Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains
aa r X i v : . [ m a t h . C A ] S e p NONZERO RADIAL SOLUTIONS FOR A CLASS OF ELLIPTICSYSTEMS WITH NONLOCAL BCS ON ANNULAR DOMAINS
GENNARO INFANTE AND PAOLAMARIA PIETRAMALA
Abstract.
We provide new results on the existence, non-existence, localization and mul-tiplicity of nontrivial solutions for systems of Hammerstein integral equations. Some of thecriteria involve a comparison with the spectral radii of some associated linear operators.We apply our results to prove the existence of multiple nonzero radial solutions for somesystems of elliptic boundary value problems subject to nonlocal boundary conditions. Ourapproach is topological and relies on the classical fixed point index. We present an exampleto illustrate our theory. Introduction
In the interesting paper [14], Do ´O, Lorca and Ubilla, motivated by the work of Lee [37]and by their previous paper [13], considered the existence of three positive solutions for thesemilinear elliptic system ∆ u + ˜ f ( | x | , u, v ) = 0 , | x | ∈ [ R , R ] , ∆ v + ˜ f ( | x | , u, v ) = 0 , | x | ∈ [ R , R ] , (1.1)subject to the non-homogenous boundary conditions (BCs) u | ∂B R = 0 and u | ∂B R = A ,v | ∂B R = 0 and v | ∂B R = A , (1.2)where x ∈ R n , 0 < R < R < ∞ , A , A > B ρ = { x ∈ R n : | x | < ρ } . Themethodology used in [14] is to seek radial solutions of the system (1.1)-(1.2), by means of anauxiliary system of Hammerstein integral equations u ( t ) = Z k ( t, s ) ˆ f ( s, u ( s ) , v ( s ) , A , A ) ds,v ( t ) = Z k ( t, s ) ˆ f ( s, u ( s ) , v ( s ) , A , A ) ds, (1.3) Mathematics Subject Classification.
Primary 45G15, secondary 34B10, 35B07, 35J57, 47H30.
Key words and phrases.
Elliptic system, annular domain, radial solution, multiplicity, non-existence,spectral radius, cone, nontrivial solution, nonlocal boundary conditions, fixed point index.Partially supported by G.N.A.M.P.A. - INdAM (Italy). here k ( t, s ) = s (1 − t ) , s ≤ t,t (1 − s ) , s > t. The integral equations in (1.3) share the same non-negative kernel and the non-homogeneousterms that occur in (1.2) are incorporated in the nonlinearities ˆ f , ˆ f (a similar idea hasbeen fruitfully employed in [15] also in the context of exterior domains). The existence ofpositive solutions of (1.3) is obtained via the well-known Krasnosel’ski˘ı-Guo Theorem on conecompressions and cone expansions (see [19]). The Krasnosel’ski˘ı-Guo Theorem and, morein general, topological methods have been used to study the existence of positive solutionsfor elliptic equations subject to homogeneous BCs on annular domains, see for example thepapers by Dunninger and Wang [11, 12], Lan and Lin [35], Lan and Webb [36], Ma [38],Wang [49] and references therein.The study of nonlocal
BCs, in the framework of ODEs, has been initiated by 1908 byPicone [41], who considered multi-point BCs. This topic has been developed by a largenumber of authors. The motivation for this type of study is driven also by the fact thatnonlocal problems occur when modelling several phenomena in engineering, physics and lifesciences. For an introduction to nonlocal problems we refer to the reviews by Whyburn [59],Conti [10], Ma [39], Ntouyas [40] and ˇStikonas [48].Nonlocal BCs have been studied also in the the context of elliptic problems, we mentionhere the papers by Amster and Maurette [3], Beals [4], Bitsadze and Samarski˘ı [5], Brow-der [6], Schechter [45], Skubachevski˘ı [46, 47], Wang [50], Ye and Ke [63]. In [51] Webbconsidered the existence of positive radial solutions for the boundary value problem (BVP) △ u + h ( | x | ) f ( u ) = 0 , | x | ∈ [ R , R ] ,u | ∂B R = 0 and ( u ( R · ) − αu ( R η · )) | ∂B = 0 , (1.4)where α > R η ∈ ( R , R ).Here we develop a theory for the existence of nonzero solutions of systems of Hammersteinintegral equations of the type u ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds,v ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds, (1.5)that is well-suited to prove the existence of nontrivial radial solutions for a class of ellipticsystems subject to nonlocal BCs, similar to the ones that occur in (1.4). With this approachthe kernels, allowed to change sign , take into account the nonlocalities in the BCs.The existence of positive solutions of systems of integral equations of the type (1.5) hasbeen widely studied, see for example [1, 8, 9, 11, 12, 17, 18, 20, 21, 33, 34, 35, 30, 61, 62]and references therein. Nonzero solutions of systems of Hammerstein integral equations were onsidered in [16]; here we improve the results of [16] in several directions: we allow differentgrowths in the nonlinearities, discuss non-existence results and provide some criteria thatinvolve the spectral radii of some suitable associated linear operators.We illustrate our theory in the special case of a system of nonlinear elliptic BVPs with non-local BCs, that generates two different kernels in the associated system of integral equations,namely ∆ u + h ( | x | ) f ( u, v ) = 0 , | x | ∈ [ R , R ] , ∆ v + h ( | x | ) f ( u, v ) = 0 , | x | ∈ [ R , R ] ,∂u∂r (cid:12)(cid:12) ∂B R = 0 and ( u ( R · ) − α u ( R η · )) | ∂B = 0 ,∂v∂r (cid:12)(cid:12) ∂B R = 0 and (cid:0) v ( R · ) − α ∂v∂r ( R ξ · ) (cid:1) | ∂B = 0 , where x ∈ R n , α , α ∈ R , 0 < R < R < ∞ , R η , R ξ ∈ ( R , R ) and ∂∂r denotes differentia-tion in the radial direction r = | x | .Here we focus the attention on the existence of solutions that are allowed to change sign ,in the spirit of the earlier works [28, 29]. The approach that we use is topological, relies onclassical fixed point index theory and we make use of ideas from the papers [16, 26, 27, 29,35, 36, 51, 55, 58]. In the last Section we present an example that illustrates the applicabilityof our results. 2. The system of integral equations
