Abstract and Applied Analysis

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Variational Methods and Critical Point Theory

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Volume 2012 |Article ID 760854 | https://doi.org/10.1155/2012/760854

Luisa Toscano, Speranza Toscano, "Dirichlet and Neumann Problems Related to Nonlinear Elliptic Systems: Solvability, Multiple Solutions, Solutions with Positive Components", Abstract and Applied Analysis, vol. 2012, Article ID 760854, 44 pages, 2012. https://doi.org/10.1155/2012/760854

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

Academic Editor: D. O'Regan
Received01 Feb 2012
Accepted02 Apr 2012
Published15 Aug 2012


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: 𝐴div𝑖𝑥,𝑢1,,𝑢𝑛,𝑢1,,𝑢𝑛=𝜆𝑖𝑏𝑖||𝑢𝑖||𝑝2𝑢𝑖+𝑑𝑖𝑥,𝑢1,,𝑢𝑛,𝑢1,,𝑢𝑛+𝑓𝑖𝑢inΩ,𝑖=0on𝜕Ωas𝑖=1,,𝑛,(1.1) and to Neumann problems (Section 3) of the type: 𝐴div𝑖𝑥,𝑢1,,𝑢𝑛,𝑢1,,𝑢𝑛=𝜆𝑖𝑏𝑖||𝑢𝑖||𝑝2𝑢𝑖+𝑑𝑖𝑥,𝑢1,,𝑢𝑛,𝑢1,,𝑢𝑛+𝑓𝑖𝐴inΩ,𝑖𝑥,𝑢1,,𝑢𝑛,𝑢1,,𝑢𝑛𝜈=𝜇𝑖̂𝑏𝑖||𝑢𝑖||𝑝2𝑢𝑖+𝑑𝑖𝑥,𝑢1,,𝑢𝑛,𝑢1,,𝑢𝑛+𝑓𝑖on𝜕Ωas𝑖=1,,𝑛,(1.2) where 𝑛1,𝜆𝑖,𝜇𝑖 are real parameters, Ω is a bounded connected open set of 𝑅𝑁 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,,𝑊𝑛 real reflexive Banach spaces (𝑛1). Let 𝑊 be the product space 𝑋𝑛=1𝑊. Let be the norm on 𝑊, the norm on 𝑊 (dual space of 𝑊), and , (resp.,) the duality between 𝑊 (dual space of 𝑊) and 𝑊 (resp. 𝑊 and 𝑊). Let us denote by “𝜕” Fréchet differential operator and by “𝜕𝑢” Fréchet differential operator with respect to 𝑢. Let 𝐴0 and 𝐷𝑗0(𝑗=1,,𝑚;𝑚1) be real functionals defined in 𝑊,𝐵 and 𝐵(=1,,𝑛) real functionals defined in 𝑊 satisfying the conditions:(𝑖11)𝐴 is lower weakly semicontinuous in 𝑊 and 𝐶1(𝑊{0}),𝐵𝐵and are weakly continuous in 𝑊 and𝐶1(𝑊),𝑝>1𝐴(𝑡𝑣)=𝑡𝑝𝐴(𝑣)forall𝑡0and forall𝑣𝑊,𝐵(𝑡𝑣)=𝑡𝑝𝐵(𝑣) and 𝐵(𝑡𝑣)=𝑡𝑝𝐵(𝑣)forall𝑡0 andforall𝑣𝑊;(𝑖12)𝐷𝑗 is weakly continuous in 𝑊 and 𝐶1(𝑊{0}),𝑞𝑗>1𝐷𝑗(𝑡𝑣)=𝑡𝑞𝑗𝐷𝑗(𝑣)forall𝑡0 and forall𝑣𝑊,1<𝑞1<<𝑞𝑚if𝑚>1.

Let𝐹=(𝐹1,,𝐹𝑛)with 𝐹𝑊,𝜆and𝜇𝑅; let us consider the following problem.

Problem (𝑃). Find 𝑢=(𝑢1,,𝑢𝑛)𝑊{0}such that 𝜕𝑢𝑖𝐴(𝑢),𝑣𝑖𝑖=𝜆𝑖𝜕𝐵𝑖𝑢𝑖,𝑣𝑖𝑖+𝜇𝑖𝜕𝐵𝑖𝑢𝑖,𝑣𝑖𝑖+𝑚𝑗=1𝜕𝑢𝑖𝐷𝑗(𝑢),𝑣𝑖𝑖+𝐹𝑖,𝑣𝑖𝑖𝑖{1,,𝑛},𝑣𝑖𝑊𝑖.(1.3) Obviously Problem (P) means to find the critical points 𝑢𝑊{0} of the Euler functional: 𝐸(𝑣)=𝐴(𝑣)𝑛=1𝜆𝐵𝑣+𝜇𝐵𝑣𝑚𝑗=1𝐷𝑗𝑣(𝑣)𝐹,𝑣𝑣=1,,𝑣𝑛𝑊,(1.4) where 𝐹,𝑣=𝑛=1𝐹,𝑣.

