣).(2.69) If 𝑣₃≡0, set 𝑣²¹=(1−𝑠)^1/𝑝𝑣¹¹,𝑣²³=𝑠^1/𝑝𝑣¹¹,𝑣^2ℓ=𝑣^1ℓ as ℓ∈{1,…,𝑛}⧵{1,3}, with𝑣²=(𝑣²¹,…,𝑣^2𝑛), it results in 𝐻_𝜆𝑣²=𝐻_𝜆(𝑣),𝐷₁𝑣²=𝐷₁(𝑣).(2.70) This method let us to find 𝑣̃.
Then, if 𝑣^0ℎ≡0(resp.𝑣_ℎ≡0) for some ℎ∈{1,…,𝑛}, with 𝑣⁰̃(resp.𝑣̃) as in (2.68) we have from (2.53) (resp. (2.54)) 𝐷₁(𝑣⁰̃)=𝑒(resp.𝐻_𝜆(𝑣̃)=𝑒). Consequently ([1], see the proof of Theorem 2.1 (resp. Theorem 4.1)) 𝑢⁰̃=𝜏⁰𝑣⁰̃(resp.𝑢̃=𝜏𝑣̃) is a weak solution of system (2.49). Therefore [6] 𝑢^0𝑖̃>0(resp.𝑢̃_𝑖>0) as 𝑖=1,…,𝑛.

Application 2.22. Let us assume 𝜆_ℓ,𝑏_ℓ, and 𝐴 as in Application 2.17, 𝐷_𝑗(𝑣)=𝑞^𝑗−1_Ω𝑑_𝑗^𝑛_ℓ=1||𝑣_ℓ||^𝛾𝑗^{𝑞𝑗/𝛾𝑗}𝑑𝑥∀𝑣=𝑣₁,…,𝑣_𝑛∈𝑊as𝑗=1,…,𝑚,(2.71) where 𝑝<𝑑𝑞₁<⋯<𝑞_𝑚<𝑝,̃1<𝛾_𝑗<𝑞_𝑗,𝑑_𝑚∈𝐿^∞(Ω),_𝑗∈𝐿^∞(Ω)⧵{0},𝑑_𝑗≤0if𝑗=1,…,𝑚−1.(2.72) Let us consider the system: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆𝑏||𝑢_𝑖||^𝑝−2𝑢_𝑖+^𝑚_𝑗=1𝑑_𝑗^𝑛_ℓ=1||𝑢_ℓ||^𝛾𝑗^{(𝑞𝑗/𝛾𝑗)−1}𝑢||𝑢_𝑖||^𝛾𝑗−2𝑢_𝑖inΩ,_𝑖=0on𝜕Ωas𝑖=1,…,𝑛,(2.73) under almost one of the conditions: 𝑑^+𝑚≢0,_Ω𝑑_𝑚𝑢^∗^𝑞𝑚𝑑𝑥<0.(2.74) By using some results ([1], Theorems 2.2 and 4.2, and Remarks 2.3 and 4.4), we can advance a proposition similar to Proposition 2.18 replacing in particular 𝑉⁺(𝐷₁) with 𝑉⁺(𝐷_𝑚) and 𝑆(𝐷₁) with𝑆(𝐷_𝑚).
Thanks to Proposition A.3 and a result of [5], for the solutions 𝑢⁰ and 𝑢 to system (2.73), we have 𝑢^0𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼0𝑖_ℓoc(Ω),𝑢_𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼𝑖_ℓoc(Ω).(2.75) We continue to analyze the properties of 𝑢⁰ and 𝑢. To this aim we recall that ([1], Theorems 2.2 and 4.2), set for each 𝑣∈𝑉⁺(𝐷_𝑚)(resp.𝑣∈𝑉^−𝜆∩𝑆(𝐷_𝑚))𝜓(𝑡,𝑣)=𝑝𝑡^𝑝−1𝐻_𝜆(𝑣)−∑^𝑚𝑗=1𝑞_𝑗𝑡^𝑞𝑗−1𝐷_𝑗(𝑣), we have: ∃∣𝑡(𝑣)>0∶𝜓(𝑡(𝑣),𝑣)=0,𝜕𝜓𝜕𝑡(𝑡(𝑣),𝑣)≠0.(2.76) Besides with 𝐸(𝑣)=(𝑡(𝑣))_𝑝𝐻_𝜆(𝑣)−∑^𝑚𝑗=1(𝑡(𝑣))^𝑞𝑗𝐷_𝑗(𝑣), it results in 𝐸𝑣₀𝐸=inf𝐸(𝑣)∶𝑣∈𝑆_𝜆∩𝑉⁺𝐷_𝑚,𝜏⁰=𝑡𝑣⁰,(2.77)𝑣=inf𝐸(𝑣)∶𝑣∈𝑉^−𝜆∩𝑆𝐷_𝑚,𝜏=𝑡𝑣.(2.78)

Proposition 2.23. When 𝛾_𝑚<𝑝≤𝛾_𝑗 as 𝑗=1,…,𝑚−1, then 𝑢^0𝑖≢0∀𝑖∈{1,…,𝑛}.(2.79)

Proof. It is sufficiently (Remark 1.1) to prove that 𝑖^ℎ16holds∀ℎ∈{1,…,𝑛}with𝔉=𝑆_𝜆∩𝑉⁺𝐷_𝑚.(2.80) Let 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑆_𝜆∩𝑉⁺(𝐷_𝑚) with 𝑣_ℎ≡0. As in Proposition 2.19, it is possible to find 𝑣_ℎ∈𝐶^∞0(Ω)⧵{0} such that with 𝑣(𝑠)=(𝑠^1/𝑝𝑣₁,…,(1−𝑠)^1/𝑝𝑣_ℎ,…,𝑠^1/𝑝𝑣_𝑛), it results in 𝐻l_𝜆im(𝑣_𝑠→1−(𝑠))=1∀𝑠∈[0,1],𝐷_𝑚(𝑣(𝑠))>0∀𝑠∈𝑠₀,10≤𝑠₀<1,𝑑𝑑𝑠𝐷^𝑗(𝑣(𝑠))∈𝑅as𝑗=1,…,𝑚−1,lim_𝑠→1−𝑑𝑑𝑠𝐷^𝑚(𝑣(𝑠))=−∞.(2.81)

Proposition 2.24. When 𝑝<𝛾_𝑚≤𝛾_𝑗 as 𝑗=1,…,𝑚−1, then 𝑢_𝑖>0∀𝑖∈{1,…,𝑛}.(2.82)

Proof. It is sufficiently [6] to prove that 𝑢_𝑖≢0forall𝑖∈{1,…,𝑛}. Reasoning by contradiction, let, for example, 𝑣₁≡0and𝑣₂≢0 such that _Ω||∇𝑣₂||^𝑝𝑑𝑥−𝜆_Ω𝑏𝑣^𝑝2𝑑𝑥<0.(2.83) Since 𝑡𝑣>0,𝜓𝑡𝑣,𝑣=0,𝜕𝜓𝜕𝑡𝑡𝑣,𝑣≠0,(2.84) there exist an open ball 𝐵of𝑊 with centre 𝑣 included in 𝑉^−𝜆 and a unique functional 𝑡^∗(𝑣) belongs to 𝐶¹(𝐵) such that 𝑡^∗(𝑣)>0,𝜓𝑡^∗(𝑣),𝑣=0∀𝑣∈𝐵.(2.85) Then, the functional 𝐸^∗(𝑣)=𝑡^∗(𝑣)^𝑝𝐻_𝜆(𝑣)−^𝑚_𝑗=1𝑡^∗(𝑣)^𝑞𝑗𝐷_𝑗(𝑣)∀𝑣∈𝐵(2.86) belongs to 𝐶¹(𝐵), and we have 𝑡(𝑣)=𝑡^∗(𝑣)∀𝑣∈𝐵∩𝑆𝐷_𝑚.(2.87) Then, for (2.78) 𝐸^∗𝑣=inf𝐸^∗(𝑣)∶𝑣∈𝐵∩𝑆𝐷_𝑚.(2.88) Now, let us remark that with 𝑣(𝑠)=((1−𝑠)^1/𝛾𝑚𝑣₂,𝑠^1/𝛾𝑚𝑣₂,𝑣₃,…,𝑣_𝑛), it results in 𝐷li_𝑚m(_𝑠→1−𝑣(𝑠))=−1∀𝑠∈[0,1],∃𝑠₀∈[0,1[∶𝑣(𝑠)∈𝐵∀𝑠∈𝑠₀,1,𝑑𝑑𝑠𝐻^𝜆(𝑣(𝑠))=+∞,lim_𝑠→1−𝑑𝑑𝑠𝐷^𝑗(𝑣(𝑠))∈𝑅as𝑗=1,…,𝑚−1.(2.89) Then, since 𝑑𝑑𝑠𝐸^∗(𝑣(𝑠))=𝑡^∗(𝑣(𝑠))^𝑝𝑑𝑑𝑠𝐻^𝜆(𝑣(𝑠))−^𝑚_𝑗=1𝑡^∗(𝑣(𝑠))^𝑞𝑗𝑑𝑑𝑠𝐷^𝑗(𝑣(𝑠))∀𝑠∈𝑠₀,1,(2.90) we havelim_𝑠→1⁻(𝑑/𝑑𝑠)𝐸^∗(𝑣(𝑠))=+∞. Consequently, ∃𝑠₁∈𝑠₀,1∶𝑑𝑑𝑠𝐸^∗(𝑣(𝑠))>0∀𝑠∈𝑠₁,1,(2.91) from which we get the contradiction: 𝐸^∗𝑣≤𝐸^∗(𝑣(𝑠))<𝐸^∗𝑣∀𝑠∈𝑠₁,1.(2.92)

Proposition 2.25. When 𝑝=𝛾₁=⋯=𝛾_𝑚, we allow that 𝑢^0𝑖>0,𝑢_𝑖>0∀𝑖∈{1,…,𝑛}.(2.93)

Proof. We reason as in Proposition 2.21, taking into account (2.77) and (2.78) ([1], see proofs of Theorems 2.2 and 4.2).

Application 2.26. Let for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝^𝑛−1_ℓ=1_Ω||∇𝑣_ℓ||^𝑝𝑑𝑥,𝐷_𝑗(𝑣)=−^𝑛_ℓ=1_Ω𝐷||𝑣_ℓ||^𝑞𝑗ℓ𝑑𝑥as𝑗=1,…,𝑚−1(𝑚≥2),_𝑚(𝑣)=𝑞^𝑚−1⎡⎢⎢⎣_Ω^𝑛_ℓ=1𝑑_ℓ||𝑣_ℓ||^𝛾^{𝑞𝑚/𝛾}𝑑𝑥−^𝑛_ℓ=1_Ω𝑑_ℓ||𝑣_ℓ||^𝑞𝑚𝑑𝑥⎤⎥⎥⎦,(2.94) where 1<𝛾<𝑝≤𝑞_𝑗ℓ,^𝑛_ℓ=1𝑑_ℓ𝑞_𝑗ℓ=𝑞_𝑗<𝑞_𝑚<𝑝,̃𝑞₁<⋯<𝑞_𝑚−1,,𝑑_ℓ∈𝐿^∞(Ω),𝑑_ℓ,𝑑_ℓ>0.(2.95) Let us consider the system: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖−^𝑚−1_𝑗=1𝑞^𝑗𝑖_ℓ≠𝑖_Ω||𝑢_ℓ||^𝑞𝑗ℓ𝑑𝑥||𝑢_𝑖||^{𝑞𝑗𝑖−2}𝑢_𝑖+^𝑛_ℓ=1𝑑_ℓ||𝑢_ℓ||^𝛾^{(𝑞𝑚/𝛾)−1}𝑢𝑑_𝑖||𝑢_𝑖||^𝛾−2𝑢_𝑖−𝑑_𝑖||𝑢_𝑖||^𝑞𝑚−2𝑢_𝑖inΩ,_𝑖=0on𝜕Ωas𝑖=1,…,𝑛.(2.96) Let us introduce the conditions: ∃Ω⁺⊆Ω∶||Ω⁺||_𝑁>0𝑑,𝑑^{𝑞𝑚/𝛾}_ℓ>𝑑_ℓinΩ⁺forsomeℓ∈{1,…,𝑛−1}⟹𝑉⁺𝐷_𝑚≠∅,(2.97)_{𝑞𝑚𝑛/𝛾}<𝑑_𝑛⟹𝐷_𝑚0,…,0,𝑐_𝑛𝑢^∗𝑛<0∀𝑐_𝑛∈𝑅⧵{0}.(2.98) Then (Propositions 2.2, 2.3 and 2.5) (2.97)and(2.98)⟹w(2i.t9h7𝜆)_ℓ⟹with𝜆_ℓ<𝜆^∗ℓ∀ℓ∈{1,…,𝑛}𝑖₁₄holds,(2.99)(2.98)⟹with𝜆_ℓ<𝜆^∗ℓ∀ℓ∈{1,…,𝑛−1}∃𝛿^∗1>0∶𝑖₁₄holdsif𝜆_𝑛<𝜆^∗𝑛+𝛿^∗1,(2.100)<𝜆^∗ℓ∀ℓ∈{1,…,𝑛−1}∃𝛿^∗2>0∶𝑖₁₅holdsif𝜆_𝑛∈𝜆^∗𝑛,𝜆^∗𝑛+𝛿^∗2.(2.101) Since ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4), we get the following proposition.

