Multiple Positive solutions of a ( p 1 , p 2 ) -Laplacian system with nonlinear BCs
aa r X i v : . [ m a t h . C A ] M a y MULTIPLE POSITIVE SOLUTIONS OF A ( p , p ) -LAPLACIAN SYSTEMWITH NONLINEAR BCS FILOMENA CIANCIARUSO AND PAOLAMARIA PIETRAMALA
Abstract.
Using the theory of fixed point index, we discuss existence, non-existence, localiza-tion and multiplicity of positive solutions for a ( p , p )-Laplacian system with nonlinear Robinand/or Dirichlet type boundary conditions. We give an example to illustrate our theory. Introduction
In the remarkable paper [39] Wang proved the existence of one positive solution of followingone-dimensional p -Laplacian equation(1.1) ( ϕ p ( u ′ )) ′ ( t ) + g ( t ) f ( u ( t )) = 0 , t ∈ (0 , , subject to one of the following three pair of nonlinear boundary conditions (BCs) u ′ (0) = 0 , u (1) + B (cid:0) u ′ (1) (cid:1) = 0 ,u (0) = B (cid:0) u ′ (0) (cid:1) , u ′ (1) = 0 .u (0) = B (cid:0) u ′ (0) (cid:1) , u (1) + B (cid:0) u ′ (1) (cid:1) = 0 . The results of [39] were extended by Karakostas [23] to the context of deviated arguments.In both cases, the existence results are obtained via a careful study of an associated integraloperator combined with the use of the Krasnosel’ski˘ı-Guo Theorem on cone compressions andcone expansions.The Krasnosel’ski˘ı-Guo Theorem, more in general, topological methods are a commonly usedtool in the study of existence of positive solutions for the p -Laplacian equation (1.1) subject todifferent BCs. This is an active area of research, for example, homogeneous Dirichlet BCs havebeen studied in [1, 5, 16, 25, 31, 37, 43, 47], homogeneous Robin BCs in [31, 43, 47], non localBCs of Dirichlet type in [3, 4, 6, 7, 9, 14, 24, 39, 41, 48] and nonlocal BCs of Robin type in[14, 30, 32, 40, 42, 48].Here we study the the one-dimensional ( p , p )-Laplacian system(1.2) ( ϕ p ( u ′ )) ′ ( t ) + g ( t ) f ( t, u ( t ) , v ( t )) = 0 , t ∈ (0 , , ( ϕ p ( v ′ )) ′ ( t ) + g ( t ) f ( t, u ( t ) , v ( t )) = 0 , t ∈ (0 , , with ϕ p i ( w ) = | w | p i − w , subject to the nonlinear boundary conditions (BCs)(1.3) u ′ (0) = 0 , u (1) + B (cid:0) u ′ (1) (cid:1) = 0 , v (0) = B (cid:0) v ′ (0) (cid:1) , v (1) = 0 . Mathematics Subject Classification.
Primary 45G15, secondary 34B18.
Key words and phrases.
Fixed point index, cone, positive solution, p-laplacian, system, nonlinear boundaryconditions.
The existence of positive solutions of systems of equations of the type (1.2) has been widelystudied, see for example [8, 28, 29, 44] under homogeneous Dirichlet BCs and [16, 22, 34, 38, 46]with homogeneous Robin or Neumann BCs. For earlier contributions on problems with nonlinearBCs we refer to [11, 12, 15, 17, 18, 19, 20, 23, 30, 33, 39] and references therein.We improve and complement the previous results in several directions: we obtain multiplicity results for ( p , p )-Laplacian system subject to nonlinear BCs, we allow different growths in thenonlinearities f and f and we also discuss non-existence results. Finally we illustrate in anexample that all the constants that occur in our results can be computed.Our approach is to seek solutions of the system (1.2)-(1.3) as fixed points of a suitable integraloperator. We make use of the classical fixed point index theory and benefit of ideas from thepapers [19, 21, 23, 39]. 2. The system of integral equations
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 ) = { } .If Ω is a open bounded subset of a cone K (in the relative topology) we denote by Ω and ∂ Ωthe closure and the boundary relative to K . When Ω is an open bounded subset of X we writeΩ K = Ω ∩ K , an open subset of K .The following Lemma summarizes some classical results regarding the fixed point index, formore details see [2, 13]. Lemma 2.1.