Let us set 𝐻𝜆𝜇(𝑣)=𝐴(𝑣)𝑛=1𝜆𝐵𝑣+𝜇𝐵𝑣𝑣𝑣=1,,𝑣𝑛𝜆𝑊,𝜆=1,,𝜆𝑛𝜇,𝜇=1,,𝜇𝑛𝑅𝑛,𝑆𝜆𝜇=𝑣𝑊𝐻𝜆𝜇(𝑣)=1,𝑉𝜆𝜇=𝑣𝑊𝐻𝜆𝜇(𝑣)<0,as𝑚1𝑉=1,,𝑚+𝐷𝑚1,,𝐷𝑚=𝑣𝑊𝑚𝑗=𝑚1𝐷𝑗,𝑆(𝑣)>0+𝐷1,,𝐷𝑚=𝑣𝑊𝑚𝑗=1𝐷𝑗,𝑆𝐷(𝑣)=1𝑗=𝑣𝑊𝐷𝑗(𝑣)=1,𝑉+(𝐹)={𝑣𝑊𝐹,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)>0𝑣𝑝𝑐(𝜆,𝜇)𝐻𝜆𝜇(𝑣)forall𝑣𝑊; (𝑖14)𝑐(𝜆,𝜇)>0𝑣𝑝𝑐(𝜆,𝜇)𝐻𝜆𝜇(𝑣)forall𝑣𝑉+(𝐷𝑚)(if𝑉+(𝐷𝑚)); (𝑖15)𝑚1{1,,𝑚}𝑉𝜆𝜇𝑆(𝐷𝑚1) 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 𝔉𝑆𝜆𝜇, we suppose(𝑖16) for each 𝑣=(𝑣1,,𝑣𝑛)𝔉 with 𝑣=0, there exist 𝑣𝑊{0} and the real functions 𝜙1,,𝜙𝑛 such that 𝜙𝐶0([0,1])𝐶1([0,1[) and 𝜙(1)=0, 𝜙𝐶1([0,1]) and 𝜙(1)=1 as , 𝑣(𝑠)=(𝜙1(𝑠)𝑣1,,𝜙(𝑠)𝑣,,𝜙𝑛(𝑠)𝑣𝑛)𝔉 for all 𝑠[𝑠0,1](0𝑠0<1), lim𝑠1(𝑑/𝑑𝑠)𝐷𝑗(𝑣(𝑠))<+ for all 𝑗{1,,𝑚}, lim𝑠1(𝑑/𝑑𝑠)𝐷𝑗(𝑣(𝑠))= for some 𝑗{1,,𝑚}.
The condition (𝑖16) assures that for the solutions 𝑢=(𝑢1,,𝑢𝑛) of Problem (P), found with the method used in the recalled theorems, we have 𝑢0 if 𝐹0.

Before showing Dirichlet problems (including the problem studied in [2] by Drábek and Pohozaev when 𝑛=1 and 𝑚=1) we give Propositions 2.22.6 which show some cases in which hypotheses (𝑖13)−(𝑖15) hold. These propositions are based on the comparison between the parameters 𝜆𝑖 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 3.13.5 in which the parameters 𝜆𝑖 and 𝜇𝑖 have compared with zero. Finally, the results in Appendix are very useful: Propositions A.1 and A.2 in order to get condition (𝑖16), 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 Ω𝑅𝑁 be an open, bounded, connected and 𝐶2,𝛽 set with 0<𝛽1. Let||𝑁 the Lebesgue measure on 𝑅𝑁,1<𝑝<,̃𝑝=𝑁𝑝/(𝑁𝑝)if𝑁>𝑝,̃𝑝=otherwise.

Let us assume𝑊𝑊=01,𝑝(Ω)𝑛(𝑛1)with𝑣=𝑛=1Ω||𝑣||𝑝𝑑𝑥1/𝑝𝑣𝑣=1,,𝑣𝑛𝐵𝑊,𝑣=𝑝1Ω𝑏||𝑣||𝑝𝑑𝑥𝑣𝑊01,𝑝(Ω)where𝑏𝐿(Ω){0},𝑏𝐵0,0.(2.1) Moreover we consider the functionals 𝐴 (as in (𝑖11)) such that̃𝑐>0𝐴(𝑣)𝑝1̃𝑐𝑣𝑝𝑣𝑊.(2.2) Let us use the notation 𝐻𝜆(𝑆𝜆and𝑉𝜆,resp.) instead of 𝐻𝜆𝜇(𝑆𝜆𝜇and𝑉𝜆𝜇,resp.).

As =1,,𝑛 let 𝜆and𝑢, respectively, the first eigenvalue and the first eigenfunction of the problem: 𝑢𝑊01,𝑝||(Ω)̃𝑐div𝑢||𝑝2𝑢=𝜃𝑏||𝑢||𝑝2𝑢inΩ.(2.3) Let us remember that [4]𝑢𝐶1,𝛼(Ω) with 0<𝛼<1,𝑢>0 in Ω;𝜆=̃𝑐Ω|𝑢|𝑝𝑑𝑥/Ω𝑏|𝑢|𝑝𝑑𝑥=min{̃𝑐Ω|𝑣|𝑝𝑑𝑥/Ω𝑏|𝑣|𝑝𝑑𝑥Ω𝑏|𝑣|𝑝𝑑𝑥>0};𝜆 is simple, that is, each eigenfunction of (2.3) related to 𝜆 is of the type 𝑐𝑢 with 𝑐𝑅{0};𝜆 is isolate, that is, there exists 𝑎>0 such that 𝜆 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 𝑏𝐿(Ω) and 𝑏+=max{𝑏,0}0 as =1,,𝑛. This holds also for the results of this section if we limit to consider only the parameters 𝜆1,,𝜆𝑛 nonnegative.

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

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

Proposition 2.2. If 𝜆<𝜆forall{1,,𝑛}, then (𝑖13) holds. Consequently, (𝑖14) holds when 𝑉+(𝐷𝑚).