We begin by stating some assumptions on the terms that occur in the system of Hammer-stein integral equations u ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds,v ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds, (2.1)namely: • For every i = 1 , f i : [0 , × ( −∞ , ∞ ) × ( −∞ , ∞ ) → [0 , ∞ ) satisfies Carath´eodoryconditions, that is, f i ( · , u, v ) is measurable for each fixed ( u, v ) and f i ( t, · , · ) is contin-uous for almost every (a.e.) t ∈ [0 , r > φ i,r ∈ L ∞ [0 , f i ( t, u, v ) ≤ φ i,r ( t ) for u, v ∈ [ − r, r ] and a. e. t ∈ [0 , . • For every i = 1 , k i : [0 , × [0 , → ( −∞ , ∞ ) is measurable, and for every τ ∈ [0 , t → τ | k i ( t, s ) − k i ( τ, s ) | = 0 for a. e. s ∈ [0 , . For every i = 1 ,
2, there exist a subinterval [ a i , b i ] ⊆ [0 , i ∈ L ∞ [0 , c i ∈ (0 , | k i ( t, s ) | ≤ Φ i ( s ) for t ∈ [0 ,
1] and a. e. s ∈ [0 , ,k i ( t, s ) ≥ c i Φ i ( s ) for t ∈ [ a i , b i ] and a. e. s ∈ [0 , . • For every i = 1 , g i Φ i ∈ L [0 , g i ≥ R b i a i Φ i ( s ) g i ( s ) ds > C [0 , × C [0 ,
1] endowed with the norm k ( u, v ) k := max {k u k ∞ , k v k ∞ } , where k w k ∞ := max {| w ( t ) | , t ∈ [0 , } .We recall that a cone K in a Banach space X is a closed convex set such that λ x ∈ K for x ∈ K and λ ≥ K ∩ ( − K ) = { } . Take˜ K i := { w ∈ C [0 ,
1] : min t ∈ [ a i ,b i ] w ( t ) ≥ c i k w k ∞ } , and consider the cone K in C [0 , × C [0 ,
1] defined by K := { ( u, v ) ∈ ˜ K × ˜ K } . For a nontrivial solution of the system (2.1) we mean a solution ( u, v ) ∈ K of (2.1) suchthat k ( u, v ) k 6 = 0. Note that the functions in ˜ K i are positive on the sub-interval [ a i , b i ] butare allowed to change sign in [0 , non-negative functions first used by Krasnosel’ski˘ı,see e.g. [31], and D. Guo, see e.g. [19].Under our assumptions, we show that the integral operator T ( u, v )( t ) := R k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds R k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds ! := T ( u, v )( t ) T ( u, v )( t ) ! , (2.2)leaves the cone K invariant and is compact. Lemma 2.1.
The operator (2.2) maps K into K and is compact.Proof. Take ( u, v ) ∈ K such that k ( u, v ) k ≤ r . Then we have, for t ∈ [0 , | T ( u, v )( t ) | ≤ Z Φ ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds and therefore k T ( u, v ) k ∞ ≤ Z Φ ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds. Then we obtain min t ∈ [ a ,b ] T ( u, v )( t ) ≥ c Z Φ ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds ≥ c k T ( u, v ) k ∞ . ence we have T ( u, v ) ∈ ˜ K . In a similar manner we proceed for T ( u, v ).Moreover, the map T is compact since, by routine arguments, the components T i are compactmaps. (cid:3) The next Lemma summarizes some classical results regarding the fixed point index, formore details see [2, 19]. If Ω is a open bounded subset of a cone K (in the relative topology)we denote by Ω and ∂ Ω the closure and the boundary relative to K . When Ω is an openbounded subset of X we write Ω K = Ω ∩ K , an open subset of K . Lemma 2.2.
Let Ω be an open bounded set with ∈ Ω K and Ω K = K . Assume that F : Ω K → K is a compact map such that x = F x for all x ∈ ∂ Ω K . Then the fixed pointindex i K ( F, Ω K ) has the following properties. (1) If there exists e ∈ K \ { } such that x = F x + λe for all x ∈ ∂ Ω K and all λ > ,then i K ( F, Ω K ) = 0 . (2) If µx = F x for all x ∈ ∂ Ω K and for every µ ≥ , then i K ( F, Ω K ) = 1 . (3) If i K ( F, Ω K ) = 0 , then F has a fixed point in Ω K . (4) Let Ω be open in X with Ω ⊂ Ω K . If i K ( F, Ω K ) = 1 and i K ( F, Ω K ) = 0 , then F hasa fixed point in Ω K \ Ω K . The same result holds if i K ( F, Ω K ) = 0 and i K ( F, Ω K ) = 1 . We use the following (relative) open bounded sets in K : K ρ ,ρ = { ( u, v ) ∈ K : k u k ∞ < ρ and k v k ∞ < ρ } , and V ρ ,ρ = { ( u, v ) ∈ K : min t ∈ [ a ,b ] u ( t ) < ρ and min t ∈ [ a ,b ] v ( t ) < ρ } . If ρ = ρ = ρ we write simply K ρ and V ρ . The set V ρ (in the context of systems) wasintroduced by the authors in [23] and is equal to the set called Ω ρ/c in [16]. Ω ρ/c is anextension to the case of systems of a set given by Lan [33].For our index calculations we make use of the following Lemma, similar to Lemma 5 of[16]. The novelty here is the use of different radii, in the spirit of the paper [9]. This choiceallows more freedom in the growth of the nonlinearities. The proof of the Lemma is similarto the corresponding one in [16] and is omitted. Lemma 2.3.
The sets defined above have the following properties: • K ρ ,ρ ⊂ V ρ ,ρ ⊂ K ρ /c ,ρ /c . • ( w , w ) ∈ ∂V ρ ,ρ iff ( w , w ) ∈ K and min t ∈ [ a i ,b i ] w i ( t ) = ρ i for some i ∈ { , } and min t ∈ [ a j ,b j ] w j ( t ) ≤ ρ j for j = i . • If ( w , w ) ∈ ∂V ρ ,ρ , then for some i ∈ { , } ρ i ≤ w i ( t ) ≤ ρ i /c i for each t ∈ [ a i , b i ] and for j = i we have ≤ w j ( t ) ≤ ρ j /c j for each t ∈ [ a j , b j ] and k w j k ∞ ≤ ρ j /c j . . Existence results
We are now able to prove a result concerning the fixed point index on the set K ρ ,ρ . Lemma 3.1.