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 point index 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 . To the system (1.2)-(1.3) we associate the following system of integral equations, which isconstructed in similar manner as in [39], where the case of a single equation is studied. u ( t ) = Z t ϕ − p (cid:16)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:17) ds + B (cid:18) ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) , ≤ t ≤ ,v ( t ) = R t ϕ − p (cid:0)R σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:1) ds + B (cid:0) ϕ − p (cid:0)R σ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:1)(cid:1) , ≤ t ≤ σ u,v , R t ϕ − p (cid:16)R sσ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:17) ds, σ u,v ≤ t ≤ , (2.1) p , p )-LAPLACIAN SYSTEM 3 where ϕ − p i ( w ) = | w | pi − sgn w and σ u,v is the smallest solution x ∈ [0 ,
1] of the equation Z x ϕ − p (cid:18)Z xs g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z x g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) = Z x ϕ − p (cid:18)Z sx g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds. By a solution of (1.2)-(1.3), we mean a solution of the system (2.1).In order to utilize the fixed point index theory we state the following assumptions on theterms that occur in the system (2.1).( C
1) For every i = 1 , f i : [0 , × [0 , ∞ ) × [0 , ∞ ) → [0 , ∞ ) satisfies Carath´eodory conditions,that is, f i ( · , u, v ) is measurable for each fixed ( u, v ) and f i ( t, · , · ) is continuous for almostevery (a.e.) t ∈ [0 , r > φ i,r ∈ L ∞ [0 ,
1] such that f i ( t, u, v ) ≤ φ i,r ( t ) for u, v ∈ [0 , r ] and a. e. t ∈ [0 , . ( C g ∈ L [0 , g ≥ < Z ϕ − p (cid:16)Z s g ( τ ) dτ (cid:17) ds < + ∞ . ( C g ∈ L [0 , g ≥ < Z / ϕ − p Z / s g ( τ ) dτ ! ds + Z / ϕ − p Z s / g ( τ ) dτ ! ds < + ∞ . ( C
4) For every i = 1 , B i : R → R is a continuous function and there exist h i , h i ≥ h i v ≤ B i ( v ) ≤ h i v for any v ≥ . Remark 2.2.
The condition (2.2) is weaker than the condition(2.3) 0 < Z ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds < + ∞ . In fact, for example, the function g ( t ) = t − , t ∈ [0 , / , t , t ∈ (1 / , , satisfies (2.2) but not satisfies (2.3). Remark 2.3.
From ( C
2) and ( C
3) follow that there exists [ a , b ] ⊂ [0 ,
1) such that R b a g ( s ) ds > a , b ] ⊂ (0 ,
1) such that R b a g ( 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 , } .Take the cones K := { w ∈ C [0 ,
1] : w ≥ , concave and nonincreasing } ,K := { w ∈ C [0 ,
1] : w ≥ , concave } . FILOMENA CIANCIARUSO AND PAOLAMARIA PIETRAMALA
It is known (see e.g. [39]) that • for w ∈ K we have w ( t ) ≥ (1 − t ) k w k ∞ , for t ∈ [0 , • for w ∈ K we have w ( t ) ≥ min { t, − t }k w k ∞ , for t ∈ [0 , K i are strictly positive on the sub-interval [ a i , b i ] and in particularwe have • for w ∈ K we have min t ∈ [0 ,b ] w ( t ) ≥ (1 − b ) k w k ∞ ; • for w ∈ K we have min t ∈ [ a ,b ] w ( t ) ≥ min { a , − b }k w k ∞ .In the following we make use of the notations: c := 1 − b , c := min { a , − b } . Consider now the cone K in C [0 , × C [0 ,
1] defined by K := { ( u, v ) ∈ K × K } . For a positive solution of the system (2.1) we mean a solution ( u, v ) ∈ K of (2.1) such that k ( u, v ) k >
0. We seek such solution as a fixed point of the following operator T .Consider the integral operator T ( u, v )( t ) := T ( u, v )( t ) T ( u, v )( t ) ! , (2.4)where T ( u, v )( t ) := Z t ϕ − p (cid:16)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:17) ds + B (cid:18) ϕ − p (cid:0) Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:1)(cid:19) and T ( u, v )( t ) := R t ϕ − p (cid:0)R σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:1) ds + B (cid:0) ϕ − p (cid:0)R σ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:1)(cid:1) , ≤ t ≤ σ u,v , R t ϕ − p (cid:16)R sσ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:17) ds, σ u,v ≤ t ≤ , From the definitions, for every ( u, v ) ∈ K we havemax t ∈ [0 , T ( u, v )( t ) = T ( u, v )( σ u,v ) . Under our assumptions, we can show that the integral operator T leaves the cone K invariantand is compact. Lemma 2.4.
