Abstract

This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type โˆ’[๐‘€1(โˆซฮฉ|โˆ‡๐‘ข|๐‘)]๐‘โˆ’1ฮ”๐‘๐‘ข=๐œ†๐น๐‘ข(๐‘ฅ,๐‘ข,๐‘ฃ), in ฮฉ, โˆ’[๐‘€2(โˆซฮฉ|โˆ‡๐‘ฃ|๐‘ž)]๐‘žโˆ’1ฮ”๐‘ž๐‘ฃ=๐œ†๐น๐‘ฃ(๐‘ฅ,๐‘ข,๐‘ฃ), in ฮฉ, ๐‘ข=๐‘ฃ=0, on ๐œ•ฮฉ. Under bounded condition on M and some novel and periodic condition on F, some new results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by means of Bonanno's multiple critical points theorems without the Palais-Smale condition and Ricceri's three critical points theorem, respectively.

1. Introduction and Preliminaries

We are concerned with the following nonlocal elliptic system of (๐‘,๐‘ž)-Kirchhoff type: โˆ’๎‚ธ๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถ๎‚น๐‘โˆ’1ฮ”๐‘๐‘ข=๐œ†๐น๐‘ข(๐‘ฅ,๐‘ข,๐‘ฃ),inโˆ’๎‚ธ๐‘€ฮฉ,2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘ž๎‚ถ๎‚น๐‘žโˆ’1ฮ”๐‘ž๐‘ฃ=๐œ†๐น๐‘ฃ(๐‘ฅ,๐‘ข,๐‘ฃ),inฮฉ,๐‘ข=๐‘ฃ=0;on๐œ•ฮฉ,(1.1) where ฮฉโŠ‚๐‘…๐‘(๐‘โ‰ฅ1) is a bounded smooth domain, ๐œ†โˆˆ(0,+โˆž), ๐‘>๐‘,๐‘ž>๐‘, ฮ”๐‘ is the ๐‘-Laplacian operator ฮ”๐‘๎‚€||||๐‘ข=divโˆ‡๐‘ข๐‘โˆ’2๎‚,โˆ‡๐‘ข(1.2) and ๐‘€๐‘–โˆถ๐‘…+โ†’๐‘…,๐‘–=1,2, are continuous functions with bounded conditions.(๐‘€)There are two positive constants ๐‘š0,๐‘š1 such that ๐‘š0โ‰ค๐‘€๐‘–(๐‘ก)โ‰ค๐‘š1,โˆ€๐‘กโ‰ฅ0,๐‘–=1,2.(1.3) Furthermore, ๐นโˆถฮฉร—๐‘…ร—๐‘…โ†’๐‘… is a function such that ๐น(๐‘ฅ,๐‘ ,๐‘ก) is measurable in ๐‘ฅ for all (๐‘ ,๐‘ก)โˆˆ๐‘…ร—๐‘… and ๐น(๐‘ฅ,๐‘ ,๐‘ก) is ๐ถ1 in (๐‘ ,๐‘ก) for a.e. ๐‘ฅโˆˆฮฉ, and ๐น๐‘ข denotes the partial derivative of ๐น with respect to ๐‘ข. Moreover, ๐น(๐‘ฅ,๐‘ ,๐‘ก) satisfies the following.(๐น1)๐น(๐‘ฅ,0,0)=0 for a.e. ๐‘ฅโˆˆฮฉ.(๐น2) There exist two positive constants ๐›พ<๐‘,๐›ฝ<๐‘ž and a positive real function ๐›ผ(๐‘ฅ)โˆˆ๐ฟโˆž(ฮฉ) such that ||||๎€ท๐น(๐‘ฅ,๐‘ ,๐‘ก)โ‰ค๐›ผ(๐‘ฅ)1+|๐‘ |๐›พ+|๐‘ก|๐›ฝ๎€ธ,fora.e.๐‘ฅโˆˆฮฉandall(๐‘ ,๐‘ก)โˆˆ๐‘…ร—๐‘….(1.4)

The system (1.1) is related to a model given by the equation of elastic strings ๐œŒ๐œ•2๐‘ข๐œ•๐‘ก2โˆ’๎‚ต๐‘ƒ0โ„Ž+๐ธ๎€œ2๐ฟ๐ฟ0|||๐œ•๐‘ข|||๐œ•๐‘ฅ2๎‚ถ๐œ•๐‘‘๐‘ฅ2๐‘ข๐œ•๐‘ฅ2=0(1.5) which was proposed by Kirchhoff [1] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings, where the parameters in (1.5) have the following meanings: ๐œŒ is the mass density, ๐‘ƒ0 is the initial tension, โ„Ž is the area of the cross-section, ๐ธ is the Young modulus of the material, and ๐ฟ is the length of the string. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations.

