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: ξ€·π΄βˆ’div𝑖π‘₯,𝑒1,…,𝑒𝑛,βˆ‡π‘’1,…,βˆ‡π‘’π‘›ξ€Έξ€Έ=πœ†π‘–π‘π‘–||𝑒𝑖||π‘βˆ’2𝑒𝑖+𝑑𝑖π‘₯,𝑒1,…,𝑒𝑛,βˆ‡π‘’1,…,βˆ‡π‘’π‘›ξ€Έ+𝑓𝑖𝑒inΞ©,𝑖=0onπœ•Ξ©as𝑖=1,…,𝑛,(1.1) and to Neumann problems (Section 3) of the type: ξ€·π΄βˆ’div𝑖π‘₯,𝑒1,…,𝑒𝑛,βˆ‡π‘’1,…,βˆ‡π‘’π‘›ξ€Έξ€Έ=πœ†π‘–π‘π‘–||𝑒𝑖||π‘βˆ’2𝑒𝑖+𝑑𝑖π‘₯,𝑒1,…,𝑒𝑛,βˆ‡π‘’1,…,βˆ‡π‘’π‘›ξ€Έ+𝑓𝑖𝐴inΞ©,𝑖π‘₯,𝑒1,…,𝑒𝑛,βˆ‡π‘’1,…,βˆ‡π‘’π‘›ξ€Έβ‹…πœˆ=πœ‡π‘–Μ‚π‘π‘–||𝑒𝑖||π‘βˆ’2𝑒𝑖+𝑑𝑖π‘₯,𝑒1,…,𝑒𝑛,βˆ‡π‘’1,…,βˆ‡π‘’π‘›ξ€Έ+𝑓𝑖onπœ•Ξ©as𝑖=1,…,𝑛,(1.2) where 𝑛β‰₯1,πœ†π‘–,πœ‡π‘– are real parameters, Ξ© is a bounded connected open set of 𝑅𝑁 with regular boundary πœ•Ξ©, and 𝜈 is the outward orthogonal unitary vector to πœ•Ξ©.

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 HΓΆlderian with their first derivatives. It is suitable to recall the problem studied in [1] with some notations and hypotheses.

Let π‘Š1,…,π‘Šπ‘› real reflexive Banach spaces (𝑛β‰₯1). Let π‘Š be the product space 𝑋𝑛ℓ=1π‘Šβ„“. Let β€–β‹…β€– be the norm on π‘Š, β€–β‹…β€–βˆ— the norm on π‘Šβˆ— (dual space of π‘Š), and βŸ¨β‹…,β‹…βŸ©β„“ (resp.βŸ¨βŸ¨β‹…,β‹…βŸ©βŸ©) the duality between π‘Šβˆ—β„“ (dual space of π‘Šβ„“) and π‘Šβ„“ (resp. π‘Šβˆ— and π‘Š). Let us denote by β€œπœ•β€ FrΓ©chet differential operator and by β€œπœ•π‘’β„“β€ FrΓ©chet differential operator with respect to 𝑒ℓ. Let 𝐴≒0 and 𝐷𝑗≒0(𝑗=1,…,π‘š;π‘šβ‰₯1) be real functionals defined in π‘Š,𝐡ℓ and 𝐡ℓ(β„“=1,…,𝑛) real functionals defined in π‘Šβ„“ satisfying the conditions:(𝑖11)𝐴 is lower weakly semicontinuous in π‘Š and 𝐢1(π‘Šβ§΅{0}),𝐡ℓ𝐡andβ„“ are weakly continuous in π‘Šβ„“ and𝐢1(π‘Šβ„“),βˆƒπ‘>1∢𝐴(𝑑𝑣)=𝑑𝑝𝐴(𝑣)forall𝑑β‰₯0and forallπ‘£βˆˆπ‘Š,𝐡ℓ(𝑑𝑣ℓ)=𝑑𝑝𝐡ℓ(𝑣ℓ) and 𝐡ℓ(𝑑𝑣ℓ)=𝑑𝑝𝐡ℓ(𝑣ℓ)forall𝑑β‰₯0 andforallπ‘£β„“βˆˆπ‘Šβ„“;(𝑖12)𝐷𝑗 is weakly continuous in π‘Š and 𝐢1(π‘Šβ§΅{0}),βˆƒπ‘žπ‘—>1βˆΆπ·π‘—(𝑑𝑣)=π‘‘π‘žπ‘—π·π‘—(𝑣)forall𝑑β‰₯0 and forallπ‘£βˆˆπ‘Š,1<π‘ž1<β‹―<π‘žπ‘šifπ‘š>1.

Let𝐹=(𝐹1,…,𝐹𝑛)with πΉβ„“βˆˆπ‘Šβ„“βˆ—,πœ†β„“andπœ‡β„“βˆˆπ‘…; let us consider the following problem.

Problem (𝑃). Find 𝑒=(𝑒1,…,𝑒𝑛)βˆˆπ‘Šβ§΅{0}such that ξ«πœ•π‘’π‘–π΄(𝑒),𝑣𝑖𝑖=πœ†π‘–ξ«πœ•π΅π‘–ξ€·π‘’π‘–ξ€Έ,𝑣𝑖𝑖+πœ‡π‘–ξ‚¬πœ•ξπ΅π‘–ξ€·π‘’π‘–ξ€Έ,𝑣𝑖𝑖+π‘šξ“π‘—=1ξ«πœ•π‘’π‘–π·π‘—(𝑒),𝑣𝑖𝑖+βŸ¨πΉπ‘–,π‘£π‘–βŸ©π‘–βˆ€π‘–βˆˆ{1,…,𝑛},βˆ€π‘£π‘–βˆˆπ‘Šπ‘–.(1.3) Obviously Problem (P) means to find the critical points π‘’βˆˆπ‘Šβ§΅{0} of the Euler functional: 𝐸(𝑣)=𝐴(𝑣)βˆ’π‘›ξ“β„“=1ξ‚ƒπœ†β„“π΅β„“ξ€·π‘£β„“ξ€Έ+πœ‡β„“ξπ΅β„“ξ€·π‘£β„“ξ€Έξ‚„βˆ’π‘šξ“π‘—=1𝐷𝑗𝑣(𝑣)βˆ’βŸ¨βŸ¨πΉ,π‘£βŸ©βŸ©βˆ€π‘£=1,…,π‘£π‘›ξ€Έβˆˆπ‘Š,(1.4) where βˆ‘βŸ¨βŸ¨πΉ,π‘£βŸ©βŸ©=𝑛ℓ=1βŸ¨πΉβ„“,π‘£β„“βŸ©β„“.

Let us set π»πœ†πœ‡(𝑣)=𝐴(𝑣)βˆ’π‘›ξ“β„“=1ξ‚ƒπœ†β„“π΅β„“ξ€·π‘£β„“ξ€Έ+πœ‡β„“ξπ΅β„“ξ€·π‘£β„“ξ€Έξ‚„ξ€·π‘£βˆ€π‘£=1,…,π‘£π‘›ξ€Έξ€·πœ†βˆˆπ‘Š,βˆ€πœ†=1,…,πœ†π‘›ξ€Έξ€·πœ‡,πœ‡=1,…,πœ‡π‘›ξ€Έβˆˆπ‘…π‘›,π‘†πœ†πœ‡=ξ€½π‘£βˆˆπ‘ŠβˆΆπ»πœ†πœ‡ξ€Ύ(𝑣)=1,π‘‰βˆ’πœ†πœ‡=ξ€½π‘£βˆˆπ‘ŠβˆΆπ»πœ†πœ‡ξ€Ύ(𝑣)<0,asπ‘š1𝑉=1,…,π‘š+ξ€·π·π‘š1,…,π·π‘šξ€Έ=ξƒ―π‘£βˆˆπ‘ŠβˆΆπ‘šξ“π‘—=π‘š1𝐷𝑗,𝑆(𝑣)>0+𝐷1,…,π·π‘šξ€Έ=ξƒ―π‘£βˆˆπ‘ŠβˆΆπ‘šξ“π‘—=1𝐷𝑗,𝑆𝐷(𝑣)=1𝑗=ξ€½π‘£βˆˆπ‘ŠβˆΆπ·π‘—(𝑣)=βˆ’1,𝑉+(𝐹)={π‘£βˆˆπ‘ŠβˆΆβŸ¨βŸ¨πΉ,π‘£βŸ©βŸ©>0}.(1.5) About Problem (P), using Lagrange multipliers and the β€œfibering method,” different existence theorems have been proved in [1]. They base on one of the following hypotheses: (𝑖13)βˆƒπ‘(πœ†,πœ‡)>0βˆΆβ€–π‘£β€–π‘β‰€π‘(πœ†,πœ‡)π»πœ†πœ‡(𝑣)forallπ‘£βˆˆπ‘Š; (𝑖14)βˆƒπ‘(πœ†,πœ‡)>0βˆΆβ€–π‘£β€–π‘β‰€π‘(πœ†,πœ‡)π»πœ†πœ‡(𝑣)forallπ‘£βˆˆπ‘‰+(π·π‘š)(if𝑉+(π·π‘š)β‰ βˆ…); (𝑖15)βˆƒπ‘š1∈{1,…,π‘š}βˆΆπ‘‰βˆ’πœ†πœ‡βˆ©π‘†(π·π‘š1) 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 π”‰βŠ†π‘†πœ†πœ‡, we suppose(π‘–β„Ž16) for each 𝑣=(𝑣1,…,𝑣𝑛)βˆˆπ”‰ with π‘£β„Ž=0, there exist π‘£β„Žβˆˆπ‘Šβ„Žβ§΅{0} and the real functions πœ™1,…,πœ™π‘› such that πœ™β„ŽβˆˆπΆ0([0,1])∩𝐢1([0,1[) and πœ™β„Ž(1)=0, πœ™β„“βˆˆπΆ1([0,1]) and πœ™β„“(1)=1 as β„“β‰ β„Ž, 𝑣(𝑠)=(πœ™1(𝑠)𝑣1,…,πœ™β„Ž(𝑠)π‘£β„Ž,…,πœ™π‘›(𝑠)𝑣𝑛)βˆˆπ”‰ for all π‘ βˆˆ[𝑠0,1](0≀𝑠0<1), limξ…žξ…žπ‘ β†’1βˆ’(𝑑/𝑑𝑠)𝐷𝑗(𝑣(𝑠))<+∞ for all π‘—βˆˆ{1,…,π‘š}, lim𝑠→1βˆ’(𝑑/𝑑𝑠)𝐷𝑗(𝑣(𝑠))=βˆ’βˆž for some π‘—βˆˆ{1,…,π‘š}.
The condition (π‘–β„Ž16) assures that for the solutions 𝑒=(𝑒1,…,𝑒𝑛) of Problem (P), found with the method used in the recalled theorems, we have π‘’β„Žβ‰ 0 if πΉβ„Žβ‰‘0.