The operator (2.4) maps K into K and is compact.Proof. Take ( u, v ) ∈ K . Then we have T ( u, v ) ∈ K . Now, we show that the map T is compact.Firstly, we show that T sends bounded sets into bounded sets. Take ( u, v ) ∈ K such that p , p )-LAPLACIAN SYSTEM 5 k ( u, v ) k ≤ r . Then, for all t ∈ [0 ,
1] we have T ( u, v )( t ) = Z t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) ≤ Z t ϕ − p (cid:18)Z s g ( τ ) φ ,r ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ≤ Z t ϕ − p (cid:18)Z g ( τ ) φ ,r ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) φ ,r ( τ ) dτ (cid:19) ≤ Z ϕ − p (cid:18)Z g ( τ ) φ ,r ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) φ ,r ( τ ) dτ (cid:19) < + ∞ . We prove now that T sends bounded sets into equicontinuous sets. Let t , t ∈ [0 , t < t ,( u, v ) ∈ K such that k ( u, v ) k ≤ r . Then we have | T ( u, v )( t ) − T ( u, v )( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t t ϕ − p (cid:18)Z g ( τ ) φ ,r ( τ ) dτ (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = C r | t − t | . Therefore we obtain | T ( u, v )( t ) − T ( u, v )( t ) | → t → t . By the Ascoli-Arzel`aTheorem we can conclude that T is a compact map. In a similar manner we proceed for T ( u, v ).Moreover, the map T is compact since the components T i are compact maps. (cid:3) Existence results
For our index calculations 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 ) < c ρ and min t ∈ [ a ,b ] v ( t ) < c ρ } and if ρ = ρ = ρ we write simply K ρ and V ρ . The set V ρ was introduced in [10] as an extensionto the case of systems of a set given by Lan [27]. The use of different radii, in the spirit of thepaper [21], allows more freedom in the growth of the nonlinearities.The following Lemma is similar to the Lemma 5 of [10] and therefore its proof is omitted. Lemma 3.1.
The sets defined above have the following properties: • K c ρ ,c ρ ⊂ V ρ ,ρ ⊂ K ρ ,ρ . • ( w , w ) ∈ ∂V ρ ,ρ iff ( w , w ) ∈ K and min t ∈ [ a i ,b i ] w i ( t ) = c i ρ i for some i ∈ { , } and min t ∈ [ a j ,b j ] w j ( t ) ≤ c j ρ j for j = i . • If ( w , w ) ∈ ∂V ρ ,ρ , then for some i ∈ { , } c i ρ i ≤ w i ( t ) ≤ ρ i for each t ∈ [ a i , b i ] and k w i k ∞ ≤ ρ i ; moreover for j = i we have k w j k ∞ ≤ ρ j . We firstly prove that the fixed point index is 1 on the set K ρ ,ρ . FILOMENA CIANCIARUSO AND PAOLAMARIA PIETRAMALA
Lemma 3.2.