When 𝜆𝜆  for some {1,,𝑛}, it is possible to fulfil (𝑖14) with an additional condition on 𝐷𝑚. Let 𝐼={1,,𝑛}. For any 𝐼𝐼 let 𝑉=𝑣𝑣=1,,𝑣𝑛𝑊𝑣0if𝐼𝐼,𝑣=𝑐𝑢if𝐼with𝑐𝑅and𝑐,0forsome(2.4) and let us suppose(𝑖21) There exists 𝐼𝐼𝐷𝑚(𝑣)<0forall𝑣𝑉.

Proposition 2.3. Let (𝑖21) holds with 𝐼𝐼. Let 𝑉+(𝐷𝑚). If we fix the parameters set (𝜆)𝐼𝐼 with 𝜆<𝜆, then there exists 𝛿>0 such that (𝑖14) also holds for any (𝜆)𝐼𝑋𝐼[𝜆,𝜆+𝛿[.

Proof. Arguing by contradiction, for any 𝑘 there exist (𝜆𝑘)𝐼𝑋𝐼[𝜆,𝜆+𝑘1[ and 𝑣𝑘=(𝑣𝑘1,,𝑣𝑘𝑛)𝑉+(𝐷𝑚) such that 𝐴𝑣𝑘𝑝1𝐼𝐼𝜆Ω𝑏||𝑣||𝑝𝑑𝑥𝑝1𝐼𝜆𝑘Ω𝑏||𝑣𝑘||𝑝𝑑𝑥<𝑘1𝑣𝑘𝑝.(2.5) Set 𝑤𝑘=𝑣𝑘1𝑣𝑘, we have 𝐷𝑚𝑤𝑘>0,̃𝑐𝐼𝐼Ω||𝑤𝑘||𝑝𝑑𝑥𝐼𝐼𝜆Ω𝑏||𝑤𝑘||𝑝𝑑𝑥+̃𝑐𝐼Ω||𝑤𝑘||𝑝𝑑𝑥𝐼𝜆𝑘Ω𝑏||𝑤𝑘||𝑝𝑑𝑥<𝑝𝑘1,(2.6) moreover, since 𝑤𝑘=1, there exists 𝑤𝑊 such that (within a subsequence) 𝑤𝑘𝑤weaklyin𝑊,𝑤𝑘𝑤stronglyin(𝐿𝑝(Ω))𝑛.(2.7) Taking into account that 𝐷𝑚 is weakly continuous in 𝑊, from (2.6) as 𝑘+ we get 𝐷𝑚(𝑤)0,(2.8)𝐼𝐼̃𝑐Ω||𝑤||𝑝𝑑𝑥𝜆Ω𝑏||𝑤||𝑝+𝑑𝑥𝐼̃𝑐Ω||𝑤||𝑝𝑑𝑥𝜆Ω𝑏||𝑤||𝑝𝑑𝑥0.(2.9) Since 𝑤0̃𝑐Ω||𝑤||𝑝𝑑𝑥𝜆Ω𝑏||𝑤||𝑝𝑑𝑥>0,̃𝑐Ω||𝑤||𝑝𝑑𝑥𝜆Ω𝑏||𝑤||𝑝𝑑𝑥0,(2.10) from (2.9), we deduce that 𝑤0𝐼𝐼,𝐼𝑐𝑅𝑤=𝑐𝑢.(2.11) Let us add that𝑐0 for some 𝐼, since if 𝑐=0forall𝐼 we have the contradiction ̃𝑐=̃𝑐lim𝑘+𝑤𝑘𝑝=0. Then 𝑤𝑉, and consequently 𝐷𝑚(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+(𝐷𝑚). Then, there exists 𝛿>0 such that (i14) also holds for any (𝜆)𝐼𝑋𝐼[𝜆,𝜆+𝛿[.

Let us pass to (𝑖15) and suppose(𝑖22) there exist𝐼𝐼 and 𝑚1{1,,𝑚}such that𝐷𝑚1(𝑣)<0and𝐴(𝑣)=̃𝑐𝑝1𝐼Ω|𝑣|𝑝𝑑𝑥 for any 𝑣𝑉.

Proposition 2.5. If (i22) holds with𝐼𝐼, then 𝑉𝜆𝐷𝑆𝑚1𝜆𝐼𝜆with𝐼𝑋𝐼𝜆𝜆,+𝐼.(2.12) Moreover, if we fix the parameters set (𝜆)𝐼𝐼 with 𝜆<𝜆, then there exists 𝛿>0 such that 𝑉𝜆𝐷𝑆𝑚1𝜆isboundedin𝑊𝐼𝑋𝐼𝜆,𝜆+𝛿𝜆𝐼.(2.13)