Later, (1.5) was developed to the form ๐‘ข๐‘ก๐‘ก๎‚ต๎€œโˆ’๐‘€ฮฉ|โˆ‡๐‘ข|2๎‚ถฮ”๐‘ข=๐‘“(๐‘ฅ,๐‘ข)inฮฉ,(1.6) where ๐‘€โˆถ๐‘…+โ†’๐‘… is a given function. After that, many people studied the nonlocal elliptic boundary value problem ๎‚ต๎€œโˆ’๐‘€ฮฉ|โˆ‡๐‘ข|2๎‚ถฮ”๐‘ข=๐‘“(๐‘ฅ,๐‘ข)inฮฉ,๐‘ข=0on๐œ•ฮฉ,(1.7) which is the stationary counterpart of (1.6). It is pointed out in [2] that (1.7) models several physical and biological systems, where ๐‘ข describes a process which depends on the average of itself (e.g., population density). By using the methods of sub and supersolutions, variational methods, and other techniques, many results of (1.7) were obtained, we can refer to [2โ€“12] and the references therein. In particular, Alves et al. [2, Theoremโ€‰โ€‰4] supposes that ๐‘€ satisfies bounded condition (๐‘€) and ๐‘“(๐‘ฅ,๐‘ก) satisfies condition ๐ด๐‘…, that is, for some ๐œˆ>2 and ๐‘…>0 such that 0<๐œˆ๐น(๐‘ฅ,๐‘ก)โ‰ค๐‘“(๐‘ฅ,๐‘ก)๐‘ก,โˆ€|๐‘ก|โ‰ฅ๐‘…,๐‘ฅโˆˆฮฉ,(๐ด๐‘…) where โˆซ๐น(๐‘ฅ,๐‘ก)=๐‘ก0๐‘“(๐‘ฅ,๐‘ )๐‘‘๐‘ ; one positive solutions for (1.7) was obtained. It is well known that condition ๐ด๐‘… plays an important role for showing the boundedness of Palais-Smale sequences. More recently, Corrรชa and Nascimento in [13] studied a nonlocal elliptic system of ๐‘-Kirchhoff type โˆ’๎‚ธ๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถ๎‚น๐‘โˆ’1ฮ”๐‘๐‘ข=๐‘“(๐‘ข,๐‘ฃ)+๐œŒ1(๐‘ฅ),inโˆ’๎‚ธ๐‘€ฮฉ,2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘๎‚ถ๎‚น๐‘โˆ’1ฮ”๐‘๐‘ฃ=๐‘”(๐‘ข,๐‘ฃ)+๐œŒ2(๐‘ฅ),inฮฉ,๐œ•๐‘ข=๐œ•๐œ‚๐œ•๐‘ฃ๐œ•๐œ‚=0,on๐œ•ฮฉ,(๐‘ƒ) where ๐œ‚ is the unit exterior vector on ๐œ•ฮฉ, and ๐‘€๐‘–,๐œŒ๐‘–(๐‘–=1,2),๐‘“,๐‘” satisfy suitable assumptions, where a special and important condition: periodic condition on nonlinearity was assumed. They obtained the existence of a weak solution for the nonlocal elliptic system of ๐‘-Kirchhoff type ๐‘ƒ under Neumann boundary condition via Ekeland's Variational Principle.

In the present paper, our objective is to consider the nonlocal elliptic system of (๐‘,๐‘ž)-Kirchhoff-type (1.1), instead of the nonlocal elliptic system of ๐‘-Kirchhoff type and single Kirchhoff type equation. Under bounded condition on ๐‘€ and some novel conditions without ๐‘ƒ๐‘† condition and periodic condition on ๐น, we will prove the existence of two solutions and three solutions of system (1.1) by means of one multiple critical points theorem without the Palais-Smale condition of Bonanno in [14] and an equivalent formulation [15, Theoremโ€‰โ€‰2.3] of Ricceri's three critical points theorem [16, Theoremโ€‰โ€‰1], respectively.

In order to state our main results, we need the following preliminaries.

Let ๐‘‹=๐‘Š01,๐‘(ฮฉ)ร—๐‘Š01,๐‘ž(ฮฉ) be the Cartesian product of two Sobolev spaces, which is a reflexive real Banach space endowed with the norm โ€–(๐‘ข,๐‘ฃ)โ€–=โ€–๐‘ขโ€–๐‘+โ€–๐‘ฃโ€–๐‘ž,(1.8) where โ€–โ‹…โ€–๐‘ and โ€–โ‹…โ€–๐‘ž denote the norms of ๐‘Š01,๐‘(ฮฉ) and ๐‘Š01,๐‘ž(ฮฉ), respectively. That is, โ€–๐‘ขโ€–๐‘=๎‚ต๎€œฮฉ|โˆ‡๐‘ข|๐‘๎‚ถ1/๐‘,โ€–๐‘ฃโ€–๐‘ž=๎‚ต๎€œฮฉ|โˆ‡๐‘ฃ|๐‘ž๎‚ถ1/๐‘ž(1.9) for all ๐‘ขโˆˆ๐‘Š01,๐‘(ฮฉ) and ๐‘ฃโˆˆ๐‘Š01,๐‘ž(ฮฉ).

Since ๐‘>๐‘ and ๐‘ž>๐‘, ๐‘Š01,๐‘(ฮฉ) and ๐‘Š01,๐‘ž(ฮฉ) are compactly embedded in ๐ถ0(ฮฉ). Let โŽงโŽชโŽจโŽชโŽฉ๐ถ=maxsup๐‘ขโˆˆ๐‘Š01,๐‘(ฮฉ)โงต{0}max๐‘ฅโˆˆฮฉ๎€ฝ||||๐‘ข(๐‘ฅ)๐‘๎€พโ€–๐‘ขโ€–๐‘๐‘,sup๐‘ฃโˆˆ๐‘Š01,๐‘ž(ฮฉ)โงต{0}max๐‘ฅโˆˆฮฉ๎€ฝ||||๐‘ฃ(๐‘ฅ)๐‘ž๎€พโ€–๐‘ฃโ€–๐‘ž๐‘žโŽซโŽชโŽฌโŽชโŽญ;(1.10) then we have ๐ถ<+โˆž. Furthermore, it is known from [17] that sup๐‘ขโˆˆ๐‘Š01,๐‘(ฮฉ)โงต{0}max๐‘ฅโˆˆฮฉ๎€ฝ||๐‘ข||(๐‘ฅ)๐‘๎€พโ€–๐‘ขโ€–๐‘โ‰ค๐‘โˆ’1/๐‘ƒโˆš๐œ‹๎‚€ฮ“๎‚€๐‘1+2๎‚๎‚1/๐‘๎‚ต๐‘โˆ’1๎‚ถ๐‘โˆ’๐‘1โˆ’1/๐‘ƒ||ฮฉ||(1/๐‘)โˆ’(1/๐‘ƒ),(1.11) where ฮ“ denotes the Gamma function and |ฮฉ| is the Lebesgue measure of ฮฉ. Additionally, (1.11) is an equality when ฮฉ is a ball.