Proposition 2.27. Under assumption (2.95) we have:(i)When (2.97) holds ((2.97) and (2.98) hold, resp.), set 𝜆₁,…,𝜆_𝑛as in (2.99) (resp. (2.100)) system (2.96) has at least two weak solutions 𝑢⁰ and −𝑢⁰ with 𝑢^0ℓ≥0 as ℓ=1,…,𝑛(𝑢⁰=𝜏⁰𝑣⁰,𝜏⁰=const.>0,𝑣⁰∈𝑆_𝜆∩𝑉⁺(𝐷_𝑚));(ii)When (2.98) holds, set 𝜆₁,…,𝜆_𝑛 as in (2.101) system (2.96) has at least two weak solutions 𝑢and −𝑢 with 𝑢_ℓ≥0 as ℓ=1,…,𝑛(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆∩𝑆(𝐷_𝑚)).
Consequently, when (2.97) and (2.98) hold, with 𝜆_ℓ<𝜆^∗ℓforallℓ∈{1,…,𝑛−1}and 𝜆_𝑛∈]𝜆^∗𝑛,𝜆^∗𝑛+min{𝛿^∗1,𝛿^∗2}[ system (2.96) has at least four different weak solutions.

We remark that (Proposition A.3, [5]) as 𝑖=1,…,𝑛:𝑢^0𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼0𝑖_ℓ𝑜𝑐(Ω),𝑢_𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼𝑖_ℓ𝑜𝑐(Ω).(2.102) Moreover, since 𝑢^0𝑖(resp.𝑢_𝑖) is a weak supersolution of the equation:−div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖−^𝑚−1_𝑗=1𝑎_𝑗𝑞_𝑗𝑖||𝑢_𝑖||^{𝑞𝑗𝑖−2}𝑢_𝑖−𝑑_𝑖||𝑢_𝑖||^𝑞𝑚−2𝑢_𝑖inΩ,(2.103) where 𝑎_𝑗=∏_ℓ≠𝑖∫_Ω(𝑢^0ℓ)^𝑞𝑗ℓ𝑑𝑥(resp.𝑎_𝑗=∏_ℓ≠𝑖∫_Ω(𝑢_ℓ)^𝑞𝑗ℓ𝑑𝑥), we have [6] 𝑢^0𝑖>0if𝑢^0𝑖≢0resp.𝑢_𝑖>0if𝑢_𝑖≢0.(2.104)

Proposition 2.28. It results in u^0𝑖>0as𝑖=1,…,𝑛,𝑢_ℓ>0.(2.105)

Proof. Since (2.104), we must show that 𝑢^0𝑖≢0as𝑖=1,…,𝑛,(2.106)𝑢_ℓ≢0.(2.107) About (2.106), it is sufficient (Remark 1.1) to prove that 𝑖^ℎ16holds∀ℎ∈{1,…,𝑛}with𝔉=𝑆_𝜆∩𝑉⁺𝐷_𝑚.(2.108) Let 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑆_𝜆∩𝑉⁺(𝐷_𝑚) with 𝑣_ℎ≡0. Let 𝕂⊆Ω be a compact set such that ||𝕂||_𝑁>0,𝜓=_ℓ≠ℎ𝑑_ℓ||𝑣_ℓ||^𝛾>0in𝕂.(2.109) From Proposition A.1, there exists 𝜑∈𝐶^∞0(Ω), with 0≤𝜑≤1, such that 𝛿=𝑝⁻¹_Ω||∇𝜑||^𝑝𝑑𝑥−𝜆_ℎ_Ω𝑏_ℎ𝜑^𝑝𝑑𝑥>0,_Ω𝜓^{𝑞𝑚/𝛾}𝑑_ℎ𝜑^𝑝𝑑𝑥>0.(2.110) Then, with 𝑣(𝑠)=(𝑠^1/𝑝𝑣₁,…,(1−𝑠)^1/𝑝𝛿^−1/𝑝𝜑,…,𝑠^1/𝑝𝑣_𝑛), we have li𝐻m_{𝜆𝑠→1−}(𝑣(𝑠))=1∀𝑠∈[0,1],∃𝑠₀∈[0,1[∶𝐷_𝑚(𝑣(𝑠))>0∀𝑠∈𝑠₀,1,𝑑𝑑𝑠𝐷^𝑗(𝑣(𝑠))∈]−∞,0]as𝑗=1,…,𝑚−1,lim_𝑠→1−𝑑𝑑𝑠𝐷^𝑚(𝑣(𝑠))=−∞.(2.111)
Let us prove (2.107). We recall that ([1], Theorem 4.2): 𝐸𝑣=inf𝐸(𝑣)∶𝑣∈𝑉^−𝜆∩𝑆𝐷_𝑚,(2.112) where 𝐸 as in Application 2.22. Reasoning by contradiction, let 𝑣_ℓ≡0. Then, 𝑣_ℓ≢0 for some ℓ≠ℓ and consequently from (2.104) ∑_ℓ≠ℓ𝑑_ℓ(𝑣_ℓ)^𝛾>0.
Let 𝜑∈𝐶^∞0(Ω), with 0≤𝜑≤1, such that ∫_Ω𝑑^{𝑞𝑚/𝛾}_ℓ𝜑^𝑞𝑚𝑑𝑥>∫_Ω𝑑_ℓ𝜑^𝑞𝑚𝑑𝑥. Let us consider the function: 𝑔(𝑠,𝜏)==𝐷𝑞_𝑚𝑚−1𝜏𝑣₁,…,𝑠𝜑,…,𝜏𝑣_𝑛⎡⎢⎢⎣_Ω⎛⎜⎜⎝𝑠_𝛾𝑑_ℓ𝜑^𝛾+𝜏^𝛾_ℓ_≠ℓ𝑑_ℓ𝑣_ℓ^𝛾⎞⎟⎟⎠^{𝑞𝑚/𝛾}𝑑𝑥−𝑠^𝑞𝑚_Ω𝑑_ℓ𝜑^𝑞𝑚𝑑𝑥−𝜏^𝑞𝑚_ℓ_≠ℓ_Ω∀𝑠≥0,∀𝑑𝜏_ℓ≥𝑣1_ℓ.^𝑞𝑚𝑑𝑥⎤⎥⎥⎦(2.113) Since 𝑔(0,1)=−1,𝜕𝑔𝜕𝑠(𝑠,𝜏)>0∀𝑠>0,∀𝜏≥1,𝑔(0,𝜏)=−𝜏^𝑞𝑚lim_𝑠→+∞<−1∀𝜏>1,𝑔(𝑠,𝜏)=+∞∀𝜏≥1,(2.114) we have ∀𝜏≥1∃∣𝑠(𝜏)≥0(𝑠(1)=0,𝑠(𝜏)>0for𝜏>1)∶𝑔(𝑠(𝜏),𝜏)=−1.(2.115) We note that lim_𝜏→1⁺𝑠(𝜏)=0. In fact, if {𝜏_𝑛}⊆]1,+∞[andlim𝜏_𝑛=1, being 𝑔(𝑠(𝜏_𝑛),𝜏_𝑛)=−1, {𝑠(𝜏_𝑛)} is bounded (else (within a subsequence) lim𝑔(𝑠(𝜏_𝑛),𝜏_𝑛)=+∞). Then (within a subsequence) lim𝑠(𝜏_𝑛)=𝜔 with 𝑔(𝜔,1)=0, from which 𝜔=0.
We add that 𝑠(𝜏) belongs to 𝐶¹(]1,+∞[), and its derivative has the form: 𝑠^(𝜏)=−1(𝑠(𝜏))^𝛾−1𝑔̃(𝑠(𝜏),𝜏)∀𝜏>1withlim_𝜏→1+𝑔̃(𝑠(𝜏),𝜏)∈]−∞,0[.(2.116) Hence, set 𝑣(𝜏)=(𝜏𝑣₁,…,𝑠(𝜏)𝜑,…,𝜏𝑣_𝑛), it results in 𝐷_𝑚(𝑣(𝜏))=−1∀𝜏≥1,lim_𝜏→1+𝑑𝑑𝜏𝐻^𝜆lim_𝜏→1+(𝑣(𝜏))=𝑝𝐻_𝜆𝑣<0,𝑑𝑑𝜏𝐷^𝑗(𝑣(𝜏))=0as𝑗=1,…,𝑚−1.(2.117)
As in Proposition 2.24, we introduce the open ball 𝐵 with centre 𝑣 included in 𝑉^−𝜆 and the functionals 𝑡^∗(𝑣) and 𝐸^∗(𝑣) belonging to 𝐶¹(𝐵). Chosen 𝜏₀>1 such that 𝑣(𝜏)∈𝐵forall𝜏∈[1,𝜏₀], we have 𝑑𝑑𝜏𝐸^∗(𝑣(𝜏))=𝑡^∗(𝑣(𝜏))^𝑝𝑑𝑑𝜏𝐻^𝜆(𝑣(𝜏))−^𝑚−1_𝑗=1𝑡^∗(𝑣(𝜏))^𝑞𝑗𝑑𝑑𝜏𝐷^𝑗(𝑣(𝜏))∀𝜏∈1,𝜏₀,(2.118) and consequently lim_𝜏→1⁺(𝑑/𝑑𝜏)𝐸^∗(𝑣(𝜏))<0. Then, taking into account (2.112), with 𝜏₁∈]1,𝜏₀] such that (𝑑/𝑑𝜏)𝐸^∗(𝑣(𝜏))<0forall𝜏∈]1,𝜏₁], we get the contradiction: 𝐸^∗𝑣≤𝐸^∗(𝑣(𝜏))<𝐸^∗𝑣∀𝜏∈1,𝜏₁.(2.119)

Proposition 2.29. If 𝑑^{𝑞𝑚ℓ/𝛾}>𝑑_ℓasℓ=1,…,𝑛−1, then 𝑢_ℓ>0asℓ=1,…,𝑛.(2.120)

Proof. In fact, 𝑢_ℓ>0asℓ=1,…,𝑛−1(Proposition2.23),𝑢_𝑛≡0⟹𝐷_𝑚𝑢>0.(2.121)

Application 2.30. Let for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝⁻¹_Ω^𝑛_ℓ=1||∇𝑣_ℓ||^𝛾^𝑝/𝛾𝑑𝑥+^𝑛_ℓ=1_Ω𝐷||∇𝑣_ℓ||^𝑝ℓ𝑑𝑥,_𝑗(𝑣)=_Ω𝜌_𝑗^𝑛_ℓ=1||𝑣_ℓ||^𝑞𝑗ℓ𝐷𝑑𝑥as𝑗=1,…,𝑚−1,_𝑚(𝑣)=𝑞^𝑚−1⎡⎢⎢⎣_Ω|||||_𝑛_ℓ=1𝑑_ℓ𝑣_ℓ|||||^𝑞𝑚−1^𝑛_ℓ=1𝑑_ℓ𝑣_ℓ𝑑𝑥−_Ω𝑑||𝑣_𝑛||^𝑞𝑚𝑑𝑥⎤⎥⎥⎦,(2.122) where 1<𝛾<𝑝,𝑝_ℓ>1,^𝑛_ℓ=1𝑝_ℓ=𝑝,𝑞_𝑗ℓ>1,^𝑛_ℓ=1𝜌𝑞_𝑗ℓ=𝑞_𝑗,𝑝<𝑞_𝑚,𝑞₁<⋯<𝑞_𝑚<𝑝,_𝑗̃∈𝐿^∞(Ω)⧵{0},𝜌_𝑗≤0,𝑑_ℓ,𝑑∈𝐿^∞(Ω),𝑑_ℓ(𝑥)≠0a.e.inΩ,𝑑>0.(2.123) Let as ℓ=1,…,𝑛𝐹_ℓ∈𝑊^−1,𝑝′(Ω)(𝑝^=𝑝/(𝑝−1)). Let ⟨⟨𝐹,𝑣⟩⟩=∑^𝑛ℓ=1⟨𝐹_ℓ,𝑣_ℓ⟩forall𝑣∈𝑊. Set 𝜂_𝑖=0 as 𝑖=1,…,𝑛−1 and 𝜂_𝑛=1, let us consider the system: −div⎛⎜⎜⎝⎡⎢⎢⎣^𝑛_ℓ=1||∇𝑢_ℓ||^𝛾^{(𝑝/𝛾)−1}||∇𝑢_𝑖||^𝛾−2+𝑝_𝑖_ℓ≠𝑖_Ω||∇𝑢_ℓ||^𝑝ℓ||∇𝑢^𝑖||^𝑝𝑖−2=𝜆_𝑖⎤⎥⎥⎦∇𝑢_𝑖⎞⎟⎟⎠𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+^𝑚−1_𝑗=1𝑞_𝑗𝑖𝜌_𝑗_ℓ≠𝑖||𝑢_ℓ||^𝑞𝑗ℓ||𝑢^𝑖||^{𝑞𝑗𝑖−2}𝑢_𝑖+|||||^𝑛_ℓ=1𝑑_ℓ𝑢_ℓ|||||^𝑞𝑚−1𝑢𝑑_𝑖−𝜂_𝑖𝑑||𝑢_𝑛||^𝑞𝑚−2𝑢_𝑛+𝐹_𝑖inΩ,_𝑖=0on𝜕Ωas𝑖=1,…,𝑛,(2.124) under at least one of the following conditions ∃Ω⁺⊆Ω∶||Ω⁺||_𝑁>0||𝑑,𝑑_ℓ>0inΩ⁺forsomeℓ∈{1,…,𝑛−1}⟹𝑉⁺𝐷_𝑚≠∅,(2.125)_𝑛||^𝑞𝑚<𝑑⟹𝐷_𝑚0,…,0,𝑐_𝑛𝑢^∗𝑛<0∀𝑐_𝑛∈𝑅⧵{0}.(2.126) Evidently, about the validity of (𝑖₁₄) we choose 𝜆₁,…,𝜆_𝑛 as in Application 2.26.