Before showing Dirichlet problems (including the problem studied in [2] by DrΓ‘bek and Pohozaev when 𝑛=1 and π‘š=1) we give Propositions 2.2–2.6 which show some cases in which hypotheses (𝑖13)βˆ’(𝑖15) hold. These propositions are based on the comparison between the parameters πœ†π‘– with suitable eigenvalues connected to 𝑝-Laplacian. About Neumann problems (including the one studied in [3] by Pohozaev and VΓ©ron when 𝑛=1) the same question is solved by Propositions 3.1–3.5 in which the parameters πœ†π‘– and πœ‡π‘– have compared with zero. Finally, the results in Appendix are very useful: Propositions A.1 and A.2 in order to get condition (π‘–β„Ž16), 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 Ξ©βŠ†π‘…π‘ be an open, bounded, connected and 𝐢2,𝛽 set with 0<𝛽≀1. Let|β‹…|𝑁 the Lebesgue measure on 𝑅𝑁,1<𝑝<∞,̃𝑝=𝑁𝑝/(π‘βˆ’π‘)if𝑁>𝑝,̃𝑝=∞otherwise.

Let us assumeξ‚€π‘Šπ‘Š=01,𝑝(Ξ©)𝑛(𝑛β‰₯1)with‖𝑣‖=𝑛ℓ=1ξ€œΞ©||βˆ‡π‘£β„“||𝑝ξƒͺ𝑑π‘₯1/π‘ξ€·π‘£βˆ€π‘£=1,…,π‘£π‘›ξ€Έπ΅βˆˆπ‘Š,ℓ𝑣ℓ=π‘βˆ’1ξ€œΞ©π‘β„“||𝑣ℓ||𝑝𝑑π‘₯βˆ€π‘£β„“βˆˆπ‘Š01,𝑝(Ξ©)whereπ‘β„“βˆˆπΏβˆž(Ξ©)⧡{0},𝑏ℓ𝐡β‰₯0,ℓ≑0.(2.1) Moreover we consider the functionals 𝐴 (as in (𝑖11)) such thatβˆƒΜƒπ‘>0∢𝐴(𝑣)β‰₯π‘βˆ’1Μƒπ‘β€–π‘£β€–π‘βˆ€π‘£βˆˆπ‘Š.(2.2) Let us use the notation π»πœ†(π‘†πœ†andπ‘‰βˆ’πœ†,resp.) instead of π»πœ†πœ‡(π‘†πœ†πœ‡andπ‘‰βˆ’πœ†πœ‡,resp.).

As β„“=1,…,𝑛 let πœ†βˆ—β„“andπ‘’βˆ—β„“, respectively, the first eigenvalue and the first eigenfunction of the problem: π‘’β„“βˆˆπ‘Š01,𝑝||(Ξ©)βˆΆβˆ’Μƒπ‘divβˆ‡π‘’β„“||π‘βˆ’2βˆ‡π‘’β„“ξ‚=πœƒπ‘β„“||𝑒ℓ||π‘βˆ’2𝑒ℓinΞ©.(2.3) Let us remember that [4]π‘’βˆ—β„“βˆˆπΆ1,𝛼ℓ(Ξ©) with 0<𝛼ℓ<1,π‘’βˆ—β„“>0 in Ξ©;πœ†βˆ—β„“βˆ«=̃𝑐Ω|βˆ‡π‘’βˆ—β„“|π‘βˆ«π‘‘π‘₯/Ω𝑏ℓ|π‘’βˆ—β„“|π‘βˆ«π‘‘π‘₯=min{̃𝑐Ω|βˆ‡π‘£β„“|π‘βˆ«π‘‘π‘₯/Ω𝑏ℓ|𝑣ℓ|π‘βˆ«π‘‘π‘₯βˆΆΞ©π‘β„“|𝑣ℓ|𝑝𝑑π‘₯>0};πœ†βˆ—β„“ is simple, that is, each eigenfunction of (2.3) related to πœ†βˆ—β„“ is of the type π‘β„“π‘’βˆ—β„“ with π‘β„“βˆˆπ‘…β§΅{0};πœ†βˆ—β„“ is isolate, that is, there exists π‘Ž>0 such that πœ†βˆ—β„“ is the only eigenvalue of (2.3) belonging to ]0,π‘Ž[.

Remark 2.1. About the results related to problem (2.3), it is sufficient to suppose π‘β„“βˆˆπΏβˆž(Ξ©) and 𝑏+β„“=max{𝑏ℓ,0}β‰’0 as β„“=1,…,𝑛. This holds also for the results of this section if we limit to consider only the parameters πœ†1,…,πœ†π‘› nonnegative.

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

Using the variational characterization of πœ†βˆ—β„“ it is easy to verify the following proposition.

Proposition 2.2. If πœ†β„“<πœ†βˆ—β„“forallβ„“βˆˆ{1,…,𝑛}, then (𝑖13) holds. Consequently, (𝑖14) holds when 𝑉+(π·π‘š)β‰ βˆ….

When πœ†β„“β‰₯πœ†βˆ—β„“β€‰β€‰for some β„“βˆˆ{1,…,𝑛}, it is possible to fulfil (𝑖14) with an additional condition on π·π‘š. Let 𝐼={1,…,𝑛}. For any πΌβˆ—βŠ†πΌ let π‘‰βˆ—=𝑣𝑣=1,…,π‘£π‘›ξ€Έβˆˆπ‘ŠβˆΆπ‘£β„“β‰‘0ifβ„“βˆˆπΌβ§΅πΌβˆ—,𝑣ℓ=π‘β„“π‘’βˆ—β„“ifβ„“βˆˆπΌβˆ—withπ‘β„“βˆˆπ‘…and𝑐ℓ,β‰ 0forsomeβ„“(2.4) and let us suppose(𝑖21) There exists πΌβˆ—βŠ†πΌβˆΆπ·π‘š(𝑣)<0forallπ‘£βˆˆπ‘‰βˆ—.

Proposition 2.3. Let (𝑖21) holds with πΌβˆ—β‰ πΌ. Let 𝑉+(π·π‘š)β‰ βˆ…. If we fix the parameters set (πœ†β„“)β„“βˆˆπΌβ§΅πΌβˆ— with πœ†β„“<πœ†βˆ—β„“, then there exists π›Ώβˆ—>0 such that (𝑖14) also holds for any (πœ†β„“)β„“βˆˆπΌβˆ—βˆˆπ‘‹β„“βˆˆπΌβˆ—[πœ†βˆ—β„“,πœ†βˆ—β„“+π›Ώβˆ—[.