Recall that (๐‘ข,๐‘ฃ)โˆˆ๐‘‹ is called a weak solution of system (1.1) if ๎‚ธ๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถ๎‚น๐‘โˆ’1๎€œฮฉ||||โˆ‡๐‘ข๐‘โˆ’2๎‚ธ๐‘€โˆ‡๐‘ขโˆ‡๐œ‘+2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘ž๎‚ถ๎‚น๐‘žโˆ’1๎€œฮฉ||||โˆ‡๐‘ฃ๐‘žโˆ’2๎€œโˆ‡๐‘ฃโˆ‡๐œ“โˆ’๐œ†ฮฉ๐น๐‘ข๎€œ(๐‘ฅ,๐‘ข,๐‘ฃ)๐œ‘(๐‘ฅ)๐‘‘๐‘ฅโˆ’๐œ†ฮฉ๐น๐‘ฃ(๐‘ฅ,๐‘ข,๐‘ฃ)๐œ“(๐‘ฅ)๐‘‘๐‘ฅ=0,(1.12) for all (๐œ‘,๐œ“)โˆˆ๐‘‹. Define the functional ๐ผโˆถ๐‘‹โ†’๐‘… given by 1๐ผ(๐‘ข,๐‘ฃ)=๐‘๎‚Š๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถ+1๐‘ž๎‚Š๐‘€2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘ž๎‚ถ๎€œโˆ’๐œ†ฮฉ๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅ(1.13) for all (๐‘ข,๐‘ฃ)โˆˆ๐‘‹, and where ๎‚Š๐‘€1(๎€œ๐‘ก)=๐‘ก0๎€บ๐‘€1(๎€ป๐‘ )๐‘โˆ’1๎‚Š๐‘€๐‘‘๐‘ ,2(๎€œ๐‘ก)=๐‘ก0๎€บ๐‘€2(๎€ป๐‘ )๐‘žโˆ’1๐‘‘๐‘ ,โˆ€๐‘กโ‰ฅ0.(1.14) By the conditions (๐‘€) and (๐น2), it is easy to see that ๐ผโˆˆ๐ถ1(๐‘‹,๐‘…) and a critical point of ๐ผ corresponds to a weak solution of the system (1.1).

Now, giving ๐‘ฅ0โˆˆฮฉ and choosing ๐‘…2>๐‘…1>0 such that ๐ต(๐‘ฅ0,๐‘…2)โІฮฉ, where ๐ต(๐‘ฅ,๐‘…)={๐‘ฆโˆˆ๐‘…๐‘โˆถ|๐‘ฆโˆ’๐‘ฅ|<๐‘…}. Next we give some notations.๐›ผ1=๐›ผ1๎€ท๐‘,๐‘,๐‘…1,๐‘…2๎€ธ=๐ถ1/๐‘ƒ๎€ท๐‘…๐‘2โˆ’๐‘…๐‘1๎€ธ1/๐‘ƒ๐‘…2โˆ’๐‘…1๎‚ต๐œ‹๐‘/2๎‚ถฮ“(1+๐‘/2)1/๐‘ƒ,๐›ผ2=๐›ผ2๎€ท๐‘,๐‘ž,๐‘…1,๐‘…2๎€ธ=๐ถ1/๐‘ž๎€ท๐‘…๐‘2โˆ’๐‘…๐‘1๎€ธ1/๐‘ž๐‘…2โˆ’๐‘…1๎‚ต๐œ‹๐‘/2ฮ“๎‚ถ(1+๐‘/2)1/๐‘ž.(1.15) Moreover, let ๐‘Ž,๐‘ be positive constants, denote๐‘Ž๐‘ฆ(๐‘ฅ)=๐‘…2โˆ’๐‘…1โŽ›โŽœโŽœโŽ๐‘…2โˆ’๎ƒฏ๐‘๎“๐‘–=1๎€ท๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘–0๎€ธ2๎ƒฐ1/2โŽžโŽŸโŽŸโŽ ๎€ท๐‘ฅ,โˆ€๐‘ฅโˆˆ๐ต0,๐‘…2๎€ธ๎€ท๐‘ฅโงต๐ต0,๐‘…1๎€ธ,๐ด(๐‘)={(๐‘ ,๐‘ก)โˆˆ๐‘…ร—๐‘…โˆถ|๐‘ |๐‘+|๐‘ก|๐‘ž๎€œโ‰ค๐‘},๐‘”(๐‘)=ฮฉsup(๐‘ ,๐‘ก)โˆˆ๐ด(๐‘)๎€œ๐น(๐‘ฅ,๐‘ ,๐‘ก)๐‘‘๐‘ฅ,๐‘˜(๐‘Ž)=๐ต(๐‘ฅ0,๐‘…2)โงต๐ต(๐‘ฅ0,๐‘…1)๎€œ๐น(๐‘ฅ,๐‘ฆ(๐‘ฅ),๐‘ฆ(๐‘ฅ))๐‘‘๐‘ฅ+๐ต(๐‘ฅ0,๐‘…1)โ„Ž๐น(๐‘ฅ,๐‘Ž,๐‘Ž)๐‘‘๐‘ฅ,(๐‘,๐‘Ž)=๐‘˜(๐‘Ž)โˆ’๐‘”(๐‘),๐‘€+๎ƒฏ๐‘š=max1๐‘โˆ’1๐‘,๐‘š1๐‘žโˆ’1๐‘ž๎ƒฐ,๐‘€โˆ’๎ƒฏ๐‘š=min0๐‘โˆ’1๐‘,๐‘š0๐‘žโˆ’1๐‘ž๎ƒฐ.(1.16)

Now we are ready to state our main results for the system (1.1)

Theorem 1.1. Assume that (๐น1)-(๐น2) hold and there are three positive constants ๐‘Ž,๐‘1,๐‘2 with ๐‘1<(๐‘Ž๐›ผ1)๐‘+(๐‘Ž๐›ผ2)๐‘žโ‰ค1<๐‘2 such that ๐‘€+๐‘”๎€ท๐‘1๎€ธ<๐‘€โˆ’โ„Ž๎€ท๐‘1๎€ธ,๐‘Ž,๐‘€+๐‘”๎€ท๐‘2๎€ธ<๐‘€โˆ’โ„Ž๎€ท๐‘1๎€ธ.,๐‘Ž(1.17) Then, for each ๎ƒฉ๐‘€๐œ†โˆˆ+๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ป๎€ท๐‘๐ถโ„Ž1๎€ธ,๐‘€,๐‘Žโˆ’๐‘1๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ป๐ถ๎ƒฏ1min๐‘”๎€ท๐‘1๎€ธ,1๐‘”๎€ท๐‘2๎€ธ,๎ƒฐ๎ƒช(1.18) there exists a positive real number ๐œŒ such that the system (1.1) has at least two weak solutions (๐‘ข๐‘–,๐‘ฃ๐‘–)โˆˆ๐‘‹(๐‘–=1,2) whose norms in ๐ถ0(ฮฉ) are less than some positive constant ๐œŒ.

