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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 760854, 44 pages
doi:10.1155/2012/760854
Research Article

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

Luisa Toscano and Speranza Toscano

Department of Mathematics and Applications, “R. Caccioppoli.,” University of Naples “Federico II”, Via Claudio 21, 80125 Naples, Italy

Received 1 February 2012; Accepted 2 April 2012

Academic Editor: D. O'Regan

Copyright © 2012 Luisa Toscano and Speranza Toscano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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: 𝐴 − d i v 𝑖 𝑥 , 𝑢 1 , … , 𝑢 𝑛 , ∇ 𝑢 1 , … , ∇ 𝑢 𝑛 = 𝜆 𝑖 𝑏 𝑖 | | 𝑢 𝑖 | | 𝑝 − 2 𝑢 𝑖 + 𝑑 𝑖 𝑥 , 𝑢 1 , … , 𝑢 𝑛 , ∇ 𝑢 1 , … , ∇ 𝑢 𝑛 + 𝑓 𝑖 𝑢 i n Ω , 𝑖 = 0 o n 𝜕 Ω a s 𝑖 = 1 , … , 𝑛 , ( 1 . 1 ) and to Neumann problems (Section 3) of the type: 𝐴 − d i v 𝑖 𝑥 , 𝑢 1 , … , 𝑢 𝑛 , ∇ 𝑢 1 , … , ∇ 𝑢 𝑛 = 𝜆 𝑖 𝑏 𝑖 | | 𝑢 𝑖 | | 𝑝 − 2 𝑢 𝑖 + 𝑑 𝑖 𝑥 , 𝑢 1 , … , 𝑢 𝑛 , ∇ 𝑢 1 , … , ∇ 𝑢 𝑛 + 𝑓 𝑖 𝐴 i n Ω , 𝑖 𝑥 , 𝑢 1 , … , 𝑢 𝑛 , ∇ 𝑢 1 , … , ∇ 𝑢 𝑛 ⋅ 𝜈 = 𝜇 𝑖 ̂ 𝑏 𝑖 | | 𝑢 𝑖 | | 𝑝 − 2 𝑢 𝑖 + 𝑑 𝑖 𝑥 , 𝑢 1 , … , 𝑢 𝑛 , ∇ 𝑢 1 , … , ∇ 𝑢 𝑛 + 𝑓 𝑖 o n 𝜕 Ω a s 𝑖 = 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 𝑊 1 , … , 𝑊 𝑛 real reflexive Banach spaces ( 𝑛 ≥ 1 ) . Let 𝑊 be the product space 𝑋 𝑛 ℓ = 1 𝑊 ℓ . Let ‖ ⋅ ‖ be the norm on 𝑊 , ‖ ⋅ ‖ ∗ the norm on 𝑊 ∗ (dual space of 𝑊 ), and ⟨ ⋅ , ⋅ ⟩ ℓ ( r e s p . ⟨ ⟨ ⋅ , ⋅ ⟩ ⟩ ) 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 “ 𝜕 𝑢 ℓ ” 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:( 𝑖 1 1 ) 𝐴 is lower weakly semicontinuous in 𝑊 and 𝐶 1 ( 𝑊 ⧵ { 0 } ) , 𝐵 ℓ 𝐵 a n d ℓ are weakly continuous in 𝑊 ℓ and 𝐶 1 ( 𝑊 ℓ ) , ∃ 𝑝 > 1 ∶ 𝐴 ( 𝑡 𝑣 ) = 𝑡 𝑝 𝐴 ( 𝑣 ) f o r a l l 𝑡 ≥ 0 and f o r a l l 𝑣 ∈ 𝑊 , 𝐵 ℓ ( 𝑡 𝑣 ℓ ) = 𝑡 𝑝 𝐵 ℓ ( 𝑣 ℓ ) and 𝐵 ℓ ( 𝑡 𝑣 ℓ ) = 𝑡 𝑝 𝐵 ℓ ( 𝑣 ℓ ) f o r a l l 𝑡 ≥ 0 and f o r a l l 𝑣 ℓ ∈ 𝑊 ℓ ;( 𝑖 1 2 ) 𝐷 𝑗 is weakly continuous in 𝑊 and 𝐶 1 ( 𝑊 ⧵ { 0 } ) , ∃ 𝑞 𝑗 > 1 ∶ 𝐷 𝑗 ( 𝑡 𝑣 ) = 𝑡 𝑞 𝑗 𝐷 𝑗 ( 𝑣 ) f o r a l l 𝑡 ≥ 0 and f o r a l l 𝑣 ∈ 𝑊 , 1 < 𝑞 1 < ⋯ < 𝑞 𝑚 i f 𝑚 > 1 .

Let 𝐹 = ( 𝐹 1 , … , 𝐹 𝑛 ) with 𝐹 ℓ ∈ 𝑊 ℓ ∗ , 𝜆 ℓ and 𝜇 ℓ ∈ 𝑅 ; let us consider the following problem.

Problem ( 𝑃 ). Find 𝑢 = ( 𝑢 1 , … , 𝑢 𝑛 ) ∈ 𝑊 ⧵ { 0 } such that 𝜕 𝑢 𝑖 𝐴 ( 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 𝜕 𝑢 𝑖 𝐷 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 ℓ 𝐵 ℓ 𝑣 ℓ + 𝜇 ℓ 𝐵 ℓ 𝑣 ℓ − 𝑚 𝑗 = 1 𝐷 𝑗 𝑣 ( 𝑣 ) − ⟨ ⟨ 𝐹 , 𝑣 ⟩ ⟩ ∀ 𝑣 = 1 , … , 𝑣 𝑛 ∈ 𝑊 , ( 1 . 4 ) where ∑ ⟨ ⟨ 𝐹 , 𝑣 ⟩ ⟩ = 𝑛 ℓ = 1 ⟨ 𝐹 ℓ , 𝑣 ℓ ⟩ ℓ .

Let us set 𝐻 𝜆 𝜇 ( 𝑣 ) = 𝐴 ( 𝑣 ) − 𝑛 ℓ = 1 𝜆 ℓ 𝐵 ℓ 𝑣 ℓ + 𝜇 ℓ 𝐵 ℓ 𝑣 ℓ 𝑣 ∀ 𝑣 = 1 , … , 𝑣 𝑛 𝜆 ∈ 𝑊 , ∀ 𝜆 = 1 , … , 𝜆 𝑛 𝜇 , 𝜇 = 1 , … , 𝜇 𝑛 ∈ 𝑅 𝑛 , 𝑆 𝜆 𝜇 = 𝑣 ∈ 𝑊 ∶ 𝐻 𝜆 𝜇 ( 𝑣 ) = 1 , 𝑉 − 𝜆 𝜇 = 𝑣 ∈ 𝑊 ∶ 𝐻 𝜆 𝜇 ( 𝑣 ) < 0 , a s 𝑚 1 𝑉 = 1 , … , 𝑚 + 𝐷 𝑚 1 , … , 𝐷 𝑚 = 𝑣 ∈ 𝑊 ∶ 𝑚 𝑗 = 𝑚 1 𝐷 𝑗 , 𝑆 ( 𝑣 ) > 0 + 𝐷 1 , … , 𝐷 𝑚 = 𝑣 ∈ 𝑊 ∶ 𝑚 𝑗 = 1 𝐷 𝑗 , 𝑆 𝐷 ( 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: ( 𝑖 1 3 ) ∃ 𝑐 ( 𝜆 , 𝜇 ) > 0 ∶ ‖ 𝑣 ‖ 𝑝 ≤ 𝑐 ( 𝜆 , 𝜇 ) 𝐻 𝜆 𝜇 ( 𝑣 ) f o r a l l 𝑣 ∈ 𝑊 ; ( 𝑖 1 4 ) ∃ 𝑐 ( 𝜆 , 𝜇 ) > 0 ∶ ‖ 𝑣 ‖ 𝑝 ≤ 𝑐 ( 𝜆 , 𝜇 ) 𝐻 𝜆 𝜇 ( 𝑣 ) f o r a l l 𝑣 ∈ 𝑉 + ( 𝐷 𝑚 ) ( i f 𝑉 + ( 𝐷 𝑚 ) ≠ ∅ ) ; ( 𝑖 1 5 ) ∃ 𝑚 1 ∈ { 1 , … , 𝑚 } ∶ 𝑉 − 𝜆 𝜇 ∩ 𝑆 ( 𝐷 𝑚 1 ) 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 𝜆 𝜇 , we suppose ( 𝑖 ℎ 1 6 ) for each 𝑣 = ( 𝑣 1 , … , 𝑣 𝑛 ) ∈ 𝔉 with 𝑣 ℎ = 0 , there exist 𝑣 ℎ ∈ 𝑊 ℎ ⧵ { 0 } and the real functions 𝜙 1 , … , 𝜙 𝑛 such that 𝜙 ℎ ∈ 𝐶 0 ( [ 0 , 1 ] ) ∩ 𝐶 1 ( [ 0 , 1 [ ) and 𝜙 ℎ ( 1 ) = 0 , 𝜙 ℓ ∈ 𝐶 1 ( [ 0 , 1 ] ) and 𝜙 ℓ ( 1 ) = 1 as ℓ ≠ ℎ , 𝑣 ( 𝑠 ) = ( 𝜙 1 ( 𝑠 ) 𝑣 1 , … , 𝜙 ℎ ( 𝑠 ) 𝑣 ℎ , … , 𝜙 𝑛 ( 𝑠 ) 𝑣 𝑛 ) ∈ 𝔉 for all 𝑠 ∈ [ 𝑠 0 , 1 ] ( 0 ≤ 𝑠 0 < 1 ) , l i m 𝑠 → 1 − ( 𝑑 / 𝑑 𝑠 ) 𝐷 𝑗 ( 𝑣 ( 𝑠 ) ) < + ∞ for all 𝑗 ∈ { 1 , … , 𝑚 } , l i m 𝑠 → 1 − ( 𝑑 / 𝑑 𝑠 ) 𝐷 𝑗 ( 𝑣 ( 𝑠 ) ) = − ∞ for some 𝑗 ∈ { 1 , … , 𝑚 } .
The condition ( 𝑖 ℎ 1 6 ) assures that for the solutions 𝑢 = ( 𝑢 1 , … , 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 ( 𝑖 1 3 )−( 𝑖 1 5 ) 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 ℎ 1 6 ), 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 < ∞ , ̃ 𝑝 = 𝑁 𝑝 / ( 𝑁 − 𝑝 ) i f 𝑁 > 𝑝 , ̃ 𝑝 = ∞ otherwise.