Proof. Let us prove (2.12). Let 𝑣𝑉with 𝑣=𝑢if 𝐼, then 𝐷𝑚1(𝑣)<0. Let 𝑤=|𝐷𝑚1(𝑣)|1𝑞𝑚1𝑣, we have 𝐷𝑚1||𝐷(𝑤)=𝑚1||(𝑣)1𝐷𝑚1𝐻(𝑣)=1,𝜆(𝑤)=𝑝1𝐼̃𝑐Ω||𝑤||𝑝𝑑𝑥𝜆Ω𝑏||𝑤||𝑝𝑑𝑥<0.(2.14) Let us prove (2.13). Arguing by contradiction, for any 𝑘 there exist (𝜆𝑘)𝐼𝑋𝐼[𝜆,𝜆+𝑘1[ with (𝜆𝑘)𝐼(𝜆)𝐼 and (𝑣𝑘,)𝑉𝜆𝑘𝑆(𝐷𝑚1), where 𝜆𝑘=𝜆 if 𝐼𝐼, such that sup𝑣𝑘,=+.(2.15) Relation (2.15) implies that there exists (𝑘)𝑘 strictly increasing such that 𝛿𝑘=𝑣𝑘,𝑘+as𝑘+.(2.16) Let 𝑤𝑘=𝛿𝑘1𝑣𝑘,𝑘, we have 𝐼𝐼̃𝑐Ω||𝑤𝑘||𝑝𝑑𝑥𝜆Ω𝑏||𝑤𝑘||𝑝+𝑑𝑥𝐼̃𝑐Ω||𝑤𝑘||𝑝𝑑𝑥𝜆𝑘Ω𝑏||𝑤𝑘||𝑝𝐷𝑑𝑥<0,𝑚1𝑤𝑘=𝛿𝑞𝑚1𝑘,𝑤𝑊(withinasubsequence)𝑤𝑘𝑤weaklyin𝑊,𝑤𝑘𝑤stronglyin(𝐿𝑝(Ω))𝑛.(2.17) Then, as 𝑘+ we get 𝐼𝐼̃𝑐Ω||𝑤||𝑝𝑑𝑥𝜆Ω𝑏||𝑤||𝑝+𝑑𝑥𝐼̃𝑐Ω||𝑤||𝑝𝑑𝑥𝜆Ω𝑏||𝑤||𝑝𝐷𝑑𝑥0,(2.18)𝑚1(𝑤)=0.(2.19) From (2.18), we get that 𝑤𝑉. Then since (𝑖22) inequality 𝐷𝑚1(𝑤)<0 holds, which contradicts (2.19).

Proposition 2.6. If (𝑖22) holds with 𝐼=𝐼, then 𝑉𝜆𝐷𝑆𝑚1𝜆𝜆=𝐼𝑋𝐼𝜆𝜆,+𝐼,𝛿>0𝑉𝜆𝐷𝑆𝑚1𝜆isboundedin𝑊𝜆=𝐼𝑋𝐼𝜆,𝜆+𝛿𝜆𝐼.(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 𝑏𝐿(Ω){0},𝑏0.

Application 2.8. Let 𝑛=1. Let us consider the problem ||||div𝑢𝑝2𝑢=𝜆1𝑏1|𝑢|𝑝2𝑢+𝑚𝑗=1𝑑𝑗|𝑢|𝑞𝑗2𝑢inΩ,𝑢=0on𝜕Ω,(2.21) where 𝑝<𝑞1<̃𝑝,𝑑1𝐿(Ω){0}if𝑚=1,𝑝<𝑞1<<𝑞𝑚<̃𝑝,𝑑𝑗𝐿𝑑(Ω){0}as𝑗=1,,𝑚,𝑗0as𝑗=1,,𝑚1if𝑚>1.(2.22) Evidently 𝐴(𝑣)=𝑝1Ω||||𝑣𝑝𝑑𝑥,𝐷𝑗(𝑣)=𝑞𝑗1Ω𝑑𝑗|𝑣|𝑞𝑗𝑑𝑥𝑣𝑊.(2.23) Let us advance the conditions: 𝑑+𝑚0𝑉+𝐷𝑚,(2.24)Ω𝑑𝑚𝑢1𝑞𝑚𝑑𝑥<0𝐷𝑚𝑐1𝑢1<0𝑐1.𝑅{0}(2.25) Let us note that (Propositions 2.2, 2.4, and 2.6) 𝑖(2.24)14holdsif𝜆1<𝜆1,(2.24)and(2.25)𝛿1𝑖>014holdsif𝜆1<𝜆1+𝛿1,(2.25)𝛿2𝑖>015holdsif𝜆1𝜆1,𝜆1+𝛿2.(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<𝜆1 [resp. (2.24) and (2.25) hold, with 𝜆1<𝜆1+𝛿1] problem (2.21) has at least two weak solutions 𝑢0 and𝑢0(𝑢0=𝜏0𝑣0,𝜏0=const.>0,𝑣0𝑆𝜆1𝑉+(𝐷𝑚)), and it results in 𝑢0𝐿(Ω)𝐶1,𝛼0𝑜𝑐(Ω),𝑢0>0;(ii)When (2.25) holds, with 𝜆1]𝜆1,𝜆1+𝛿2[ problem (2.21) has at least two weak solutions 𝑢𝑎𝑛𝑑𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣𝑉𝜆1𝑆(𝐷𝑚)), and it results in 𝑢𝐿(Ω)𝐶1,𝛼𝑜𝑐(Ω),𝑢>0.
Consequently, when (2.24) and (2.25) hold, with 𝜆1]𝜆1,𝜆1+min{𝛿1,𝛿2}[ 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: ||div𝑢𝑖||𝑝2𝑢𝑖=𝜆𝑖𝑏𝑖||𝑢𝑖||𝑝2𝑢𝑖+|||||𝑛=1𝑑𝑢|||||𝑞12𝑛=1𝑑𝑢𝑑𝑖𝑑𝑖||𝑢𝑖||𝑞12𝑢𝑖𝑢inΩ,𝑖=0on𝜕Ωas𝑖=1,,𝑛,(2.27) where 1<𝑞1<̃𝑝,𝑞1𝑝,𝑑,𝑑𝐿(Ω),𝑑,𝑑>0.(2.28) System (2.27) is included among Problem (P) with: 𝐴(𝑣)=𝑝𝑛1=1Ω||𝑣||𝑝𝐷𝑑𝑥,1(𝑣)=𝑞11Ω|||||𝑛=1𝑑𝑣|||||𝑞1𝑑𝑥𝑛=1Ω𝑑||𝑣||𝑞1𝑣𝑑𝑥𝑣=1,,𝑣𝑛𝑊.(2.29) Let us advance the conditions (compatible): 𝑑𝑞1<𝑑{1,,𝑛}𝐷10,,𝑐𝑖𝑢𝑖,,0<0as𝑖=1,,𝑛,𝑐𝑖𝑅{0},(2.30) there exist Ω+Ω and a constant ̃𝑐𝑗>0 such that |Ω+|𝑁>0 and 𝑗𝑑+̃𝑐𝑗𝑑𝑗𝑞1>𝑗𝑑+̃𝑐𝑞1𝑗𝑑𝑗inΩ+𝑉+𝐷1.(PropositionA.1)(2.31) Then (Propositions 2.2, 2.3, and 2.5) 𝑖(2.31)14holdsif𝜆<𝜆{1,,𝑛},(2.32) and set 𝑖{1,,𝑛}(2.30)and(2.31)with𝜆<𝜆𝑖𝛿1𝑖>014holdsif𝜆𝑖<𝜆𝑖+𝛿1(,(2.33)2.30)with𝜆<𝜆𝑖𝛿2𝑖>015holdsif𝜆𝑖𝜆𝑖,𝜆𝑖+𝛿2.(2.34)