Assume that (I ρ ,ρ ) there exist ρ , ρ > such that for every i = 1 , f ρ ,ρ i < m i where f ρ ,ρ i = sup n f i ( t, u, v ) ρ i : ( t, u, v ) ∈ [0 , × [ − ρ , ρ ] × [ − ρ , ρ ] o and m i = sup t ∈ [0 , Z | k i ( t, s ) | g i ( s ) ds. Then i K ( T, K ρ ,ρ ) = 1 .Proof. We show that λ ( u, v ) = T ( u, v ) for every ( u, v ) ∈ ∂K ρ ,ρ and for every λ ≥
1; thisensures that the index is 1 on K ρ ,ρ . In fact, if this does not happen, there exist λ ≥ u, v ) ∈ ∂K ρ ,ρ such that λ ( u, v ) = T ( u, v ). Assume, without loss of generality, that k u k ∞ = ρ and k v k ∞ ≤ ρ . Then λu ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds. Taking the absolute value we have λ | u ( t ) | = (cid:12)(cid:12)(cid:12)Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds (cid:12)(cid:12)(cid:12) , and then the supremum over [0 ,
1] gives λρ ≤ sup t ∈ [0 , Z | k ( t, s ) | g ( s ) f ( s, u ( s ) , v ( s )) ds ≤ ρ f ρ ,ρ sup t ∈ [0 , Z | k ( t, s ) | g ( s ) ds = ρ f ρ ,ρ m . Using the hypothesis (3.1) we obtain λρ < ρ . This contradicts the fact that λ ≥ (cid:3) Remark 3.2.
Take ω ∈ L ([0 , × [0 , ω + ( t, s ) = max { ω ( t, s ) , } , ω − ( s ) = max {− ω ( t, s ) , } . Then we have (cid:12)(cid:12)(cid:12)Z ω ( t, s ) ds (cid:12)(cid:12)(cid:12) ≤ max nZ ω + ( t, s ) ds, Z ω − ( t, s ) ds o ≤ Z | ω ( t, s ) | ds, since ω = ω + − ω − and | ω | = ω + + ω − . sing the inequality above, it is possible to relax the growth assumptions on the nonlin-earities f i . This is done by replacing the quantity 1 m i withsup t ∈ [0 , n max nZ k + i ( t, s ) g i ( s ) ds, Z k − i ( t, s ) g i ( s ) ds oo ;this idea has been used, in the case of one equation, in [27].We give a first Lemma that shows that the index is 0 on a set V ρ ,ρ . Lemma 3.3.
Assume that (I ρ ,ρ ) there exist ρ , ρ > such that for every i = 1 , f i, ( ρ ,ρ ) > M i , where f , ( ρ ,ρ ) = inf n f ( t, u, v ) ρ : ( t, u, v ) ∈ [ a , b ] × [ ρ , ρ /c ] × [ − ρ /c , ρ /c ] o ,f , ( ρ ,ρ ) = inf n f ( t, u, v ) ρ : ( t, u, v ) ∈ [ a , b ] × [ − ρ /c , ρ /c ] × [ ρ , ρ /c ] o , M i = inf t ∈ [ a i ,b i ] Z b i a i k i ( t, s ) g i ( s ) ds. Then i K ( T, V ρ ,ρ ) = 0 .Proof. Let e ( t ) ≡ t ∈ [0 , e, e ) ∈ K . We prove that( u, v ) = T ( u, v ) + λ ( e, e ) for ( u, v ) ∈ ∂V ρ ,ρ and λ ≥ . In fact, if this does not happen, there exist ( u, v ) ∈ ∂V ρ ,ρ and λ ≥ u, v ) = T ( u, v ) + λ ( e, e ). Without loss of generality, we can assume that for all t ∈ [ a , b ] we have ρ ≤ u ( t ) ≤ ρ /c , min u ( t ) = ρ and − ρ /c ≤ v ( t ) ≤ ρ /c . Then, for t ∈ [ a , b ], we obtain u ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + λe ( t ) , and therefore u ( t ) ≥ Z b a k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + λ. Taking the minimum over [ a , b ] gives ρ = min t ∈ [ a ,b ] u ( t ) ≥ ρ f , ( ρ ,ρ ) M + λ. Using the hypothesis (3.2) we obtain ρ > ρ + λ , a contradiction. (cid:3) n the following Lemma we exploit an idea that was used in [26] and we provide a resultof index 0 on V ρ ,ρ of a different flavour; here we control the growth of just one nonlinearity f i , at the cost of having to deal with a larger domain. Nonlinearities with different growthswere considered, with different approaches, in [8, 43, 44, 60] . Lemma 3.4.
Assume that (I ρ ,ρ ) ⋆ there exist ρ , ρ > such that for some i ∈ { , } we have (3.3) f ∗ i, ( ρ ,ρ ) > M i , where f ∗ , ( ρ ,ρ ) = inf n f ( t, u, v ) ρ : ( t, u, v ) ∈ [ a , b ] × [0 , ρ /c ] × [ − ρ /c , ρ /c ] o .f ∗ , ( ρ ,ρ ) = inf n f ( t, u, v ) ρ : ( t, u, v ) ∈ [ a , b ] × [ − ρ /c , ρ /c ] × [0 , ρ /c ] o . Then i K ( T, V ρ ,ρ ) = 0 .Proof. Suppose that the condition (3.3) holds for i = 1. Let ( u, v ) ∈ ∂V ρ ,ρ and λ ≥ u, v ) = T ( u, v ) + λ ( e, e ). So for all t ∈ [ a , b ] we have min u ( t ) ≤ ρ , 0 ≤ u ( t ) ≤ ρ /c and − ρ /c ≤ v ( t ) ≤ ρ /c and for t ∈ [ a , b ], min v ( t ) ≤ ρ . For t ∈ [ a , b ], as in the proofof Lemma 3.3, we have u ( t ) ≥ Z b a k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + λ. Taking the minimum over [ a , b ] givesmin t ∈ [ a ,b ] u ( t ) ≥ ρ f ∗ , ( ρ ,ρ ) M + λ. Using the hypothesis (3.3) we obtain ρ > ρ + λ , a contradiction. (cid:3) We now state a result regarding the existence of at least one, two or three nontrivialsolutions. The proof follows by the properties of fixed point index and is omitted. Notethat, by expanding the lists in conditions ( S ) , ( S ), it is possible to state results for four ormore nontrivial solutions, see for example the paper [32]. Theorem 3.5.