Let us assume 𝑊 𝑊 = 0 1 , 𝑝 ( Ω ) 𝑛 ( 𝑛 ≥ 1 ) w i t h ‖ 𝑣 ‖ = 𝑛 ℓ = 1 Ω | | ∇ 𝑣 ℓ | | 𝑝 𝑑 𝑥 1 / 𝑝 𝑣 ∀ 𝑣 = 1 , … , 𝑣 𝑛 𝐵 ∈ 𝑊 , ℓ 𝑣 ℓ = 𝑝 − 1 Ω 𝑏 ℓ | | 𝑣 ℓ | | 𝑝 𝑑 𝑥 ∀ 𝑣 ℓ ∈ 𝑊 0 1 , 𝑝 ( Ω ) w h e r e 𝑏 ℓ ∈ 𝐿 ∞ ( Ω ) ⧵ { 0 } , 𝑏 ℓ 𝐵 ≥ 0 , ℓ ≡ 0 . ( 2 . 1 ) Moreover we consider the functionals 𝐴 (as in ( 𝑖 1 1 )) such that ∃ ̃ 𝑐 > 0 ∶ 𝐴 ( 𝑣 ) ≥ 𝑝 − 1 ̃ 𝑐 ‖ 𝑣 ‖ 𝑝 ∀ 𝑣 ∈ 𝑊 . ( 2 . 2 ) Let us use the notation 𝐻 𝜆 ( 𝑆 𝜆 a n d 𝑉 − 𝜆 , r e s p . ) instead of 𝐻 𝜆 𝜇 ( 𝑆 𝜆 𝜇 a n d 𝑉 − 𝜆 𝜇 , r e s p . ) .

As ℓ = 1 , … , 𝑛 let 𝜆 ∗ ℓ a n d 𝑢 ∗ ℓ , respectively, the first eigenvalue and the first eigenfunction of the problem: 𝑢 ℓ ∈ 𝑊 0 1 , 𝑝 | | ( Ω ) ∶ − ̃ 𝑐 d i v ∇ 𝑢 ℓ | | 𝑝 − 2 ∇ 𝑢 ℓ = 𝜃 𝑏 ℓ | | 𝑢 ℓ | | 𝑝 − 2 𝑢 ℓ i n Ω . ( 2 . 3 ) Let us remember that [4] 𝑢 ∗ ℓ ∈ 𝐶 1 , 𝛼 ℓ ( Ω ) with 0 < 𝛼 ℓ < 1 , 𝑢 ∗ ℓ > 0 in Ω ; 𝜆 ∗ ℓ ∫ = ̃ 𝑐 Ω | ∇ 𝑢 ∗ ℓ | 𝑝 ∫ 𝑑 𝑥 / Ω 𝑏 ℓ | 𝑢 ∗ ℓ | 𝑝 ∫ 𝑑 𝑥 = m i n { ̃ 𝑐 Ω | ∇ 𝑣 ℓ | 𝑝 ∫ 𝑑 𝑥 / Ω 𝑏 ℓ | 𝑣 ℓ | 𝑝 ∫ 𝑑 𝑥 ∶ Ω 𝑏 ℓ | 𝑣 ℓ | 𝑝 𝑑 𝑥 > 0 } ; 𝜆 ∗ ℓ is simple, that is, each eigenfunction of (2.3) related to 𝜆 ∗ ℓ is of the type 𝑐 ℓ 𝑢 ∗ 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 = m a x { 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 𝜆 1 , … , 𝜆 𝑛 nonnegative.

Let us start by presenting some sufficient conditions such that ( 𝑖 1 3 ) , ( 𝑖 1 4 ) , and ( 𝑖 1 5 ) hold.

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

Proposition 2.2. If 𝜆 ℓ < 𝜆 ∗ ℓ f o r a l l ℓ ∈ { 1 , … , 𝑛 } , then ( 𝑖 1 3 ) holds. Consequently, ( 𝑖 1 4 ) holds when 𝑉 + ( 𝐷 𝑚 ) ≠ ∅ .

When 𝜆 ℓ ≥ 𝜆 ∗ ℓ   for some ℓ ∈ { 1 , … , 𝑛 } , it is possible to fulfil ( 𝑖 1 4 ) with an additional condition on 𝐷 𝑚 . Let 𝐼 = { 1 , … , 𝑛 } . For any 𝐼 ∗ ⊆ 𝐼 let 𝑉 ∗ = 𝑣 𝑣 = 1 , … , 𝑣 𝑛 ∈ 𝑊 ∶ 𝑣 ℓ ≡ 0 i f ℓ ∈ 𝐼 ⧵ 𝐼 ∗ , 𝑣 ℓ = 𝑐 ℓ 𝑢 ∗ ℓ i f ℓ ∈ 𝐼 ∗ w i t h 𝑐 ℓ ∈ 𝑅 a n d 𝑐 ℓ , ≠ 0 f o r s o m e ℓ ( 2 . 4 ) and let us suppose ( 𝑖 2 1 ) There exists 𝐼 ∗ ⊆ 𝐼 ∶ 𝐷 𝑚 ( 𝑣 ) < 0 f o r a l l 𝑣 ∈ 𝑉 ∗ .