Assume that (I ρ ,ρ ) there exist ρ , ρ > such that for every i = 1 , f ρ ,ρ i < ϕ p i ( m i ) where f ρ ,ρ i = sup n f i ( t, u, v ) ρ p i − i : ( t, u, v ) ∈ [0 , × [0 , ρ ] × [0 , ρ ] , o , m = Z ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) dτ (cid:19) , m = max (Z ϕ − p Z s g ( τ ) dτ ! ds + h ϕ − p Z g ( τ ) dτ ! , Z ϕ − p Z s g ( τ ) dτ ! 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 ).Firstly we assume that k u k ∞ = ρ and k v k ∞ ≤ ρ .Then we have λu ( t ) = Z t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) ≤ Z t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) = ρ Z t ϕ − p ( Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ) ds + h ϕ − p ( Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ) ! . Taking t = 0 gives λu (0) = λρ ≤ ρ (cid:18)Z ϕ − p (cid:18)Z s g ( τ ) f ρ ,ρ dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) f ρ ,ρ dτ (cid:19)(cid:19) = ρ ϕ − p ( f ρ ,ρ ) (cid:18)Z ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) dτ (cid:19)(cid:19) = ρ m ϕ − p ( f ρ ,ρ ) . Using the hypothesis (3.1) and the strictly monotonicity of ϕ − p we obtain λρ < ρ . Thiscontradicts the fact that λ ≥ k v k ∞ = ρ and k u k ∞ ≤ ρ .Then we have λρ = k T ( u, v ) k ∞ = T ( u, v )( σ u,v ) . p , p )-LAPLACIAN SYSTEM 7 If σ u,v ≤
12 , we have λρ = k T ( u, v ) k ∞ = T ( u, v )( σ u,v )= Z σ u,v ϕ − p (cid:18)Z σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z σ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) ≤ Z ϕ − p Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ ! ds + h ϕ − p (cid:18)Z σ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ≤ Z ϕ − p Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ ! ds + h ϕ − p Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ ! = ρ Z ϕ − p Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! ds + h ϕ − p Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! ≤ ρ ϕ − p ( f ρ ,ρ ) Z ϕ − p Z s g ( τ ) dτ ! ds + h ϕ − p Z g ( τ ) dτ ! . If σ u,v >
12 , we have λρ = k T ( u, v ) k ∞ = T ( u, v )( σ u,v )= Z σ u,v ϕ − p Z sσ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ ! ds ≤ Z ϕ − p Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ ! ds = ρ Z ϕ − p Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! ds ≤ ρ ϕ − p ( f ρ ,ρ ) Z ϕ − p Z s g ( τ ) dτ ! ds. Then, in both cases, we have λρ = k T ( u, v ) k ∞ = T ( u, v )( σ u,v ) ≤ ρ ϕ − p ( f ρ ,ρ ) × max ( Z ϕ − p Z s g ( τ ) dτ ! ds + h ϕ − p Z g ( τ ) dτ ! , Z ϕ − p Z s g ( τ ) dτ ! ds ) = ρ ϕ − p ( f ρ ,ρ ) 1 m . Using the hypothesis (3.1) and the strictly monotonicity of ϕ − p we obtain λρ < ρ . Thiscontradicts the fact that λ ≥ (cid:3) 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 ) > ϕ p i ( M i ) , where f , ( ρ ,ρ ) = inf n f ( t, u, v ) ρ p − : ( t, u, v ) ∈ [0 , b ] × [ c ρ , ρ ] × [0 , ρ ] o ,f , ( ρ ,ρ ) = inf n f ( t, u, v ) ρ p − : ( t, u, v ) ∈ [ a , b ] × [0 , ρ ] × [ c ρ , ρ ] o , FILOMENA CIANCIARUSO AND PAOLAMARIA PIETRAMALA M = Z b ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z b g ( τ ) dτ (cid:19) , and M = 12 min a ≤ ν ≤ b (cid:26)Z νa ϕ − p ( Z νs g ( τ ) dτ ) ds + Z b ν ϕ − p ( Z sν g ( τ ) dτ ) ds + h ϕ − p ( Z νa g ( τ ) dτ ) (cid:27) . 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 ). We examine the two cases:Case (1): c ρ ≤ u ( t ) ≤ ρ for t ∈ [0 , b ] and 0 ≤ v ( t ) ≤ ρ for t ∈ [0 , t ∈ [0 , b ], we have ρ ≥ u ( t )= Z t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) + λ ≥ Z b t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) + λ ≥ Z b t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + h ϕ − p (cid:18)Z b g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) + λ = ρ Z b t ϕ − p Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! ds + ρ h ϕ − p Z b g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! + λ. For t = 0 we obtain ρ ≥ ρ ϕ − p ( f , ( ρ ,ρ ) ) (cid:18)Z b ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z b g ( τ ) dτ (cid:19)(cid:19) + λ> ρ ϕ − p (cid:0) f , ( ρ ,ρ ) (cid:1) M + λ. Using the hypothesis (3.2) we obtain ρ > ρ + λ , a contradiction.Case (2): 0 ≤ u ( t ) ≤ ρ for t ∈ [0 ,