The system (2.1) has at least one nontrivial solution in K if one of thefollowing conditions holds. ( S ) For i = 1 , there exist ρ i , r i ∈ (0 , ∞ ) with ρ i /c i < r i such that (I ρ ,ρ ) [ or (I ρ ,ρ ) ⋆ ] , (I r ,r ) hold. ( S ) For i = 1 , there exist ρ i , r i ∈ (0 , ∞ ) with ρ i < r i such that (I ρ ,ρ ) , (I r ,r ) hold.The system (2.1) has at least two nontrivial solutions in K if one of the following conditionsholds. S ) For i = 1 , there exist ρ i , r i , s i ∈ (0 , ∞ ) with ρ i /c i < r i < s i such that (I ρ ,ρ ) , [ or (I ρ ,ρ ) ⋆ ] , (I r ,r ) and (I s ,s ) hold. ( S ) For i = 1 , there exist ρ i , r i , s i ∈ (0 , ∞ ) with ρ i < r i and r i /c i < s i such that (I ρ ,ρ ) , (I r ,r ) and (I s ,s ) hold.The system (2.1) has at least three nontrivial solutions in K if one of the following con-ditions holds. ( S ) For i = 1 , there exist ρ i , r i , s i , σ i ∈ (0 , ∞ ) with ρ i /c i < r i < s i and s i /c i < σ i suchthat (I ρ ,ρ ) [ or (I ρ ,ρ ) ⋆ ] , (I r ,r ) , (I s ,s ) and (I σ ,σ ) hold. ( S ) For i = 1 , there exist ρ i , r i , s i , σ i ∈ (0 , ∞ ) with ρ i < r i and r i /c i < s i < σ i such that (I ρ ,ρ ) , (I r ,r ) , (I s ,s ) and (I σ ,σ ) hold. In the case of [ a , b ] = [ a , b ] we can relax the assumptions on the nonlinearities f i . Inthe following two Lemmas we provide a modification of the conditions (I ρ ,ρ ) and (I ρ ,ρ ) ⋆ ,similar to the one in [16]. An analogous of the Theorem 3.5 holds in this case, we omit thestatement of this result. Lemma 3.6.
Assume that [ a , b ] = [ a , b ] =: [ a, b ] and that (I ρ ,ρ ) there exist ρ , ρ > such that for every i = 1 , f i, ( ρ ,ρ ) > M i , where f , ( ρ ,ρ ) = inf n f ( t, u, v ) ρ : ( t, u, v ) ∈ [ a, b ] × [ ρ , ρ /c ] × [0 , ρ /c ] o ,f , ( ρ ,ρ ) = inf n f ( t, u, v ) ρ : ( t, u, v ) ∈ [ a, b ] × [0 , ρ /c ] × [ ρ , ρ /c ] o . Then i K ( T, V ρ ,ρ ) = 0 .Proof. As in the proof of Lemma 3.3 suppose that there exist ( u, v ) ∈ ∂V ρ ,ρ and λ ≥ u, v ) = T ( u, v ) + λ ( e, e ). Without loss of generality, we can assume that for all t ∈ [ a, b ] we have ρ ≤ u ( t ) ≤ ρ /c , min u ( t ) = ρ and 0 ≤ v ( t ) ≤ ρ /c . Then, for t ∈ [ a, b ], we obtain u ( t ) ≥ Z ba k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + λ. Taking the minimum over [ a, b ] gives ρ = min t ∈ [ a,b ] u ( t ) ≥ ρ f , ( ρ ,ρ ) M + λ. Using the hypothesis (3.4) we obtain ρ > ρ + λ , a contradiction. (cid:3) emma 3.7. Assume that [ a , b ] = [ a , b ] =: [ a, b ] and that (I ρ ,ρ ) ⋆ there exist ρ , ρ > such that for some i ∈ { , } we have (3.5) f ∗ i, ( ρ ,ρ ) > M i , where f ∗ i, ( ρ ,ρ ) = inf n f i ( t, u, v ) ρ i : ( t, u, v ) ∈ [ a, b ] × [0 , ρ /c ] × [0 , ρ /c ] o . Then i K ( T, V ρ ,ρ ) = 0 .Proof. Suppose that the condition (3.5) holds for i = 1. Let ( u, v ) ∈ ∂V ρ ,ρ and λ ≥ u, v ) = T ( u, v ) + λ ( e, e ). So for all t ∈ [ a, b ] we have min u ( t ) ≤ ρ , 0 ≤ u ( t ) ≤ ρ /c ,0 ≤ v ( t ) ≤ ρ /c and min v ( t ) ≤ ρ . Now, the proof follows as the one of Lemma 3.4. (cid:3) Non-existence results
We now show a non-existence result for problem (2.1).
Theorem 4.1.
Assume that one of the following conditions holds. (1)
For i = 1 , , (4.1) f i ( t, u , u ) < m i | u i | for every t ∈ [0 , and u i = 0 . (2) For i = 1 , , (4.2) f i ( t, u , u ) > M i u i for every t ∈ [ a i , b i ] and u i > . (3) There exists i ∈ { , } such that (4.1) is verified for f i and for j = i condition (4.2) is verified for f j .Then there is no nontrivial solution of the system (2.1) in K .Proof. (1) Assume, on the contrary, that there exists ( u, v ) ∈ K such that ( u, v ) = T ( u, v )and ( u, v ) = (0 , k u k ∞ = 0. Then, for t ∈ [0 , | u ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | k ( t, s ) | g ( s ) f ( s, u ( s ) , v ( s )) ds
In order to state our eigenvalue comparison results, we consider, in a similar way as in[27], the following operators on C [0 , × C [0 , L ( u, v )( t ) := R | k ( t, s ) | g ( s ) u ( s ) ds R | k ( t, s ) | g ( s ) v ( s ) ds ! := L ( u )( t ) L ( v )( t ) ! , and L + ( u, v )( t ) := R b a k +1 ( t, s ) g ( s ) u ( s ) ds R b a k +2 ( t, s ) g ( s ) v ( s ) ds ! := L +1 ( u )( t ) L +2 ( v )( t ) ! . We denote by P the cone of positive functions, namely P := { w ∈ C [0 ,
1] : w ( t ) ≥ , t ∈ [0 , } . Theorem 5.1.