Proposition 2.3. Let ( 𝑖 2 1 ) 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 ( 𝑖 1 4 ) 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 𝑘 ℓ ) ℓ ∈ 𝐼 ∗ ∈ 𝑋 ℓ ∈ 𝐼 ∗ [ 𝜆 ∗ ℓ , 𝜆 ∗ ℓ + 𝑘 − 1 [ and 𝑣 𝑘 = ( 𝑣 𝑘 1 , … , 𝑣 𝑘 𝑛 ) ∈ 𝑉 + ( 𝐷 𝑚 ) such that 𝐴 𝑣 𝑘 − 𝑝 − 1 ℓ ∈ 𝐼 ⧵ 𝐼 ∗ 𝜆 ℓ Ω 𝑏 ℓ | | 𝑣 ℓ | | 𝑝 𝑑 𝑥 − 𝑝 − 1 ℓ ∈ 𝐼 ∗ 𝜆 𝑘 ℓ Ω 𝑏 ℓ | | 𝑣 𝑘 ℓ | | 𝑝 𝑑 𝑥 < 𝑘 − 1 ‖ ‖ 𝑣 𝑘 ‖ ‖ 𝑝 . ( 2 . 5 ) Set 𝑤 𝑘 = ‖ 𝑣 𝑘 ‖ − 1 𝑣 𝑘 , we have 𝐷 𝑚 𝑤 𝑘 > 0 , ̃ 𝑐 ℓ ∈ 𝐼 ⧵ 𝐼 ∗ Ω | | ∇ 𝑤 𝑘 ℓ | | 𝑝 𝑑 𝑥 − ℓ ∈ 𝐼 ⧵ 𝐼 ∗ 𝜆 ℓ Ω 𝑏 ℓ | | 𝑤 𝑘 ℓ | | 𝑝 𝑑 𝑥 + ̃ 𝑐 ℓ ∈ 𝐼 ∗ Ω | | ∇ 𝑤 𝑘 ℓ | | 𝑝 𝑑 𝑥 − ℓ ∈ 𝐼 ∗ 𝜆 𝑘 ℓ Ω 𝑏 ℓ | | 𝑤 𝑘 ℓ | | 𝑝 𝑑 𝑥 < 𝑝 𝑘 − 1 , ( 2 . 6 ) moreover, since ‖ 𝑤 𝑘 ‖ = 1 , there exists 𝑤 ∈ 𝑊 such that (within a subsequence) 𝑤 𝑘 ⟶ 𝑤 w e a k l y i n 𝑊 , 𝑤 𝑘 ⟶ 𝑤 s t r o n g l y i n ( 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 ∗ ̃ 𝑐 Ω | | ∇ 𝑤 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 ℓ Ω 𝑏 ℓ | | 𝑤 ℓ | | 𝑝 + 𝑑 𝑥 ℓ ∈ 𝐼 ∗ ̃ 𝑐 Ω | | ∇ 𝑤 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 ∗ ℓ Ω 𝑏 ℓ | | 𝑤 ℓ | | 𝑝 𝑑 𝑥 ≤ 0 . ( 2 . 9 ) Since 𝑤 ℓ ≢ 0 ⟹ ̃ 𝑐 Ω | | ∇ 𝑤 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 ℓ Ω 𝑏 ℓ | | 𝑤 ℓ | | 𝑝 𝑑 𝑥 > 0 , ̃ 𝑐 Ω | | ∇ 𝑤 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 ∗ ℓ Ω 𝑏 ℓ | | 𝑤 ℓ | | 𝑝 𝑑 𝑥 ≥ 0 , ( 2 . 1 0 ) from (2.9), we deduce that 𝑤 ℓ ≡ 0 ∀ ℓ ∈ 𝐼 ⧵ 𝐼 ∗ , ∀ ℓ ∈ 𝐼 ∗ ∃ 𝑐 ℓ ∈ 𝑅 ∶ 𝑤 ℓ = 𝑐 ℓ 𝑢 ∗ ℓ . ( 2 . 1 1 ) Let us add that 𝑐 ℓ ≠ 0 for some ℓ ∈ 𝐼 ∗ , since if 𝑐 ℓ = 0 f o r a l l ℓ ∈ 𝐼 ∗ we have the contradiction ̃ 𝑐 = ̃ 𝑐 l i m 𝑘 → + ∞ ‖ 𝑤 𝑘 ‖ 𝑝 = 0 . Then 𝑤 ∈ 𝑉 ∗ , and consequently 𝐷 𝑚 ( 𝑤 ) < 0 from ( 𝑖 2 1 ). This last inequality contradicts (2.8).

In the same way the following propositions can be proved.

Proposition 2.4. Let ( 𝑖 2 1 ) 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 ( 𝑖 1 5 ) and suppose( 𝑖 2 2 ) there exist 𝐼 ∗ ⊆ 𝐼 and 𝑚 1 ∈ { 1 , … , 𝑚 } such that 𝐷 𝑚 1 ( 𝑣 ) < 0 and 𝐴 ( 𝑣 ) = ̃ 𝑐 𝑝 − 1 ∑ ℓ ∈ 𝐼 ∗ ∫ Ω | ∇ 𝑣 ℓ | 𝑝 𝑑 𝑥 for any 𝑣 ∈ 𝑉 ∗ .

Proposition 2.5. If (i22) holds with 𝐼 ∗ ≠ 𝐼 , then 𝑉 − 𝜆 𝐷 ∩ 𝑆 𝑚 1 𝜆 ≠ ∅ ∀ ℓ ℓ ∈ 𝐼 𝜆 w i t h ℓ ℓ ∈ 𝐼 ∗ ∈ 𝑋 ℓ ∈ 𝐼 ∗ 𝜆 ∗ ℓ ⧵ 𝜆 , + ∞ ∗ ℓ ℓ ∈ 𝐼 ∗ . ( 2 . 1 2 ) 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 𝐷 ∩ 𝑆 𝑚 1 𝜆 i s b o u n d e d i n 𝑊 ∀ ℓ ℓ ∈ 𝐼 ∗ ∈ 𝑋 ℓ ∈ 𝐼 ∗ 𝜆 ∗ ℓ , 𝜆 ∗ ℓ + 𝛿 ∗ ⧵ 𝜆 ∗ ℓ ℓ ∈ 𝐼 ∗ . ( 2 . 1 3 )

Proof. Let us prove (2.12). Let 𝑣 ∈ 𝑉 ∗ with 𝑣 ℓ = 𝑢 ∗ ℓ if ℓ ∈ 𝐼 ∗ , then 𝐷 𝑚 1 ( 𝑣 ) < 0 . Let 𝑤 = | 𝐷 𝑚 1 ( 𝑣 ) | − 1 ⧵ 𝑞 𝑚 1 𝑣 , we have 𝐷 𝑚 1 | | 𝐷 ( 𝑤 ) = 𝑚 1 | | ( 𝑣 ) − 1 𝐷 𝑚 1 𝐻 ( 𝑣 ) = − 1 , 𝜆 ( 𝑤 ) = 𝑝 − 1 ℓ ∈ 𝐼 ∗ ̃ 𝑐 Ω | | ∇ 𝑤 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 ℓ Ω 𝑏 ℓ | | 𝑤 ℓ | | 𝑝 𝑑 𝑥 < 0 . ( 2 . 1 4 ) Let us prove (2.13). Arguing by contradiction, for any 𝑘 ∈ ℕ there exist ( 𝜆 𝑘 ℓ ) ℓ ∈ 𝐼 ∗ ∈ 𝑋 ℓ ∈ 𝐼 ∗ [ 𝜆 ∗ ℓ , 𝜆 ∗ ℓ + 𝑘 − 1 [ with ( 𝜆 𝑘 ℓ ) ℓ ∈ 𝐼 ∗ ≠ ( 𝜆 ∗ ℓ ) ℓ ∈ 𝐼 ∗ and ( 𝑣 𝑘 , ℎ ) ℎ ∈ ℕ ⊆ 𝑉 − 𝜆 𝑘 ∩ 𝑆 ( 𝐷 𝑚 1 ) , where 𝜆 𝑘 ℓ = 𝜆 ℓ if ℓ ∈ 𝐼 ⧵ 𝐼 ∗ , such that s u p ℎ ∈ ℕ ‖ ‖ 𝑣 𝑘 , ℎ ‖ ‖ = + ∞ . ( 2 . 1 5 ) Relation (2.15) implies that there exists ( ℎ 𝑘 ) 𝑘 ∈ ℕ ⊆ ℕ strictly increasing such that 𝛿 𝑘 = ‖ ‖ 𝑣 𝑘 , ℎ 𝑘 ‖ ‖ ⟶ + ∞ a s 𝑘 ⟶ + ∞ . ( 2 . 1 6 ) Let 𝑤 𝑘 = 𝛿 𝑘 − 1 𝑣 𝑘 , ℎ 𝑘 , we have ℓ ∈ 𝐼 ⧵ 𝐼 ∗ ̃ 𝑐 Ω | | ∇ 𝑤 𝑘 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 ℓ Ω 𝑏 ℓ | | 𝑤 𝑘 ℓ | | 𝑝 + 𝑑 𝑥 ℓ ∈ 𝐼 ∗ ̃ 𝑐 Ω | | ∇ 𝑤 𝑘 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 𝑘 ℓ Ω 𝑏 ℓ | | 𝑤 𝑘 | | 𝑝 𝐷 𝑑 𝑥 < 0 , 𝑚 1 𝑤 𝑘 = − 𝛿 − 𝑞 𝑚 1 𝑘 , ∃ 𝑤 ∈ 𝑊 ∶ ( w i t h i n a s u b s e q u e n c e ) 𝑤 𝑘 ⟶ 𝑤 w e a k l y i n 𝑊 , 𝑤 𝑘 ⟶ 𝑤 s t r o n g l y i n ( 𝐿 𝑝 ( Ω ) ) 𝑛 . ( 2 . 1 7 ) Then, as 𝑘 → + ∞ we get ℓ ∈ 𝐼 ⧵ 𝐼 ∗ ̃ 𝑐 Ω | | ∇ 𝑤 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 ℓ Ω 𝑏 ℓ | | 𝑤 ℓ | | 𝑝 + 𝑑 𝑥 ℓ ∈ 𝐼 ∗ ̃ 𝑐 Ω | | ∇ 𝑤 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 ∗ ℓ Ω 𝑏 ℓ | | 𝑤 ℓ | | 𝑝 𝐷 𝑑 𝑥 ≤ 0 , ( 2 . 1 8 ) 𝑚 1 ( 𝑤 ) = 0 . ( 2 . 1 9 ) From (2.18), we get that 𝑤 ∈ 𝑉 ∗ . Then since ( 𝑖 2 2 ) inequality 𝐷 𝑚 1 ( 𝑤 ) < 0 holds, which contradicts (2.19).