Proof. Arguing by contradiction, for any π‘˜βˆˆβ„• there exist (πœ†π‘˜β„“)β„“βˆˆπΌβˆ—βˆˆπ‘‹β„“βˆˆπΌβˆ—[πœ†βˆ—β„“,πœ†βˆ—β„“+π‘˜βˆ’1[ and π‘£π‘˜=(π‘£π‘˜1,…,π‘£π‘˜π‘›)βˆˆπ‘‰+(π·π‘š) such that π΄ξ€·π‘£π‘˜ξ€Έβˆ’π‘βˆ’1ξ“β„“βˆˆπΌβ§΅πΌβˆ—πœ†β„“ξ€œΞ©π‘β„“||𝑣ℓ||𝑝𝑑π‘₯βˆ’π‘βˆ’1ξ“β„“βˆˆπΌβˆ—πœ†π‘˜β„“ξ€œΞ©π‘β„“||π‘£π‘˜β„“||𝑝𝑑π‘₯<π‘˜βˆ’1β€–β€–π‘£π‘˜β€–β€–π‘.(2.5) Set π‘€π‘˜=β€–π‘£π‘˜β€–βˆ’1π‘£π‘˜, we have π·π‘šξ€·π‘€π‘˜ξ€Έξ“>0,Μƒπ‘β„“βˆˆπΌβ§΅πΌβˆ—ξ€œΞ©||βˆ‡π‘€π‘˜β„“||𝑝𝑑π‘₯βˆ’β„“βˆˆπΌβ§΅πΌβˆ—πœ†β„“ξ€œΞ©π‘β„“||π‘€π‘˜β„“||𝑝𝑑π‘₯+Μƒπ‘β„“βˆˆπΌβˆ—ξ€œΞ©||βˆ‡π‘€π‘˜β„“||𝑝𝑑π‘₯βˆ’β„“βˆˆπΌβˆ—πœ†π‘˜β„“ξ€œΞ©π‘β„“||π‘€π‘˜β„“||𝑝𝑑π‘₯<π‘π‘˜βˆ’1,(2.6) moreover, since β€–π‘€π‘˜β€–=1, there exists π‘€βˆˆπ‘Š such that (within a subsequence) π‘€π‘˜βŸΆπ‘€weaklyinπ‘Š,π‘€π‘˜βŸΆπ‘€stronglyin(𝐿𝑝(Ξ©))𝑛.(2.7) Taking into account that π·π‘š is weakly continuous in π‘Š, from (2.6) as π‘˜β†’+∞ we get π·π‘šξ“(𝑀)β‰₯0,(2.8)β„“βˆˆπΌβ§΅πΌβˆ—ξ‚Έξ€œΜƒπ‘Ξ©||βˆ‡π‘€β„“||𝑝𝑑π‘₯βˆ’πœ†β„“ξ€œΞ©π‘β„“||𝑀ℓ||𝑝+𝑑π‘₯β„“βˆˆπΌβˆ—ξ‚Έξ€œΜƒπ‘Ξ©||βˆ‡π‘€β„“||𝑝𝑑π‘₯βˆ’πœ†βˆ—β„“ξ€œΞ©π‘β„“||𝑀ℓ||𝑝𝑑π‘₯≀0.(2.9) Since π‘€β„“ξ€œβ‰’0βŸΉΜƒπ‘Ξ©||βˆ‡π‘€β„“||𝑝𝑑π‘₯βˆ’πœ†β„“ξ€œΞ©π‘β„“||𝑀ℓ||π‘ξ€œπ‘‘π‘₯>0,̃𝑐Ω||βˆ‡π‘€β„“||𝑝𝑑π‘₯βˆ’πœ†βˆ—β„“ξ€œΞ©π‘β„“||𝑀ℓ||𝑝𝑑π‘₯β‰₯0,(2.10) from (2.9), we deduce that 𝑀ℓ≑0βˆ€β„“βˆˆπΌβ§΅πΌβˆ—,βˆ€β„“βˆˆπΌβˆ—βˆƒπ‘β„“βˆˆπ‘…βˆΆπ‘€β„“=π‘β„“π‘’βˆ—β„“.(2.11) Let us add that𝑐ℓ≠0 for some β„“βˆˆπΌβˆ—, since if 𝑐ℓ=0forallβ„“βˆˆπΌβˆ— we have the contradiction ̃𝑐=̃𝑐limπ‘˜β†’+βˆžβ€–π‘€π‘˜β€–π‘=0. Then π‘€βˆˆπ‘‰βˆ—, and consequently π·π‘š(𝑀)<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 πΌβˆ—=𝐼. Let 𝑉+(π·π‘š)β‰ βˆ…. Then, there exists π›Ώβˆ—>0 such that (i14) also holds for any (πœ†β„“)β„“βˆˆπΌβˆˆπ‘‹β„“βˆˆπΌ[πœ†βˆ—β„“,πœ†βˆ—β„“+π›Ώβˆ—[.

Let us pass to (𝑖15) and suppose(𝑖22) there existπΌβˆ—βŠ†πΌ and π‘š1∈{1,…,π‘š}such thatπ·π‘š1(𝑣)<0and𝐴(𝑣)=Μƒπ‘π‘βˆ’1βˆ‘β„“βˆˆπΌβˆ—βˆ«Ξ©|βˆ‡π‘£β„“|𝑝𝑑π‘₯ for any π‘£βˆˆπ‘‰βˆ—.

Proposition 2.5. If (i22) holds withπΌβˆ—β‰ πΌ, then π‘‰βˆ’πœ†ξ€·π·βˆ©π‘†π‘š1ξ€Έξ€·πœ†β‰ βˆ…βˆ€β„“ξ€Έβ„“βˆˆπΌξ€·πœ†withβ„“ξ€Έβ„“βˆˆπΌβˆ—βˆˆπ‘‹β„“βˆˆπΌβˆ—ξ€Ίπœ†βˆ—β„“ξ€Ίβ§΅πœ†,+βˆžξ€½ξ€·βˆ—β„“ξ€Έβ„“βˆˆπΌβˆ—ξ€Ύ.(2.12) Moreover, if we fix the parameters set (πœ†β„“)β„“βˆˆπΌβ§΅πΌβˆ— with πœ†β„“<πœ†βˆ—β„“, then there exists π›Ώβˆ—>0 such that π‘‰βˆ’πœ†ξ€·π·βˆ©π‘†π‘š1ξ€Έξ€·πœ†isboundedinπ‘Šβˆ€β„“ξ€Έβ„“βˆˆπΌβˆ—βˆˆπ‘‹β„“βˆˆπΌβˆ—ξ€Ίπœ†βˆ—β„“,πœ†βˆ—β„“+π›Ώβˆ—ξ€Ίβ§΅πœ†ξ€½ξ€·βˆ—β„“ξ€Έβ„“βˆˆπΌβˆ—ξ€Ύ.(2.13)

Proof. Let us prove (2.12). Let π‘£βˆˆπ‘‰βˆ—with 𝑣ℓ=π‘’βˆ—β„“if β„“βˆˆπΌβˆ—, then π·π‘š1(𝑣)<0. Let 𝑀=|π·π‘š1(𝑣)|βˆ’1β§΅π‘žπ‘š1𝑣, we have π·π‘š1||𝐷(𝑀)=π‘š1||(𝑣)βˆ’1π·π‘š1𝐻(𝑣)=βˆ’1,πœ†(𝑀)=π‘βˆ’1ξ“β„“βˆˆπΌβˆ—ξ‚Έξ€œΜƒπ‘Ξ©||βˆ‡π‘€β„“||𝑝𝑑π‘₯βˆ’πœ†β„“ξ€œΞ©π‘β„“||𝑀ℓ||𝑝𝑑π‘₯<0.(2.14) Let us prove (2.13). Arguing by contradiction, for any π‘˜βˆˆβ„• there exist (πœ†π‘˜β„“)β„“βˆˆπΌβˆ—βˆˆπ‘‹β„“βˆˆπΌβˆ—[πœ†βˆ—β„“,πœ†βˆ—β„“+π‘˜βˆ’1[ with (πœ†π‘˜β„“)β„“βˆˆπΌβˆ—β‰ (πœ†βˆ—β„“)β„“βˆˆπΌβˆ— and (π‘£π‘˜,β„Ž)β„Žβˆˆβ„•βŠ†π‘‰βˆ’πœ†π‘˜βˆ©π‘†(π·π‘š1), where πœ†π‘˜β„“=πœ†β„“ if β„“βˆˆπΌβ§΅πΌβˆ—, such that supβ„Žβˆˆβ„•β€–β€–π‘£π‘˜,β„Žβ€–β€–=+∞.(2.15) Relation (2.15) implies that there exists (β„Žπ‘˜)π‘˜βˆˆβ„•βŠ†β„• strictly increasing such that π›Ώπ‘˜=β€–β€–π‘£π‘˜,β„Žπ‘˜β€–β€–βŸΆ+∞asπ‘˜βŸΆ+∞.(2.16) Let π‘€π‘˜=π›Ώπ‘˜βˆ’1π‘£π‘˜,β„Žπ‘˜, we have ξ“β„“βˆˆπΌβ§΅πΌβˆ—ξ‚Έξ€œΜƒπ‘Ξ©||βˆ‡π‘€π‘˜β„“||𝑝𝑑π‘₯βˆ’πœ†β„“ξ€œΞ©π‘β„“||π‘€π‘˜β„“||𝑝+𝑑π‘₯β„“βˆˆπΌβˆ—ξ‚Έξ€œΜƒπ‘Ξ©||βˆ‡π‘€π‘˜β„“||𝑝𝑑π‘₯βˆ’πœ†π‘˜β„“ξ€œΞ©π‘β„“||π‘€π‘˜||𝑝𝐷𝑑π‘₯<0,π‘š1ξ€·π‘€π‘˜ξ€Έ=βˆ’π›Ώβˆ’π‘žπ‘š1π‘˜,βˆƒπ‘€βˆˆπ‘ŠβˆΆ(withinasubsequence)π‘€π‘˜βŸΆπ‘€weaklyinπ‘Š,π‘€π‘˜βŸΆπ‘€stronglyin(𝐿𝑝(Ξ©))𝑛.(2.17) Then, as π‘˜β†’+∞ we get ξ“β„“βˆˆπΌβ§΅πΌβˆ—ξ‚Έξ€œΜƒπ‘Ξ©||βˆ‡π‘€β„“||𝑝𝑑π‘₯βˆ’πœ†β„“ξ€œΞ©π‘β„“||𝑀ℓ||𝑝+𝑑π‘₯β„“βˆˆπΌβˆ—ξ‚Έξ€œΜƒπ‘Ξ©||βˆ‡π‘€β„“||𝑝𝑑π‘₯βˆ’πœ†βˆ—β„“ξ€œΞ©π‘β„“||𝑀ℓ||𝑝𝐷𝑑π‘₯≀0,(2.18)π‘š1(𝑀)=0.(2.19) From (2.18), we get that π‘€βˆˆπ‘‰βˆ—. Then since (𝑖22) inequality π·π‘š1(𝑀)<0 holds, which contradicts (2.19).