The operators L and L + are compact and map P × P into ( P × P ) ∩ K .Proof. Note that the operators L and L + map P × P into P × P (because they have anon-negative integral kernel) and are compact. We now show that they map P × P into( P × P ) ∩ K . Firstly, we do this for the operator L .We observe that for every i = 1 , t ∈ [0 , | k i ( t, s ) | ≤ Φ i ( s ) , nd that, for t ∈ [ a i , b i ] , | k i ( t, s ) | = k i ( t, s ) ≥ c i Φ i ( t ) . Thus, with a similar proof as the one in Lemma 2.1, we obtain, for ( u, v ) ∈ P × P and t ∈ [0 , L ( u, v ) ∈ K . A similar proof works for L + , since for every i = 1 , t ∈ [0 , | k + i ( t, s ) | ≤ | k i ( t, s ) | ≤ Φ i ( s ) , and, for t ∈ [ a i , b i ] , k + i ( t, s ) = k i ( t, s ) ≥ c i Φ i ( t ) . (cid:3) We recall that λ is an eigenvalue of a linear operator Γ with corresponding eigenfunction ϕ if ϕ = 0 and λϕ = Γ ϕ . The reciprocals of nonzero eigenvalues are called characteristic values of Γ. We will denote the spectral radius of Γ by r (Γ) := lim n →∞ k Γ n k n and its principalcharacteristic value (the reciprocal of the spectral radius) by µ (Γ) = 1 /r (Γ).The following Theorem is analogous to the ones in [56, 58] and is proven by using the factsthat the considered operators leave P × P invariant, that P × P is reproducing, combinedwith the well-known Krein-Rutman Theorem. Theorem 5.2.
For i = 1 , , the spectral radius of L i is nonzero and is an eigenvalue of L i with an eigenfunction in P . A similar result holds for L + i . Remark 5.3.
As a consequence of the two previous theorems, we have that the abovementioned eigenfunction is in P ∩ ˜ K i .We consider the following operator on C [ a , b ] × C [ a , b ]:¯ L + ( u, v )( t ) := R b a k +1 ( t, s ) g ( s ) u ( s ) ds R b a k +2 ( t, s ) g ( s ) v ( s ) ds ! := ¯ L +1 ( u )( t )¯ L +2 ( v )( t ) ! . In the recent papers [54, 55], Webb developed an elegant theory valid for u -positive linearoperators. It turns out that our operators ¯ L + i fit within this setting and, in particular, satisfythe assumptions of Theorem 3 . . L + i . Theorem 5.4.
Suppose that there exist w ∈ C [ a i , b i ] \ { } , w ≥ and λ > such that λw ( t ) ≥ ¯ L + i w ( t ) , for t ∈ [ a i , b i ] . Then we have r ( ¯ L + i ) ≤ λ . Theorem 5.5.
Assume that (I + ) there exist ε > and ρ > such that one of the following conditions holds: f ( t, u, v ) ≥ ( µ ( L +1 ) + ε ) u, for ( t, u, v ) ∈ [ a , b ] × [0 , ρ ] × [ − ρ , ρ ]; f ( t, u, v ) ≥ ( µ ( L +2 ) + ε ) v, for ( t, u, v ) ∈ [ a , b ] × [ − ρ , ρ ] × [0 , ρ ] . Then i K ( T, K ρ ) = 0 for each ρ ∈ (0 , ρ ] .Proof. Let ρ ∈ (0 , ρ ]. We show that ( u, v ) = T ( u, v ) + λ ( ϕ , ϕ ) for all ( u, v ) in ∂K ρ and λ ≥
0, where ϕ i ∈ ˜ K i ∩ P is the eigenfunction of L + i with k ϕ i k ∞ = 1 corresponding to theeigenvalue 1 /µ ( L + i ). This implies that i K ( T, K ρ ) = 0.Assume, on the contrary, that there exist ( u, v ) ∈ ∂K ρ and λ ≥ u, v ) = T ( u, v ) + λ ( ϕ , ϕ ).We distinguish two cases. Firstly we discuss the case λ >
0. Suppose that (5.1) holds. Thisimplies that, for t ∈ [ a , b ] , we have u ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + λϕ ( t ) ≥ Z b a k +1 ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + λϕ ( t ) ≥ ( µ ( L +1 ) + ε ) Z b a k +1 ( t, s ) g ( s ) u ( s ) ds + λϕ ( t ) >µ ( L +1 ) Z b a k +1 ( t, s ) g ( s ) u ( s ) ds + λϕ ( t )= µ ( L +1 ) L +1 u ( t ) + λϕ ( t ) . Moreover, we have u ( t ) ≥ λϕ ( t ) and then L +1 u ( t ) ≥ λL +1 ϕ ( t ) ≥ λµ ( L +1 ) ϕ ( t ) in such a waythat we obtain u ( t ) ≥ µ ( L +1 ) L +1 u ( t ) + λϕ ( t ) ≥ λϕ ( t ) , for t ∈ [ a , b ] . By iteration, we deduce that, for t ∈ [ a , b ], we get u ( t ) ≥ nλϕ ( t ) for every n ∈ N , a contradiction because k u k ∞ ≤ ρ .Now we consider the case λ = 0. We have, for t ∈ [ a , b ], u ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds ≥ Z b a k +1 ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds ≥ ( µ ( L +1 ) + ε ) L +1 u ( t ) . Since L +1 ϕ ( t ) = r ( L +1 ) ϕ ( t ) for t ∈ [0 , t ∈ [ a , b ],¯ L +1 ϕ ( t ) = L +1 ϕ ( t ) = r ( L +1 ) ϕ ( t ) , nd we obtain r ( ¯ L +1 ) ≥ r ( L +1 ). On the other hand, we have, for t ∈ [ a , b ], u ( t ) ≥ ( µ ( L +1 ) + ε ) L +1 u ( t ) = ( µ ( L +1 ) + ε ) ¯ L +1 u ( t ) . where u ( t ) >
0. Thus, utilizing Theorem 5.4, we have r ( ¯ L +1 ) ≤ µ ( L +1 ) + ε and therefore r ( L +1 ) ≤ µ ( L +1 ) + ε and thus µ ( L +1 ) + ε ≤ µ ( L +1 ), a contradiction. (cid:3) Remark 5.6.