Proposition 2.6. If ( 𝑖 2 2 ) holds with 𝐼 ∗ = 𝐼 , then 𝑉 − 𝜆 𝐷 ∩ 𝑆 𝑚 1 𝜆 ≠ ∅ ∀ 𝜆 = ℓ ℓ ∈ 𝐼 ∈ 𝑋 ℓ ∈ 𝐼 𝜆 ∗ ℓ ⧵ 𝜆 , + ∞ ∗ ℓ ℓ ∈ 𝐼 , ∃ 𝛿 ∗ > 0 ∶ 𝑉 − 𝜆 𝐷 ∩ 𝑆 𝑚 1 𝜆 i s b o u n d e d i n 𝑊 ∀ 𝜆 = ℓ ℓ ∈ 𝐼 ∈ 𝑋 ℓ ∈ 𝐼 𝜆 ∗ ℓ , 𝜆 ∗ ℓ + 𝛿 ∗ ⧵ 𝜆 ∗ ℓ ℓ ∈ 𝐼 . ( 2 . 2 0 )
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 ̃ 𝑐 = 1 , and we suppose 𝑏 ℓ ∈ 𝐿 ∞ ( Ω ) ⧵ { 0 } , 𝑏 ℓ ≥ 0 .

Application 2.8. Let 𝑛 = 1 . Let us consider the problem | | | | − d i v ∇ 𝑢 𝑝 − 2 ∇ 𝑢 = 𝜆 1 𝑏 1 | 𝑢 | 𝑝 − 2 𝑢 + 𝑚 𝑗 = 1 𝑑 𝑗 | 𝑢 | 𝑞 𝑗 − 2 𝑢 i n Ω , 𝑢 = 0 o n 𝜕 Ω , ( 2 . 2 1 ) where 𝑝 < 𝑞 1 < ̃ 𝑝 , 𝑑 1 ∈ 𝐿 ∞ ( Ω ) ⧵ { 0 } i f 𝑚 = 1 , 𝑝 < 𝑞 1 < ⋯ < 𝑞 𝑚 < ̃ 𝑝 , 𝑑 𝑗 ∈ 𝐿 ∞ 𝑑 ( Ω ) ⧵ { 0 } a s 𝑗 = 1 , … , 𝑚 , 𝑗 ≤ 0 a s 𝑗 = 1 , … , 𝑚 − 1 i f 𝑚 > 1 . ( 2 . 2 2 ) Evidently 𝐴 ( 𝑣 ) = 𝑝 − 1 Ω | | | | ∇ 𝑣 𝑝 𝑑 𝑥 , 𝐷 𝑗 ( 𝑣 ) = 𝑞 𝑗 − 1 Ω 𝑑 𝑗 | 𝑣 | 𝑞 𝑗 𝑑 𝑥 ∀ 𝑣 ∈ 𝑊 . ( 2 . 2 3 ) Let us advance the conditions: 𝑑 + 𝑚 ≢ 0 ⟹ 𝑉 + 𝐷 𝑚 ≠ ∅ , ( 2 . 2 4 ) Ω 𝑑 𝑚 𝑢 ∗ 1 𝑞 𝑚 𝑑 𝑥 < 0 ⟹ 𝐷 𝑚 𝑐 1 𝑢 ∗ 1 < 0 ∀ 𝑐 1 . ∈ 𝑅 ⧵ { 0 } ( 2 . 2 5 ) Let us note that (Propositions 2.2, 2.4, and 2.6) 𝑖 ( 2 . 2 4 ) ⟹ 1 4 h o l d s i f 𝜆 1 < 𝜆 ∗ 1 , ( 2 . 2 4 ) a n d ( 2 . 2 5 ) ⟹ ∃ 𝛿 ∗ 1 𝑖 > 0 ∶ 1 4 h o l d s i f 𝜆 1 < 𝜆 ∗ 1 + 𝛿 ∗ 1 , ( 2 . 2 5 ) ⟹ ∃ 𝛿 ∗ 2 𝑖 > 0 ∶ 1 5 h o l d s i f 𝜆 1 ∈ 𝜆 ∗ 1 , 𝜆 ∗ 1 + 𝛿 ∗ 2 . ( 2 . 2 6 )