Proposition 2.6. If (𝑖22) holds with πΌβˆ—=𝐼, then π‘‰βˆ’πœ†ξ€·π·βˆ©π‘†π‘š1ξ€Έξ€·πœ†β‰ βˆ…βˆ€πœ†=β„“ξ€Έβ„“βˆˆπΌβˆˆπ‘‹β„“βˆˆπΌξ€Ίπœ†βˆ—β„“ξ€Ίβ§΅πœ†,+βˆžξ€½ξ€·βˆ—β„“ξ€Έβ„“βˆˆπΌξ€Ύ,βˆƒπ›Ώβˆ—>0βˆΆπ‘‰βˆ’πœ†ξ€·π·βˆ©π‘†π‘š1ξ€Έξ€·πœ†isboundedinπ‘Šβˆ€πœ†=β„“ξ€Έβ„“βˆˆπΌβˆˆπ‘‹β„“βˆˆπΌξ€Ίπœ†βˆ—β„“,πœ†βˆ—β„“+π›Ώβˆ—ξ€Ίβ§΅πœ†ξ€½ξ€·βˆ—β„“ξ€Έβ„“βˆˆπΌξ€Ύ.(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 ̃𝑐=1, and we suppose π‘β„“βˆˆπΏβˆž(Ξ©)⧡{0},𝑏ℓβ‰₯0.

Application 2.8. Let 𝑛=1. Let us consider the problem ξ‚€||||βˆ’divβˆ‡π‘’π‘βˆ’2ξ‚βˆ‡π‘’=πœ†1𝑏1|𝑒|π‘βˆ’2𝑒+π‘šξ“π‘—=1𝑑𝑗|𝑒|π‘žπ‘—βˆ’2𝑒inΞ©,𝑒=0onπœ•Ξ©,(2.21) where 𝑝<π‘ž1<̃𝑝,𝑑1∈𝐿∞(Ξ©)⧡{0}ifπ‘š=1,𝑝<π‘ž1<β‹―<π‘žπ‘š<̃𝑝,π‘‘π‘—βˆˆπΏβˆžπ‘‘(Ξ©)⧡{0}as𝑗=1,…,π‘š,𝑗≀0as𝑗=1,…,π‘šβˆ’1ifπ‘š>1.(2.22) Evidently 𝐴(𝑣)=π‘βˆ’1ξ€œΞ©||||βˆ‡π‘£π‘π‘‘π‘₯,𝐷𝑗(𝑣)=π‘žπ‘—βˆ’1ξ€œΞ©π‘‘π‘—|𝑣|π‘žπ‘—π‘‘π‘₯βˆ€π‘£βˆˆπ‘Š.(2.23) Let us advance the conditions: 𝑑+π‘šξ€·β‰’0βŸΉπ‘‰+ξ€·π·π‘šξ€Έξ€Έξ€œβ‰ βˆ…,(2.24)Ξ©π‘‘π‘šξ€·π‘’βˆ—1ξ€Έπ‘žπ‘šξ€·π‘‘π‘₯<0βŸΉπ·π‘šξ€·π‘1π‘’βˆ—1ξ€Έ<0βˆ€π‘1ξ€Έ.βˆˆπ‘…β§΅{0}(2.25) Let us note that (Propositions 2.2, 2.4, and 2.6) 𝑖(2.24)βŸΉξ€·ξ€·14ξ€Έholdsifπœ†1<πœ†βˆ—1ξ€Έ,(ξ€·2.24)and(2.25)βŸΉβˆƒπ›Ώβˆ—1𝑖>0∢14ξ€Έholdsifπœ†1<πœ†βˆ—1+π›Ώβˆ—1ξ€Έ,ξ€·(2.25)βŸΉβˆƒπ›Ώβˆ—2𝑖>0∢15ξ€Έholdsifπœ†1βˆˆξ€»πœ†βˆ—1,πœ†βˆ—1+π›Ώβˆ—2.ξ€Ίξ€Έ(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 πœ†1<πœ†βˆ—1 [resp. (2.24) and (2.25) hold, with πœ†1<πœ†βˆ—1+π›Ώβˆ—1] problem (2.21) has at least two weak solutions 𝑒0 andβˆ’π‘’0(𝑒0=𝜏0𝑣0,𝜏0=const.>0,𝑣0βˆˆπ‘†πœ†1βˆ©π‘‰+(π·π‘š)), and it results in 𝑒0∈𝐿∞(Ξ©)∩𝐢1,𝛼0β„“π‘œπ‘(Ξ©),𝑒0>0;(ii)When (2.25) holds, with πœ†1∈]πœ†βˆ—1,πœ†βˆ—1+π›Ώβˆ—2[ problem (2.21) has at least two weak solutions π‘’π‘Žπ‘›π‘‘βˆ’π‘’(𝑒=πœπ‘£,𝜏=const.>0,π‘£βˆˆπ‘‰βˆ’πœ†1βˆ©π‘†(π·π‘š)), and it results in π‘’βˆˆπΏβˆž(Ξ©)∩𝐢1,π›Όβ„“π‘œπ‘(Ξ©),𝑒>0.
Consequently, when (2.24) and (2.25) hold, with πœ†1∈]πœ†βˆ—1,πœ†βˆ—1+min{π›Ώβˆ—1,π›Ώβˆ—2}[ problem (2.21) has at least four different weak solutions.

Remark 2.10. Our results include the ones of DrΓ‘bek and Pohozaev [2] when π‘š=1.