Note that condition (5.1) holds, for example, if µ ( L +1 ) < lim inf u → + inf t ∈ [ a ,b ] f ( t, u, v ) u , uniformly w.r.t. v ∈ R . A similar type of condition has been used in [8].
Theorem 5.7.
Assume that (I ∞ ) there exists R > such that the following conditions hold: (5.2) f ( t, u, v ) ≥ ( µ ( L +1 ) + ε ) u, for ( t, u, v ) ∈ [ a , b ] × [ cR , + ∞ ) × R ; f ( t, u, v ) ≥ ( µ ( L +2 ) + ε ) v, for ( t, u, v ) ∈ [ a , b ] × R × [ cR , + ∞ ) . Then i K ( T, K R ) = 0 for each R ≥ R .Proof. Let R ≥ R . We show that ( u, v ) = T ( u, v ) + λ ( ϕ , ϕ ) for all ( u, v ) in ∂K R and λ ≥
0, where ϕ i ∈ ˜ K i ∩ P is the eigenfunction of L + i with k ϕ i k ∞ = 1 corresponding to theeigenvalue 1 /µ ( L + i ). This implies that i K ( T, K R ) = 0.Assume, on the contrary, that there exist ( u, v ) ∈ ∂K R and λ ≥ u, v ) = T ( u, v ) + λ ( ϕ , ϕ ).Suppose that k u k ∞ = R and k v k ∞ ≤ R . We have u ( t ) ≥ c k u k ∞ = cR ≥ cR for t ∈ [ a , b ],thus condition (5.2) holds. Hence, we have f ( t, u ( t ) , v ( t )) ≥ ( µ ( L +1 ) + ε ) u ( t ) for t ∈ [ a , b ].This implies, proceeding as in the proof of Theorem 5.5 for the case λ >
0, that for t ∈ [ a , b ] u ( t ) ≥ µ ( L +1 ) L +1 u ( t ) + λϕ ( t ) ≥ λϕ ( t ) . Then u ( t ) ≥ nλϕ ( t ) for every n ∈ N , a contradiction because k u k ∞ = R .The proof in the case λ = 0 is treated as in the proof of Theorem 5.5. (cid:3) Theorem 5.8.
Assume that (I + ) there exist ε > and ρ > such that the following conditions hold: f ( t, u, v ) ≤ ( µ ( L ) − ε ) | u | , for all ( t, u, v ) ∈ [0 , × [ − ρ , ρ ] × [ − ρ , ρ ]; f ( t, u, v ) ≤ ( µ ( L ) − ε ) | v | , for all ( t, u, v ) ∈ [0 , × [ − ρ , ρ ] × [ − ρ , ρ ] . Then i K ( T, K ρ ) = 1 for each ρ ∈ (0 , ρ ] . roof. Let ρ ∈ (0 , ρ ]. We prove that T ( u, v ) = λ ( u, v ) for ( u, v ) ∈ ∂K ρ and λ ≥
1, whichimplies i K ( T, K ρ ) = 1. In fact, if we assume otherwise, then there exists ( u, v ) ∈ ∂K ρ and λ ≥ λ ( u, v ) = T ( u, v ). Therefore, | u ( t ) | ≤ λ | u ( t ) | = | T ( u, v )( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | k ( t, s ) | g ( s ) f ( s, u ( s ) , v ( s )) ds ≤ ( µ ( L ) − ε ) Z | k ( t, s ) | g ( s ) | u ( s ) | ds =( µ ( L ) − ε ) L | u | ( t ) . Thus, we have that, for t ∈ [0 , | u ( t ) | ≤ ( µ ( L ) − ε ) L [( µ ( L ) − ε ) L | u | ( t )]=( µ ( L ) − ε ) L | u | ( t ) ≤ · · · ≤ ( µ ( L ) − ε ) n L n | u | ( t ) , thus, taking the norms, 1 ≤ ( µ ( L ) − ε ) n k L n k , and then1 ≤ ( µ ( L ) − ε ) lim n →∞ k L n k n = µ ( L ) − εµ ( L ) < , a contradiction. (cid:3) Theorem 5.9.
Assume that (I ∞ ) there exist ε > and R > such that the following conditions hold: f ( t, u, v ) ≤ ( µ ( L ) − ε ) | u | , for | u | ≥ R , | v | ≥ R , and a.e. t ∈ [0 , f ( t, u, v ) ≤ ( µ ( L ) − ε ) | v | , for | u | ≥ R , | v | ≥ R , and a.e. t ∈ [0 , . Then there exists R such that i K ( T, K R ) = 1 for each R > R .Proof. Since the functions f i satisfy Carath´eodory condition, there exists φ i,R ∈ L ∞ [0 , f i ( t, u, v ) ≤ φ i,R ( t ) for u, v ∈ [ − R , R ] and a. e. t ∈ [0 , . Hence, we have(5.3) f ( t, u, v ) ≤ ( µ ( L ) − ε ) | u | + φ ,R ( t ) for all u, v ∈ R and a.e. t ∈ [0 , , and f ( t, u, v ) ≤ ( µ ( L ) − ε ) | v | + φ ,R ( t ) for all u, v ∈ R and a. e. t ∈ [0 , . Denote by Id the identity operator. Since for i = 1 , µ ( L i ) − ε ) L i havespectral radius less than one, we have that the operators (Id − ( µ ( L i ) − ε ) L i ) − exist and arebounded. Moreover, from the Neumann series expression,(Id − ( µ ( L i ) − ε ) L i ) − = ∞ X k =0 (( µ ( L i ) − ε ) L i ) k we obtain that (Id − ( µ ( L i ) − ε ) L i ) − map P into P , since the operators L i have this property. ake for i = 1 , C i := Z Φ i ( s ) g i ( s ) φ i,R ( s ) ds, and R := max {k (Id − ( µ ( L i ) − ε ) L i ) − C i k ∞ , i = 1 , } ∈ R . Now we prove that for each
R > R , T ( u, v ) = λ ( u, v ) for all ( u, v ) ∈ ∂K R and λ ≥ i K ( T, K R ) = 1. Otherwise there exist ( u, v ) ∈ ∂K R and λ ≥ λ ( u, v ) = T ( u, v ). Suppose that k u k ∞ = R and k v k ∞ ≤ R .From the inequality (5.3), we have, for t ∈ [0 , | u ( t ) | ≤ λ | u ( t ) | = | T ( u, v )( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | k ( t, s ) | g ( s ) f ( s, u ( s ) , v ( s )) ds ≤ ( µ ( L ) − ε ) Z | k ( t, s ) | g ( s ) | u ( s ) | ds + C i =( µ ( L i ) − ε ) L | u | ( t ) + C , which implies (Id − ( µ ( L ) − ε ) L ) | u | ( t ) ≤ C . Since (Id − ( µ ( L ) − ε ) L ) − is non-negative, we have | u ( t ) | ≤ (Id − ( µ ( L ) − ε ) L ) − C ≤ R . Therefore, we have k u k ∞ ≤ R < R , a contradiction. (cid:3) The index results in Sections 2 and 5 can be combined in order to establish results onexistence of multiple nontrivial solutions for the system (2.1), we refer to [35] for similarstatements. 6.