Theorem 1.2. Assume that (๐น1)-(๐น2) hold and there are two positive constants ๐‘Ž,๐‘, with (๐‘Ž๐›ผ1)๐‘+(๐‘Ž๐›ผ2)๐‘ž>๐‘๐‘€+/๐‘€โˆ’ such that
(๐น3)๐น(๐‘ฅ,๐‘ ,๐‘ก)โ‰ฅ0 for a.e. ๐‘ฅโˆˆฮฉโงต๐ต(๐‘ฅ0,๐‘…1) and all (๐‘ ,๐‘ก)โˆˆ[0,๐‘Ž]ร—[0,๐‘Ž];
(๐น4)[(๐‘Ž๐›ผ1)๐‘+(๐‘Ž๐›ผ2)๐‘ž]|ฮฉ|sup(๐‘ฅ,๐‘ ,๐‘ก)โˆˆฮฉร—๐ด(๐‘๐‘€+/๐‘€โˆ’)โˆซ๐น(๐‘ฅ,๐‘ ,๐‘ก)<๐‘๐ต(๐‘ฅ0,๐‘…1)๐น(๐‘ฅ,๐‘Ž,๐‘Ž)๐‘‘๐‘ฅ.
Then there exist an open interval ฮ›โІ[0,+โˆž] and a positive real number ๐œŒ such that, for each ๐œ†โˆˆฮ›, the system (1.1) has at least three weak solutions ๐‘ค๐‘–=(๐‘ข๐‘–,๐‘ฃ๐‘–)โˆˆ๐‘‹(๐‘–=1,2,3) whose norms โ€–๐‘ค๐‘–โ€– are less than ๐œŒ.

2. Proofs of Main Results

Before proving the results, we state one multiple critical points theorem without the Palais-Smale condition of Bonanno in [13] and an equivalent formulation [14, Theoremโ€‰โ€‰2.3] of Ricceri's three critical points theorem [15, Theoremโ€‰โ€‰1], which are our main tools.

Theorem 2.1 (see [14, Theoremโ€‰โ€‰2.1]). Let ๐‘‹ be a reflexive real Banach space, and let ฮจ,ฮฆโˆถ๐‘‹โ†’๐‘… be two sequentially weakly lower semicontinuous functions. Assume that ฮจ is (strongly) continuous and satisfies limโ€–๐‘ขโ€–โ†’โˆžฮจ(๐‘ข)=+โˆž. Assume also that there exist two constants ๐‘Ÿ1 and ๐‘Ÿ2 such that (๐‘—)inf๐‘‹ฮจ<๐‘Ÿ1<๐‘Ÿ2; (๐‘—๐‘—)๐œ‘1(๐‘Ÿ1)<๐œ‘2(๐‘Ÿ1,๐‘Ÿ2);(๐‘—๐‘—๐‘—)๐œ‘1(๐‘Ÿ2)<๐œ‘2(๐‘Ÿ1,๐‘Ÿ2); where ๐œ‘1(๐‘Ÿ)=inf๐‘ขโˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)ฮฆ(๐‘ข)โˆ’inf๐‘ขโˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)๐‘คฮฆ(๐‘ข),๐œ‘๐‘Ÿโˆ’ฮจ(๐‘ข)2๎€ท๐‘Ÿ1,๐‘Ÿ2๎€ธ=inf๐‘ขโˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ1)sup๐‘ฃโˆˆฮจโˆ’1[๐‘Ÿ1,๐‘Ÿ2)ฮฆ(๐‘ข)โˆ’ฮฆ(๐‘ฃ).ฮจ(๐‘ฃ)โˆ’ฮจ(๐‘ข)(2.1) Then, for each ๎ƒฉ1๐œ†โˆˆ๐œ‘2๎€ท๐‘Ÿ1,๐‘Ÿ2๎€ธ๎ƒฏ1,min๐œ‘1๎€ท๐‘Ÿ1๎€ธ,1๐œ‘2๎€ท๐‘Ÿ2๎€ธ,๎ƒฐ๎ƒช(2.2) the functional ฮจ+๐œ†ฮฆ has two local minima which lie in ฮจโˆ’1(โˆ’โˆž,๐‘Ÿ1) and ฮจโˆ’1[๐‘Ÿ1,๐‘Ÿ2), respectively.

Theorem 2.2 (see [15, Theoremโ€‰โ€‰2.3]). Let ๐‘‹ be a separable and reflexive real Banach space. ฮจโˆถ๐‘‹โ†’๐‘ is a continuously Gรขteaux differentiable and sequentially weakly lower semicontinuous functional whose Gรขteaux derivative admits a continuous inverse on ๐‘‹โˆ—; ฮฆโˆถ๐‘‹โ†’๐‘… is a continuously Gรขteaux differentiable functional whose Gรขteaux derivative is compact. Suppose that (i)limโ€–๐‘ขโ€–โ†’โˆž(ฮจ(๐‘ข)+๐œ†ฮฆ(๐‘ข))=+โˆž for each ๐œ†>0;(ii)There are a real number ๐‘Ÿ, and ๐‘ข0,๐‘ข1โˆˆ๐‘‹ such that ฮจ(๐‘ข0)<๐‘Ÿ<ฮจ(๐‘ข1);(iii)inf๐‘ขโˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ]ฮฆ(๐‘ข)>((ฮจ(๐‘ข1)โˆ’๐‘Ÿ)ฮฆ(๐‘ข0)+(๐‘Ÿโˆ’ฮจ(๐‘ข0))ฮฆ(๐‘ข1))/(ฮจ(๐‘ข1)โˆ’ฮจ(๐‘ข0)).
Then there exist an open interval ฮ›โІ[0,+โˆž] and a positive real number ๐œŒ such that, for each ๐œ†โˆˆฮ›, the equation ฮจ๎…ž(๐‘ข)+๐œ†ฮฆ๎…ž(๐‘ข)=0 has at least three weak solutions whose norms in ๐‘‹ are less than ๐œŒ.

First, we give one basic lemma.