Proposition 2.31 (see [1], Theorem 3.2). Under assumptions (2.123), (2.125) ((2.125) and (2.126), resp.), if 𝐹≢0 and ‖𝐹‖_∗ is sufficiently small, for 𝜆₁,…,𝜆_𝑛 as in (2.99) (resp. (2.100)) system (2.124) has at least one weak solution 𝑢̃(𝑢̃=𝜏̃𝑣̃, 𝜏̃=const.>0,𝑣̃∈𝑆_𝜆∩𝑉⁺(𝐷_𝑚)).

Let us note that 𝑢_ℎ̃≢0evenif𝐹_ℎ≡0since𝐹_ℎ≡0,𝑢_ℎ̃≡0⟹^𝑛_ℓ=1𝑑_ℓ𝑢_ℓ̃≡0⟹𝐷_𝑚(𝑢)̃≤0.(2.127)

Application 2.32. Let 𝜆₁=⋯=𝜆_𝑛=0, and for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝⁻¹_Ω^𝑛_ℓ=1||∇𝑣_ℓ𝐷||^𝛾+𝑎||𝑣_ℓ||^𝛾^𝑝/𝛾𝑑𝑥,_𝑗(𝑣)=𝑞^𝑗−1_Ω𝑑_𝑗^𝑛_ℓ=1||𝑣_ℓ||^𝛾𝑗^{𝑞𝑗/𝛾𝑗}𝑑𝑥as𝑗=1,…,𝑚,with𝑚>2,(2.128) under one of the following assumptions: 𝑎∈𝐿1^∞<(Ω𝛾)_𝑗,𝑎≥0,𝑑_𝑗∈𝐿^∞(Ω)⧵{0}with𝑑₁≤0,𝑑_𝑗≥0as𝑗≥2,𝑎∈𝐿_∞<𝛾<𝑝<𝑞₂<⋯<𝑞_𝑚<𝑝̃as𝑗≥2,𝛾≤𝛾₁<𝑞₁<𝑞₂;(2.129)(Ω),𝑎1≥<0,𝛾_𝑗as𝑗=1,…,𝑚𝑑_𝑗∈𝐿^∞(Ω)⧵{0},𝑑_𝑗≥0,<𝛾<𝑝<𝑞₁<⋯<𝑞_𝑚<𝑝.̃(2.130) Set 𝐹 as in Application 2.30. Let us consider the system: −div⎛⎜⎜⎝^𝑛_ℓ=1||∇𝑢_ℓ||^𝛾+𝑎||𝑢_ℓ||^𝛾^{(𝑝/𝛾)−1}||∇𝑢_𝑖||^𝛾−2∇𝑢_𝑖=−_𝑛⎞⎟⎟⎠_ℓ=1||∇𝑢_ℓ||^𝛾+𝑎||𝑢_ℓ||^𝛾^{(𝑝/𝛾)−1}𝑎||𝑢_𝑖||^𝛾−2𝑢_𝑖+^𝑚_𝑗=1𝑑_𝑗^𝑛_ℓ=1||𝑢_ℓ||^𝛾𝑗^{(𝑞𝑗/𝛾𝑗)−1}𝑢||𝑢_𝑖||^𝛾𝑗−2𝑢_𝑖+𝐹_𝑖inΩ,_𝑖=0on𝜕Ωas𝑖=1,…,𝑛.(2.131) Let us verify that (2.129)resp.(2.130)⟹𝑖^ℎ16rehsopl.d𝔉s∀=ℎ𝑆∈_𝜆{1,…,𝑛}with𝔉=𝑆_𝜆∩𝑉⁺𝐷₂,…,𝐷_𝑚∩𝑉⁺𝐷₁,…,𝐷_𝑚.(2.132) Let 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝔉 with, for example, 𝑣₁≡0. Let 𝑗₀∈{2,…,𝑚}(resp.𝑗₀∈{1,…,𝑚}) and ℓ₀∈{2,…,𝑚} such that 𝑑_𝑗0𝑣_ℓ0≢0. Let us suppose ℓ₀=2 and set 𝑣(𝑠)=((1−𝑠)^1/𝛾𝑣₂,𝑠^1/𝛾𝑣₂,𝑣₃,…,𝑣_𝑛). Then, 𝐴(𝑣(𝑠)li)m=_𝑠→1−1∀𝑠∈[0,1],∃𝑠₀∈[0,1[∶𝐷_𝑗0(𝑣(𝑠))>0∀𝑠∈𝑠₀,1,𝑑𝑑𝑠𝐷^𝑗0(𝑣(𝑠))=−∞,lim^{𝑠→1}₋𝑑𝑑𝑠𝐷^𝑗(𝑣(𝑠))<+∞as𝑗≠𝑗₀.(2.133)

Proposition 2.33. Under assumption (2.129) (resp. (2.130)), system (2.131) with 𝐹≡0 has at least two weak solutions 𝑢⁰ and −𝑢⁰, and we have as 𝑖=1,…,𝑛: 𝑢^0𝑖∈𝐿^∞(Ω),𝑢^0𝑖≥0,𝑢^0𝑖≢0.(2.134) Consequently, 𝑎≡0⟹𝑢^0𝑖∈𝐶^1,𝛼0𝑖_ℓ𝑜𝑐(Ω),𝑎≡0and(2.129)holdswith𝑝≤𝛾₁resp.(2.130)holds⟹𝑢^0𝑖>0.(2.135)

Proof. The statement is due to ([1], Theorem 2.2, Remark 2.3), [5], Proposition A.3, [6].

Proposition 2.34 (see [1], Theorems 3.1, 3.2). Under assumption (2.129) (resp. (2.130)), system (2.131) with 𝐹≢0 and ‖𝐹‖_∗ sufficiently small has at least two different weak solutions 𝑉𝑢¹₊and𝑢²(𝑢^𝑖=𝜏^𝑖𝑣^𝑖,𝜏^𝑖=const.>0,𝑣¹∈𝑉⁺(𝐹)∩𝑆_𝜆,𝑣²∈𝑆_𝜆∩𝑉⁺(𝐷₂,…,𝐷_𝑚)[resp.𝑣²∈𝑆_𝜆∩(𝐷₁,…,𝐷_𝑚)]), and we have 𝑢^2ℎ≢0 even if 𝐹_ℎ≡0.

Remark 2.35. If ⋃^𝑚𝑗=2{𝑥∈Ω∶𝑑_𝑗(𝑥)>0}[resp.⋃^𝑚𝑗=1{𝑥∈Ω∶𝑑_𝑗(𝑥)>0}]=Ω (within a set with measure equal to zero), with the same reasoning used about (2.132), we get that 𝑖^ℎ16holds∀ℎ∈{1,…,𝑛}with𝔉=𝑉⁺(𝐹)∩𝑆_𝜆,(2.136) hence, 𝑢^1ℎ≢0 even if 𝐹_ℎ≡0.

3. Neumann Problems

Let Ω⊆𝑅^𝑁 be an open, bounded, and connected 𝐶^0,1 set. Let |⋅|_𝑁,𝑝and𝑝̃ as in Section 2, 𝜎 the measure on 𝜕Ω,𝜈 the outward unit normal to 𝜕Ω,𝑝̂=(𝑁−1)𝑝/(𝑁−𝑝) if 𝑝<𝑁,𝑝̂=∞ if 𝑝≥𝑁. Let us assume𝑊=𝑊^1,𝑝(Ω)^𝑛(𝑛≥1)with‖𝑣‖=^𝑛_ℓ=1_Ω||∇𝑣_ℓ𝐵||^𝑝+||𝑣_ℓ||^𝑝𝑑𝑥^1/𝑝∀𝑣=𝑣₁,…,𝑣_𝑛∈𝑊,_ℓ𝐵𝑣_ℓ=𝑝⁻¹_Ω𝑏_ℓ||𝑣_ℓ||^𝑝𝑑𝑥∀𝑣_ℓ∈𝑊^1,𝑝(Ω),where𝑏_ℓ∈𝐿^∞(Ω)⧵{0},_ℓ𝑣_ℓ=𝑝⁻¹_𝜕Ω𝑏̂_ℓ||𝑣_ℓ||^𝑝𝑑𝜎∀𝑣_ℓ∈𝑊^1,𝑝(Ω),where𝑏̂_ℓ∈𝐿^∞(𝜕Ω)⧵{0}.(3.1) We note that for each 𝑣_ℓ∈𝑊^1,𝑝(Ω) we set 𝛾₀(𝑣_ℓ)=𝑣_ℓ where 𝛾₀ is the trace operator from 𝑊^1,𝑝(Ω) into 𝑊^{1−(1/𝑝),𝑝}(𝜕Ω). Morever we consider the functionals 𝐴 (as in (𝑖₁₁)) such that∃𝑐̃>0∶𝐴(𝑣)≥𝑝⁻¹𝑐̃^𝑛_ℓ=1_Ω||∇𝑣_ℓ||^𝑝𝑑𝑥∀𝑣∈𝑊.(3.2) It is easy to verify the following.

Proposition 3.1. Let 𝑏_ℓ,𝑏̂_ℓ≥0 as ℓ=1,…,𝑛. Then, 𝑖₁₃holdsif𝜆_ℓ,𝜇_ℓ≤0,𝜆_ℓ+𝜇_ℓ<0asℓ=1,…,𝑛.(3.3)

Let us set 𝐼={1,…,𝑛} and for each 𝐼^∗⊆𝐼𝐶^∗=𝑐=𝑐₁,…,𝑐_𝑛∈𝑅^𝑛∶𝑐_ℓ=0ifℓ∈𝐼⧵𝐼^∗,𝑐_ℓ≠0forsomeℓ∈𝐼^∗.(3.4) Let us introduce the conditions:(𝑖₃₁)there exists 𝐼^∗⊆𝐼∶𝐷_𝑚(𝑐)<0forall𝑐∈𝐶^∗;(𝑖₃₂)there exist 𝐼^∗⊆𝐼 and 𝑚₁∈{1,…,𝑚}∶𝐷_𝑚1(𝑐)<0 and 𝐴(𝑐)=0forall𝑐∈𝐶^∗.

Proposition 3.2. Let (𝑖₃₁) holds with 𝐼^∗≠𝐼. Let 𝑉⁺(𝐷_𝑚)≠∅. Let 𝑏_ℓ,𝑏̂_ℓ≥0 as ℓ∈𝐼⧵𝐼^∗. Then with 𝜆_ℓ,𝜇_ℓ≤0 and 𝜆_ℓ+𝜇_ℓ<0 as ℓ∈𝐼⧵𝐼^∗∃𝛿^∗>0: (𝑖₁₄) holds if|𝜆_ℓ|,|𝜇_ℓ|≤𝛿^∗ as ℓ∈𝐼^∗.

Proof. Reasoning by contradiction, for each 𝑘∈ℕ there exist 𝜆^𝑘ℓ,𝜇^𝑘ℓ∈[−𝑘⁻¹,𝑘⁻¹], with ℓ∈𝐼^∗, and 𝑣^𝑘=(𝑣^𝑘1,…,𝑣^𝑘𝑛)∈𝑉⁺(𝐷_𝑚) such that ‖‖𝑣^𝑘‖‖^𝑝>𝑘⎧⎪⎨⎪⎩𝐴𝑣^𝑘−_{ℓ∈𝐼⧵𝐼∗}𝑝⁻¹𝜆^ℓ_Ω𝑏_ℓ||𝑣^𝑘ℓ||^𝑝𝑑𝑥+𝜇_ℓ_𝜕Ω−𝑏̂_ℓ||𝑣^𝑘ℓ||^𝑝𝑑𝜎_{ℓ∈𝐼∗}𝑝⁻¹𝜆^𝑘ℓ_Ω𝑏_ℓ||𝑣^𝑘ℓ||^𝑝𝑑𝑥+𝜇^𝑘ℓ_𝜕Ω𝑏̂_ℓ||𝑣^𝑘ℓ||^𝑝𝑑𝜎,(3.5) then, set 𝑤^𝑘=‖𝑣^𝑘‖⁻¹𝑣^𝑘, we have 𝐷_𝑚𝑤^𝑘>0,𝑝⁻¹⎧⎪⎨⎪⎩𝑐̃^𝑛_ℓ=1_Ω||∇𝑤^𝑘ℓ||^𝑝𝑑𝑥−_{ℓ∈𝐼⧵𝐼∗}𝜆^ℓ_Ω𝑏_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝑥+𝜇_ℓ_𝜕Ω<𝑘₋₁𝑏̂_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝜎⎫⎪⎬⎪⎭+_{ℓ∈𝐼∗}𝑝⁻¹𝜆^𝑘ℓ_Ω𝑏_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝑥+𝜇^𝑘ℓ_𝜕Ω𝑏̂_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝜎.(3.6) Since ‖𝑤^𝑘‖=1, there exists 𝑤∈𝑊 such that (within a subsequence) 𝑤^𝑘⟶𝑤weaklyin𝑊,𝑤^𝑘⟶𝑤stronglyin(𝐿^𝑝(Ω))^𝑛,𝑤^𝑘⟶𝑤stronglyin(𝐿^𝑝(𝜕Ω))^𝑛.(3.7) Consequently, from (3.6), passing to limit as 𝑘→+∞, we get 𝐷_𝑚(𝑤)≥0,^𝑛_ℓ=1_Ω||∇𝑤_ℓ||^𝑝𝑑𝑥=0,_{ℓ∈𝐼⧵𝐼∗}𝜆^ℓ_Ω𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥+𝜇_ℓ_𝜕Ω𝑏̂_ℓ||𝑤_ℓ||^𝑝𝑑𝜎=0,(3.8) from which 𝑤=0, and then the contradiction 0=lim_𝑘→+∞‖𝑤^𝑘‖=1.

Proposition 3.3. Let (𝑖₃₁) holds with 𝐼^∗=𝐼. Let 𝑉⁺(𝐷_𝑚)≠∅. Then, ∃𝛿^∗>0∶𝑖₁₄holdsif||𝜆_ℓ||,||𝜇_ℓ||≤𝛿^∗asℓ=1,…,𝑛.(3.9)

The proof as in Proposition 3.2.