An auxiliary system of ODEs
We now present some results regarding the following system of ODEs u ′′ ( t ) + g ( t ) f ( t, u ( t ) , v ( t )) = 0 , a.e. on [0 , ,v ′′ ( t ) + g ( t ) f ( t, u ( t ) , v ( t )) = 0 , a.e. on [0 , , (6.1)with the BCs u ′ (0) = 0 , α u ( η ) = u (1) , < η < ,v ′ (0) = 0 , v (1) = α v ′ ( ξ ) , < ξ < . (6.2)Here we focus on the case α <
0, 0 < α < − ξ , that leads to the case of solutions thatare positive on some sub-intervals of [0 ,
1] and are allowed to change sign elsewhere. o the system (6.1)-(6.2) we associate the system of Hammerstein integral equations u ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds,v ( t ) = Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds, (6.3)where the Green’s functions are given by(6.4) k ( t, s ) = 11 − α (1 − s ) − α − α ( η − s ) , s ≤ η , s > η − t − s, s ≤ t, , s > t, and(6.5) k ( t, s ) = (1 − s ) − α , s ≤ ξ , s > ξ − t − s, s ≤ t, , s > t. The Green’s function k has been studied in [29], where it was shown that we may takeΦ ( s ) = 1 − s, arbitrary [ a , b ] ⊂ [0 , η ] and c = (1 − η ) / (1 − α ).Regarding k , this has been studied in [22]; we may takeΦ ( s ) = 1 − s, arbitrary [ a , b ] ⊂ [0 , ξ ] and c = 1 − α − ξ .The results of the previous Sections, for example Theorem 3.5, can be applied to thesystem (6.3).6.1. Optimal intervals.
We now assume that g = g ≡ a i , b i ]such that M i ( a i , b i ) = (cid:16) inf t ∈ [ a i ,b i ] Z b i a i k i ( t, s ) ds (cid:17) − is a minimum. This type of problem has been tackled in the past in the case of second andhigher order BVPs in [7, 24, 25, 42, 52, 53, 57].Since in [0 , × [0 ,
1] the kernel k is non-positive only for1 − α η − α ≤ t ≤ ≤ s ≤ − α − α t + 1 α , by direct calculation, we have Z | k ( t, s ) | ds = − t / − α ( η − α η + 12 ) − η / ϑ ( t ) , ≤ t ≤ − α η − α , − α + 2 − α t + 2 α t + − α − α η + 2 − α (1 − α ) =: ϑ ( t ) , − α η − α ≤ t ≤ , nd therefore we obtain1 /m = sup t ∈ [0 , Z | k ( t, s ) | ds = − α ( η − α η + 12 ) − η / ϑ (0) , if − α η + α + 1 ≥ , − α + 2 − α + 2 α + − α − α η + 2 − α (1 − α ) = ϑ (1) , if − α η + α + 1 ≤ . Firstly we note that 1 − α η − α ≥ η . For arbitrary 0 ≤ a < b ≤ η , the kernel k is a positive,non-increasing function of t . Thus we have1 /M ( a, b ) = min t ∈ [ a,b ] Z ba k ( t, s ) ds = Z ba k ( b, s ) ds. Note that inf ≤ a
1] is non-positive only for1 − α ≤ t ≤ ≤ s ≤ ξ ;by direct calculation, we have Z | k ( t, s ) | ds = − t / − α ξ + 1 / θ ( t ) , ≤ t ≤ − α , − t / ξt − ξ + α ξ + 1 / θ ( t ) , − α ≤ t ≤ , and therefore we obtain1 /m = sup t ∈ [0 , Z | k ( t, s ) | ds = max { θ (0) , θ (1) } = θ (0) = − α ξ + 1 / . For arbitrary 0 ≤ a < b ≤ ξ , the kernel k is a positive, non-increasing function of t . Thuswe have 1 /M ( a, b ) = min t ∈ [ a,b ] Z ba k ( t, s ) ds = Z ba k ( b, s ) ds. ote that inf ≤ a
We now turn back our attention to the systems of BVPs∆ u + h ( | x | ) f ( u, v ) = 0 , | x | ∈ [ R , R ] , ∆ v + h ( | x | ) f ( u, v ) = 0 , | x | ∈ [ R , R ] ,∂u∂r (cid:12)(cid:12) ∂B R = 0 and ( u ( R · ) − α u ( R η · )) | ∂B = 0 ,∂v∂r (cid:12)(cid:12) ∂B R = 0 and (cid:0) v ( R · ) − α ∂v∂r ( R ξ · ) (cid:1) | ∂B = 0 , (7.1)where x ∈ R n , α <
0, 0 < α <
1, 0 < R < R < ∞ , R η , R ξ ∈ ( R , R ).Consider in R n , n ≥