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 𝜆 1 < 𝜆 ∗ 1 [resp. (2.24) and (2.25) hold, with 𝜆 1 < 𝜆 ∗ 1 + 𝛿 ∗ 1 ] problem (2.21) has at least two weak solutions 𝑢 0 and − 𝑢 0 ( 𝑢 0 = 𝜏 0 𝑣 0 , 𝜏 0 = c o n s t . > 0 , 𝑣 0 ∈ 𝑆 𝜆 1 ∩ 𝑉 + ( 𝐷 𝑚 ) ), and it results in 𝑢 0 ∈ 𝐿 ∞ ( Ω ) ∩ 𝐶 1 , 𝛼 0 ℓ 𝑜 𝑐 ( Ω ) , 𝑢 0 > 0 ;(ii)When (2.25) holds, with 𝜆 1 ∈ ] 𝜆 ∗ 1 , 𝜆 ∗ 1 + 𝛿 ∗ 2 [ problem (2.21) has at least two weak solutions 𝑢 𝑎 𝑛 𝑑 − 𝑢 ( 𝑢 = 𝜏 𝑣 , 𝜏 = c o n s t . > 0 , 𝑣 ∈ 𝑉 − 𝜆 1 ∩ 𝑆 ( 𝐷 𝑚 ) ), and it results in 𝑢 ∈ 𝐿 ∞ ( Ω ) ∩ 𝐶 1 , 𝛼 ℓ 𝑜 𝑐 ( Ω ) , 𝑢 > 0 .
Consequently, when (2.24) and (2.25) hold, with 𝜆 1 ∈ ] 𝜆 ∗ 1 , 𝜆 ∗ 1 + m i n { 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: | | − d i v ∇ 𝑢 𝑖 | | 𝑝 − 2 ∇ 𝑢 𝑖 = 𝜆 𝑖 𝑏 𝑖 | | 𝑢 𝑖 | | 𝑝 − 2 𝑢 𝑖 + | | | | | 𝑛 ℓ = 1 𝑑 ℓ 𝑢 ℓ | | | | | 𝑞 1 − 2 𝑛 ℓ = 1 𝑑 ℓ 𝑢 ℓ 𝑑 𝑖 − 𝑑 𝑖 | | 𝑢 𝑖 | | 𝑞 1 − 2 𝑢 𝑖 𝑢 i n Ω , 𝑖 = 0 o n 𝜕 Ω a s 𝑖 = 1 , … , 𝑛 , ( 2 . 2 7 ) where 1 < 𝑞 1 < ̃ 𝑝 , 𝑞 1 ≠ 𝑝 , 𝑑 ℓ , 𝑑 ℓ ∈ 𝐿 ∞ ( Ω ) , 𝑑 ℓ , 𝑑 ℓ > 0 . ( 2 . 2 8 ) System (2.27) is included among Problem (P) with: 𝐴 ( 𝑣 ) = 𝑝 𝑛 − 1 ℓ = 1 Ω | | ∇ 𝑣 ℓ | | 𝑝 𝐷 𝑑 𝑥 , 1 ( 𝑣 ) = 𝑞 1 − 1 ⎡ ⎢ ⎢ ⎣ Ω | | | | | 𝑛 ℓ = 1 𝑑 ℓ 𝑣 ℓ | | | | | 𝑞 1 𝑑 𝑥 − 𝑛 ℓ = 1 Ω 𝑑 ℓ | | 𝑣 ℓ | | 𝑞 1 ⎤ ⎥ ⎥ ⎦ 𝑣 𝑑 𝑥 ∀ 𝑣 = 1 , … , 𝑣 𝑛 ∈ 𝑊 . ( 2 . 2 9 ) Let us advance the conditions (compatible): 𝑑 𝑞 1 ℓ < 𝑑 ℓ ∀ ℓ ∈ { 1 , … , 𝑛 } ⟹ 𝐷 1 0 , … , 𝑐 𝑖 𝑢 ∗ 𝑖 , … , 0 < 0 a s 𝑖 = 1 , … , 𝑛 , 𝑐 𝑖 ∈ 𝑅 ⧵ { 0 } , ( 2 . 3 0 ) there exist Ω + ⊆ Ω and a constant ̃ 𝑐 𝑗 > 0 such that | Ω + | 𝑁 > 0 and ℓ ≠ 𝑗 𝑑 ℓ + ̃ 𝑐 𝑗 𝑑 𝑗 𝑞 1 > ℓ ≠ 𝑗 𝑑 ℓ + ̃ 𝑐 𝑞 1 𝑗 𝑑 𝑗 i n Ω + ⟹ 𝑉 + 𝐷 1 . ≠ ∅ ( P r o p o s i t i o n A . 1 ) ( 2 . 3 1 ) Then (Propositions 2.2, 2.3, and 2.5) 𝑖 ( 2 . 3 1 ) ⟹ 1 4 h o l d s i f 𝜆 ℓ < 𝜆 ∗ ℓ ∀ ℓ ∈ { 1 , … , 𝑛 } , ( 2 . 3 2 ) and set 𝑖 ∈ { 1 , … , 𝑛 } ( 2 . 3 0 ) a n d ( 2 . 3 1 ) ⟹ w i t h 𝜆 ℓ < 𝜆 ∗ ℓ ∀ ℓ ≠ 𝑖 ∃ 𝛿 ∗ 1 𝑖 > 0 ∶ 1 4 h o l d s i f 𝜆 𝑖 < 𝜆 ∗ 𝑖 + 𝛿 ∗ 1 ( , ( 2 . 3 3 ) 2 . 3 0 ) ⟹ w i t h 𝜆 ℓ < 𝜆 ∗ ℓ ∀ ℓ ≠ 𝑖 ∃ 𝛿 ∗ 2 𝑖 > 0 ∶ 1 5 h o l d s i f 𝜆 𝑖 ∈ 𝜆 ∗ 𝑖 , 𝜆 ∗ 𝑖 + 𝛿 ∗ 2 . ( 2 . 3 4 )

Taking into account that 𝐷 1 ( 𝑣 1 , … , 𝑣 𝑛 ) ≤ 𝐷 1 ( | 𝑣 1 | , … , | 𝑣 𝑛 | ) and 𝐷 1 ( − 𝑣 ) = 𝐷 1 ( 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 𝜆 1 , … , 𝜆 𝑛 as in (2.32) (resp. (2.33)) system (2.27) has at least two weak solutions 𝑢 0 and − 𝑢 0 with 𝑢 0 ℓ ≥ 0 as ℓ = 1 , … , 𝑛 ( 𝑢 0 = 𝜏 0 𝑣 0 , 𝜏 0 = c o n s t . > 0 , 𝑣 0 ∈ 𝑆 𝜆 ∩ 𝑉 + ( 𝐷 1 ) ) ; (ii)When (2.30) holds, choosing 𝜆 1 , … , 𝜆 𝑛 as in (2.34) system (2.27) has at least two weak solutions 𝑢 and − 𝑢 ( 𝑢 = 𝜏 𝑣 , 𝜏 = c o n s t . > 0 , 𝑣 ∈ 𝑉 − 𝜆 ∩ 𝑆 ( 𝐷 1 ) ) .
Consequently, when (2.30) and (2.31) hold, with 𝜆 ℓ < 𝜆 ∗ ℓ f o r a l l ℓ ≠ 𝑖 and 𝜆 𝑖 ∈ ] 𝜆 ∗ 𝑖 , 𝜆 ∗ 𝑖 + m i n { 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: 𝑢 0 𝑖 ≢ 0 a s 𝑖 = 1 , … , 𝑛 , ∃ ℎ , 𝑘 ∈ { 1 , … , 𝑛 } ∶ 𝑢 ℎ ≢ 0 , 𝑢 𝑘 ≢ 0 . ( 2 . 3 5 )

Proposition 2.14. If 𝑝 < 𝑞 1 , then as 𝑖 = 1 , … , 𝑛 : 𝑢 0 𝑖 ∈ 𝐿 ∞ ( Ω ) ∩ 𝐶 1 , 𝛼 0 𝑖 ℓ 𝑜 𝑐 ( Ω ) , 𝑢 0 𝑖 > 0 . ( 2 . 3 6 )

Proof. It is easy to prove that 𝑛 𝑖 = 1 Ω | | ∇ 𝑢 0 𝑖 | | 𝑝 − 2 ∇ 𝑢 0 𝑖 ⋅ ∇ 𝑣 𝑖 𝑑 𝑥 ≤ Ω 𝑔 𝑛 𝑖 = 1 𝑢 0 𝑖 𝑝 − 1 𝑛 𝑖 = 1 𝑣 𝑖 𝑣 𝑑 𝑥 ∀ 𝑣 = 1 , … , 𝑣 𝑛 ∈ 𝑊 0 1 , 𝑝 ( Ω ) ∩ 𝐿 ∞ ( Ω ) 𝑛 w i t h 𝑣 𝑖 ≥ 0 , ( 2 . 3 7 ) where 𝑔 ∈ 𝐿 𝑞 1 / ( 𝑞 1 − 𝑝 ) ( Ω ) . Then (Proposition A.3) 𝑢 0 𝑖 ∈ 𝐿 ∞ ( Ω ) and consequently [5] 𝑢 0 𝑖 ∈ 𝐶 1 , 𝛼 0 𝑖 ℓ o c ( Ω ) .
Let us note that 𝑢 0 𝑖 is a weak supersolution to the equation: | | − d i v ∇ 𝑢 𝑖 | | 𝑝 − 2 ∇ 𝑢 𝑖 = 𝜆 𝑖 𝑏 𝑖 | | 𝑢 𝑖 | | 𝑝 − 2 𝑢 𝑖 − 𝑑 𝑖 | | 𝑢 𝑖 | | 𝑞 1 − 2 𝑢 𝑖 i n Ω . ( 2 . 3 8 ) Then, since (2.35), it must be [6] 𝑢 0 𝑖 > 0 .

Let us continue the analysis of system (2.27) under the condition: ℓ ≠ 𝑖 𝑑 ℓ 𝑞 1 𝑑 < m i n 1 𝑑 , … , 𝑛 ∀ 𝑖 ∈ { 1 , … , 𝑛 } , ( 2 . 3 9 ) then 𝐷 1 𝑐 1 𝑢 ∗ 1 , … , 𝑐 𝑛 𝑢 ∗ 𝑛 𝑐 < 0 ∀ 1 , … , 𝑐 𝑛 ∈ 𝑅 𝑛 ⧵ { 0 } w i t h 𝑐 𝑖 = 0 f o r a t l e a s t o n e 𝑖 ∈ { 1 , … , 𝑛 } . ( 2 . 4 0 ) Hence (Proposition 2.5) if 𝐼 ∗ ⊆ 𝐼 and 𝐼 ∗ ≠ 𝐼 : ( 2 . 3 9 ) ⟹ a s 𝜆 ℓ < 𝜆 ∗ ℓ ∀ ℓ ∈ 𝐼 ⧵ 𝐼 ∗ ∃ 𝛿 ∗ 𝑖 > 0 ∶ 1 5 𝜆 h o l d s i f 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 𝜆 1 , … , 𝜆 𝑛 as in (2.41) system (2.27) has at least two weak solutions 𝑢 and − 𝑢 ( 𝑢 = 𝜏 𝑣 , 𝜏 = c o n s t . > 0 , 𝑣 ∈ 𝑉 − 𝜆 ∩ 𝑆 ( 𝐷 1 ) ) with 𝑢 𝑖 ≢ 0 as 𝑖 = 1 , … , 𝑛 .