Proposition 3.4. Let (𝑖₃₂) holds with 𝐼^∗≠𝐼. Let ∫_Ω𝑏_ℓ𝑑𝑥,∫_𝜕Ω𝑏̂_ℓ𝑑𝜎>0 as ℓ∈𝐼^∗. Then, 𝑉^−𝜆𝜇∩𝑆𝐷_𝑚1≠∅∀𝜆_ℓ,𝜇_ℓ_ℓ∈𝐼with𝜆_ℓ,𝜇_ℓ≥0∀ℓ∈𝐼^∗,𝜆_ℓ+𝜇_ℓ>0forsomeℓ∈𝐼^∗.(3.10) Moreover, if 𝑏_ℓ,𝑏̂_ℓ≥0 as ℓ∈𝐼⧵𝐼^∗, we have with 𝜆_ℓ,𝜇_ℓ≤0 and 𝜆_ℓ+𝜇_ℓ<0 as ℓ∈𝐼⧵𝐼^∗∃𝛿^∗>0∶(𝑖₁₅) holds if 𝜆_ℓ,𝜇_ℓ∈[0,𝛿^∗]∀ℓ∈𝐼^∗ and 𝜆_ℓ+𝜇_ℓ>0 for some ℓ∈𝐼^∗.

Proof. The first statement is evident. Let us prove the second one. Reasoning by contradiction, for each 𝑘∈ℕ there exist 𝜆^𝑘ℓ,𝜇^𝑘ℓ∈[0,𝑘⁻¹], with ℓ∈𝐼^∗ and 𝜆^𝑘ℓ+𝜇^𝑘ℓ>0 for some ℓ∈𝐼^∗, and a sequence (𝑣^𝑘,ℎ)_ℎ∈ℕ such that 𝑣^𝑘,ℎ_ℎ∈ℕ⊆𝑉^−𝜆_𝑘𝜇𝑘∩𝑆𝐷_𝑚1𝜆^𝑘ℓ=𝜆_ℓ,𝜇^𝑘ℓ=𝜇_ℓasℓ∈𝐼⧵𝐼^∗,sup_ℎ‖‖𝑣^𝑘,ℎ‖‖=+∞.(3.11)
Let {ℎ_𝑘}⊆ℕbe a strictly increasing sequence such that ‖𝑣^𝑘,ℎ𝑘‖→+∞ as 𝑘→+∞. Let 𝑤^𝑘=‖𝑣^𝑘,ℎ𝑘‖⁻¹𝑣^𝑘,ℎ𝑘. Then, 𝐷_𝑚1(𝑤^𝑘)=−‖𝑣^𝑘,ℎ𝑘‖^−𝑞𝑚1 and 𝑝⁻¹⎧⎪⎨⎪⎩𝑐̃^𝑛_ℓ=1_Ω||∇𝑤^𝑘ℓ||^𝑝𝑑𝑥−_{ℓ∈𝐼⧵𝐼∗}𝜆^ℓ_Ω𝑏_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝑥+𝜇_ℓ_𝜕Ω<𝑝₋₁𝑏̂_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝜎⎫⎪⎬⎪⎭_ℓ_∈𝐼∗𝜆^𝑘ℓ_Ω𝑏_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝑥+𝜇^𝑘ℓ_𝜕Ω𝑏̂_ℓ||𝑤^𝑘ℓ||^𝑝𝑑𝜎,(3.12) moreover, there exists 𝑤∈𝑊 such that (within a subsequence) 𝑤^𝑘⟶𝑤we𝑤ak_𝑘lyin𝑊,𝑤^𝑘⟶𝑤stronglyin(𝐿^𝑝(Ω))^𝑛,⟶𝑤stronglyin(𝐿^𝑝(𝜕Ω))^𝑛.(3.13) Consequently, 𝐷_𝑚1(𝑤)=0,^𝑛_ℓ=1_Ω||∇𝑤_ℓ||^𝑝𝑑𝑥=0,_{ℓ∈𝐼⧵𝐼∗}𝜆^ℓ_Ω𝑏_ℓ||𝑤_ℓ||^𝑝𝑑𝑥+𝜇_ℓ_𝜕Ω𝑏̂_ℓ||𝑤_ℓ||^𝑝𝑑𝜎=0,(3.14) then 𝑤=0, and the contradiction 0=lim_𝑘→+∞‖𝑤^𝑘‖=1.

Proposition 3.5. Let (𝑖₃₂) holds with 𝐼^∗=𝐼. Let ∫_Ω𝑏_ℓ𝑑𝑥,∫_𝜕Ω𝑏̂_ℓ𝑑𝜎>0 as ℓ=1,…,𝑛. Then, ∃𝛿_∗𝑉^−𝜆𝜇∩𝑆𝐷_𝑚1≠∅if𝜆_ℓ,𝜇_ℓ≥0∀ℓ∈𝐼and𝜆_ℓ+𝜇_ℓ>0forsomeℓ∈𝐼,>0∶𝑖₁₅holdsif𝜆_ℓ,𝜇_ℓ∈0,𝛿^∗∀ℓ∈𝐼and𝜆_ℓ+𝜇_ℓ>0forsomeℓ∈𝐼.(3.15)

The proof as in Proposition 3.4.

Remark 3.6. It is suitable to make some clarifications.(i)The assumption “𝑏_ℓ,𝑏̂_ℓ≥0” (see Propositions 3.1, 3.2, and 3.4) can be replaced by “𝑏_ℓ,𝑏̂_ℓ do not change sign.” In this case we can choose 𝜆_ℓ and 𝜇_ℓ such that 𝜆_ℓ𝑏_ℓ≤0,𝜇_ℓ𝑏̂_ℓ≤0 and |𝜆_ℓ|+|𝜇_ℓ|>0.(ii)The assumption “∫_Ω𝑏_ℓ𝑑𝑥,∫_𝜕Ω𝑏̂_ℓ𝑑𝜎>0” (see Propositions 3.4 and 3.5) can be replaced by “∫_Ω𝑏_ℓ𝑑𝑥,∫_𝜕Ω𝑏̂_ℓ𝑑𝜎≠0”. In this case, we can choose 𝜆_ℓ and 𝜇_ℓ such that 𝜆_ℓ∫_Ω𝑏_ℓ𝑑𝑥,𝜇_ℓ∫_𝜕Ω𝑏̂_ℓ𝑑𝜎≥0 and |𝜆_ℓ|+|𝜇_ℓ|>0 for some ℓ, with |𝜆_ℓ|,|𝜇_ℓ|≤𝛿^∗ instead of 𝜆_ℓ,𝜇_ℓ∈[0,𝛿^∗].(iii)When for each ℓ∈{1,…,𝑛}𝑏_ℓ,𝑏̂_ℓ do not change sign, then the conclusion of the Proposition 3.2 [resp. Proposition 3.3] holds even if 𝜆_ℓ𝑏_ℓ,𝜇_ℓ𝑏̂_ℓ≤0 and |𝜆_ℓ|+|𝜇_ℓ|>𝛿^∗ as ℓ∈𝐼^∗(resp. as ℓ=1,…,𝑛).
In order to simplify the presentation of the applications, we suppose in the next 𝑏_ℓ∈𝐿^∞(Ω)⧵{0} and𝑏̂_ℓ∈𝐿^∞(𝜕Ω)⧵{0}, while the additional assumptions on 𝑏_ℓ,𝑏̂_ℓ and the assumptions on ∫_Ω𝑏_ℓ𝑑𝑥,∫_𝜕Ω𝑏̂_ℓ𝑑𝜎 (the same of Propositions 3.1, 3.2, 3.4, and 3.5) will be pointed out case by case.

Passing to the applications (with 𝑛>1), we recall that in [3] Pohozaev and Véron in the case 𝑛=1 have studied the Neumann problem:−div||∇𝑢||^𝑝−2∇𝑢||∇=𝑢||𝜆𝑏_𝑝−2(𝑥)|𝑢|^𝑝−2𝑢+𝑐(𝑥)|𝑢|^𝑠−2𝑢+𝑎(𝑥)|𝑢|^𝑞−2𝑢inΩ,𝜕𝑢𝜕𝜈=𝑘(𝑥)|𝑢|^𝑟−2𝑢on𝜕Ω.(3.16) The existence theorems proved by these authors can be got by using some results of ([1], Theorems 2.1, 2.2, 4.1, and 4.2; Remarks 2.1, 2.3, 4.1, and 4.4), Propositions 3.3 and 3.5.

Application 3.7. Let for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝^𝑛−1_ℓ=1_Ω||∇𝑣_ℓ||^𝑝𝑑𝑥,𝐷₁(𝑣)=𝑞¹⁻¹⎡⎢⎢⎣_𝜕Ω^𝑛_ℓ=1𝑑_ℓ||𝑣_ℓ||^𝛾^𝑞1/𝛾𝑑𝜎−^𝑛_ℓ=1_𝜕Ω𝑑_ℓ||𝑣_ℓ||^𝑞1𝑑𝜎⎤⎥⎥⎦,(3.17) where 1<𝛾<𝑞₁<𝑝,̂𝑞₁≠𝑝,𝑑_ℓ,𝑑_ℓ∈𝐿^∞(𝜕Ω),𝑑_ℓ,𝑑_ℓ>0.(3.18) Let us consider the system: ||∇𝑢_𝑖−div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖inΩ,||^𝑝−2𝜕𝑢_𝑖𝜕𝜈=𝜇^𝑖𝑏̂_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+^𝑛_ℓ=1𝑑_ℓ||𝑢_ℓ||^𝛾^{(𝑞1/𝛾)−1}𝑑_𝑖||𝑢_𝑖||^𝛾−2𝑢_𝑖−𝑑_𝑖||𝑢_𝑖||^𝑞1−2𝑢_𝑖on𝜕Ωas𝑖=1,…,𝑛.(3.19) Let us introduce the conditions: _𝜕Ω^𝑛_ℓ=1𝑑_ℓ^𝑞1/𝛾∃Γ⊆𝜕Ω∶𝜎(Γ𝑑)𝜎><0,_𝜕Ω𝑑𝑑𝜎_𝑛𝑑=min𝑑₁,…,𝑑_𝑛,(3.20)_ℓ=1𝑑_ℓ^𝑞1/𝛾>^𝑛_ℓ=1𝑑_ℓonΓ,(3.21)_Ω𝑏_ℓ𝑑𝑥>0,_𝜕Ω𝑏̂_ℓ𝑑𝜎>0asℓ=1,…,𝑛.(3.22) Evidently (3.20) ⇒𝐷₁(𝑐)<0forall𝑐∈𝑅^𝑛⧵{0}. Moreover (3.21) ⇒𝑉⁺(𝐷₁)≠∅ (Proposition A.2). Hence (Propositions 3.3 and 3.5) (3.⟹20)an∃d𝛿_∗1(3.21)(3.⟹20)an∃d𝛿_∗2(>3.202∶)𝑖₁₄holdsif||𝜆_ℓ||,||𝜇_ℓ||≤𝛿^∗1∀ℓ∈{1,…,𝑛},(3.23)>0𝑖₁₅holdsif𝜆_ℓ,𝜇_ℓ∈0,𝛿^∗2∀ℓ∈{1,…,𝑛},𝜆_ℓ+𝜇_ℓ>0forsomeℓ.(3.24)

Proposition 3.8 (see ([1], Theorems 2.1 and 4.1; Remarks 2.1 and 4.1); Proposition A.4; [5, 6]). Under assumption (3.18), we have:(i)When (3.20) and (3.21) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.23) system (3.19) has at least two weak solutions 𝑢⁰ and −𝑢⁰(𝑢⁰=𝜏⁰𝑣⁰,𝜏⁰=const.>0,𝑣⁰∈𝑆_𝜆𝜇∩𝑉⁺(𝐷₁)), and it results in 𝑢^0𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼0𝑖_ℓ𝑜𝑐(Ω),𝑢^0𝑖≥0as𝑖=1,…,𝑛,𝑢^0𝑖>0if𝑢^0𝑖≢0;(3.25)(ii)When (3.20) and (3.22) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.24) system (3.19) has at least two weak solutions 𝑢 and −𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆𝜇∩𝑆(𝐷₁)), and it results in𝑢_𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼𝑖_ℓ𝑜𝑐(Ω),𝑢_𝑖≥0as𝑖=1,…,𝑛,𝑢_𝑖>0if𝑢_𝑖≢0.(3.26) Consequently, when (3.20)–(3.22) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.24) and min{𝛿^∗1,𝛿^∗2} instead of 𝛿^∗2 system (3.19) has at least four different weak solutions.

Proposition 3.9. If 𝛾<𝑝<𝑞₁, then 𝑢^0𝑖>0as𝑖=1,…𝑛.