Application 2.11. Let us consider the system: ξ‚€||βˆ’divβˆ‡π‘’π‘–||π‘βˆ’2βˆ‡π‘’π‘–ξ‚=πœ†π‘–π‘π‘–||𝑒𝑖||π‘βˆ’2𝑒𝑖+|||||𝑛ℓ=1𝑑ℓ𝑒ℓ|||||π‘ž1βˆ’2𝑛ℓ=1𝑑ℓ𝑒ℓξƒͺπ‘‘π‘–βˆ’ξ‚π‘‘π‘–||𝑒𝑖||π‘ž1βˆ’2𝑒𝑖𝑒inΞ©,𝑖=0onπœ•Ξ©as𝑖=1,…,𝑛,(2.27) where 1<π‘ž1<̃𝑝,π‘ž1≠𝑝,𝑑ℓ,ξ‚π‘‘β„“βˆˆπΏβˆž(Ξ©),𝑑ℓ,𝑑ℓ>0.(2.28) System (2.27) is included among Problem (P) with: 𝐴(𝑣)=π‘π‘›βˆ’1ℓ=1ξ€œΞ©||βˆ‡π‘£β„“||𝑝𝐷𝑑π‘₯,1(𝑣)=π‘ž1βˆ’1βŽ‘βŽ’βŽ’βŽ£ξ€œΞ©|||||𝑛ℓ=1𝑑ℓ𝑣ℓ|||||π‘ž1𝑑π‘₯βˆ’π‘›ξ“β„“=1ξ€œΞ©ξ‚π‘‘β„“||𝑣ℓ||π‘ž1⎀βŽ₯βŽ₯βŽ¦ξ€·π‘£π‘‘π‘₯βˆ€π‘£=1,…,π‘£π‘›ξ€Έβˆˆπ‘Š.(2.29) Let us advance the conditions (compatible): π‘‘π‘ž1β„“<ξ‚π‘‘β„“ξ€·βˆ€β„“βˆˆ{1,…,𝑛}⟹𝐷1ξ€·0,…,π‘π‘–π‘’βˆ—π‘–ξ€Έ,…,0<0as𝑖=1,…,𝑛,π‘π‘–ξ€Έβˆˆπ‘…β§΅{0},(2.30) there exist Ξ©+βŠ†Ξ© and a constant ̃𝑐𝑗>0 such that |Ξ©+|𝑁>0 and ℓ≠𝑗𝑑ℓ+̃𝑐𝑗𝑑𝑗ξƒͺπ‘ž1>ℓ≠𝑗𝑑ℓ+Μƒπ‘π‘ž1𝑗𝑑𝑗inΞ©+ξ€·βŸΉπ‘‰+𝐷1ξ€Έξ€Έ.β‰ βˆ…(PropositionA.1)(2.31) Then (Propositions 2.2, 2.3, and 2.5) 𝑖(2.31)βŸΉξ€·ξ€·14ξ€Έholdsifπœ†β„“<πœ†βˆ—β„“ξ€Έβˆ€β„“βˆˆ{1,…,𝑛},(2.32) and set π‘–βˆˆ{1,…,𝑛}ξ€·(2.30)and(2.31)⟹withπœ†β„“<πœ†βˆ—β„“βˆ€β„“β‰ π‘–βˆƒπ›Ώβˆ—1𝑖>0∢14ξ€Έholdsifπœ†π‘–<πœ†βˆ—π‘–+π›Ώβˆ—1ξ€Έ(ξ€·,(2.33)2.30)⟹withπœ†β„“<πœ†βˆ—β„“βˆ€β„“β‰ π‘–βˆƒπ›Ώβˆ—2𝑖>0∢15ξ€Έholdsifπœ†π‘–βˆˆξ€»πœ†βˆ—π‘–,πœ†βˆ—π‘–+π›Ώβˆ—2ξ€Ίξ€Έ.(2.34)

Taking into account that 𝐷1(𝑣1,…,𝑣𝑛)≀𝐷1(|𝑣1|,…,|𝑣𝑛|) and 𝐷1(βˆ’π‘£)=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 πœ†1,…,πœ†π‘› as in (2.32) (resp. (2.33)) system (2.27) has at least two weak solutions 𝑒0 and βˆ’π‘’0 with 𝑒0β„“β‰₯0 as β„“=1,…,𝑛(𝑒0=𝜏0𝑣0,𝜏0=const.>0,𝑣0βˆˆπ‘†πœ†βˆ©π‘‰+(𝐷1)); (ii)When (2.30) holds, choosingπœ†1,…,πœ†π‘› as in (2.34) system (2.27) has at least two weak solutions 𝑒 and βˆ’π‘’(𝑒=πœπ‘£,𝜏=const.>0,π‘£βˆˆπ‘‰βˆ’πœ†βˆ©π‘†(𝐷1)).
Consequently, when (2.30) and (2.31) hold, with πœ†β„“<πœ†βˆ—β„“forallℓ≠𝑖 and πœ†π‘–βˆˆ]πœ†βˆ—π‘–,πœ†βˆ—π‘–+min{π›Ώβˆ—1,π›Ώβˆ—2}[ system (2.27) has at least four different weak solutions.

The following proposition is obvious.

Proposition 2.13. The following relations hold: 𝑒0𝑖≒0as𝑖=1,…,𝑛,βˆƒβ„Ž,π‘˜βˆˆ{1,…,𝑛}βˆΆπ‘’β„Žβ‰’0,π‘’π‘˜β‰’0.(2.35)

Proposition 2.14. If 𝑝<π‘ž1, then as 𝑖=1,…,𝑛: 𝑒0π‘–βˆˆπΏβˆž(Ξ©)∩𝐢1,𝛼0π‘–β„“π‘œπ‘(Ξ©),𝑒0𝑖>0.(2.36)

Proof. It is easy to prove that 𝑛𝑖=1ξ€œΞ©||βˆ‡π‘’0𝑖||π‘βˆ’2βˆ‡π‘’0π‘–β‹…βˆ‡π‘£π‘–ξ€œπ‘‘π‘₯≀Ω𝑔𝑛𝑖=1𝑒0𝑖ξƒͺπ‘βˆ’1𝑛𝑖=1𝑣𝑖ξƒͺ𝑣𝑑π‘₯βˆ€π‘£=1,…,π‘£π‘›ξ€Έβˆˆξ‚€π‘Š01,𝑝(Ξ©)∩𝐿∞(Ω)𝑛with𝑣𝑖β‰₯0,(2.37) where π‘”βˆˆπΏπ‘ž1/(π‘ž1βˆ’π‘)(Ξ©). Then (Proposition A.3) 𝑒0π‘–βˆˆπΏβˆž(Ξ©) and consequently [5] 𝑒0π‘–βˆˆπΆ1,𝛼0𝑖ℓoc(Ξ©).
Let us note that 𝑒0𝑖 is a weak supersolution to the equation: ξ‚€||βˆ’divβˆ‡π‘’π‘–||π‘βˆ’2βˆ‡π‘’π‘–ξ‚=πœ†π‘–π‘π‘–||𝑒𝑖||π‘βˆ’2π‘’π‘–βˆ’ξ‚π‘‘π‘–||𝑒𝑖||π‘ž1βˆ’2𝑒𝑖inΞ©.(2.38) Then, since (2.35), it must be [6] 𝑒0𝑖>0.

Let us continue the analysis of system (2.27) under the condition: ℓ≠𝑖𝑑ℓξƒͺπ‘ž1𝑑<min1𝑑,…,π‘›ξ‚‡βˆ€π‘–βˆˆ{1,…,𝑛},(2.39) then 𝐷1𝑐1π‘’βˆ—1,…,π‘π‘›π‘’βˆ—π‘›ξ€Έξ€·π‘<0βˆ€1,…,π‘π‘›ξ€Έβˆˆπ‘…π‘›β§΅{0}with𝑐𝑖=0foratleastoneπ‘–βˆˆ{1,…,𝑛}.(2.40) Hence (Proposition 2.5) if πΌβˆ—βŠ†πΌ and πΌβˆ—β‰ πΌ: ξ‚΅(2.39)⟹asπœ†β„“<πœ†βˆ—β„“βˆ€β„“βˆˆπΌβ§΅πΌβˆ—βˆƒπ›Ώβˆ—ξ€·π‘–>0∢15ξ€Έξ€·πœ†holdsifβ„“ξ€Έβ„“βˆˆπΌβˆ—βˆˆπ‘‹β„“βˆˆπΌβˆ—ξ€Ίπœ†βˆ—β„“,πœ†βˆ—β„“+π›Ώβˆ—ξ€Ίβ§΅ξ€·πœ†βˆ—β„“ξ€Έβ„“βˆˆπΌβˆ—ξ‚Ά.(2.41)

Proposition 2.15. Under assumptions (2.28) and (2.39), choosing πœ†1,…,πœ†π‘› as in (2.41) system (2.27) has at least two weak solutions 𝑒andβˆ’π‘’(𝑒=πœπ‘£,𝜏=const.>0,π‘£βˆˆπ‘‰βˆ’πœ†βˆ©π‘†(𝐷1)) with 𝑒𝑖≒0 as 𝑖=1,…,𝑛.

Proof. Thanks to ([1], Theorem 4.1), there exists π‘£βˆˆπ‘‰βˆ’πœ†βˆ©π‘†(𝐷1) such that π»πœ†ξ€·π‘£ξ€Έξ€½π»=infπœ†(𝑣)βˆΆπ‘£βˆˆπ‘‰βˆ’πœ†ξ€·π·βˆ©π‘†1ξ€Έξ€Ύ=𝑒,𝑒=πœπ‘£isaweaksolutionofsystem(2.27),(2.42) where 𝜏=(βˆ’π‘π‘ž1βˆ’1𝑒)1/(π‘ž1βˆ’π‘).
Reasoning by contradiction, let, for example, 𝑒1≑0. Since βˆ’1=𝐷1(𝑣)≀𝐷1(0,|𝑣2|,…,|𝑣𝑛|) and from (2.39) 𝐷1(0,|𝑣2|,…,|𝑣𝑛|)<0, setting 𝛿=|𝐷1(0,|𝑣2|,…,|𝑣𝑛|)|βˆ’1/π‘ž1 we have 𝐷1ξ€·||0,𝛿𝑣2||||,…,𝛿𝑣𝑛||ξ€Έ=βˆ’1,π»πœ†ξ€·||0,𝛿𝑣2||||,…,𝛿𝑣𝑛||ξ€Έ=π›Ώπ‘π»πœ†ξ€·π‘£ξ€Έβ‰€π»πœ†ξ€·π‘£ξ€Έ,(2.43) then π»πœ†(0,𝛿|𝑣2|,…,𝛿|𝑣𝑛|)=π»πœ†(𝑣). This implies that ([1], see the proof of Theorem 4.1) (0,πœπ›Ώ|𝑣2|,…,πœπ›Ώ|𝑣𝑛|) is a weak solution of system (2.27). Then (βˆ‘π‘›β„“=2𝑑ℓ|𝑣ℓ|)π‘ž1βˆ’1≑0from which 𝑒ℓ≑0too as β„“=2,…,𝑛.
Condition (2.39) holds in particular when 𝑛ℓ=1𝑑ℓξƒͺπ‘ž1𝑑<min1𝑑,…,𝑛.(2.44)