Proof. Thanks to ([1], Theorem 4.1), there exists 𝑣 ∈ 𝑉 − 𝜆 ∩ 𝑆 ( 𝐷 1 ) such that 𝐻 𝜆 𝑣 𝐻 = i n f 𝜆 ( 𝑣 ) ∶ 𝑣 ∈ 𝑉 − 𝜆 𝐷 ∩ 𝑆 1 = 𝑒 , 𝑢 = 𝜏 𝑣 i s a w e a k s o l u t i o n o f s y s t e m ( 2 . 2 7 ) , ( 2 . 4 2 ) where 𝜏 = ( − 𝑝 𝑞 1 − 1 𝑒 ) 1 / ( 𝑞 1 − 𝑝 ) .
Reasoning by contradiction, let, for example, 𝑢 1 ≡ 0 . Since − 1 = 𝐷 1 ( 𝑣 ) ≤ 𝐷 1 ( 0 , | 𝑣 2 | , … , | 𝑣 𝑛 | ) and from (2.39) 𝐷 1 ( 0 , | 𝑣 2 | , … , | 𝑣 𝑛 | ) < 0 , setting 𝛿 = | 𝐷 1 ( 0 , | 𝑣 2 | , … , | 𝑣 𝑛 | ) | − 1 / 𝑞 1 we have 𝐷 1 | | 0 , 𝛿 𝑣 2 | | | | , … , 𝛿 𝑣 𝑛 | | = − 1 , 𝐻 𝜆 | | 0 , 𝛿 𝑣 2 | | | | , … , 𝛿 𝑣 𝑛 | | = 𝛿 𝑝 𝐻 𝜆 𝑣 ≤ 𝐻 𝜆 𝑣 , ( 2 . 4 3 ) then 𝐻 𝜆 ( 0 , 𝛿 | 𝑣 2 | , … , 𝛿 | 𝑣 𝑛 | ) = 𝐻 𝜆 ( 𝑣 ) . This implies that ([1], see the proof of Theorem 4.1) ( 0 , 𝜏 𝛿 | 𝑣 2 | , … , 𝜏 𝛿 | 𝑣 𝑛 | ) is a weak solution of system (2.27). Then ( ∑ 𝑛 ℓ = 2 𝑑 ℓ | 𝑣 ℓ | ) 𝑞 1 − 1 ≡ 0 from which 𝑢 ℓ ≡ 0 too as ℓ = 2 , … , 𝑛 .
Condition (2.39) holds in particular when 𝑛 ℓ = 1 𝑑 ℓ 𝑞 1 𝑑 < m i n 1 𝑑 , … , 𝑛 . ( 2 . 4 4 )

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 < 𝑞 1 𝑢 𝑖 ∈ 𝐿 ∞ ( Ω ) ∩ 𝐶 1 , 𝑎 𝑖 ℓ 𝑜 𝑐 ( Ω ) , 𝑢 𝑖 > 0 a s 𝑖 = 1 , … , 𝑛 . ( 2 . 4 5 )

Proof. Set 𝛿 = | 𝐷 1 ( | 𝑣 1 | , … , | 𝑣 𝑛 | ) | − 1 / 𝑞 1 , as in Proposition 2.15 ( 𝜏 𝛿 | 𝑣 1 | , … , 𝜏 𝛿 | 𝑣 𝑛 | ) is a weak solution to system (2.27).
Let us add that since (2.44) ⇒ 𝐷 1 ( 𝑐 1 𝑢 ∗ 1 , … , 𝑐 𝑛 𝑢 ∗ 𝑛 ) < 0 f o r a l l ( 𝑐 1 , … , 𝑐 𝑛 ) ∈ 𝑅 𝑛 ⧵ { 0 } , there exists (Proposition 2.6) 𝛿 ∗ ∗ > 0 such that 𝑖 1 5 𝜆 h o l d s i f ℓ ℓ ∈ 𝐼 ∈ 𝑛 𝑋 ℓ = 1 𝜆 ∗ ℓ , 𝜆 ∗ ℓ + 𝛿 ∗ ∗ ⧵ 𝜆 ∗ ℓ ℓ ∈ 𝐼 . ( 2 . 4 6 ) Then the existence of 𝑢 is assured also choosing 𝜆 1 , … , 𝜆 𝑛 as in (2.46), and the conclusions of Proposition 2.16 hold.

Application 2.17. Let us set 𝜆 1 = ⋯ = 𝜆 𝑛 = 𝜆 , 𝑏 1 = ⋯ = 𝑏 𝑛 = 𝑏 t h e n 𝜆 ∗ 1 = ⋯ = 𝜆 ∗ 𝑛 = 𝜆 ∗ , 𝑢 ∗ 1 = ⋯ = 𝑢 ∗ 𝑛 = 𝑢 ∗ , 𝐴 ( 𝑣 ) = 𝑝 𝑛 − 1 ℓ = 1 Ω | | ∇ 𝑣 ℓ | | 𝑝 𝑑 𝑥 , 𝐷 1 ( 𝑣 ) = 𝑞 1 − 1 Ω 𝑑 1 𝑛 ℓ = 1 | | 𝑣 ℓ | | 𝛾 𝑞 1 / 𝛾 𝑣 𝑑 𝑥 , ∀ 𝑣 = 1 , … , 𝑣 𝑛 ∈ 𝑊 , ( 2 . 4 7 ) where 1 < 𝛾 < 𝑞 1 < ̃ 𝑝 , 𝑞 1 ≠ 𝑝 , 𝑑 1 ∈ 𝐿 ∞ ( Ω ) . ( 2 . 4 8 ) Let us consider the system: | | − d i v ∇ 𝑢 𝑖 | | 𝑝 − 2 ∇ 𝑢 𝑖 = | | 𝑢 𝜆 𝑏 𝑖 | | 𝑝 − 2 𝑢 𝑖 + 𝑑 1 𝑛 ℓ = 1 | | 𝑢 ℓ | | 𝛾 ( 𝑞 1 / 𝛾 ) − 1 | | 𝑢 𝑖 | | 𝛾 − 2 𝑢 𝑖 𝑢 i n Ω , 𝑖 = 0 o n 𝜕 Ω a s 𝑖 = 1 , … , 𝑛 . ( 2 . 4 9 ) We advance the conditions 𝑑 + 1 ≢ 0 ⟹ 𝑉 + 𝐷 1 ≠ ∅ , ( 2 . 5 0 ) Ω 𝑑 1 𝑢 ∗ 𝑞 1 𝑑 𝑥 < 0 ⟹ 𝐷 1 𝑐 1 𝑢 ∗ , … , 𝑐 𝑛 𝑢 ∗ 𝑐 < 0 ∀ 1 , … , 𝑐 𝑛 ∈ 𝑅 𝑛 ⧵ . { 0 } ( 2 . 5 1 ) Therefore, 𝑖 ( 2 . 5 0 ) ⟹ 1 4 h o l d s i f 𝜆 < 𝜆 ∗ ( P r o p o s i t i o n 2 . 2 ) , ( 2 . 5 0 ) a n d ( 2 . 5 1 ) ⟹ ∃ 𝛿 ∗ 1 𝑖 > 0 ∶ 1 4 h o l d s i f 𝜆 < 𝜆 ∗ + 𝛿 ∗ 1 ( P r o p o s i t i o n 2 . 4 ) , ( 2 . 5 1 ) ⟹ ∃ 𝛿 ∗ 2 𝑖 > 0 ∶ 1 5 h o l d s i f 𝜆 𝜆 ∈ ∗ , 𝜆 ∗ + 𝛿 ∗ 2 ( P r o p o s i t i o n 2 . 6 ) . ( 2 . 5 2 ) 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 ∗ ( r e s p . 𝜆 < 𝜆 ∗ + 𝛿 ∗ 1 ) system (2.49) has at least two weak solutions 𝑢 0 and − 𝑢 0 with 𝑢 0 ℓ ≥ 0 as ℓ = 1 , … , 𝑛 ( 𝑢 0 = 𝜏 0 𝑣 0 , 𝜏 0 = c o n s t . > 0 , 𝑣 0 ∈ 𝑆 𝜆 ∩ 𝑉 + ( 𝐷 1 ) ) ;(ii)When (2.51) holds , i f 𝜆 ∈ ] 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 = 𝜏 𝑣 , 𝜏 = c o n s t . > 0 , 𝑣 ∈ 𝑉 − 𝜆 ∩ 𝑆 ( 𝐷 1 ) ) .
Consequently, when (2.50) and (2.51) hold, with 𝜆 ∈ ] 𝜆 ∗ , 𝜆 ∗ + m i n { 𝛿 ∗ 1 , 𝛿 ∗ 2 } [ system (2.49) has at least four different weak solutions.