Proof. It is sufficient (Remark 1.1) to verify that 𝑖^ℎ16holdsasℎ=1,…,𝑛with𝔉=𝑆_𝜆𝜇∩𝑉⁺𝐷₁.(3.27)
Let 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑉⁺(𝐷₁)∩𝑆_𝜆𝜇. Let, for example, 𝑣₁≡0. Since ∫_𝜕Ω(∑_ℓ≠1𝑑_ℓ|𝑣_ℓ|^𝛾)^𝑞1/𝛾𝑑𝜎>0, there exists Γ⁺⊆𝜕Ω such that 𝜎Γ⁺>0,_ℓ≠1𝑑_ℓ||𝑣_ℓ||^𝛾>0onΓ⁺.(3.28) Let 𝕂⊆Ω a compact set and Ω′ an open set such that ||𝕂||_𝑁>0,𝕂⊆Ω^,Ω^⊆Ω.(3.29) Since Propositions A.1 and A.2, there exist a compact set Γ⁺⊆Γ⁺, with 𝜎(Γ⁺)>0, and (𝜑_1𝜀)_0<𝜀<𝜀0,(𝜑_2𝜀)_0<𝜀<𝜀0⊆𝐶^∞0(𝑅^𝑁) such that 0≤𝜑_1𝜀≤1,supp𝜑_1𝜀⊆Ω^,𝜑_1𝜀⟶𝜒stronglyin𝐿^𝑠(Ω)_Ω0≤𝜑_2𝜀||∇𝜑_1𝜀||^𝑠𝑑𝑥⟶+∞as𝜀⟶0⁺∀𝑠∈[1,+∞[,≤1,supp𝜑_2𝜀⊆𝑅^𝑁⧵Ω′,𝜑_2𝜀⟶𝜒stronglyin𝐿^𝑠(𝜕Ω),_Ω𝜑^𝑠2𝜀𝑑𝑥⟶0as𝜀⟶0⁺∀𝑠∈[1,+∞[,(3.30) where 𝜒(resp.𝜒) is the characteristic function of 𝕂(resp.Γ⁺). Let us choose 𝜀 such that 𝛿=𝑝⁻¹_Ω||∇𝜑_𝜀||^𝑝𝑑𝑥−𝜆₁_Ω𝑏₁𝜑^𝑝𝜀𝑑𝑥−𝜇₁_𝜕Ω𝑏̂₁𝜑^𝑝𝜀𝑑𝜎>0,_𝜕Ω_ℓ≠1𝑑_ℓ||𝑣_ℓ||^𝛾^{(𝑞1/𝛾)−1}𝑑₁𝜑^𝛾𝜀𝑑𝜎>0𝜑_𝜀=𝜑_1𝜀+𝜑_2𝜀,(3.31) and we set 𝑣(𝑠)=((1−𝑠)^1/𝑝𝛿^−1/𝑝𝜑_𝜀,𝑠^1/𝑝𝑣₂,…,𝑠^1/𝑝𝑣_𝑛). Then, 𝐻_𝜆𝜇(𝑣(𝑠))=1∀𝑠∈[0,1l]im,_𝑠→1−∃𝑠₀∈[0,1[∶𝐷₁(𝑣(𝑠))>0∀𝑠∈𝑠₀,1,𝑑𝑑𝑠𝐷¹(𝑣(𝑠))=−∞.(3.32)

Proposition 3.10. If 𝜆𝑑^{𝑞1ℓ/𝛾}<𝑑_ℓ𝑎𝑠ℓ=1,…,𝑛,(3.33)_ℓ+𝜇_ℓ>0𝑎𝑠ℓ=1,…,𝑛,(3.34) then 𝑢_𝑖>0𝑎𝑠𝑖=1,…,𝑛.

Proof. We recall that ([1], Theorem 4.1) 𝐻_𝜆𝜇𝑣=inf𝐻_𝜆𝜇(𝑣)∶𝑣∈𝑉^−𝜆𝜇∩𝑆𝐷₁.(3.35) Reasoning by contradiction let, for example, 𝑣₁≡0. As 𝑐₁=const.>0 and 𝑔𝜏(𝑠,𝜏)=𝐷₁(𝑠𝑐₁,𝜏𝑣₂,…,𝜏𝑣_𝑛)=𝑞¹⁻¹[∫_𝜕Ω(𝑑₁𝑠^𝛾𝑐^𝛾1+𝜏^𝛾∑_ℓ≠1𝑑_ℓ(𝑣_ℓ)^𝛾)^𝑞1⧵𝛾𝑑𝜎−𝑠^𝑞1𝑐^𝑞1₁∫_𝜕Ω𝑑₁𝑑𝜎−_𝑞1∑_ℓ≠1∫_𝜕Ω𝑑_ℓ(𝑣_ℓ)^𝑞1𝑑𝜎] for all 𝑠,𝜏≥0, we have 𝑔(0,𝜏)=−𝜏^𝑞1>−1forall𝜏∈]0,1[ and since (3.33) lim_𝑠→+∞𝑔(𝑠,𝜏)=−∞forall𝜏≥0. Thenforall𝜏∈]0,1[, it is possible to choose 𝑠(𝜏)>0 such that 𝑔(𝑠(𝜏),𝜏)=−1. Let us add that there exist 𝑠₀>0 and 𝜏₀∈]0,1[ such that (𝜕𝑔/𝜕𝑠)(𝑠,𝜏)>0forall(𝑠,𝜏)∈]0,𝑠₀[x]𝜏₀,1[.
Let now {𝜏_𝑛}⊆]𝜏₀,1[ and lim𝜏_𝑛=1. Since 𝑔(𝑠(𝜏_𝑛),𝜏_𝑛)=−1,{𝑠(𝜏_𝑛)} is necessarily bounded. Then (within a subsequence) lim𝑠(𝜏_𝑛)=𝜔≥𝑠₀. Consequently, from the inequality: 𝐻_𝜆𝜇𝑣≤𝐻_𝜆𝜇𝑣𝜏_𝑛,where𝑣𝜏_𝑛=𝑠𝜏_𝑛𝑐₁,𝜏_𝑛𝑣₂,…,𝜏_𝑛𝑣_𝑛∈𝑉^−𝜆𝜇∩𝑆𝐷₁,(3.36) as 𝑛→+∞ and from (3.34), we get the contradiction: 𝐻_𝜆𝜇𝑣≤−𝑝⁻¹𝜔^𝑝𝑐^𝑝1𝜆¹_Ω𝑏₁𝑑𝑥+𝜇₁_𝜕Ω𝑏̂₁𝑑𝜎+𝐻_𝜆𝜇𝑣<𝐻_𝜆𝜇𝑣.(3.37)

Remark 3.11. Let us note that the conditions (3.20), (3.21), and (3.33) are compatible.

Application 3.12. Let for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝⁻¹^𝑛−1_ℓ=1_Ω||∇𝑣_ℓ||^𝑝𝑑𝑥+_Ω||∇𝑣^𝑛𝐷||^𝛾+_𝜕Ω||𝑣_𝑛||^𝛾𝑑𝜎^𝑝/𝛾𝑑𝑥₁,(𝑣)=𝑞¹⁻¹^𝑛−1_ℓ=1_Ω𝜌_ℓ||𝑣_ℓ+𝑣_𝑛||^𝑞1−1𝑣_ℓ+𝑣_𝑛𝑑𝑥−^𝑛_ℓ=1_𝜕Ω𝑑_ℓ||𝑣_ℓ||^𝑞1𝑑𝜎,(3.38) where 1<𝛾<𝑝,1<𝑞₁<𝑝,̂𝑞₁≠𝑝,𝜌_ℓ∈𝐿^∞(Ω),𝜌_ℓ>0,𝑑_ℓ∈𝐿^∞(𝜕Ω),𝑑_ℓ>0.(3.39) Let us consider the system: −−ddiivv||∇||𝑢∇_𝑖||𝑢^𝑝−2_𝑛∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝜌_𝑖||𝑢_𝑖+𝑢_𝑛||^𝑞1−1inΩas𝑖=1,…,𝑛−1,||^𝛾+_𝜕Ω||𝑢_𝑛||^𝛾𝑑𝜎^{(𝑝/𝛾)−1}||∇𝑢_𝑛||^𝛾−2∇𝑢_𝑛=𝜆^𝑛𝑏_𝑛||𝑢_𝑛||^𝑝−2𝑢_𝑛+^𝑛−1_ℓ=1||∇𝑢_𝑖𝜌_ℓ||𝑢_ℓ+𝑢_𝑛||^𝑞1−1inΩ,||^𝑝−2𝜕𝑢_𝑖𝜕𝜈=𝜇^𝑖||∇𝑢_𝑛𝑏̂_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖−𝑑_𝑖||𝑢_𝑖||^𝑞1−2𝑢_𝑖on𝜕Ωas𝑖=1,…,𝑛−1,||^𝛾+_𝜕Ω||𝑢_𝑛||^𝛾𝑑𝜎^{(𝑝/𝛾)−1}||∇𝑢_𝑛||^𝛾−2𝜕𝑢_𝑛=𝜇_𝑛𝜕𝜈𝑏̂_𝑛||𝑢_𝑛||^𝑝−2𝑢_𝑛−_Ω||∇𝑢^𝑛||^𝛾+_𝜕Ω||𝑢_𝑛||^𝛾𝑑𝜎^{(𝑝/𝛾)−1}𝑑𝑥||𝑢_𝑛||^𝛾−2𝑢_𝑛−𝑑_𝑛||𝑢_𝑛||^𝑞1−2𝑢_𝑛on𝜕Ω.(3.40) Pointing out that 𝑉⁺(𝐷₁)≠∅, we advance the conditions _Ω^𝑛−1_ℓ=1𝜌_ℓ𝑑𝑥<_𝜕Ω𝑑𝑑𝜎𝑑=min𝑑₁,…,𝑑_𝑛,(3.41)_Ω𝑏_ℓ𝑑𝑥>0,𝑏_𝜕Ω𝑏̂_ℓ𝑑𝜎>0asℓ=1,…,𝑛−1,(3.42)_𝑛≥0,𝑏̂_𝑛≥0.(3.43) Taking into account that (3.41)⟹𝐷₁𝑐₁,…,𝑐_𝑛−1,0<0∀𝑐₁,…,𝑐_𝑛−1∈𝑅^𝑛−1⧵{0},(3.44) we have (Propositions 3.2 and 3.4) (3.⟹41)anwdith(3𝜆.4_𝑛3)(3.⟹41)-(3w.4i3th)𝜆_𝑛,𝜇_𝑛≤0,𝜆_𝑛+𝜇_𝑛<0∃𝛿^∗1>0∶𝑖₁₄holdsif||𝜆_ℓ||,||𝜇_ℓ||≤𝛿^∗1asℓ=1,…,𝑛−1,(3.45)if𝜆_ℓ,𝜇_𝑛≤0,𝜆_𝑛+𝜇_𝑛<0∃𝛿^∗2>0∶𝑖₁₅holds,𝜇_ℓ∈0,𝛿^∗2asℓ=1,…,𝑛−1and𝜆_ℓ+𝜇_ℓ>0forsomeℓ).(3.46)

Proposition 3.13 . (see ([1], Theorems 2.1 and 4.1; Remark 2.1); Proposition A.4; [5, 6]). Under assumption (3.39), we have(i)When (3.41) and (3.43) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.45) system (3.40) has at least one weak solution 𝑢⁰ (𝑢⁰=𝜏⁰𝑣⁰,𝜏⁰=const.>0,𝑣⁰∈𝑆_𝜆𝜇∩𝑉⁺(𝐷₁)), and it results in 𝑢^0𝑖∈𝐿^∞(Ω𝑢)∩𝐶^1,𝛼0𝑖_ℓ𝑜𝑐(Ω),𝑢^0𝑖>0as𝑖=1,…,𝑛−1,_0𝑛∈𝐿^∞(Ω),𝑢^0𝑛≥0,𝑢^0𝑛≢0;(3.47)(ii)When (3.41)–(3.43) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.46) system (3.40) has at least one weak solution 𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆𝜇∩𝑆(𝐷₁)), and it results in 𝑢_𝑖≢0 as 𝑖=1,…,𝑛.
Consequently, when (3.41)–(3.43) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.46) and min{𝛿^∗1,𝛿^∗2} instead of 𝛿^∗2 system (3.40) has at least two different weak solutions.

About the properties of 𝑢^0𝑖 and 𝑢_𝑖 expressed by Proposition 3.13, it is necessary to remark that if𝑢=(𝑢₁,…,𝑢_𝑛)isanontrivialweaksolutiontosystem (3.40), then 𝑢_𝑖≢0 as 𝑖=1,…,𝑛. In fact,𝑢_𝑛≡0⟹𝑢_𝑖≡0as𝑖=1,…,𝑛−1,𝑢_𝑖≡0forsome𝑖∈{1,…,𝑛−1}⟹𝑢_𝑛≡0.(3.48)

Application 3.14. Let 𝑛=2 and for any 𝑣=(𝑣₁,𝑣₂)∈𝑊: 𝐴(𝑣)=𝑝²⁻¹_ℓ=1_Ω||∇𝑣_ℓ||^𝑝𝑑𝑥,𝐷_𝑗(𝑣)=𝑞^𝑗−1_Ω𝜌_𝑗|||||₂_ℓ=1𝑑_𝑗ℓ||𝑣_ℓ||^𝛾𝑗|||||^{𝑞𝑗/𝛾𝑗}𝐷𝑑𝑥as𝑗=1,…,𝑚−1,_𝑚(𝑣)=𝑞^𝑚−1_𝜕Ω𝜌_𝑚²_ℓ=1||𝑣_ℓ||^𝛾𝑚^{𝑞𝑚/𝛾𝑚}𝑑𝜎,(3.49) where 1<𝛾_𝑗<𝑞_𝑗as𝑗=𝑑1,…,𝑚,𝑝<𝑞₁<⋯<𝑞_𝑚<𝑝,̂𝜌_𝑗∈𝐿^∞(Ω),𝜌_𝑗<0,_𝑗ℓ∈𝐿^∞(Ω)⧵{0},𝜌_𝑚∈𝐿^∞(𝜕Ω).(3.50) Let us consider the system: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+^𝑚−1_𝑗=1𝜌_𝑗|||||₂_ℓ=1𝑑_𝑗ℓ||𝑢_ℓ||^𝛾𝑗|||||^{(𝑞𝑗/𝛾𝑗)−2}²_ℓ=1𝑑_𝑗ℓ||𝑢_ℓ||^𝛾𝑗𝑑^𝑗𝑖||∇𝑢_𝑖||𝑢_𝑖||^𝛾𝑗−2𝑢_𝑖inΩ,||^𝑝−2𝜕𝑢_𝑖𝜕𝜈=𝜇^𝑖𝑏̂_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝜌_𝑚²_ℓ=1||𝑢_ℓ||^𝛾𝑚^{(𝑞𝑚/𝛾𝑚)−1}||𝑢_𝑖||^𝛾𝑚−2𝑢_𝑖on𝜕Ωas𝑖=1,2.(3.51) Let us introduce the conditions: 𝜌^+𝑚≢0⟹𝑉⁺𝐷_𝑚≠∅(PropositionA.2),(3.52)_𝜕Ω𝜌_𝑚𝑑𝜎<0⟹𝐷_𝑚𝑐₁,𝑐₂<0∀𝑐₁,𝑐₂∈𝑅²⧵{0},(3.53)_Ω𝑏_ℓ𝑑𝑥>0,_𝜕Ω𝑏̂_ℓ𝑑𝜎>0asℓ=1,2,(3.54) we have (Propositions 3.3 and 3.5) ((33..⟹5532))aann∃dd𝛿_∗2((33..5543))⟹∃𝛿^∗1>0∶𝑖₁₄holdsif||𝜆_ℓ||,||𝜇_ℓ||≤𝛿^∗1asℓ=1,2,(3.55)>0∶𝑖₁₅holdsif𝜆_ℓ,𝜇_ℓ∈0,𝛿^∗2asℓ=1,2,𝜆_ℓ+𝜇_ℓ>0forsomeℓ.(3.56)