Proposition 2.16. Replacing in Proposition 2.15 (2.39) with (2.44), it is right to say that 𝑒𝑖β‰₯0 and 𝑒𝑖≒0 as 𝑖=1,…,𝑛. Consequently, if 𝑝<π‘ž1π‘’π‘–βˆˆπΏβˆž(Ξ©)∩𝐢1,π‘Žπ‘–β„“π‘œπ‘(Ξ©),𝑒𝑖>0as𝑖=1,…,𝑛.(2.45)

Proof. Set 𝛿=|𝐷1(|𝑣1|,…,|𝑣𝑛|)|βˆ’1/π‘ž1, as in Proposition 2.15(πœπ›Ώ|𝑣1|,…,πœπ›Ώ|𝑣𝑛|) is a weak solution to system (2.27).
Let us add that since (2.44)⇒𝐷1(𝑐1π‘’βˆ—1,…,π‘π‘›π‘’βˆ—π‘›)<0forall(𝑐1,…,𝑐𝑛)βˆˆπ‘…π‘›β§΅{0}, there exists (Proposition 2.6) π›Ώβˆ—βˆ—>0 such that 𝑖15ξ€Έξ€·πœ†holdsifβ„“ξ€Έβ„“βˆˆπΌβˆˆπ‘›π‘‹β„“=1ξ€Ίπœ†βˆ—β„“,πœ†βˆ—β„“+π›Ώβˆ—βˆ—ξ€Ίβ§΅πœ†ξ€½ξ€·βˆ—β„“ξ€Έβ„“βˆˆπΌξ€Ύ.(2.46) Then the existence of 𝑒 is assured also choosing πœ†1,…,πœ†π‘› as in (2.46), and the conclusions of Proposition 2.16 hold.

Application 2.17. Let us set πœ†1=β‹―=πœ†π‘›=πœ†,𝑏1=β‹―=𝑏𝑛=𝑏thenπœ†βˆ—1=β‹―=πœ†βˆ—π‘›=πœ†βˆ—,π‘’βˆ—1=β‹―=π‘’βˆ—π‘›=π‘’βˆ—ξ€Έ,𝐴(𝑣)=π‘π‘›βˆ’1ℓ=1ξ€œΞ©||βˆ‡π‘£β„“||𝑝𝑑π‘₯,𝐷1(𝑣)=π‘ž1βˆ’1ξ€œΞ©π‘‘1𝑛ℓ=1||𝑣ℓ||𝛾ξƒͺπ‘ž1/𝛾𝑣𝑑π‘₯,βˆ€π‘£=1,…,π‘£π‘›ξ€Έβˆˆπ‘Š,(2.47) where 1<𝛾<π‘ž1<̃𝑝,π‘ž1≠𝑝,𝑑1∈𝐿∞(Ξ©).(2.48) Let us consider the system: ξ‚€||βˆ’divβˆ‡π‘’π‘–||π‘βˆ’2βˆ‡π‘’π‘–ξ‚=||π‘’πœ†π‘π‘–||π‘βˆ’2𝑒𝑖+𝑑1𝑛ℓ=1||𝑒ℓ||𝛾ξƒͺ(π‘ž1/𝛾)βˆ’1||𝑒𝑖||π›Ύβˆ’2𝑒𝑖𝑒inΞ©,𝑖=0onπœ•Ξ©as𝑖=1,…,𝑛.(2.49) We advance the conditions 𝑑+1ξ€·β‰’0βŸΉπ‘‰+𝐷1ξ€Έξ€Έξ€œβ‰ βˆ…,(2.50)Ω𝑑1ξ€·π‘’βˆ—ξ€Έπ‘ž1𝑑π‘₯<0⟹𝐷1𝑐1π‘’βˆ—,…,π‘π‘›π‘’βˆ—ξ€Έξ€·π‘<0βˆ€1,…,π‘π‘›ξ€Έβˆˆπ‘…π‘›β§΅ξ€Έ.{0}(2.51) Therefore, 𝑖(2.50)⟹14ξ€Έholdsifπœ†<πœ†βˆ—ξ‚ξ‚€(Proposition2.2),(2.50)and(2.51)βŸΉβˆƒπ›Ώβˆ—1𝑖>0∢14ξ€Έholdsifπœ†<πœ†βˆ—+π›Ώβˆ—1(Proposition2.4),(2.51)βŸΉβˆƒπ›Ώβˆ—2𝑖>0∢15ξ€Έholdsifξ€»πœ†πœ†βˆˆβˆ—,πœ†βˆ—+π›Ώβˆ—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 πœ†<πœ†βˆ—(resp.πœ†<πœ†βˆ—+π›Ώβˆ—1) system (2.49) has at least two weak solutions 𝑒0and βˆ’π‘’0 with 𝑒0β„“β‰₯0 as β„“=1,…,𝑛(𝑒0=𝜏0𝑣0,𝜏0=const.>0,𝑣0βˆˆπ‘†πœ†βˆ©π‘‰+(𝐷1));(ii)When (2.51) holds,ifπœ†βˆˆ]πœ†βˆ—,πœ†βˆ—+π›Ώβˆ—2[system (2.49) has at least two weak solutions 𝑒 and βˆ’π‘’ with 𝑒ℓβ‰₯0 as β„“=1,…,𝑛(𝑒=πœπ‘£,𝜏=const.>0,π‘£βˆˆπ‘‰βˆ’πœ†βˆ©π‘†(𝐷1)).
Consequently, when (2.50) and (2.51) hold, withπœ†βˆˆ]πœ†βˆ—,πœ†βˆ—+min{π›Ώβˆ—1,π›Ώβˆ—2}[ system (2.49) has at least four different weak solutions.

In order to establish some properties of 𝑒0 and 𝑒 it is useful to recall that ([1], Theorems 2.1 and 4.1)𝐷1𝑣0𝐷=sup1(𝑣)βˆΆπ‘£βˆˆπ‘†πœ†βˆ©π‘‰+𝐷1=𝑒,𝜏0=ξ€·π‘ž1π‘βˆ’1𝑒1/(π‘βˆ’π‘ž1),𝐻(2.53)πœ†ξ€·π‘£ξ€Έξ€½π»=infπœ†(𝑣)βˆΆπ‘£βˆˆπ‘‰βˆ’πœ†ξ€·π·βˆ©π‘†1ξ€Έξ€Ύ=𝑒,ξ€·πœ=βˆ’π‘π‘ž1βˆ’1𝑒1/(π‘ž1βˆ’π‘).(2.54)

Proposition 2.19. When 𝑝<π‘ž1, we have 𝑒0π‘–βˆˆπΏβˆž(Ξ©)∩𝐢1,𝛼0π‘–β„“π‘œπ‘(Ξ©),(2.55) besides 𝑒0𝑖≒0βˆ€π‘–βˆˆ{1,…,𝑛}if𝛾<𝑝.(2.56)

Proof. The relation 𝑒0π‘–βˆˆπΏβˆž(Ξ©) comes from Proposition A.3. Then [5] 𝑒0π‘–βˆˆπΆ1,𝛼0𝑖ℓoc(Ξ©).
About (2.56), it is sufficiently (Remark 1.1) to prove that ξ€·π‘–β„Ž16ξ€Έholdsβˆ€β„Žβˆˆ{1,…,𝑛}with𝔉=π‘†πœ†βˆ©π‘‰+𝐷1ξ€Έ.(2.57)
Let 𝑣=(𝑣1,…,𝑣𝑛)βˆˆπ‘†πœ†βˆ©π‘‰+(𝐷1) with π‘£β„Žβ‰‘0. Since π‘£βˆˆπ‘‰+𝐷1ξ€ΈβŸΉξƒ©||𝕂||βˆƒacompactsetπ•‚βŠ†Ξ©βˆΆπ‘>0,𝑑1>0andπœ“=β„“β‰ β„Ž||𝑣ℓ||𝛾ξƒͺ>0in𝕂,(2.58) let (Proposition A.1) (πœ‘πœ€)0<πœ€<πœ€0βŠ†πΆβˆž0(Ξ©) with 0β‰€πœ‘πœ€β‰€1 such that πœ‘πœ€βŸΆπœ’stronglyin𝐿𝑠(ξ€œΞ©),Ξ©||βˆ‡πœ‘πœ€||𝑠𝑑π‘₯⟢+∞asπœ€βŸΆ0+[[,βˆ€π‘ βˆˆ1,+∞(2.59) where πœ’ is the characteristic function of 𝕂. Set πœ€ such that ξ€œΞ©π‘‘1πœ“(π‘ž1/𝛾)βˆ’1πœ‘π›Ύπœ€π‘‘π‘₯>0,𝛿=π‘βˆ’1ξ‚Έξ€œΞ©||βˆ‡πœ‘πœ€||𝑝𝑑π‘₯βˆ’πœ†ξ€œΞ©π‘πœ‘π‘πœ€ξ‚Ήπ‘‘π‘₯>0,(2.60) with 𝑣(𝑠)=(𝑠1/𝑝𝑣1,…,(1βˆ’π‘ )1/π‘π›Ώβˆ’1/π‘πœ‘πœ€,…,𝑠1/𝑝𝑣𝑛) it results in π»πœ†(𝑣(𝑠))=π›Ώβˆ’1(1βˆ’π‘ )π‘βˆ’1ξ‚Έξ€œΞ©||βˆ‡πœ‘πœ€||𝑝𝑑π‘₯βˆ’πœ†ξ€œΞ©π‘πœ‘π‘πœ€ξ‚Ήπ‘‘π‘₯+π‘ π»πœ†[],(𝑣)=1βˆ€π‘ βˆˆ0,1βˆƒπ‘ 0∈[[0,1∢𝐷1𝑠(𝑣(𝑠))>0βˆ€π‘ βˆˆ0ξ€»,1,lim𝑠→1βˆ’π‘‘π·π‘‘π‘ 1(𝑣(𝑠))=βˆ’βˆž.(2.61)