In order to establish some properties of 𝑢 0 and 𝑢 it is useful to recall that ([1], Theorems 2.1 and 4.1) 𝐷 1 𝑣 0 𝐷 = s u p 1 ( 𝑣 ) ∶ 𝑣 ∈ 𝑆 𝜆 ∩ 𝑉 + 𝐷 1 = 𝑒 , 𝜏 0 = 𝑞 1 𝑝 − 1 𝑒 1 / ( 𝑝 − 𝑞 1 ) , 𝐻 ( 2 . 5 3 ) 𝜆 𝑣 𝐻 = i n f 𝜆 ( 𝑣 ) ∶ 𝑣 ∈ 𝑉 − 𝜆 𝐷 ∩ 𝑆 1 = 𝑒 , 𝜏 = − 𝑝 𝑞 1 − 1 𝑒 1 / ( 𝑞 1 − 𝑝 ) . ( 2 . 5 4 )

Proposition 2.19. When 𝑝 < 𝑞 1 , we have 𝑢 0 𝑖 ∈ 𝐿 ∞ ( Ω ) ∩ 𝐶 1 , 𝛼 0 𝑖 ℓ 𝑜 𝑐 ( Ω ) , ( 2 . 5 5 ) besides 𝑢 0 𝑖 ≢ 0 ∀ 𝑖 ∈ { 1 , … , 𝑛 } i f 𝛾 < 𝑝 . ( 2 . 5 6 )

Proof. The relation 𝑢 0 𝑖 ∈ 𝐿 ∞ ( Ω ) comes from Proposition A.3. Then [5] 𝑢 0 𝑖 ∈ 𝐶 1 , 𝛼 0 𝑖 ℓ o c ( Ω ) .
About (2.56), it is sufficiently (Remark 1.1) to prove that 𝑖 ℎ 1 6 h o l d s ∀ ℎ ∈ { 1 , … , 𝑛 } w i t h 𝔉 = 𝑆 𝜆 ∩ 𝑉 + 𝐷 1 . ( 2 . 5 7 )
Let 𝑣 = ( 𝑣 1 , … , 𝑣 𝑛 ) ∈ 𝑆 𝜆 ∩ 𝑉 + ( 𝐷 1 ) with 𝑣 ℎ ≡ 0 . Since 𝑣 ∈ 𝑉 + 𝐷 1 ⟹ | | 𝕂 | | ∃ a c o m p a c t s e t 𝕂 ⊆ Ω ∶ 𝑁 > 0 , 𝑑 1 > 0 a n d 𝜓 = ℓ ≠ ℎ | | 𝑣 ℓ | | 𝛾 > 0 i n 𝕂 , ( 2 . 5 8 ) let (Proposition A.1) ( 𝜑 𝜀 ) 0 < 𝜀 < 𝜀 0 ⊆ 𝐶 ∞ 0 ( Ω ) with 0 ≤ 𝜑 𝜀 ≤ 1 such that 𝜑 𝜀 ⟶ 𝜒 s t r o n g l y i n 𝐿 𝑠 ( Ω ) , Ω | | ∇ 𝜑 𝜀 | | 𝑠 𝑑 𝑥 ⟶ + ∞ a s 𝜀 ⟶ 0 + [ [ , ∀ 𝑠 ∈ 1 , + ∞ ( 2 . 5 9 ) where 𝜒 is the characteristic function of 𝕂 . Set 𝜀 such that Ω 𝑑 1 𝜓 ( 𝑞 1 / 𝛾 ) − 1 𝜑 𝛾 𝜀 𝑑 𝑥 > 0 , 𝛿 = 𝑝 − 1 Ω | | ∇ 𝜑 𝜀 | | 𝑝 𝑑 𝑥 − 𝜆 Ω 𝑏 𝜑 𝑝 𝜀 𝑑 𝑥 > 0 , ( 2 . 6 0 ) with 𝑣 ( 𝑠 ) = ( 𝑠 1 / 𝑝 𝑣 1 , … , ( 1 − 𝑠 ) 1 / 𝑝 𝛿 − 1 / 𝑝 𝜑 𝜀 , … , 𝑠 1 / 𝑝 𝑣 𝑛 ) it results in 𝐻 𝜆 ( 𝑣 ( 𝑠 ) ) = 𝛿 − 1 ( 1 − 𝑠 ) 𝑝 − 1 Ω | | ∇ 𝜑 𝜀 | | 𝑝 𝑑 𝑥 − 𝜆 Ω 𝑏 𝜑 𝑝 𝜀 𝑑 𝑥 + 𝑠 𝐻 𝜆 [ ] , ( 𝑣 ) = 1 ∀ 𝑠 ∈ 0 , 1 ∃ 𝑠 0 ∈ [ [ 0 , 1 ∶ 𝐷 1 𝑠 ( 𝑣 ( 𝑠 ) ) > 0 ∀ 𝑠 ∈ 0 , 1 , l i m 𝑠 → 1 − 𝑑 𝐷 𝑑 𝑠 1 ( 𝑣 ( 𝑠 ) ) = − ∞ . ( 2 . 6 1 )

Proposition 2.20. When 𝑝 < 𝑞 1 , we have 𝑢 𝑖 ∈ 𝐿 ∞ ( Ω ) ∩ 𝐶 1 , 𝛼 𝑖 ℓ 𝑜 𝑐 ( Ω ) , ( 2 . 6 2 ) 𝑢 𝑖 > 0 ∀ 𝑖 ∈ { 1 , … , 𝑛 } i f 𝑝 < 𝛾 . ( 2 . 6 3 )

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, 𝑣 1 ≡ 0 . We note that 𝑣 ∈ 𝑉 − 𝜆 ⟹ ∃ ℓ ∈ { 2 , … , 𝑛 } ∶ Ω | | ∇ 𝑣 ℓ | | 𝑝 𝑑 𝑥 − 𝜆 Ω 𝑏 𝑣 𝑝 ℓ 𝑑 𝑥 < 0 . ( 2 . 6 4 ) Let us suppose ℓ = 2 and set 𝑣 ( 𝑠 ) = ( ( 1 − 𝑠 ) 1 / 𝛾 𝑣 2 , 𝑠 1 / 𝛾 𝑣 2 , 𝑣 3 , … , 𝑣 𝑛 ) . Then 𝐷 1 [ ] ( 𝑣 ( 𝑠 ) ) = − 1 ∀ 𝑠 ∈ 0 , 1 , ∃ 𝑠 0 ∈ [ [ 0 , 1 ∶ 𝐻 𝜆 𝑠 ( 𝑣 ( 𝑠 ) ) < 0 ∀ 𝑠 ∈ 0 , , 1 l i m 𝑠 → 1 − 𝑑 𝐻 𝑑 𝑠 𝜆 ( 𝑣 ( 𝑠 ) ) = + ∞ . ( 2 . 6 5 ) Set 𝑠 1 ∈ [ 𝑠 0 , 1 [ such that ( 𝑑 / 𝑑 𝑠 ) 𝐻 𝜆 ( 𝑣 ( 𝑠 ) ) > 0 f o r a l l 𝑠 ∈ [ 𝑠 1 , 1 [ and taking into account (2.54), we get the contradiction: 𝐻 𝜆 𝑣 ≤ 𝐻 𝜆 ( 𝑣 ( 𝑠 ) ) < 𝐻 𝜆 𝑣 𝑠 ∀ 𝑠 ∈ 1 , 1 . ( 2 . 6 6 )

Proposition 2.21. When 𝛾 = 𝑝 < 𝑞 1 , we allow that as 𝑖 = 1 , … , 𝑛 : 𝑢 0 𝑖 > 0 , 𝑢 𝑖 > 0 . ( 2 . 6 7 )