Proposition 3.15 (see ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4); Proposition A.4; [5]). Under assumption (3.50), we have(i)When (3.52) and (3.53) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.55) system (3.51) has at least two weak solutions 𝑢⁰ and −𝑢⁰(𝑢⁰=𝜏⁰𝑣⁰,𝜏⁰=const.>0,𝑣⁰∈𝑆_𝜆𝜇∩𝑉⁺(𝐷_𝑚)), and it results in 𝑢^0𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼0𝑖_ℓ𝑜𝑐(Ω),𝑢^0𝑖≥0as𝑖=1,2;(3.57)(ii)When (3.53) and (3.54) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.56) system (3.51) has at least two weak solutions 𝑢 and −𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆𝜇∩𝑆(𝐷_𝑚)), and it results in 𝑢_𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼𝑖_ℓ𝑜𝑐(Ω),𝑢_𝑖≥0as𝑖=1,2.(3.58)
Consequently, when (3.52)–(3.54) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.56), and min{𝛿^∗1,𝛿^∗2} instead of 𝛿^∗2 system (3.51) has at least four different weak solutions.

Proposition 3.16. Under the assumption 𝑝≤2𝛾_𝑗 and 𝑑_𝑗1⋅𝑑_𝑗2<0 as 𝑗=1,…,𝑚−1, we have(i)if 𝛾_𝑗0<𝑝 for some 𝑗₀∈{1,…,𝑚}, then 𝑢^0𝑖>0 as 𝑖=1,2;(ii)if 𝛾_𝑗0<𝛾_𝑚≤𝑝 for some 𝑗₀∈{1,…,𝑚−1}, then𝑢_𝑖>0as𝑖=1,2.

Proof. First of all 𝑢^0𝑖 is a weak supersolution to the equation: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+^𝑚−1_𝑗=1𝜌_𝑗||𝑑_𝑗1𝑢⁰¹^𝛾𝑗+𝑑_𝑗2𝑢⁰²^𝛾𝑗||^{(𝑞𝑗/𝛾𝑗)−2}𝑑^2𝑗𝑖||𝑢_𝑖||^{2𝛾𝑗−2}𝑢_𝑖inΩ.(3.59) Also, 𝑢_𝑖 has a similar property. Then [6] it is sufficient to verify that 𝑢^0𝑖≢0,(3.60)𝑢_𝑖≢0.(3.61) About (3.60), let us prove (Remark 1.1) that 𝑖^ℎ16holdsasℎ=1,2with𝔉=𝑆_𝜆𝜇∩𝑉⁺𝐷_𝑚.(3.62) Let 𝑣=(𝑣₁,𝑣₂)∈𝑉⁺(𝐷_𝑚)∩𝑆_𝜆𝜇. Let, for example, 𝑣₁≡0. Let 𝕂⊆ΩΩacompactset∶||𝕂||_𝑁>0,𝑣₂≢0in𝕂,_Γ⊆𝜕Ωan∶o𝜎pe(nΓ)se>t∶0,𝕂𝜌⊆_𝑚Ω^,Ω^⊆Ω,||𝑣₂||>0onΓ.(3.63) Since Propositions A.1 and A.2, there exists 𝜑∈𝐶^∞0(𝑅^𝑁), with 0≤𝜑≤1 and supp𝜑⊆Ω^∪(𝑅^𝑁⧵Ω^), such that _Ω𝜌_𝑗||𝑑_𝑗2||𝑣₂||^𝛾𝑗||𝛿=^{(𝑞𝑗/𝛾𝑗)−2}||𝑣𝑝₂||^𝛾𝑗₋₁𝜑^𝛾𝑗𝑑_𝑗1𝑑_𝑗2𝑑𝑥>0as𝑗=1,…,𝑚−1,_𝜕Ω𝜌_𝑚||𝑣₂||^{𝑞𝑚−𝛾𝑚}𝜑^𝛾𝑚𝑑𝜎>0,_Ω||∇𝜑||^𝑝𝑑𝑥−𝜆₁_Ω𝑏₁𝜑^𝑝𝑑𝑥−𝜇₁_𝜕Ω𝑏̂₁𝜑^𝑝𝑑𝜎>0.(3.64) Then with 𝑣(𝑠)=((1−𝑠)^1/𝑝𝛿^−1/𝑝𝜑,𝑠^1/𝑝𝑣₂), we have 𝐻_𝜆𝜇(𝑣l(i𝑠m))_𝑠→1−=1∀𝑠∈[0,1],∃𝑠₀∈[0,1[∶𝐷_𝑚(𝑣(𝑠))>0∀𝑠∈𝑠₀,1,𝑑𝑑𝑠𝐷^𝑗0(𝑣(𝑠))=−∞,lim_𝑠→1−𝑑𝑑𝑠𝐷^𝑗(𝑣(𝑠))<+∞as𝑗≠𝑗₀.(3.65) Passing to (3.61), let us introduce the function 𝜓(𝑡,𝑣)=𝑝𝑡^𝑝−1𝐻_𝜆𝜇(𝑣)−∑^𝑚𝑗=1𝑞_𝑗𝑡^𝑞𝑗−1𝐷_𝑗(𝑣), and let us remember that ([1], Theorem 4.2) ∀𝑣𝐸∈𝑉^−𝜆𝜇∩𝑆𝐷_𝑚∃∣𝑡(𝑣)>0∶𝜓(𝑡(𝑣),𝑣)=0,𝑣=inf𝐸(𝑣)∶𝑣∈𝑉^−𝜆𝜇∩𝑆𝐷_𝑚,(3.66) where 𝐸(𝑣)=(𝑡(𝑣))_𝑝𝐻_𝜆𝜇(𝑣)−∑^𝑚𝑗=1(𝑡(𝑣))^𝑞𝑗𝐷_𝑗(𝑣).
Reasoning by contradiction, let us set, for example, 𝑣₁≡0 and set 𝑣(𝑠)=((1−𝑠)^1/𝛾𝑚𝑣₂,𝑠^1/𝛾𝑚𝑣₂). Since li𝐷m_{𝑚𝑠→1−}(𝑣(𝑠))=−1∀𝑠∈[0,1],∃𝑠₀∈[0,1[∶𝐻_𝜆𝜇(𝑣(𝑠))<0∀𝑠∈𝑠₀,1,𝑑𝑑𝑠𝐷^𝑗0(𝑣(𝑠))=−∞,lim_𝑠→1−𝑑𝑑𝑠𝐷^𝑗(𝑣(𝑠))<+∞as𝑗∈{1,…,𝑚−1}⧵𝑗₀,(3.67) as in Proposition 2.24, we get the contradiction: 𝐸𝑣≤𝐸(𝑣(𝑠))<𝐸𝑣∀𝑠∈𝑠₁,1𝑠₀≤𝑠₁<1.(3.68)

Application 3.17. Let 𝑛>2 and set for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝^𝑛−1_ℓ=1_Ω𝐷||∇𝑣_ℓ||^𝑝𝑑𝑥,_𝑗(𝑣)=𝑞^𝑗𝑛−1_{ℓ=1ℓ≠𝑗}_Ω𝐷𝜌_𝑗||𝑑_𝑗𝑗||𝑣_𝑗||^𝛾𝑗+𝑑_𝑗ℓ||𝑣_ℓ||^𝛾𝑗||^{𝑞𝑗/𝛾𝑗}𝑑𝑥as𝑗=1,…,𝑛,_𝑛+1(𝑣)=𝑞^−1𝑛+1_𝜕Ω𝜌_𝑛+1^𝑛_ℓ=1||𝑣_ℓ||^𝛾𝑛+1^{𝑞𝑛+1/𝛾𝑛+1}𝑑𝜎,(3.69) where 1<𝛾_𝑗<𝑞_𝑗as𝑗=1,…𝑑,𝑛+1,𝑝<𝑞₁<⋯<𝑞_𝑛+1<𝑝,̂𝜌_𝑗∈𝐿^∞(Ω),𝜌_𝑗<0,_𝑗ℓ∈𝐿^∞(Ω)⧵{0},𝜌_𝑛+1∈𝐿^∞(𝜕Ω).(3.70) Let us consider the system: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+_ℓ≠𝑖𝜌_𝑖||𝑑_𝑖𝑖||𝑢_𝑖||^𝛾𝑖+𝑑_𝑖ℓ||𝑢_ℓ||^𝛾𝑖||^{(𝑞𝑖/𝛾𝑖)−2}𝑑_𝑖𝑖||𝑢_𝑖||^𝛾𝑖+𝑑_𝑖ℓ||𝑢_ℓ||^𝛾𝑖𝑑_𝑖𝑖||𝑢_𝑖||^𝛾𝑖−2𝑢_𝑖||∇𝑢_𝑖+_𝑗≠𝑖𝜌_𝑗||𝑑_𝑗𝑗||𝑢_𝑗||^𝛾𝑗+𝑑_𝑗𝑖||𝑢_𝑖||^𝛾𝑗||^{(𝑞𝑗/𝛾𝑗)−2}𝑑_𝑗𝑗||𝑢_𝑗||^𝛾𝑗+𝑑_𝑗𝑖||𝑢_𝑖||^𝛾𝑗𝑑_𝑗𝑖||𝑢_𝑖||^𝛾𝑗−2𝑢_𝑖inΩ,||^𝑝−2𝜕𝑢_𝑖𝜕𝜈=𝜇^𝑖𝑏̂_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝜌_𝑛+1^𝑛_ℓ=1||𝑢_ℓ||^𝛾𝑛+1^{(𝑞𝑛+1/𝛾𝑛+1)−1}||𝑢_𝑖||^{𝛾𝑛+1−2}𝑢_𝑖on𝜕Ωas𝑖=1,…,𝑛.(3.71) Let us make the assumptions: 𝜌^+𝑛+1≢0,_𝜕Ω𝜌_𝑛+1𝑑𝜎<0,_Ω𝑏_ℓ𝑑𝑥>0,_𝜕Ω𝑏̂_ℓ𝑑𝜎>0asℓ=1,…,𝑛.(3.72) About Neumann’s problem (3.71), we have an existence result similar to the one of Proposition 3.15 related to system (3.51). About the positive sign of the components of the weak solutions 𝑢⁰ and 𝑢 to system (3.71), as in Proposition 3.16, we show.Proposition 3.18. Under the assumption 𝑝≤2𝛾_𝑗 as 𝑗=1,…,𝑛 and 𝑑_𝑗𝑗⋅𝑑_𝑗ℓ<0 as 𝑗,ℓ∈{1,…,𝑛} with ℓ≠𝑗, we have(i)if either 𝛾_𝑛+1<𝑝 or 𝛾_𝑗<𝑝 for all 𝑗∈{1,…,𝑛}⧵{𝑗₀} for some 𝑗₀, then 𝑢^0𝑖>0 as 𝑖=1,…,𝑛;(ii)if 𝛾_𝑗<𝛾_n+1≤𝑝 for all 𝑗∈{1,…,𝑛}⧵{𝑗₀} for some 𝑗₀, then 𝑢_𝑖>0 as 𝑖=1,…,𝑛.The following remark deals also with Application 3.14.Remark 3.19. Making in (3.50) (resp. (3.70)) the change 𝑞₁<⋯<𝑞_𝑚<𝑝resp.𝑞₁<⋯<𝑞_𝑛+1<𝑝,(3.73) system (3.51) (resp. (3.71)) has at least the two weak solutions 𝑢and−𝑢 ([1], Theorem 4.2; Remark 4.4). The components of 𝑢 keep the properties that Propositions 3.15 and 3.16 (Proposition 3.15 and Proposition 3.18 resp.) underline.