Proposition 2.20. When 𝑝<π‘ž1, we have π‘’π‘–βˆˆπΏβˆž(Ξ©)∩𝐢1,π›Όπ‘–β„“π‘œπ‘(Ξ©),(2.62)𝑒𝑖>0βˆ€π‘–βˆˆ{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 𝑒𝑖≒0 as 𝑖=1,…,𝑛. Reasoning by contradiction, let, for example, 𝑣1≑0. We note that π‘£βˆˆπ‘‰βˆ’πœ†βŸΉξ‚΅ξ€œβˆƒβ„“βˆˆ{2,…,𝑛}∢Ω||βˆ‡π‘£β„“||𝑝𝑑π‘₯βˆ’πœ†ξ€œΞ©π‘π‘£π‘β„“ξ‚Άπ‘‘π‘₯<0.(2.64) Let us suppose β„“=2 and set 𝑣(𝑠)=((1βˆ’π‘ )1/𝛾𝑣2,𝑠1/𝛾𝑣2,𝑣3,…,𝑣𝑛). Then 𝐷1[](𝑣(𝑠))=βˆ’1βˆ€π‘ βˆˆ0,1,βˆƒπ‘ 0∈[[0,1βˆΆπ»πœ†ξ€Ίπ‘ (𝑣(𝑠))<0βˆ€π‘ βˆˆ0ξ€»,,1lim𝑠→1βˆ’π‘‘π»π‘‘π‘ πœ†(𝑣(𝑠))=+∞.(2.65) Set 𝑠1∈[𝑠0,1[ such that (𝑑/𝑑𝑠)π»πœ†(𝑣(𝑠))>0forallπ‘ βˆˆ[𝑠1,1[ and taking into account (2.54), we get the contradiction: π»πœ†ξ€·π‘£ξ€Έβ‰€π»πœ†(𝑣(𝑠))<π»πœ†ξ€·π‘£ξ€Έξ€Ίπ‘ βˆ€π‘ βˆˆ1ξ€Ί,1.(2.66)

Proposition 2.21. When 𝛾=𝑝<π‘ž1, we allow that as 𝑖=1,…,𝑛: 𝑒0𝑖>0,𝑒𝑖>0.(2.67)

Proof. The assumption 𝛾=𝑝 implies that ξ€·π‘£βˆ€π‘£=1,…,π‘£π‘›ξ€Έβˆˆπ‘Šβ§΅{0}withπ‘£β„ŽβˆƒΜƒξ€·Μƒπ‘£β‰‘0forsomeβ„Žβˆˆ{1,…,𝑛},𝑣=1̃𝑣,…,π‘›ξ€ΈΜƒπ‘£βˆˆπ‘ŠβˆΆβ„“β‰’0asβ„“=1,…,𝑛,π»πœ†(̃𝑣)=π»πœ†(𝑣),𝐷1(̃𝑣)=𝐷1(𝑣).(2.68) Let, for example, 𝑣1≑0 and 𝑣2β‰’0. Set π‘ βˆˆ]0,1[ and 𝑣11=(1βˆ’π‘ )1/𝑝𝑣2,𝑣12=𝑠1/𝑝𝑣2,𝑣1β„“=𝑣ℓ asβ„“>2, with 𝑣1=(𝑣11,…,𝑣1𝑛), we have π»πœ†ξ€·π‘£1ξ€Έ=π»πœ†(𝑣),𝐷1𝑣1ξ€Έ=𝐷1(𝑣).(2.69) If 𝑣3≑0, set 𝑣21=(1βˆ’π‘ )1/𝑝𝑣11,𝑣23=𝑠1/𝑝𝑣11,𝑣2β„“=𝑣1β„“ as β„“βˆˆ{1,…,𝑛}⧡{1,3}, with𝑣2=(𝑣21,…,𝑣2𝑛), it results in π»πœ†ξ€·π‘£2ξ€Έ=π»πœ†(𝑣),𝐷1𝑣2ξ€Έ=𝐷1(𝑣).(2.70) This method let us to find ̃𝑣.
Then, if 𝑣0β„Žβ‰‘0(resp.π‘£β„Žβ‰‘0) for some β„Žβˆˆ{1,…,𝑛}, with ̃𝑣0Μƒ(resp.𝑣) as in (2.68) we have from (2.53) (resp. (2.54)) 𝐷1(̃𝑣0)=𝑒(resp.π»πœ†(̃𝑣)=𝑒). Consequently ([1], see the proof of Theorem 2.1 (resp. Theorem 4.1)) ̃𝑒0=𝜏0̃𝑣0Μƒ(resp.𝑒=πœΜƒπ‘£) is a weak solution of system (2.49). Therefore [6] ̃𝑒0𝑖̃>0(resp.𝑒𝑖>0) as 𝑖=1,…,𝑛.

Application 2.22. Let us assume πœ†β„“,𝑏ℓ, and 𝐴 as in Application 2.17, 𝐷𝑗(𝑣)=π‘žπ‘—βˆ’1ξ€œΞ©π‘‘π‘—ξƒ©π‘›ξ“β„“=1||𝑣ℓ||𝛾𝑗ξƒͺπ‘žπ‘—/𝛾𝑗𝑣𝑑π‘₯βˆ€π‘£=1,…,π‘£π‘›ξ€Έβˆˆπ‘Šas𝑗=1,…,π‘š,(2.71) where 𝑝<π‘ž1<β‹―<π‘žπ‘š<̃𝑝,1<𝛾𝑗<π‘žπ‘—,π‘‘π‘šβˆˆπΏβˆžπ‘‘(Ξ©),π‘—βˆˆπΏβˆž(Ξ©)⧡{0},𝑑𝑗≀0if𝑗=1,…,π‘šβˆ’1.(2.72) Let us consider the system: ξ‚€||βˆ’divβˆ‡π‘’π‘–||π‘βˆ’2βˆ‡π‘’π‘–ξ‚=||π‘’πœ†π‘π‘–||π‘βˆ’2𝑒𝑖+π‘šξ“π‘—=1𝑑𝑗𝑛ℓ=1||𝑒ℓ||𝛾𝑗ξƒͺ(π‘žπ‘—/𝛾𝑗)βˆ’1||𝑒𝑖||π›Ύπ‘—βˆ’2𝑒𝑖𝑒inΞ©,𝑖=0onπœ•Ξ©as𝑖=1,…,𝑛,(2.73) under almost one of the conditions: 𝑑+π‘šξ€œβ‰’0,Ξ©π‘‘π‘šξ€·π‘’βˆ—ξ€Έπ‘žπ‘šπ‘‘π‘₯<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+(π·π‘š) and 𝑆(𝐷1) with𝑆(π·π‘š).
Thanks to Proposition A.3 and a result of [5], for the solutions 𝑒0 and 𝑒 to system (2.73), we have 𝑒0π‘–βˆˆπΏβˆž(Ξ©)∩𝐢1,𝛼0𝑖ℓoc(Ξ©),π‘’π‘–βˆˆπΏβˆž(Ξ©)∩𝐢1,𝛼𝑖ℓoc(Ξ©).(2.75) We continue to analyze the properties of 𝑒0 and 𝑒. To this aim we recall that ([1], Theorems 2.2 and 4.2), set for each π‘£βˆˆπ‘‰+(π·π‘š)(resp.π‘£βˆˆπ‘‰βˆ’πœ†βˆ©π‘†(π·π‘š))πœ“(𝑑,𝑣)=π‘π‘‘π‘βˆ’1π»πœ†βˆ‘(𝑣)βˆ’π‘šπ‘—=1π‘žπ‘—π‘‘π‘žπ‘—βˆ’1𝐷𝑗(𝑣), we have: βˆƒβˆ£π‘‘(𝑣)>0βˆΆπœ“(𝑑(𝑣),𝑣)=0,πœ•πœ“πœ•π‘‘(𝑑(𝑣),𝑣)β‰ 0.(2.76) Besides with 𝐸(𝑣)=(𝑑(𝑣))π‘π»πœ†βˆ‘(𝑣)βˆ’π‘šπ‘—=1(𝑑(𝑣))π‘žπ‘—π·π‘—(𝑣), it results in 𝐸𝑣0𝐸=inf(𝑣)βˆΆπ‘£βˆˆπ‘†πœ†βˆ©π‘‰+ξ€·π·π‘šξ€Έξ‚‡,𝜏0𝑣=𝑑0ξ€Έ,𝐸(2.77)𝑣=inf𝐸(𝑣)βˆΆπ‘£βˆˆπ‘‰βˆ’πœ†ξ€·π·βˆ©π‘†π‘šξ€Έξ‚‡,ξ€·πœ=𝑑𝑣.(2.78)