Proof. The assumption 𝛾 = 𝑝 implies that 𝑣 ∀ 𝑣 = 1 , … , 𝑣 𝑛 ∈ 𝑊 ⧵ { 0 } w i t h 𝑣 ℎ ∃ ̃ ̃ 𝑣 ≡ 0 f o r s o m e ℎ ∈ { 1 , … , 𝑛 } , 𝑣 = 1 ̃ 𝑣 , … , 𝑛 ̃ 𝑣 ∈ 𝑊 ∶ ℓ ≢ 0 a s ℓ = 1 , … , 𝑛 , 𝐻 𝜆 ( ̃ 𝑣 ) = 𝐻 𝜆 ( 𝑣 ) , 𝐷 1 ( ̃ 𝑣 ) = 𝐷 1 ( 𝑣 ) . ( 2 . 6 8 ) Let, for example, 𝑣 1 ≡ 0 and 𝑣 2 ≢ 0 . Set 𝑠 ∈ ] 0 , 1 [ and 𝑣 1 1 = ( 1 − 𝑠 ) 1 / 𝑝 𝑣 2 , 𝑣 1 2 = 𝑠 1 / 𝑝 𝑣 2 , 𝑣 1 ℓ = 𝑣 ℓ as ℓ > 2 , with 𝑣 1 = ( 𝑣 1 1 , … , 𝑣 1 𝑛 ) , we have 𝐻 𝜆 𝑣 1 = 𝐻 𝜆 ( 𝑣 ) , 𝐷 1 𝑣 1 = 𝐷 1 ( 𝑣 ) . ( 2 . 6 9 ) If 𝑣 3 ≡ 0 , set 𝑣 2 1 = ( 1 − 𝑠 ) 1 / 𝑝 𝑣 1 1 , 𝑣 2 3 = 𝑠 1 / 𝑝 𝑣 1 1 , 𝑣 2 ℓ = 𝑣 1 ℓ as ℓ ∈ { 1 , … , 𝑛 } ⧵ { 1 , 3 } , with 𝑣 2 = ( 𝑣 2 1 , … , 𝑣 2 𝑛 ) , it results in 𝐻 𝜆 𝑣 2 = 𝐻 𝜆 ( 𝑣 ) , 𝐷 1 𝑣 2 = 𝐷 1 ( 𝑣 ) . ( 2 . 7 0 ) This method let us to find ̃ 𝑣 .
Then, if 𝑣 0 ℎ ≡ 0 ( r e s p . 𝑣 ℎ ≡ 0 ) for some ℎ ∈ { 1 , … , 𝑛 } , with ̃ 𝑣 0 ̃ ( r e s p . 𝑣 ) as in (2.68) we have from (2.53) (resp. (2.54)) 𝐷 1 ( ̃ 𝑣 0 ) = 𝑒 ( r e s p . 𝐻 𝜆 ( ̃ 𝑣 ) = 𝑒 ) . Consequently ([1], see the proof of Theorem 2.1 (resp. Theorem 4.1)) ̃ 𝑢 0 = 𝜏 0 ̃ 𝑣 0 ̃ ( r e s p . 𝑢 = 𝜏 ̃ 𝑣 ) is a weak solution of system (2.49). Therefore [6] ̃ 𝑢 0 𝑖 ̃ > 0 ( r e s p . 𝑢 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 Ω 𝑑 𝑗 𝑛 ℓ = 1 | | 𝑣 ℓ | | 𝛾 𝑗 𝑞 𝑗 / 𝛾 𝑗 𝑣 𝑑 𝑥 ∀ 𝑣 = 1 , … , 𝑣 𝑛 ∈ 𝑊 a s 𝑗 = 1 , … , 𝑚 , ( 2 . 7 1 ) where 𝑝 < 𝑞 1 < ⋯ < 𝑞 𝑚 < ̃ 𝑝 , 1 < 𝛾 𝑗 < 𝑞 𝑗 , 𝑑 𝑚 ∈ 𝐿 ∞ 𝑑 ( Ω ) , 𝑗 ∈ 𝐿 ∞ ( Ω ) ⧵ { 0 } , 𝑑 𝑗 ≤ 0 i f 𝑗 = 1 , … , 𝑚 − 1 . ( 2 . 7 2 ) Let us consider the system: | | − d i v ∇ 𝑢 𝑖 | | 𝑝 − 2 ∇ 𝑢 𝑖 = | | 𝑢 𝜆 𝑏 𝑖 | | 𝑝 − 2 𝑢 𝑖 + 𝑚 𝑗 = 1 𝑑 𝑗 𝑛 ℓ = 1 | | 𝑢 ℓ | | 𝛾 𝑗 ( 𝑞 𝑗 / 𝛾 𝑗 ) − 1 | | 𝑢 𝑖 | | 𝛾 𝑗 − 2 𝑢 𝑖 𝑢 i n Ω , 𝑖 = 0 o n 𝜕 Ω a s 𝑖 = 1 , … , 𝑛 , ( 2 . 7 3 ) under almost one of the conditions: 𝑑 + 𝑚 ≢ 0 , Ω 𝑑 𝑚 𝑢 ∗ 𝑞 𝑚 𝑑 𝑥 < 0 . ( 2 . 7 4 ) 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 + ( 𝐷 1 ) with 𝑉 + ( 𝐷 𝑚 ) and 𝑆 ( 𝐷 1 ) with 𝑆 ( 𝐷 𝑚 ) .
Thanks to Proposition A.3 and a result of [5], for the solutions 𝑢 0 and 𝑢 to system (2.73), we have 𝑢 0 𝑖 ∈ 𝐿 ∞ ( Ω ) ∩ 𝐶 1 , 𝛼 0 𝑖 ℓ o c ( Ω ) , 𝑢 𝑖 ∈ 𝐿 ∞ ( Ω ) ∩ 𝐶 1 , 𝛼 𝑖 ℓ o c ( Ω ) . ( 2 . 7 5 ) We continue to analyze the properties of 𝑢 0 and 𝑢 . To this aim we recall that ([1], Theorems 2.2 and 4.2), set for each 𝑣 ∈ 𝑉 + ( 𝐷 𝑚 ) ( r e s p . 𝑣 ∈ 𝑉 − 𝜆 ∩ 𝑆 ( 𝐷 𝑚 ) ) 𝜓 ( 𝑡 , 𝑣 ) = 𝑝 𝑡 𝑝 − 1 𝐻 𝜆 ∑ ( 𝑣 ) − 𝑚 𝑗 = 1 𝑞 𝑗 𝑡 𝑞 𝑗 − 1 𝐷 𝑗 ( 𝑣 ) , we have: ∃ ∣ 𝑡 ( 𝑣 ) > 0 ∶ 𝜓 ( 𝑡 ( 𝑣 ) , 𝑣 ) = 0 , 𝜕 𝜓 𝜕 𝑡 ( 𝑡 ( 𝑣 ) , 𝑣 ) ≠ 0 . ( 2 . 7 6 ) Besides with 𝐸 ( 𝑣 ) = ( 𝑡 ( 𝑣 ) ) 𝑝 𝐻 𝜆 ∑ ( 𝑣 ) − 𝑚 𝑗 = 1 ( 𝑡 ( 𝑣 ) ) 𝑞 𝑗 𝐷 𝑗 ( 𝑣 ) , it results in 𝐸 𝑣 0 𝐸 = i n f ( 𝑣 ) ∶ 𝑣 ∈ 𝑆 𝜆 ∩ 𝑉 + 𝐷 𝑚 , 𝜏 0 𝑣 = 𝑡 0 , 𝐸 ( 2 . 7 7 ) 𝑣 = i n f 𝐸 ( 𝑣 ) ∶ 𝑣 ∈ 𝑉 − 𝜆 𝐷 ∩ 𝑆 𝑚 , 𝜏 = 𝑡 𝑣 . ( 2 . 7 8 )

Proposition 2.23. When 𝛾 𝑚 < 𝑝 ≤ 𝛾 𝑗 as 𝑗 = 1 , … , 𝑚 − 1 , then 𝑢 0 𝑖 ≢ 0 ∀ 𝑖 ∈ { 1 , … , 𝑛 } . ( 2 . 7 9 )

Proof. It is sufficiently (Remark 1.1) to prove that 𝑖 ℎ 1 6 h o l d s ∀ ℎ ∈ { 1 , … , 𝑛 } w i t h 𝔉 = 𝑆 𝜆 ∩ 𝑉 + 𝐷 𝑚 . ( 2 . 8 0 ) Let 𝑣 = ( 𝑣 1 , … , 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