Lemma 2.3. Assume that (๐‘€) and (๐น2) hold; let 1ฮจ(๐‘ข,๐‘ฃ)=๐‘๎‚Š๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถ+1๐‘ž๎‚Š๐‘€2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘ž๎‚ถ๎€œ,ฮฆ(๐‘ข,๐‘ฃ)=โˆ’ฮฉ๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅ,(2.3) for all (๐‘ข,๐‘ฃ)โˆˆ๐‘‹. Then ฮจ and ฮฆ are continuously Gรขteaux differentiable and sequentially weakly lower semicontinuous functionals. Moreover, the Gรขteaux derivative of ฮจ admits a continuous inverse on ๐‘‹โˆ— and the Gรขteaux derivative of ฮฆ is compact.

Proof. By condition (๐‘€), it is easy to see that ฮจ is continuously Gรขteaux differentiable. Moreover, the Gรขteaux derivative of ฮจ admits a continuous inverse on ๐‘‹โˆ—. Thanks to ๐‘>๐‘,๐‘ž>๐‘, and (๐น2), ฮฆ is continuously Gรขteaux differentiable and sequentially weakly lower semicontinuous functional whose Gรขteaux derivative is compact. Next We will prove that ฮจ is a sequentially weakly lower semicontinuous functional. Indeed, for any (๐‘ข๐‘›,๐‘ฃ๐‘›)โˆˆ๐‘‹ with (๐‘ข๐‘›,๐‘ฃ๐‘›)โ‡€(๐‘ข,๐‘ฃ) in ๐‘‹, then ๐‘ข๐‘›โ‡€๐‘ข in ๐‘Š01,๐‘(ฮฉ) and ๐‘ฃ๐‘›โ‡€๐‘ฃ in ๐‘Š01,๐‘ž(ฮฉ). Therefore, liminf๐‘›โ†’โˆžโ€–โ€–๐‘ข๐‘›โ€–โ€–๐‘โ‰ฅโ€–๐‘ขโ€–๐‘,liminf๐‘›โ†’โˆžโ€–โ€–๐‘ฃ๐‘›โ€–โ€–๐‘žโ‰ฅโ€–๐‘ฃโ€–๐‘ž(2.4) due to the weakly lower semicontinuity of norm. Hence by virtue of the continuity and monotonicity of ๎‚Š๐‘€1 and ๎‚Š๐‘€2, we conclude that ๎‚Š๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถโ‰ค๎‚Š๐‘€1๎‚ตliminf๐‘›โ†’โˆž๎€œฮฉ||โˆ‡๐‘ข๐‘›||๐‘๎‚ถโ‰คliminf๐‘›โ†’โˆž๎‚Š๐‘€1๎‚ต๎€œฮฉ||โˆ‡๐‘ข๐‘›||๐‘๎‚ถ,๎‚Š๐‘€2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘ž๎‚ถโ‰ค๎‚Š๐‘€2๎‚ตliminf๐‘›โ†’โˆž๎€œฮฉ||โˆ‡๐‘ฃ๐‘›||๐‘ž๎‚ถโ‰คliminf๐‘›โ†’โˆž๎‚Š๐‘€2๎‚ต๎€œฮฉ||โˆ‡๐‘ฃ๐‘›||๐‘ž๎‚ถ,(2.5) Consequently, ฮจ is a sequentially weakly lower semicontinuous functional.