Taking into account that 𝐷1(𝑣1,,𝑣𝑛)𝐷1(|𝑣1|,,|𝑣𝑛|) 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,,𝜆𝑛 as in (2.32) (resp. (2.33)) system (2.27) has at least two weak solutions 𝑢0 and 𝑢0 with 𝑢00 as =1,,𝑛(𝑢0=𝜏0𝑣0,𝜏0=const.>0,𝑣0𝑆𝜆𝑉+(𝐷1)); (ii)When (2.30) holds, choosing𝜆1,,𝜆𝑛 as in (2.34) system (2.27) has at least two weak solutions 𝑢 and 𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣𝑉𝜆𝑆(𝐷1)).
Consequently, when (2.30) and (2.31) hold, with 𝜆<𝜆forall𝑖 and 𝜆𝑖]𝜆𝑖,𝜆𝑖+min{𝛿1,𝛿2}[ system (2.27) has at least four different weak solutions.

The following proposition is obvious.

Proposition 2.13. The following relations hold: 𝑢0𝑖0as𝑖=1,,𝑛,,𝑘{1,,𝑛}𝑢0,𝑢𝑘0.(2.35)

Proposition 2.14. If 𝑝<𝑞1, then as 𝑖=1,,𝑛: 𝑢0𝑖𝐿(Ω)𝐶1,𝛼0𝑖𝑜𝑐(Ω),𝑢0𝑖>0.(2.36)

Proof. It is easy to prove that 𝑛𝑖=1Ω||𝑢0𝑖||𝑝2𝑢0𝑖𝑣𝑖𝑑𝑥Ω𝑔𝑛𝑖=1𝑢0𝑖𝑝1𝑛𝑖=1𝑣𝑖𝑣𝑑𝑥𝑣=1,,𝑣𝑛𝑊01,𝑝(Ω)𝐿(Ω)𝑛with𝑣𝑖0,(2.37) where 𝑔𝐿𝑞1/(𝑞1𝑝)(Ω). Then (Proposition A.3) 𝑢0𝑖𝐿(Ω) and consequently [5] 𝑢0𝑖𝐶1,𝛼0𝑖oc(Ω).
Let us note that 𝑢0𝑖 is a weak supersolution to the equation: ||div𝑢𝑖||𝑝2𝑢𝑖=𝜆𝑖𝑏𝑖||𝑢𝑖||𝑝2𝑢𝑖𝑑𝑖||𝑢𝑖||𝑞12𝑢𝑖inΩ.(2.38) Then, since (2.35), it must be [6] 𝑢0𝑖>0.

Let us continue the analysis of system (2.27) under the condition: 𝑖𝑑𝑞1𝑑<min1𝑑,,𝑛𝑖{1,,𝑛},(2.39) then 𝐷1𝑐1𝑢1,,𝑐𝑛𝑢𝑛𝑐<01,,𝑐𝑛𝑅𝑛{0}with𝑐𝑖=0foratleastone𝑖{1,,𝑛}.(2.40) Hence (Proposition 2.5) if 𝐼𝐼 and 𝐼𝐼: (2.39)as𝜆<𝜆𝐼𝐼𝛿𝑖>015𝜆holdsif𝐼𝑋𝐼𝜆,𝜆+𝛿𝜆𝐼.(2.41)

Proposition 2.15. Under assumptions (2.28) and (2.39), choosing 𝜆1,,𝜆𝑛 as in (2.41) system (2.27) has at least two weak solutions 𝑢and𝑢(𝑢=𝜏𝑣,𝜏=const.>0,𝑣𝑉𝜆𝑆(𝐷1)) with 𝑢𝑖0 as 𝑖=1,,𝑛.