Application 3.20. Let for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝^𝑛−1_ℓ=1_Ω𝐷||∇𝑣_ℓ||^𝑝𝑑𝑥,_𝑗(𝑣)=𝑞^𝑗−1_𝜕Ω𝐷𝜌_𝑗||𝑑_𝑗1||𝑣_𝑗||^𝛾𝑗+𝑑_𝑗2||𝑣_𝑛||^𝛾𝑗||^{𝑞𝑗/𝛾𝑗}𝑑𝜎as𝑗=1,…,𝑛−1,_𝑛(𝑣)=_𝜕Ω||𝑣_𝑛||𝛾̂^𝑛𝑑𝜎_Ω𝜌_𝑛||𝑣_𝑛||^𝛾𝑛𝑑𝑥,(3.74) where 1<𝛾_𝑗<𝑞_𝑗<𝑝̂𝑝a<s𝑗𝑞₁=1,…,𝑛−1,1<𝛾_𝑛<𝑝,̃1<𝛾̂_𝑛<𝑝,̂𝜌_𝑗<⋯<𝑞_𝑛−1<𝑞_𝑛=𝛾_𝑛+𝛾̂_𝑛,∈𝐿^∞(𝜕Ω),𝜌_𝑗<0,𝑑_𝑗1,𝑑_𝑗2∈𝐿^∞(𝜕Ω)⧵{0},𝜌_𝑛∈𝐿^∞(Ω).(3.75) Let us consider the system: −−ddiivv||∇||∇𝑢𝑢_𝑖𝑛||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2inΩas𝑖=1,…,𝑛−1,||^𝑝−2∇𝑢_𝑛=𝜆^𝑛𝑏_𝑛||𝑢_𝑛||^𝑝−2𝑢_𝑛+𝛾_𝑛_𝜕Ω||∇𝑢_𝑖||𝑢_𝑛||𝛾̂^𝑛𝑑𝜎𝜌_𝑛||𝑢_𝑛||^𝛾𝑛−2𝑢_𝑛inΩ,||^𝑝−2𝜕𝑢_𝑖𝜕𝜈=𝜇^𝑖𝑏̂_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝜌_𝑖||𝑑_𝑖1||𝑢_𝑖||^𝛾𝑖+𝑑_𝑖2||𝑢_𝑛||^𝛾𝑖||^{(𝑞𝑖/𝛾𝑖)−2}||∇𝑢_𝑛×𝑑_𝑖1||𝑢_𝑖||^𝛾𝑖+𝑑_𝑖2||𝑢_𝑛||^𝛾𝑖𝑑_𝑖1||𝑢_𝑖||^𝛾𝑖−2𝑢_𝑖on𝜕Ωas𝑖=1,…,𝑛−1,||^𝑝−2𝜕𝑢_𝑛𝜕𝜈=𝜇^𝑛𝑏̂_𝑛||𝑢_𝑛||^𝑝−2𝑢_𝑛+^𝑛−1_𝑗=1𝜌_𝑗||𝑑_𝑗1||𝑢_𝑗||^𝛾𝑗+𝑑_𝑗2||𝑢_𝑛||^𝛾𝑗||^{(𝑞𝑗/𝛾𝑗)−2}×𝑑_𝑗1||𝑢_𝑗||^𝛾𝑗+𝑑_𝑗2||𝑢_𝑛||^𝛾𝑗𝑑_𝑗2||𝑢_𝑛||^𝛾𝑗−2𝑢_𝑛+𝛾̂_𝑛_Ω𝜌_𝑛||𝑢_𝑛||^𝛾𝑛𝑑𝑥||𝑢_𝑛||𝛾̂^𝑛−2𝑢_𝑛on𝜕Ω.(3.76) Let us introduce the conditions: 𝜌^+𝑛≢0⟹𝑉⁺𝐷_𝑛≠∅(PropositionsA.1andA.2),(3.77)_Ω𝜌_𝑛𝑑𝑥<0𝑏⟹𝐷_𝑛0,….,0,𝑐_𝑛<0∀𝑐_𝑛∈𝑅^𝑛⧵{0},(3.78)_ℓ≥0,𝑏̂_ℓ≥0asℓ=1,…,𝑛−1,(3.79)_Ω𝑏_𝑛𝑑𝑥>0,_𝜕Ω𝑏̂_𝑛𝑑𝜎>0.(3.80) We have (Propositions 3.2 and 3.4) (3.⟹77)-(3w.7i9th)𝜆_ℓ𝑖,𝜇_ℓ≤0,𝜆_ℓ+𝜇_ℓ<0𝑎𝑠ℓ=1,…,𝑛−1∃𝛿^∗1>0∶₁₄(3.⟹78)-(3w.8i0th)h𝜆o_ℓldsif||𝜆_𝑛||,||𝜇_𝑛||≤𝛿^∗1,(3.81)∃𝛿_∗2,𝜇_ℓ≤0,𝜆_ℓ+𝜇_ℓ<0asℓ=1,…,𝑛−1>0∶𝑖₁₅holdsif𝜆_𝑛,𝜇_𝑛∈0,𝛿^∗2and𝜆_𝑛+𝜇_𝑛>0.(3.82)

Proposition 3.21 (see ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4); Proposition A.4; [5, 6]). Under assumption (3.75), we have(i)When (3.77)–(3.79) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.81), system (3.76) has at least two weak solutions 𝑢⁰ and −𝑢⁰(𝑢⁰=𝜏⁰𝑣⁰,𝜏⁰=const.>0,𝑣⁰∈𝑆_𝜆𝜇∩𝑉⁺(𝐷_𝑛)), and it results in 𝑢^0𝑖≥0(𝑖=1,…,𝑛), 𝑢^0𝑛≢0. If 𝛾_𝑛<𝑝̂, then𝑢𝑢^0𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼0𝑖_ℓ𝑜𝑐(Ω)(𝑖=1,…,𝑛),_0𝑖≢0⟹𝑢^0𝑖>0(𝑖=1,…,𝑛−1);(3.83)(ii)When (3.78)–(3.80) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.82), system (3.76) has at least two weak solutions 𝑢 and −𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣∈𝑉^−𝜆𝜇∩𝑆(𝐷_𝑛)), and it results in 𝑢_𝑖≥0(𝑖=1,…,𝑛), 𝑢_𝑛≢0. If 𝛾_𝑛<𝑝̂, then𝑢_𝑖∈𝐿^∞(Ω)∩𝐶^1,𝛼𝑖_ℓ𝑜𝑐(Ω)(𝑖=1,…,𝑛),𝑢_𝑖≢0⇒𝑢_𝑖>0(𝑖=1,…,𝑛−1).(3.84) Consequently, when (3.77)–(3.80) hold, with 𝜆_ℓ,𝜇_ℓ as in (3.82), and min{𝛿^∗1,𝛿^∗2} instead of 𝛿^∗2 system (3.76) has at least four different weak solutions. Obviously, 𝑢^0𝑛>0 and 𝑢_𝑛>0 if 𝑝≤𝛾_𝑛<𝑝̂.

The following proposition gives a sufficient condition to𝑢^0𝑖>0as𝑖=1,…,𝑛−1,(3.85)𝑢_𝑖>0as𝑖=1,…,𝑛−1.(3.86)

Proposition 3.22. Let 𝛾_𝑛<𝑝̂. If 𝛾_𝑗<𝑝 and 𝑑_𝑗1⋅𝑑_𝑗2<0 as 𝑗=1,…,𝑛−1, then (3.85) and (3.86) hold.

Proof. Since 𝑣₁,…,𝑣_𝑛∈𝑉⁺𝐷_𝑛⟹∃Γ⊆𝜕Ω∶𝜎(Γ)>0,||𝑣_𝑛||>0onΓ,(3.87) using Propositions A.1 and A.2, we can verify that 𝑖^ℎ16holdsasℎ=1,…,𝑛−1with𝔉=𝑆_𝜆𝜇∩𝑉⁺𝐷_𝑛,(3.88) from which (Remark 1.1) we get (3.85).
Let us prove (3.86). Reasoning by contradiction, let us set, for example, 𝑣₁≡0. If 𝑣(𝑠)=((1−𝑠)^1/𝑝𝑣_𝑛,𝑣₂,…,𝑣_𝑛), we have 𝐷_𝑛(𝑣(𝑠))=−1∀𝑠∈[0,1l]i,m_𝑠→1−∃𝑠₀∈[0,1[∶𝐻_𝜆𝜇(𝑣(𝑠))<0∀𝑠∈𝑠₀,1,𝑑𝑑𝑠𝐷¹(𝑣(𝑠))=−∞.(3.89) Then as in Proposition 3.16, we get a contradiction.

Remark 3.23. Making in (3.75) the change: 1<𝛾_𝑗<𝑞_𝑗as𝑗=1,…,𝑛−1,1<𝛾_𝑛,1<𝛾̂_𝑛,𝑞₁<⋯<𝑞_𝑛=𝛾_𝑛+𝛾̂_𝑛<𝑝,(3.90) system (3.76) has at least the two weak solutions 𝑢and−𝑢 ([1], Theorem 4.2; Remark 4.4). The components of 𝑢, all bounded, are locally Hölderian with their first derivatives. If 𝑑_𝑗1⋅𝑑_𝑗2<0 as 𝑗=1,…,𝑛−1, then (3.86) holds.

Application 3.24. Let for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝^𝑛−1_ℓ=1_Ω||∇𝑣^ℓ𝐷||^𝛾+_Ω||𝑣_ℓ||^𝛾𝑑𝑥^𝑝/𝛾𝑑𝑥,_𝑗(𝑣)=𝑞^𝑗−1_𝜕Ω^𝑛_ℓ=1||𝑣_ℓ||^𝛾^{𝑞𝑗/𝛾}𝑑𝜎as𝑗=1,…,𝑚−1,𝐷_𝑚(𝑣)=𝑞^𝑚−1_Ω^𝑛_ℓ=1||𝑣_ℓ||^𝛾^{𝑞𝑚/𝛾}𝑑𝑥,(3.91) where 1<𝛾<𝑞₁<⋯<𝑞_𝑚<𝑝.(3.92) Let us consider the system: −div||∇𝑢_𝑖||^𝛾+_Ω||𝑢_𝑖||^𝛾𝑑𝑥^{(𝑝/𝛾)−1}||∇𝑢_𝑖||^𝛾−2∇𝑢_𝑖=𝜆_𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖−_Ω||∇𝑢^𝑖||^𝛾+_Ω||𝑢_𝑖||^𝛾𝑑𝑥^{(𝑝/𝛾)−1}𝑑𝑥||𝑢_𝑖||^𝛾−2𝑢_𝑖+^𝑛_ℓ=1||𝑢_ℓ||^𝛾^{(𝑞𝑚/𝛾)−1}||∇𝑢_𝑖||𝑢_𝑖||^𝛾−2𝑢_𝑖inΩ,||^𝛾+_Ω||𝑢_𝑖||^𝛾𝑑𝑥^{(𝑝/𝛾)−1}||∇𝑢_𝑖||^𝛾−2𝜕𝑢_𝑖=𝜇_𝑖𝜕𝜈𝑏̂_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+^𝑚−1_𝑗=1^𝑛_ℓ=1||𝑢_ℓ||^𝛾^{(𝑞𝑗/𝛾)−1}||𝑢_𝑖||^𝛾−2𝑢_𝑖on𝜕Ωas𝑖=1,…𝑛.(3.93) We advance the condition: 𝑏_ℓ≥0,𝑏̂_ℓ≥0asℓ=1,…𝑛,(3.94) and we note that (Proposition 3.1) (3.94)⟹𝑖₁₃holdsif𝜆_ℓ,𝜇_ℓ≤0,𝜆_ℓ+𝜇_ℓ<0asℓ=1,…,𝑛.(3.95)

Proposition 3.25. Under conditions (3.92) and (3.94), with 𝜆_ℓ,𝜇_ℓ as in (3.95), system (3.93) has at least two weak solutions 𝑢⁰ and −𝑢⁰(𝑢⁰=𝜏⁰𝑣⁰,𝜏⁰=const.>0,𝑣⁰∈𝑆⁺(𝐷₁,…,𝐷_𝑚)), and it results in 𝑢^0𝑖∈𝐿^∞(Ω),𝑢^0𝑖≥0,𝑢^0𝑖≢0as𝑖=1,…,𝑛.(3.96)

Proof. We recall that ([1], Section 2), set 𝜓(𝑡,𝑣)=𝑝𝑡^𝑝−1𝐻_𝜆𝜇(𝑣)−∑^𝑚𝑗=1𝑞_𝑗𝑡^𝑞𝑗−1𝐷_𝑗(𝑣), we have ∀𝑣∈t𝑉he⁺fu𝐷n₁c,ti…on,a𝐷l𝑡_𝑚(𝑣)is∃𝐶∣₁𝑡(𝑣)>0∶𝜓(𝑡(𝑣),𝑣)=0,in𝑉⁺𝐷₁,…,𝐷_𝑚.(3.97) We introduce the functional 𝐸(𝑣)=(𝑡(𝑣))_𝑝𝐻_𝜆𝜇(𝑣)−∑^𝑚𝑗=1(𝑡(𝑣))^𝑞𝑗𝐷_𝑗(𝑣) which is 𝐶¹ in 𝑉⁺(𝐷₁,…,𝐷_𝑚). We still remember that ([1], Theorem 2.3; Remark 2.5) ∃𝑣⁰∈𝑆⁺𝐸𝐷𝑣₁,…,𝐷_𝑚,with𝑣^0𝑖≥0as𝑖=1,…,𝑛,suchthat₀𝑢₀=inf𝐸(𝑣)∶𝑣∈𝑆⁺𝐷₁,…,𝐷_𝑚,=𝑡𝑣⁰𝑣⁰isaweaksolutiontosystem(3.93).(3.98) The property 𝑢^0𝑖∈𝐿^∞(Ω) is due to Proposition A.4. Let us verify that 𝑢^0𝑖≢0 as 𝑖=1,…,𝑛. Reasoning by contradiction, let us set, for example, 𝑣⁰¹≡0 and 𝑣⁰²≢0. As 𝑣(𝑠)=((1−𝑠)^1/𝛾𝑣⁰²,𝑠^1/𝛾𝑣⁰²,𝑣⁰³,…,𝑣^0𝑛), we have ^𝑚_𝑗=1𝐷_𝑗(𝑣(𝑠))=1∀𝑠∈[0,1],𝑑𝑑𝑠𝐻^𝜆𝜇(𝑣(𝑠))_𝑠=1>0.(3.99) Then, since (𝑑/𝑑𝑠)𝐸(𝑣(𝑠))=(𝑡(𝑣(𝑠)))^𝑝(𝑑/𝑑𝑠)𝐻_𝜆𝜇(𝑣(𝑠)),there exists 𝑠₀∈[0,1[ such that (𝑑/𝑑𝑠)𝐸(𝑣(𝑠))>0forall𝑠∈[𝑠₀,1], from which the contradiction: 𝐸𝑣₀≤𝐸(𝑣(𝑠))<𝐸𝑣⁰,∀𝑠∈𝑠₀,1.(3.100)