Proof of Theorem 1.1. Let 1ฮจ(๐‘ข,๐‘ฃ)=๐‘๎‚Š๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถ+1๐‘ž๎‚Š๐‘€2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘ž๎‚ถ๎€œ,ฮฆ(๐‘ข,๐‘ฃ)=โˆ’ฮฉ๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅ(2.6) for all (๐‘ข,๐‘ฃ)โˆˆ๐‘‹. Under condition (๐‘€), by a simple computation, we have ๐‘€โˆ’๎€ทโ€–๐‘ขโ€–๐‘๐‘+โ€–๐‘ฃโ€–๐‘ž๐‘ž๎€ธโ‰คฮจ(๐‘ข,๐‘ฃ)โ‰ค๐‘€+๎€ทโ€–๐‘ขโ€–๐‘๐‘+โ€–๐‘ฃโ€–๐‘ž๐‘ž๎€ธ.(2.7)
Therefore, (2.7) implies that limโ€–(๐‘ข,๐‘ฃ)โ€–โ†’+โˆžฮจ(๐‘ข,๐‘ฃ)=+โˆž.(2.8)
Put ๐‘Ÿ1=๐‘€โˆ’๐‘1๐ถ๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ป,๐‘Ÿ2=๐‘€โˆ’๐‘2๐ถ๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ป.(2.9)
Denote ๐œ‘1(๐‘Ÿ)=inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)ฮฆ(๐‘ข,๐‘ฃ)โˆ’inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)๐‘คฮฆ(๐‘ข,๐‘ฃ),๐œ‘๐‘Ÿโˆ’ฮจ(๐‘ข,๐‘ฃ)2๎€ท๐‘Ÿ1,๐‘Ÿ2๎€ธ=inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ1)sup(๐‘ข1,๐‘ฃ1)โˆˆฮจโˆ’1[๐‘Ÿ1,๐‘Ÿ2)๎€ท๐‘ขฮฆ(๐‘ข,๐‘ฃ)โˆ’ฮฆ1,๐‘ฃ1๎€ธฮจ๎€ท๐‘ข1,๐‘ฃ1๎€ธ,โˆ’ฮจ(๐‘ข,๐‘ฃ)(2.10) and ฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)๐‘ค is the closure of ฮจโˆ’1(โˆ’โˆž,๐‘Ÿ) in the weak topology.
Set ๐‘ค0โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ(๐‘ฅ)=0,๐‘ฅโˆˆ๎€ท๐‘ฅฮฉโงต๐ต0,๐‘…2๎€ธ,๐‘Ž๐‘…2โˆ’๐‘…1โŽ›โŽœโŽœโŽ๐‘…2โˆ’๎ƒฏ๐‘๎“๐‘–=1๎€ท๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘–0๎€ธ๎ƒฐ1/2โŽžโŽŸโŽŸโŽ ๎€ท๐‘ฅ,๐‘ฅโˆˆ๐ต0,๐‘…2๎€ธ๎€ท๐‘ฅโงต๐ต0,๐‘…1๎€ธ,๎€ท๐‘ฅ๐‘Ž,๐‘ฅโˆˆ๐ต0,๐‘…1๎€ธ.(2.11) Then (๐‘ข0,๐‘ฃ0)โˆˆ๐‘‹, where ๐‘ข0(๐‘ฅ)=๐‘ฃ0(๐‘ฅ)=๐‘ค0(๐‘ฅ) and โ€–โ€–๐‘ข0โ€–โ€–๐‘๐‘=โ€–โ€–๐‘ค0โ€–โ€–๐‘๐‘=๎€ท๐‘Ž๐›ผ1๎€ธ๐‘๐ถ,โ€–โ€–๐‘ฃ0โ€–โ€–๐‘ž๐‘ž=โ€–โ€–๐‘ค0โ€–โ€–๐‘ž๐‘ž=๎€ท๐‘Ž๐›ผ1๎€ธ๐‘ž๐ถ.(2.12) Consequently, (2.7) and (2.12) imply that ๐‘Ÿ1๎€ท๐‘ข<ฮจ0,๐‘ฃ0๎€ธ<๐‘Ÿ2.(2.13) Furthermore, (2.13) implies that ๐œ‘2๎€ท๐‘Ÿ1,๐‘Ÿ2๎€ธ=inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ1)sup(๐‘ข1,๐‘ฃ1)โˆˆฮจโˆ’1[๐‘Ÿ1,๐‘Ÿ2)ฮฆ๎€ท๐‘ข(๐‘ข,๐‘ฃ)โˆ’ฮฆ1,๐‘ฃ1๎€ธฮจ๎€ท๐‘ข1,๐‘ฃ1๎€ธโˆ’ฮจ(๐‘ข,๐‘ฃ)โ‰ฅinf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ1)๎€ท๐‘ขฮฆ(๐‘ข,๐‘ฃ)โˆ’ฮฆ0,๐‘ฃ0๎€ธฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธ.โˆ’ฮจ(๐‘ข,๐‘ฃ)(2.14) On the other hand, by (๐น1), (1.17), and, (2.11), one has ๎€œฮฉ๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธ๎€ท๐‘๐‘‘๐‘ฅ=๐‘˜(๐‘Ž)>โ„Ž1๎€ธ>๐‘€,๐‘Ž+๐‘€โˆ’๐‘”๎€ท๐‘1๎€ธ๎€ท๐‘>๐‘”1๎€ธ=๎€œฮฉsup๎€ท๐‘(๐‘ ,๐‘ก)โˆˆ๐ด1๎€ธ๐น(๐‘ฅ,๐‘ ,๐‘ก)๐‘‘๐‘ฅ.(2.15) For each (๐‘ข,๐‘ฃ)โˆˆ๐‘‹ with ฮจ(๐‘ข,๐‘ฃ)โ‰ค๐‘Ÿ1, and ๐‘ฅโˆˆฮฉ, by (2.7), we conclude ||||๐‘ข(๐‘ฅ)๐‘+||||๐‘ฃ(๐‘ฅ)๐‘ž๎€ทโ€–โ‰ค๐ถ๐‘ขโ€–๐‘๐‘+โ€–๐‘ฃโ€–๐‘ž๐‘ž๎€ธโ‰ค๐ถ๐‘Ÿ1๐‘€โˆ’=๐‘1๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ปโ‰ค๐‘1.