Proof. Thanks to ([1], Theorem 4.1), there exists 𝑣𝑉𝜆𝑆(𝐷1) such that 𝐻𝜆𝑣𝐻=inf𝜆(𝑣)𝑣𝑉𝜆𝐷𝑆1=𝑒,𝑢=𝜏𝑣isaweaksolutionofsystem(2.27),(2.42) where 𝜏=(𝑝𝑞11𝑒)1/(𝑞1𝑝).
Reasoning by contradiction, let, for example, 𝑢10. Since 1=𝐷1(𝑣)𝐷1(0,|𝑣2|,,|𝑣𝑛|) and from (2.39) 𝐷1(0,|𝑣2|,,|𝑣𝑛|)<0, setting 𝛿=|𝐷1(0,|𝑣2|,,|𝑣𝑛|)|1/𝑞1 we have 𝐷1||0,𝛿𝑣2||||,,𝛿𝑣𝑛||=1,𝐻𝜆||0,𝛿𝑣2||||,,𝛿𝑣𝑛||=𝛿𝑝𝐻𝜆𝑣𝐻𝜆𝑣,(2.43) then 𝐻𝜆(0,𝛿|𝑣2|,,𝛿|𝑣𝑛|)=𝐻𝜆(𝑣). This implies that ([1], see the proof of Theorem 4.1) (0,𝜏𝛿|𝑣2|,,𝜏𝛿|𝑣𝑛|) is a weak solution of system (2.27). Then (𝑛=2𝑑|𝑣|)𝑞110from which 𝑢0too as =2,,𝑛.
Condition (2.39) holds in particular when 𝑛=1𝑑𝑞1𝑑<min1𝑑,,𝑛.(2.44)

Proposition 2.16. Replacing in Proposition 2.15 (2.39) with (2.44), it is right to say that 𝑢𝑖0 and 𝑢𝑖0 as 𝑖=1,,𝑛. Consequently, if 𝑝<𝑞1𝑢𝑖𝐿(Ω)𝐶1,𝑎𝑖𝑜𝑐(Ω),𝑢𝑖>0as𝑖=1,,𝑛.(2.45)

Proof. Set 𝛿=|𝐷1(|𝑣1|,,|𝑣𝑛|)|1/𝑞1, as in Proposition 2.15(𝜏𝛿|𝑣1|,,𝜏𝛿|𝑣𝑛|) is a weak solution to system (2.27).
Let us add that since (2.44)𝐷1(𝑐1𝑢1,,𝑐𝑛𝑢𝑛)<0forall(𝑐1,,𝑐𝑛)𝑅𝑛{0}, there exists (Proposition 2.6) 𝛿>0 such that 𝑖15𝜆holdsif𝐼𝑛𝑋=1𝜆,𝜆+𝛿𝜆𝐼.(2.46) Then the existence of 𝑢 is assured also choosing 𝜆1,,𝜆𝑛 as in (2.46), and the conclusions of Proposition 2.16 hold.

Application 2.17. Let us set 𝜆1==𝜆𝑛=𝜆,𝑏1==𝑏𝑛=𝑏then𝜆1==𝜆𝑛=𝜆,𝑢1==𝑢𝑛=𝑢,𝐴(𝑣)=𝑝𝑛1=1Ω||𝑣||𝑝𝑑𝑥,𝐷1(𝑣)=𝑞11Ω𝑑1𝑛=1||𝑣||𝛾𝑞1/𝛾𝑣𝑑𝑥,𝑣=1,,𝑣𝑛𝑊,(2.47) where 1<𝛾<𝑞1<̃𝑝,𝑞1𝑝,𝑑1𝐿(Ω).(2.48) Let us consider the system: ||div𝑢𝑖||𝑝2𝑢𝑖=||𝑢𝜆𝑏𝑖||𝑝2𝑢𝑖+𝑑1𝑛=1||𝑢||𝛾(𝑞1/𝛾)1||𝑢𝑖||𝛾2𝑢𝑖𝑢inΩ,𝑖=0on𝜕Ωas𝑖=1,,𝑛.(2.49) We advance the conditions 𝑑+10𝑉+𝐷1,(2.50)Ω𝑑1𝑢𝑞1𝑑𝑥<0𝐷1𝑐1𝑢,,𝑐𝑛𝑢𝑐<01,,𝑐𝑛𝑅𝑛.{0}(2.51) Therefore, 𝑖(2.50)14holdsif𝜆<𝜆(Proposition2.2),(2.50)and(2.51)𝛿1𝑖>014holdsif𝜆<𝜆+𝛿1(Proposition2.4),(2.51)𝛿2𝑖>015holdsif𝜆𝜆,𝜆+𝛿2(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+𝛿1) system (2.49) has at least two weak solutions 𝑢0and 𝑢0 with 𝑢00 as =1,,𝑛(𝑢0=𝜏0𝑣0,𝜏0=const.>0,𝑣0𝑆𝜆𝑉+(𝐷1));(ii)When (2.51) holds,if𝜆]𝜆,𝜆+𝛿2[system (2.49) has at least two weak solutions 𝑢 and 𝑢 with 𝑢0 as =1,,𝑛(𝑢=𝜏𝑣,𝜏=const.>0,𝑣𝑉𝜆𝑆(𝐷1)).
Consequently, when (2.50) and (2.51) hold, with𝜆]𝜆,𝜆+min{𝛿1,𝛿2}[ 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)𝐷1𝑣0𝐷=sup1(𝑣)𝑣𝑆𝜆𝑉+𝐷1=𝑒,𝜏0=𝑞1𝑝1𝑒1/(𝑝𝑞1),𝐻(2.53)𝜆𝑣𝐻=inf𝜆(𝑣)𝑣𝑉𝜆𝐷𝑆1=𝑒,𝜏=𝑝𝑞11𝑒1/(𝑞1𝑝).(2.54)