Proposition 2.23. When π›Ύπ‘š<𝑝≀𝛾𝑗 as 𝑗=1,…,π‘šβˆ’1, then 𝑒0𝑖≒0βˆ€π‘–βˆˆ{1,…,𝑛}.(2.79)

Proof. It is sufficiently (Remark 1.1) to prove that ξ€·π‘–β„Ž16ξ€Έholdsβˆ€β„Žβˆˆ{1,…,𝑛}with𝔉=π‘†πœ†βˆ©π‘‰+ξ€·π·π‘šξ€Έ.(2.80) Let 𝑣=(𝑣1,…,𝑣𝑛)βˆˆπ‘†πœ†βˆ©π‘‰+(π·π‘š) with π‘£β„Žβ‰‘0. As in Proposition 2.19, it is possible to find π‘£β„ŽβˆˆπΆβˆž0(Ξ©)⧡{0} such that with 𝑣(𝑠)=(𝑠1/𝑝𝑣1,…,(1βˆ’π‘ )1/π‘π‘£β„Ž,…,𝑠1/𝑝𝑣𝑛), it results in π»πœ†[](𝑣(𝑠))=1βˆ€π‘ βˆˆ0,1,π·π‘šξ€Ίπ‘ (𝑣(𝑠))>0βˆ€π‘ βˆˆ0,1ξ€»ξ€·0≀𝑠0ξ€Έ,<1lim𝑠→1βˆ’π‘‘π·π‘‘π‘ π‘—(𝑣(𝑠))βˆˆπ‘…as𝑗=1,…,π‘šβˆ’1,lim𝑠→1βˆ’π‘‘π·π‘‘π‘ π‘š(𝑣(𝑠))=βˆ’βˆž.(2.81)

Proposition 2.24. When 𝑝<π›Ύπ‘šβ‰€π›Ύπ‘— as 𝑗=1,…,π‘šβˆ’1, then 𝑒𝑖>0βˆ€π‘–βˆˆ{1,…,𝑛}.(2.82)

Proof. It is sufficiently [6] to prove that 𝑒𝑖≒0forallπ‘–βˆˆ{1,…,𝑛}. Reasoning by contradiction, let, for example, 𝑣1≑0and𝑣2β‰’0 such that ξ€œΞ©||βˆ‡π‘£2||𝑝𝑑π‘₯βˆ’πœ†ξ€œΞ©π‘π‘£π‘2𝑑π‘₯<0.(2.83) Since 𝑑𝑣𝑑>0,πœ“π‘£ξ€Έ,𝑣=0,πœ•πœ“ξ€·π‘‘ξ€·πœ•π‘‘π‘£ξ€Έ,𝑣≠0,(2.84) there exist an open ball 𝐡ofπ‘Š with centre 𝑣 included in π‘‰βˆ’πœ† and a unique functional π‘‘βˆ—(𝑣) belongs to 𝐢1(𝐡) such that π‘‘βˆ—ξ€·π‘‘(𝑣)>0,πœ“βˆ—ξ€Έξ‚(𝑣),𝑣=0βˆ€π‘£βˆˆπ΅.(2.85) Then, the functional πΈβˆ—ξ€·π‘‘(𝑣)=βˆ—ξ€Έ(𝑣)π‘π»πœ†(𝑣)βˆ’π‘šξ“π‘—=1ξ€·π‘‘βˆ—ξ€Έ(𝑣)π‘žπ‘—π·π‘—ξ‚π΅(𝑣)βˆ€π‘£βˆˆ(2.86) belongs to 𝐢1(𝐡), and we have 𝑑(𝑣)=π‘‘βˆ—ξ‚ξ€·π·(𝑣)βˆ€π‘£βˆˆπ΅βˆ©π‘†π‘šξ€Έ.(2.87) Then, for (2.78) πΈβˆ—ξ€·π‘£ξ€Έξ‚†πΈ=infβˆ—ξ‚ξ€·π·(𝑣)βˆΆπ‘£βˆˆπ΅βˆ©π‘†π‘šξ€Έξ‚‡.(2.88) Now, let us remark that with 𝑣(𝑠)=((1βˆ’π‘ )1/π›Ύπ‘šπ‘£2,𝑠1/π›Ύπ‘šπ‘£2,𝑣3,…,𝑣𝑛), it results in π·π‘š[](𝑣(𝑠))=βˆ’1βˆ€π‘ βˆˆ0,1,βˆƒπ‘ 0∈[[𝑠0,1βˆΆπ‘£(𝑠)βˆˆπ΅βˆ€π‘ βˆˆ0ξ€»,,1lim𝑠→1βˆ’π‘‘π»π‘‘π‘ πœ†(𝑣(𝑠))=+∞,lim𝑠→1βˆ’π‘‘π·π‘‘π‘ π‘—(𝑣(𝑠))βˆˆπ‘…as𝑗=1,…,π‘šβˆ’1.(2.89) Then, since π‘‘πΈπ‘‘π‘ βˆ—ξ€·π‘‘(𝑣(𝑠))=βˆ—ξ€Έ(𝑣(𝑠))π‘π‘‘π»π‘‘π‘ πœ†(𝑣(𝑠))βˆ’π‘šξ“π‘—=1ξ€·π‘‘βˆ—ξ€Έ(𝑣(𝑠))π‘žπ‘—π‘‘π·π‘‘π‘ π‘—ξ€Ίπ‘ (𝑣(𝑠))βˆ€π‘ βˆˆ0ξ€Ί,,1(2.90) we havelim𝑠→1βˆ’(𝑑/𝑑𝑠)πΈβˆ—(𝑣(𝑠))=+∞. Consequently, βˆƒπ‘ 1βˆˆξ€Ίπ‘ 0ξ€ΊβˆΆπ‘‘,1πΈπ‘‘π‘ βˆ—ξ€Ίπ‘ (𝑣(𝑠))>0βˆ€π‘ βˆˆ1ξ€Ί,1,(2.91) from which we get the contradiction: πΈβˆ—ξ€·π‘£ξ€Έβ‰€πΈβˆ—(𝑣(𝑠))<πΈβˆ—ξ€·π‘£ξ€Έξ€Ίπ‘ βˆ€π‘ βˆˆ1ξ€Ί,1.(2.92)

Proposition 2.25. When 𝑝=𝛾1=β‹―=π›Ύπ‘š, we allow that 𝑒0𝑖>0,𝑒𝑖>0βˆ€π‘–βˆˆ{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,…,𝑣𝑛)βˆˆπ‘Š: 𝐴(𝑣)=π‘π‘›βˆ’1ℓ=1ξ€œΞ©||βˆ‡π‘£β„“||𝑝𝑑π‘₯,𝐷𝑗(𝑣)=βˆ’π‘›ξ‘β„“=1ξ€œΞ©||𝑣ℓ||π‘žπ‘—β„“π·π‘‘π‘₯as𝑗=1,…,π‘šβˆ’1(π‘šβ‰₯2),π‘š(𝑣)=π‘žπ‘šβˆ’1βŽ‘βŽ’βŽ’βŽ£ξ€œΞ©ξƒ©π‘›ξ“β„“=1𝑑ℓ||𝑣ℓ||𝛾ξƒͺπ‘žπ‘š/𝛾𝑑π‘₯βˆ’π‘›ξ“β„“=1ξ€œΞ©ξ‚π‘‘β„“||𝑣ℓ||π‘žπ‘šβŽ€βŽ₯βŽ₯⎦,𝑑π‘₯(2.94) where 1<𝛾<π‘β‰€π‘žπ‘—β„“,𝑛ℓ=1π‘žπ‘—β„“=π‘žπ‘—<π‘žπ‘š<̃𝑝,π‘ž1<β‹―<π‘žπ‘šβˆ’1,𝑑ℓ,ξ‚π‘‘β„“βˆˆπΏβˆž(Ξ©),𝑑ℓ,𝑑ℓ