(2.16) Therefore, the combination of (2.15) and (2.16) implies ฮฆ๎€ท๐‘ข(๐‘ข,๐‘ฃ)โˆ’ฮฆ0,๐‘ฃ0๎€ธฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธ=โˆซโˆ’ฮจ(๐‘ข,๐‘ฃ)ฮฉ๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธโˆซ๐‘‘๐‘ฅโˆ’ฮฉ๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธโ‰ฅโˆซโˆ’ฮจ(๐‘ข,๐‘ฃ)ฮฉ๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธโˆซ๐‘‘๐‘ฅโˆ’ฮฉsup|๐‘ข(๐‘ฅ)|๐‘+|๐‘ฃ(๐‘ฅ)|๐‘žโ‰ค๐‘1๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธโ‰ฅโˆซโˆ’ฮจ(๐‘ข,๐‘ฃ)ฮฉ๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธโˆซ๐‘‘๐‘ฅโˆ’ฮฉsup|๐‘ข(๐‘ฅ)|๐‘+|๐‘ฃ(๐‘ฅ)|๐‘žโ‰ค๐‘1๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธโ‰ฅโˆซฮฉ๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธโˆซ๐‘‘๐‘ฅโˆ’ฮฉsup|๐‘ข(๐‘ฅ)|๐‘+|๐‘ฃ(๐‘ฅ)|๐‘žโ‰ค๐‘1๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅ๐‘€+๎€ทโ€–โ€–๐‘ข0โ€–โ€–๐‘๐‘+โ€–โ€–๐‘ฃ0โ€–โ€–๐‘ž๐‘ž๎€ธ=๐ถ๐‘€+๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ปโ„Ž๎€ท๐‘1๎€ธ.,๐‘Ž(2.17) By (2.14) and (2.17), we have ๐œ‘2๎€ท๐‘Ÿ1,๐‘Ÿ2๎€ธโ‰ฅ๐ถ๐‘€+๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ปโ„Ž๎€ท๐‘1๎€ธ.,๐‘Ž(2.18) Similarly, for every (๐‘ข,๐‘ฃ)โˆˆ๐‘‹ such that ฮจ(๐‘ข,๐‘ฃ)โ‰ค๐‘Ÿ, where ๐‘Ÿ is a positive real number, and ๐‘ฅโˆˆฮฉ, one has ||||๐‘ข(๐‘ฅ)๐‘+||||๐‘ฃ(๐‘ฅ)๐‘ž๎€ทโ‰ค๐ถโ€–๐‘ขโ€–๐‘๐‘+โ€–๐‘ฃโ€–๐‘ž๐‘ž๎€ธโ‰ค๐ถ๐‘Ÿ๐‘€โˆ’.(2.19) By virtue of ฮจ being sequentially weakly lower semicontinuous, then ฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)๐‘ค=ฮจโˆ’1(โˆ’โˆž,๐‘Ÿ]. Consequently, ๐œ‘1(๐‘Ÿ)=inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)ฮฆ(๐‘ข,๐‘ฃ)โˆ’inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)๐‘คฮฆ(๐‘ข,๐‘ฃ)โ‰ค๐‘Ÿโˆ’ฮจ(๐‘ข,๐‘ฃ)ฮฆ(0,0)โˆ’inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)๐‘คฮฆ(๐‘ข,๐‘ฃ)โ‰ค๐‘Ÿโˆ’ฮจ(0,0)โˆ’inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)๐‘คฮฆ(๐‘ข,๐‘ฃ)๐‘Ÿโ‰คโˆซฮฉsup|๐‘ข(๐‘ฅ)|๐‘+|๐‘ฃ(๐‘ฅ)|๐‘žโ‰ค๐ถ๐‘Ÿ/๐‘€โˆ’๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅ๐‘Ÿ.(2.20) It implies that ๐œ‘1๎€ท๐‘Ÿ1๎€ธโ‰ค๐‘”๎€ท๐‘1๎€ธ๐‘Ÿ1=๐ถ๐‘€โˆ’๐‘1๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ป๐‘”๎€ท๐‘1๎€ธ<๐ถ๐‘€+๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ปโ„Ž๎€ท๐‘1๎€ธ,,๐‘Ž(2.21)๐œ‘1๎€ท๐‘Ÿ2๎€ธโ‰ค๐‘”๎€ท๐‘2๎€ธ๐‘Ÿ2=๐ถ๐‘€โˆ’๐‘2๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ป๐‘”๎€ท๐‘2๎€ธ<๐ถ๐‘€+๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ปโ„Ž๎€ท๐‘1๎€ธ.,๐‘Ž(2.22) By (2.18)โ€“(2.22), we conclude ๐œ‘1๎€ท๐‘Ÿ1๎€ธโ‰ค๐œ‘2๎€ท๐‘Ÿ1,๐‘Ÿ2๎€ธ,๐œ‘1๎€ท๐‘Ÿ2๎€ธโ‰ค๐œ‘2๎€ท๐‘Ÿ1,๐‘Ÿ2๎€ธ.(2.23) Therefore, the conditions (๐‘—), (๐‘—๐‘—), and (๐‘—๐‘—๐‘—) in Theorem 2.1 are satisfied. Consequently, by Lemma 2.3 and above facts, the functional ฮจ+๐œ†ฮฆ has two local minima (๐‘ข1,๐‘ฃ1),(๐‘ข2,๐‘ฃ2)โˆˆ๐‘‹, which lie in ฮจโˆ’1(โˆ’โˆž,๐‘Ÿ1) and ฮจโˆ’1[๐‘Ÿ1,๐‘Ÿ2), respectively. Since ๐ผ=ฮจ+๐œ†ฮฆโˆˆ๐ถ1, (๐‘ข1,๐‘ฃ1),(๐‘ข2,๐‘ฃ2)โˆˆ๐‘‹ are the solutions of the equation ฮจโ€ฒ(๐‘ข,๐‘ฃ)+๐œ†ฮฆ๎…ž(๐‘ข,๐‘ฃ)=0.(2.24) Then (๐‘ข1,๐‘ฃ1),(๐‘ข2,๐‘ฃ2)โˆˆ๐‘‹ are the weak solutions of system (1.1).
Since ฮจ(๐‘ข๐‘–,๐‘ฃ๐‘–)<๐‘Ÿ2,๐‘–=1,2, by (1.10) and (2.7), ||๐‘ข๐‘–||(๐‘ฅ)๐‘+||๐‘ฃ๐‘–||(๐‘ฅ)๐‘žโ‰ค๐ถ๐‘Ÿ2๐‘€โˆ’โ‰ค๐‘2,๐‘–=1,2;(2.25) which implies there exists a positive real number ๐œŒ such that the norms of (๐‘ข๐‘–,๐‘ฃ๐‘–)โˆˆ๐‘‹(๐‘–=1,2) in ๐ถ0(ฮฉ) are less than some positive constant ๐œŒ. This completes the proof of Theorem 1.1.