Proposition 2.19. When 𝑝<𝑞1, we have 𝑢0𝑖𝐿(Ω)𝐶1,𝛼0𝑖𝑜𝑐(Ω),(2.55) besides 𝑢0𝑖0𝑖{1,,𝑛}if𝛾<𝑝.(2.56)

Proof. The relation 𝑢0𝑖𝐿(Ω) comes from Proposition A.3. Then [5] 𝑢0𝑖𝐶1,𝛼0𝑖oc(Ω).
About (2.56), it is sufficiently (Remark 1.1) to prove that 𝑖16holds{1,,𝑛}with𝔉=𝑆𝜆𝑉+𝐷1.(2.57)
Let 𝑣=(𝑣1,,𝑣𝑛)𝑆𝜆𝑉+(𝐷1) with 𝑣0. Since 𝑣𝑉+𝐷1||𝕂||acompactset𝕂Ω𝑁>0,𝑑1>0and𝜓=||𝑣||𝛾>0in𝕂,(2.58) let (Proposition A.1) (𝜑𝜀)0<𝜀<𝜀0𝐶0(Ω) with 0𝜑𝜀1 such that 𝜑𝜀𝜒stronglyin𝐿𝑠(Ω),Ω||𝜑𝜀||𝑠𝑑𝑥+as𝜀0+[[,𝑠1,+(2.59) where 𝜒 is the characteristic function of 𝕂. Set 𝜀 such that Ω𝑑1𝜓(𝑞1/𝛾)1𝜑𝛾𝜀𝑑𝑥>0,𝛿=𝑝1Ω||𝜑𝜀||𝑝𝑑𝑥𝜆Ω𝑏𝜑𝑝𝜀𝑑𝑥>0,(2.60) with 𝑣(𝑠)=(𝑠1/𝑝𝑣1,,(1𝑠)1/𝑝𝛿1/𝑝𝜑𝜀,,𝑠1/𝑝𝑣𝑛) it results in 𝐻𝜆(𝑣(𝑠))=𝛿1(1𝑠)𝑝1Ω||𝜑𝜀||𝑝𝑑𝑥𝜆Ω𝑏𝜑𝑝𝜀𝑑𝑥+𝑠𝐻𝜆[],(𝑣)=1𝑠0,1𝑠0[[0,1𝐷1𝑠(𝑣(𝑠))>0𝑠0,1,lim𝑠1𝑑𝐷𝑑𝑠1(𝑣(𝑠))=.(2.61)

Proposition 2.20. When 𝑝<𝑞1, we have 𝑢𝑖𝐿(Ω)𝐶1,𝛼𝑖𝑜𝑐(Ω),(2.62)𝑢𝑖>0𝑖{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 𝑢𝑖0 as 𝑖=1,,𝑛. Reasoning by contradiction, let, for example, 𝑣10. We note that 𝑣𝑉𝜆{2,,𝑛}Ω||𝑣||𝑝𝑑𝑥𝜆Ω𝑏𝑣𝑝𝑑𝑥<0.(2.64) Let us suppose =2 and set 𝑣(𝑠)=((1𝑠)1/𝛾𝑣2,𝑠1/𝛾𝑣2,𝑣3,,𝑣𝑛). Then 𝐷1[](𝑣(𝑠))=1𝑠0,1,𝑠0[[0,1𝐻𝜆𝑠(𝑣(𝑠))<0𝑠0,,1lim𝑠1𝑑𝐻𝑑𝑠𝜆(𝑣(𝑠))=+.(2.65) Set 𝑠1[𝑠0,1[ such that (𝑑/𝑑𝑠)𝐻𝜆(𝑣(𝑠))>0forall𝑠[𝑠1,1[ and taking into account (2.54), we get the contradiction: 𝐻𝜆𝑣𝐻𝜆(𝑣(𝑠))<𝐻𝜆𝑣𝑠𝑠1,1.(2.66)

Proposition 2.21. When 𝛾=𝑝<𝑞1, we allow that as 𝑖=1,,𝑛: 𝑢0𝑖>0,𝑢𝑖>0.(2.67)