Application 3.26. Let for each 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊: 𝐴(𝑣)=𝑝^𝑛−1_ℓ=1_Ω||∇𝑣_ℓ||^𝑝𝑑𝑥,𝐷₁(𝑣)=𝑞¹⁻¹_Ω𝜌|||||^𝑛_ℓ=1𝑑_ℓ𝑣_ℓ|||||^𝑞1𝐷𝑑𝑥,₂(𝑣)=_Ω^𝑛_ℓ=1𝑑_ℓ||𝑣_ℓ||^𝛾𝑑𝑥_𝜕Ω𝜌̂^𝑛_ℓ=1𝑑_ℓ||𝑣_ℓ||𝛾̂𝑑𝜎,(3.101) where 1<𝑝𝛾<<𝑞𝑝₂,̃1<𝛾̂<𝑝,̂1<𝑞₁<min𝑝,̃𝑞₂=𝛾+𝛾̂,assomeℓ,𝑑_ℓ,𝜌,𝑑_ℓ∈𝐿^∞(Ω)⧵{0},𝜌≤0,𝜌𝑑_ℓ≢0∈𝐿^∞(Ω)⧵{0},𝑑_ℓ≥0,𝜌̂∈𝐿^∞(Ω)⧵{0},𝑑_ℓ=const.>0.(3.102) Let as ℓ=1,…,𝑛𝐹_ℓ=𝑓_ℓ+𝑓_ℓ, where 𝑓_ℓ∈𝐿^𝑝′(Ω)(𝑝^=𝑝/(𝑝−1)) and 𝑓_ℓ∈(𝑊^{1−(1/𝑝),𝑝}(𝜕Ω))^∗ (dual space of 𝑊^{1−(1/𝑝),𝑝}(𝜕Ω)). Let ⟨⟨𝐹,𝑣⟩⟩=∑^𝑛ℓ=1⟨𝐹_ℓ,𝑣_ℓ⟩forall𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑊. Let us consider the system: −div||∇𝑢_𝑖||^𝑝−2∇𝑢_𝑖=𝜆^𝑖𝑏_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝜌|||||^𝑛_ℓ=1𝑑_ℓ𝑢_ℓ|||||^𝑞1−2^𝑛_ℓ=1𝑑_ℓ𝑢_ℓ𝑑^𝑖+𝛾_𝜕Ω𝜌̂^𝑛_ℓ=1||∇𝑢_𝑖𝑑_ℓ||𝑢_ℓ||𝛾̂𝑑𝜎𝑑_𝑖||𝑢_𝑖||^𝛾−2𝑢_𝑖+𝑓_𝑖inΩ,||^𝑝−2𝜕𝑢_𝑖=𝜇_𝑖𝜕𝜈𝑏̂_𝑖||𝑢_𝑖||^𝑝−2𝑢_𝑖+𝛾̂_Ω^𝑛_ℓ=1𝑑_ℓ||𝑢_ℓ||^𝛾𝑑𝑥𝜌̂𝑑^𝑖||𝑢_𝑖||𝛾̂⁻²𝑢_𝑖+𝑓_𝑖on𝜕Ωas𝑖=1,…,𝑛.(3.103) Let us introduce the conditions: (𝜌̂)⁺≢0⟹𝑉⁺𝐷₂≠∅(PropositionA.1andA.2)_𝜕Ω,𝜌̂𝑑𝜎<0⟹𝐷₂(𝑐)<0∀𝑐∈𝑅^𝑛⧵{0},(3.104) and let us note that (Proposition 3.3) (3.104)⟹∃𝛿^∗>0∶𝑖₁₄holdsif||𝜆_ℓ||,||𝜇_ℓ||≤𝛿^∗asℓ=1,…,𝑛.(3.105)

Proposition 3.27. Under assumptions (3.102) and (3.104), if 𝐹≢0 and ‖𝐹‖_∗ is sufficiently small, then with 𝜆_ℓ,𝜇_ℓ as in (3.105), system (3.103) has at least one weak solution 𝑢̃(𝑢̃=𝜏̃𝑣,̃𝜏̃=const.>0,𝑣̃∈𝑆_𝜆𝜇∩𝑉⁺(𝐷₂)). When 𝛾<𝑝≤𝑞₁, it results in 𝑢_ℎ̃≢0evenif𝐹_ℎ≡0.(3.106)

Proof. The existence of 𝑢̃ is due to ([1], Theorem 3.2). About (3.106), it is sufficiently (Remark 1.1) to verify that 𝑖^ℎ16holdsasℎ=1,…,𝑛with𝔉=𝑆_𝜆𝜇∩𝑉⁺𝐷₂.(3.107) Let 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑉⁺(𝐷₂)∩𝑆_𝜆𝜇 with, for example, 𝑣₁≡0. Let 𝜓=∑_ℓ≠1𝑑_ℓ𝑣_ℓ. Let 𝕂⊆Ω be a compact set having positive measure such that 𝑑₁>0in𝕂if𝜌𝑑₁𝜓≡0,either𝜌𝑑₁𝜓>0in𝕂or𝜌𝑑₁𝜓<0in𝕂if𝜌𝑑₁𝜓≢0.(3.108) Proposition A.1 lets us choose 𝜑∈𝐶^∞0(𝑅^𝑁) satisfying the following conditions: 𝛿=𝑝⁻¹_Ω||∇𝜑||^𝑝𝑑𝑥−𝜆₁_Ω𝑏₁||𝜑||^𝑝𝑑𝑥>0,_Ω𝑑₁𝜑^𝛾𝑑𝑥>0if𝜌𝑑₁𝜓≡0,_Ω𝜌𝑑₁||𝜓||^𝑞1−2𝜓𝜑𝑑𝑥>0if𝜌𝑑₁𝜓≢0.(3.109) Then with 𝑣(𝑠)=((1−𝑠)^1/𝑝𝛿^−1/𝑝𝜑,𝑠^1/𝑝𝑣₂,…,𝑠^1/𝑝𝑣_𝑛), we have 𝐻_𝜆𝜇(𝑣l(i𝑠m))_𝑠→1−=1∀𝑠∈[0,1],𝐷₂(𝑣(𝑠))>0∀𝑠∈𝑠₀,10≤𝑠₀<1,𝑑𝑑𝑠𝐷¹(𝑣(𝑠))∈𝑅,lim_𝑠→1−𝑑𝑑𝑠𝐷²lim_𝑠→1−(𝑣(𝑠))=−∞if𝜌𝑑₁𝜓≡0,𝑑𝑑𝑠𝐷¹(𝑣(𝑠))=−∞,lim_𝑠→1−𝑑𝑑𝑠𝐷²(𝑣(𝑠))∈𝑅if𝜌𝑑₁𝜓≢0.(3.110)
Now we replace conditions (3.104) with the following: 𝜌̂≥0,𝑏_ℓ≥0,𝑏̂_ℓ≥0asℓ=1,…,𝑛.(3.111)

Proposition 3.28. Under assumptions (3.102) and (3.111), if 𝐹≢0 and ‖𝐹‖_∗ is sufficiently small, then with 𝜆_ℓ,𝜇_ℓ≤0 and𝜆_ℓ+𝜇_ℓ<0 as ℓ=1,…,𝑛 system (3.103) has at least two different weak solution 𝑢¹ and 𝑢² (𝑢^𝑖=𝜏^𝑖𝑣^𝑖,𝜏^𝑖=const.>0,𝑣¹∈𝑆_𝜆𝜇∩𝑉⁺(𝐹),𝑣²∈𝑆_𝜆𝜇∩𝑉⁺(𝐷₂)). When 𝛾<𝑝≤𝑞₁, it results in 𝑢^2ℎ≢0evenif𝐹_ℎ≡0.(3.112)

Proof. The existence of 𝑢¹ and 𝑢² is due to ([1], Theorems 3.1, 3.2, and 3.3; Remark 3.1). Relation (3.112) is proved as in Proposition 3.27.

Appendix

In this appendix, we present some results used previously. The first one is trivial. The second one is easy to prove. It is possible to show the third one and the fourth one with the technique developed by Drabek in ([7, Lemma 3.2]). The symbols 𝜎,𝑝,̂and𝑝̃ are the same introduced in Section 3.

Proposition A.1. Let Ω be an open set of 𝑅^𝑁. Let 𝕂⊆Ω be a compact set with |𝕂|_𝑁>0. If Ω′ is an open set such that 𝕂⊆Ω^⊆Ω, then there exists a family of functions (𝜑_𝜀)_0<𝜀<𝜀0⊆𝐶^∞0(Ω) such that 0≤𝜑_𝜀≤1,supp𝜑_𝜀⊆Ω^,𝜑_𝜀⟶𝜒stronglyin𝐿^𝑠(Ω),_Ω||∇𝜑_𝜀||^𝑠𝑑𝑥⟶+∞as𝜀⟶0⁺∀𝑠∈[1,+∞[,(A.1) where 𝜒 is the characteristic function of 𝕂.

Proposition A.2. Let Ω⊆𝑅^𝑁 be an open, bounded, connected and 𝐶^0,1 set. Let 𝑈 be an open neighborhood of 𝜕Ω. If Γ is a subset of 𝜕Ω with 𝜎(Γ)>0, then there exist a compact set Γ⊆Γ with 𝜎(Γ)>0 and a family of functions (𝜑_𝜀)_0<𝜀<𝜀0⊆𝐶^∞0(𝑅^𝑁) such that 0≤𝜑_𝜀≤1,supp𝜑_𝜀⊆𝑈,𝜑_𝜀⟶𝜒stronglyin𝐿^𝑠(𝜕Ω),_𝑅𝑁𝜑^𝑠𝜀𝑑𝑥⟶0as𝜀⟶0⁺∀𝑠∈[1,+∞[,(A.2) where 𝜒 is the characteristic function of Γ.

Let Ω⊆𝑅^𝑁 be an open, bounded, connected and 𝐶^0,1 set. Let as 𝑖=1,…,𝑛𝐴_𝑖(𝑥,𝜉,𝜂¹,…,𝜂^𝑛) be a Carathèodory function into 𝑅^𝑁 defined for 𝑥∈Ω,for 𝜉∈𝑅^𝑛 and for (𝜂¹,…,𝜂^𝑛)∈(𝑅^𝑁)^𝑛 such that 𝐴_𝑖𝑥,𝜉,𝜂¹,…,𝜂^𝑛⋅𝜂^𝑖≥𝑐₀||𝜂^𝑖||^𝑝,(A.3) where 1<𝑝<+∞, 𝑐₀=const.>0.

Proposition A.3. Let (𝑢₁,…,𝑢_𝑛)∈(𝑊^01,𝑝(Ω))^𝑛𝑤𝑖𝑡ℎ𝑢_𝑖≥0. If there exist 𝑟∈]𝑝,𝑝[̃ and 𝑔∈𝐿^{𝑟/(𝑟−𝑝)}(Ω) with 𝑔≥0 such that ^𝑛_𝑖=1_Ω𝐴_𝑖𝑥,𝑢₁,…,𝑢_𝑛,∇𝑢₁,…,∇𝑢_𝑛⋅∇𝑣_𝑖𝑑𝑥≤_Ω𝑔^𝑛_𝑖=1𝑢_𝑖^𝑝−1^𝑛_𝑖=1𝑣_𝑖∀𝑣𝑑𝑥₁,…,𝑣_𝑛∈𝑊^01,𝑝(Ω)∩𝐿^∞(Ω)^𝑛with𝑣_𝑖≥0,(A.4) then 𝑢_𝑖∈𝐿^∞(Ω)as𝑖=1,…𝑛.

Proposition A.4. Let (𝑢₁,…,𝑢_𝑛)∈(𝑊^1,𝑝(Ω))^𝑛witℎ𝑢_𝑖≥0. If there exist 𝑟∈]𝑝,𝑝[̂,𝑔∈𝐿^{𝑟/(𝑟−𝑝)}(Ω) with 𝑔≥0, 𝑔̂∈𝐿^{𝑟/(𝑟−𝑝)}(𝜕Ω) with 𝑔̂≥0 such that ^𝑛_𝑖=1_Ω≤𝐴_𝑖𝑥,𝑢₁,…,𝑢_𝑛,∇𝑢₁,…,∇𝑢_𝑛⋅∇𝑣_𝑖𝑑𝑥_Ω𝑔1+^𝑛_𝑖=1𝑢_𝑖^𝑝−1^𝑛_𝑖=1𝑣_𝑖𝑑𝑥+_𝜕Ω𝑔̂1+^𝑛_𝑖=1𝑢_𝑖^𝑝−1^𝑛_𝑖=1𝑣_𝑖∀𝑣𝑑𝜎₁,…,𝑣_𝑛∈𝑊^1,𝑝(Ω)∩𝐿^∞(Ω)^𝑛with𝑣_𝑖≥0,(A.5) then 𝑢_𝑖∈𝐿^∞(Ω)as𝑖=1,…,𝑛.

Remark A.5. If 𝑔̂≡0, we can suppose 𝑟∈]𝑝,𝑝[̃.

Acknowledgment

This paper is supported by the Second University of Naples.