2, the equation(7.2) △ w + h ( | x | ) f ( w ) = 0 , for a.e. | x | ∈ [ R , R ] . with the BCs ∂w∂r (cid:12)(cid:12) ∂B R = 0 and ( w ( R · ) − α w ( R η · )) | ∂B = 0 , or ∂w∂r (cid:12)(cid:12) ∂B R = 0 and (cid:0) w ( R · ) − α ∂w∂r ( R ξ · ) (cid:1) | ∂B = 0 . In order to establish the existence of radial solutions w = w ( r ), r = | x | , we proceed as in[33, 35, 36] and we rewrite (7.2) in the form(7.3) w ′′ ( r ) + n − r w ′ ( r ) + h ( r ) f ( w ( r )) = 0 a.e. on [ R , R ] . Set w ( t ) = w ( r ( t )) where, for n ≥ r ( t ) = ( γ + ( β − γ ) t ) − / ( n − , for t ∈ [0 , , with γ = R − ( n − and β = R − ( n − , and for n = 2, r ( t ) = R − t R t , for t ∈ [0 , . ake for n ≥ φ ( t ) = (( β − γ ) / ( n − ( γ + ( β − γ ) t ) − n − / ( n − , and for n = 2 φ ( t ) = (cid:0) R (1 − t ) log R R (cid:1) . Then the equation (7.3) becomes w ′′ ( t ) + φ ( t ) h ( r ( t )) f ( w ( t )) = 0 , a.e. on [0 , , subject to the BCs w ′ (0) = 0 , αw ( η ) = w (1) , < η < , or w ′ (0) = 0 , αw ′ ( ξ ) = w (1) , < ξ < . Thus, to the system (7.1) we can associate the system of Hammerstein integral equations u ( t ) = Z k ( t, s ) g ( s ) f ( u ( s ) , v ( s )) ds,v ( t ) = Z k ( t, s ) g ( s ) f ( u ( s ) , v ( s )) ds, (7.4)where k is as in (6.4), k is as in (6.5) and g i ( t ) := φ ( t ) h i ( r ( t )) . The results of the previous Sections can be applied to the system (7.4), yielding resultsfor the system (7.1), we refer to [35, 36] for the results that may be stated.We illustrate in the following example that all the constants that occur in the Theorem 3.5can be computed.
Example 7.1.
Consider in R , the system of BVPs∆ u + f ( u, v ) = 0 , | x | ∈ [1 , e ] , ∆ v + f ( u, v ) = 0 , | x | ∈ [1 , e ] ,∂u∂r (cid:12)(cid:12) ∂B e = 0 and ( u ( · )+ u ( √ · )) | ∂B = 0 ,∂v∂r (cid:12)(cid:12) ∂B e = 0 and (cid:0) v ( · ) − ∂v∂r ( √ e · ) (cid:1) | ∂B = 0 . (7.5)To the system (7.5) we associate the system of second order ODEs u ′′ ( t ) + e (1 − t ) f ( u ( t ) , v ( t )) = 0 , t ∈ [0 , ,v ′′ ( t ) + e (1 − t ) f ( u ( t ) , v ( t )) = 0 , t ∈ [0 , ,u ′ (0) = 0 , u (1 /
2) + u (1) = 0 ,v ′ (0) = 0 , v ′ (1 /
4) = 4 v (1) . ow we have1 m = sup t ∈ [0 , Z | k ( t, s ) | g ( s ) ds = max n sup t ∈ [0 , / (cid:8) − e (cid:0) − t + 32 t + 192 t − (cid:1)(cid:9) , sup t ∈ [1 / , / (cid:8) − e (cid:0) t + 864 t − t + 19 − t (cid:1)(cid:9) , sup t ∈ [3 / , (cid:8) t + 54 e t + 15 / t e − t e + 467384 e − te − (cid:9)o , and1 m = sup t ∈ [0 , Z | k ( t, s ) | g ( s ) ds = max n sup t ∈ [0 , / (cid:8) − e (cid:0) − t + 64 t + 384 t − (cid:1)(cid:9) , sup t ∈ [1 / , / (cid:8) − e (cid:0) − t + 64 t + 384 t − (cid:1)(cid:9) , sup t ∈ [3 / , (cid:8) − e (cid:0) − t − t + 64 t + 384 t (cid:1)(cid:9)o . We fix [ a , b ] = [ a , b ] = [0 , / M = inf t ∈ [0 , / Z / k ( t, s ) g ( s ) ds = inf t ∈ [0 , / n − e −
377 + 256 t − t + 1536 t ) o , and1 M = inf t ∈ [0 , / Z / k ( t, s ) g ( s ) ds = inf t ∈ [0 , / n − e (cid:0) −
377 + 256 t − t + 1536 t (cid:1)o . By direct computation, we get c = 14 ; m = 38465 e ; M = 38437 e ; c = 12 ; m = 768155 e ; M = 38437 e . Let us now consider f ( u, v ) = 14 ( | u | + | v | + 1) , f ( u, v ) = 13 ( | u | + v ) . hen, with the choice of ρ = 1 / ρ = 1 / r = r = 1, s = 3 and s = 5, we obtaininf n f ( u, v ) : ( u, v ) ∈ [0 , ρ ] × [ − ρ , ρ ] o = f (0 , > M ρ , sup n f ( u, v ) : ( u, v ) ∈ [ − r , r ] × [ − r , r ] o = f (1 , < m r , sup n f ( u, v ) : ( u, v ) ∈ [ − r , r ] × [ − r , r ] o = f (1 , < m r , inf n f ( u, v ) : ( u, v ) ∈ [ s , s ] × [ − s , s ] o = f ( s , > M s , inf n f ( u, v ) : ( u, v ) ∈ [ − s , s ] × [ s , s ] o = f (0 , s ) > M s . Thus the conditions (I ρ ,ρ ) ⋆ , (I r ,r ) and (I s ,s ) are satisfied; therefore the system (7.5) hasat least two nontrivial solutions. References [1] R. P. Agarwal, D. O’Regan and P. J. Y. Wong,
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E-mail address : [email protected] Paolamaria Pietramala, Dipartimento di Matematica e Informatica, Universit`a dellaCalabria, 87036 Arcavacata di Rende, Cosenza, Italy
E-mail address : [email protected]@unical.it