Proof. The assumption 𝛾=𝑝 implies that 𝑣𝑣=1,,𝑣𝑛𝑊{0}with𝑣̃̃𝑣0forsome{1,,𝑛},𝑣=1̃𝑣,,𝑛̃𝑣𝑊0as=1,,𝑛,𝐻𝜆(̃𝑣)=𝐻𝜆(𝑣),𝐷1(̃𝑣)=𝐷1(𝑣).(2.68) Let, for example, 𝑣10 and 𝑣20. Set 𝑠]0,1[ and 𝑣11=(1𝑠)1/𝑝𝑣2,𝑣12=𝑠1/𝑝𝑣2,𝑣1=𝑣 as>2, with 𝑣1=(𝑣11,,𝑣1𝑛), we have 𝐻𝜆𝑣1=𝐻𝜆(𝑣),𝐷1𝑣1=𝐷1(𝑣).(2.69) If 𝑣30, set 𝑣21=(1𝑠)1/𝑝𝑣11,𝑣23=𝑠1/𝑝𝑣11,𝑣2=𝑣1 as {1,,𝑛}{1,3}, with𝑣2=(𝑣21,,𝑣2𝑛), it results in 𝐻𝜆𝑣2=𝐻𝜆(𝑣),𝐷1𝑣2=𝐷1(𝑣).(2.70) This method let us to find ̃𝑣.
Then, if 𝑣00(resp.𝑣0) for some {1,,𝑛}, with ̃𝑣0̃(resp.𝑣) as in (2.68) we have from (2.53) (resp. (2.54)) 𝐷1(̃𝑣0)=𝑒(resp.𝐻𝜆(̃𝑣)=𝑒). Consequently ([1], see the proof of Theorem 2.1 (resp. Theorem 4.1)) ̃𝑢0=𝜏0̃𝑣0̃(resp.𝑢=𝜏̃𝑣) is a weak solution of system (2.49). Therefore [6] ̃𝑢0𝑖̃>0(resp.𝑢𝑖>0) as 𝑖=1,,𝑛.

Application 2.22. Let us assume 𝜆,𝑏, and 𝐴 as in Application 2.17, 𝐷𝑗(𝑣)=𝑞𝑗1Ω𝑑𝑗𝑛=1||𝑣||𝛾𝑗𝑞𝑗/𝛾𝑗𝑣𝑑𝑥𝑣=1,,𝑣𝑛𝑊as𝑗=1,,𝑚,(2.71) where 𝑝<𝑞1<<𝑞𝑚<̃𝑝,1<𝛾𝑗<𝑞𝑗,𝑑𝑚𝐿𝑑(Ω),𝑗𝐿(Ω){0},𝑑𝑗0if𝑗=1,,𝑚1.(2.72) Let us consider the system: ||div𝑢𝑖||𝑝2𝑢𝑖=||𝑢𝜆𝑏𝑖||𝑝2𝑢𝑖+𝑚𝑗=1𝑑𝑗𝑛=1||𝑢||𝛾𝑗(𝑞𝑗/𝛾𝑗)1||𝑢𝑖||𝛾𝑗2𝑢𝑖𝑢inΩ,𝑖=0on𝜕Ωas𝑖=1,,𝑛,(2.73) under almost one of the conditions: 𝑑+𝑚0,Ω𝑑𝑚𝑢𝑞𝑚𝑑𝑥<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+(𝐷𝑚) and 𝑆(𝐷1) with𝑆(𝐷𝑚).
Thanks to Proposition A.3 and a result of [5], for the solutions 𝑢0 and 𝑢 to system (2.73), we have 𝑢0𝑖𝐿(Ω)𝐶1,𝛼0𝑖oc(Ω),𝑢𝑖𝐿(Ω)𝐶1,𝛼𝑖oc(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 𝑣𝑉+(𝐷𝑚)(resp.𝑣𝑉𝜆𝑆(𝐷𝑚))𝜓(𝑡,𝑣)=𝑝𝑡𝑝1𝐻𝜆(𝑣)𝑚𝑗=1𝑞𝑗𝑡𝑞𝑗1𝐷𝑗(𝑣), we have: 𝑡(𝑣)>0𝜓(𝑡(𝑣),𝑣)=0,𝜕𝜓𝜕𝑡(𝑡(𝑣),𝑣)0.(2.76) Besides with 𝐸(𝑣)=(𝑡(𝑣))𝑝𝐻𝜆(𝑣)𝑚𝑗=1(𝑡(𝑣))𝑞𝑗𝐷𝑗(𝑣), it results in 𝐸𝑣0𝐸=inf(𝑣)𝑣𝑆𝜆𝑉+𝐷𝑚,𝜏0𝑣=𝑡0,𝐸(2.77)𝑣=inf𝐸(𝑣)𝑣𝑉𝜆𝐷𝑆𝑚,𝜏=𝑡𝑣.(2.78)

Proposition 2.23. When 𝛾𝑚<𝑝𝛾𝑗 as 𝑗=1,,𝑚1, then 𝑢0𝑖0𝑖{1,,𝑛}.(2.79)

Proof. It is sufficiently (Remark 1.1) to prove that 𝑖16holds{1,,𝑛}with𝔉=𝑆𝜆𝑉+𝐷𝑚.(2.80) Let 𝑣=(𝑣1,,𝑣𝑛)𝑆𝜆𝑉+(𝐷𝑚) with 𝑣0. As in Proposition 2.19, it is possible to find 𝑣𝐶0(Ω){0} such that with 𝑣(𝑠)=(𝑠1/𝑝𝑣1,,(1𝑠)1/𝑝𝑣,,𝑠1/𝑝𝑣𝑛), it results in 𝐻𝜆[](𝑣(𝑠))=1𝑠0,1,𝐷𝑚𝑠(𝑣(𝑠))>0𝑠0,10𝑠0,<1lim𝑠1𝑑𝐷𝑑𝑠𝑗(𝑣(𝑠))𝑅as𝑗=1,,𝑚1,lim𝑠1𝑑𝐷𝑑𝑠𝑚(𝑣(𝑠))=.(2.81)

Proposition 2.24. When 𝑝<𝛾𝑚𝛾𝑗 as 𝑗=1,,𝑚1, then 𝑢𝑖>0𝑖{1,,𝑛}.(2.82)

Proof. It is sufficiently [6] to prove that 𝑢𝑖0forall𝑖{1,,𝑛}. Reasoning by contradiction, let, for example,