Proof of Theorem 1.2. Let 1ฮจ(๐‘ข,๐‘ฃ)=๐‘๎‚Š๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถ+1๐‘ž๎‚Š๐‘€2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘ž๎‚ถ๎€œ,ฮฆ(๐‘ข,๐‘ฃ)=โˆ’ฮฉ๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅ(2.26) for all (๐‘ข,๐‘ฃ)โˆˆ๐‘‹. By (๐น2) and (2.7), we have 1ฮจ(๐‘ข,๐‘ฃ)+๐œ†ฮฆ(๐‘ข,๐‘ฃ)=๐‘๎‚Š๐‘€1๎‚ต๎€œฮฉ||||โˆ‡๐‘ข๐‘๎‚ถ+1๐‘ž๎‚Š๐‘€2๎‚ต๎€œฮฉ||||โˆ‡๐‘ฃ๐‘ž๎‚ถ๎€œโˆ’๐œ†ฮฉ๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅโ‰ฅ๐‘€โˆ’๎€ทโ€–๐‘ขโ€–๐‘๐‘+โ€–๐‘ฃโ€–๐‘ž๐‘ž๎€ธ๎€œโˆ’๐œ†ฮฉ๎‚€||||๐›ผ(๐‘ฅ)1+๐‘ข(๐‘ฅ)๐›พ+||||๐‘ฃ(๐‘ฅ)๐›ฝ๎‚๐‘‘๐‘ฅโ‰ฅ๐‘€โˆ’๎€ทโ€–๐‘ขโ€–๐‘๐‘+โ€–๐‘ฃโ€–๐‘ž๐‘ž๎€ธโˆ’๐œ†โ€–๐›ผโ€–โˆž๎‚€||ฮฉ||+๐‘˜1โ€–๐‘ขโ€–๐›พ๐‘+๐‘˜2โ€–๐‘ฃโ€–๐›ฝ๐‘ž๎‚,(2.27) where ๐‘˜1,๐‘˜1 are positive constants. Since ๐›พ<๐‘,๐›ฝ<๐‘ž, (2.27) implies that limโ€–(๐‘ข,๐‘ฃ)โ€–โ†’+โˆž(ฮจ(๐‘ข,๐‘ฃ)+๐œ†ฮฆ(๐‘ข,๐‘ฃ))=+โˆž.(2.28) The same as in (2.11), defining a function ๐‘ค0(๐‘ฅ), and letting ๐‘ข0(๐‘ฅ)=๐‘ฃ0(๐‘ฅ)=๐‘ค0(๐‘ฅ), then (2.12) is also satisfied. Choosing ๐‘Ÿ=๐‘๐‘€+/๐ถ, by (2.7), (2.12), and (๐‘Ž๐›ผ1)๐‘+(๐‘Ž๐›ผ2)๐‘ž>๐‘๐‘€+/๐‘€โˆ’, we conclude ฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธโ‰ฅ๐‘€โˆ’๎€ทโ€–โ€–๐‘ข0โ€–โ€–๐‘๐‘+โ€–โ€–๐‘ฃ0โ€–โ€–๐‘ž๐‘ž๎€ธ=๐‘€โˆ’๐ถ๎€บ๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ป>๐‘€โˆ’๐ถ๐‘๐‘€+๐‘€โˆ’=๐‘Ÿ.(2.29) By (๐น3) and the definitions of ๐‘ข0 and ๐‘ฃ0, one has ||ฮฉ||sup๎€ท(๐‘ฅ,๐‘ ,๐‘ก)โˆˆฮฉร—๐ด๐‘๐‘€+/๐‘€โˆ’๎€ธ๐‘๐น(๐‘ฅ,๐‘ ,๐‘ก)<๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€œ๐ต(๐‘ฅ0,๐‘…1)=๐น(๐‘ฅ,๐‘Ž,๐‘Ž)๐‘‘๐‘ฅ๐‘๐‘€+๐ถโˆซ๐ต(๐‘ฅ0,๐‘…1)๐น(๐‘ฅ,๐‘Ž,๐‘Ž)๐‘‘๐‘ฅ๐‘€+๎€ท๎€ท๐‘Ž๐›ผ1๎€ธ๐‘+๎€ท๐‘Ž๐›ผ2๎€ธ๐‘ž๎€ธโ‰ค/๐ถ๐‘๐‘€+๐ถโˆซฮฉโงต๐ต(๐‘ฅ0,๐‘…1)๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธโˆซ๐‘‘๐‘ฅ+๐ต(๐‘ฅ0,๐‘…1)๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธ๐‘‘๐‘ฅ๐‘€+๎€ทโ€–โ€–๐‘ข0โ€–โ€–๐‘๐‘+โ€–โ€–๐‘ฃ0โ€–โ€–๐‘ž๐‘ž๎€ธโ‰ค๐‘๐‘€+๐ถโˆซฮฉ๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธ๐‘‘๐‘ฅฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธ.(2.30) For every (๐‘ข,๐‘ฃ)โˆˆ๐‘‹ such that ฮจ(๐‘ข,๐‘ฃ)โ‰ค๐‘Ÿ, and ๐‘ฅโˆˆฮฉ, one has ||||๐‘ข(๐‘ฅ)๐‘+||||๐‘ฃ(๐‘ฅ)๐‘ž๎€ทโ‰ค๐ถโ€–๐‘ขโ€–๐‘๐‘+โ€–๐‘ฃโ€–๐‘ž๐‘ž๎€ธโ‰ค๐ถ๐‘Ÿ๐‘€โˆ’=๐ถ๐‘€โˆ’๐‘๐‘€+๐ถ=๐‘๐‘€+๐‘€โˆ’.(2.31) By the combination of (2.30) and (2.31), we have sup(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)(โˆ’ฮฆ(๐‘ข,๐‘ฃ))=sup{(๐‘ข,๐‘ฃ)โˆฃฮจ(๐‘ข,๐‘ฃ)โ‰ค๐‘Ÿ}๎€œฮฉ๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅโ‰คsup{(๐‘ข,๐‘ฃ)||๐‘ข(๐‘ฅ)|๐‘+|๐‘ฃ(๐‘ฅ)|๐‘žโ‰ค๐‘๐‘€+/๐‘€โˆ’}๎€œฮฉโ‰ค๎€œ๐น(๐‘ฅ,๐‘ข,๐‘ฃ)๐‘‘๐‘ฅฮฉsup๎€ท(๐‘ ,๐‘ก)โˆˆ๐ด๐‘๐‘€+/๐‘€โˆ’๎€ธโ‰ค||ฮฉ||๐น(๐‘ฅ,๐‘ ,๐‘ก)๐‘‘๐‘ฅsup๎€ท(๐‘ฅ,๐‘ ,๐‘ก)โˆˆฮฉร—๐ด๐‘๐‘€+/๐‘€โˆ’๎€ธโ‰ค๐น(๐‘ฅ,๐‘ ,๐‘ก)๐‘๐‘€+๐ถโˆซฮฉ๐น๎€ท๐‘ฅ,๐‘ข0,๐‘ฃ0๎€ธ๐‘‘๐‘ฅฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธ๎€ท๐‘ข=๐‘Ÿโˆ’ฮฆ0,๐‘ฃ0๎€ธฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธ.(2.32) Therefore, inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1(โˆ’โˆž,๐‘Ÿ)ฮฆ๎€ท๐‘ขฮฆ(๐‘ข,๐‘ฃ)>๐‘Ÿ0,๐‘ฃ0๎€ธฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธ.(2.33) Note that ฮฆ(0,0)=ฮจ(0,0)=0, we conclude that inf(๐‘ข,๐‘ฃ)โˆˆฮจโˆ’1](โˆ’โˆž,๐‘Ÿ๎€ทฮจ๎€ท๐‘ขฮฆ(๐‘ข,๐‘ฃ)>0,๐‘ฃ0๎€ธ๎€ธฮฆ๎€ท๐‘ขโˆ’๐‘Ÿ(0,0)+(๐‘Ÿโˆ’ฮจ(0,0))ฮฆ0,๐‘ฃ0๎€ธฮจ๎€ท๐‘ข0,๐‘ฃ0๎€ธ.โˆ’ฮจ(0,0)(2.34) Hence, by Lemma 2.3 and above facts, ฮจ and ฮฆ satisfy all conditions of Theorem 2.2; then the conclusion directly follows from Theorem 2.2.

Acknowledgments

The authors would like to thank the referee for the useful suggestions. This work is supported in part by the National Natural Science Foundation of China (10961028), Yunnan NSF Grant no. 2010CD086, and the Foundation of young teachers of Qujing Normal University (2009QN018).