1] and c ρ ≤ v ( t ) ≤ ρ .We distinguish three cases:Case (i) 0 < σ u,v ≤ a . p , p )-LAPLACIAN SYSTEM 9 Therefore we get ρ ≥ v ( σ u,v ) = T ( u, v )( σ u,v ) + λ = Z σ u,v ϕ − p Z sσ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ ! ds + λ ≥ Z b a ϕ − p (cid:18)Z sa g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + λ = ρ Z b a ϕ − p Z sa g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! ds + λ ≥ ρ ϕ − p ( f , ( ρ ,ρ ) ) (cid:18)Z b a ϕ − p (cid:18)Z sa g ( τ ) dτ (cid:19) ds (cid:19) + λ ≥ ρ ϕ − p ( f , ( ρ ,ρ ) ) 1 M + λ. Using the hypothesis (3.2) we obtain ρ > ρ + λ , a contradiction.Case (ii) σ u,v ≥ b . ρ ≥ v ( σ u,v ) = T ( u, v )( σ u,v ) + λ = Z σ u,v ϕ − p (cid:18)Z σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z σ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) + λ ≥ Z b a ϕ − p (cid:18)Z b s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + h ϕ − p (cid:18)Z b a g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) + λ = ρ Z b a ϕ − p Z b s g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! ds + ρ h ϕ − p Z b a g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! + λ ≥ ρ ϕ − p ( f , ( ρ ,ρ ) ) (cid:18)Z b a ϕ − p (cid:18)Z b s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z b a g ( τ ) dτ (cid:19)(cid:19) + λ ≥ ρ ϕ − p ( f , ( ρ ,ρ ) ) 1 M + λ. Using the hypothesis (3.2) we obtain ρ > ρ + λ , a contradiction.Case (iii) a < σ u,v < b .2 ρ ≥ v ( σ u,v ) = 2 λ + 2 T ( u, v )( σ u,v ) = 2 λ + Z σ u,v ϕ − p (cid:18)Z σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z σ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) + Z σ u,v ϕ − p Z sσ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ ! ds ≥ λ + Z σ u,v a ϕ − p (cid:18)Z σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + h (cid:18) ϕ − p (cid:18)Z σ u,v a g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) + Z b σ u,v ϕ − p Z sσ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ ! ds = 2 λ + ρ " Z σ u,v a ϕ − p Z σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! ds + h ϕ − p Z σ u,v a g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! + Z b σ u,v ϕ − p Z sσ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) ρ p − dτ ! ds ≥ λ + ρ ϕ − p ( f , ( ρ ,ρ ) ) " Z σ u,v a ϕ − p (cid:18)Z σ u,v s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z σ u,v a g ( τ ) dτ (cid:19) + Z b σ u,v ϕ − p Z sσ u,v g ( τ ) dτ ! ds ≥ λ + 2 ρ ϕ − p ( f , ( ρ ,ρ ) ) 1 M . Using the hypothesis (3.2) we obtain ρ > λ + ρ , a contradiction. (cid:3) Remark 3.4.
We point out that a stronger, but easier to check, hypothesis than (3.2) is f i, ( ρ ,ρ ) > ϕ p i ( ˜ M i ) , where 1˜ M = Z b ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds and 1˜ M = 12 min a ≤ ν ≤ b (cid:26)Z νa ϕ − p ( Z νs g ( τ ) dτ ) ds + Z b ν ϕ − p ( Z sν g ( τ ) dτ ) ds (cid:27) . In the following Lemma we exploit an idea that was used in [19, 21] and we provide a resultof index 0 controlling the growth of just one nonlinearity f i , at the cost of having to deal with alarger domain. Nonlinearities with different growths were considered for examples in [35, 36, 45]. Lemma 3.5.
Assume that (I ρ ,ρ ) ⋆ there exist ρ , ρ > such that for some i ∈ { , } we have (3.3) f ∗ i, ( ρ ,ρ ) > ϕ p i ( M i ) , p , p )-LAPLACIAN SYSTEM 11 where f ∗ i, ( ρ ,ρ ) = inf n f i ( t, u, v ) ρ p i − i : ( t, u, v ) ∈ [ a i , b i ] × [0 , ρ ] × [0 , ρ ] 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 ). Thus we proceed as in the proof of Lemma 3.3. (cid:3) The proof of the next result regarding the existence of at least one, two or three positivesolutions follows by the properties of fixed point index and is omitted. It is possible to stateresults for four or more positive solutions, in a similar way as in [26], by expanding the lists inconditions ( S ) , ( S ). Theorem 3.6.
The system (2.1) has at least one positive solution in K if one of the followingconditions holds. ( S ) For i = 1 , there exist ρ i , r i ∈ (0 , ∞ ) with ρ 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 < c i r i such that (I ρ ,ρ ) , (I r ,r ) hold.The system (2.1) has at least two positive solutions in K if one of the following conditions holds. ( S ) For i = 1 , there exist ρ i , r i , s i ∈ (0 , ∞ ) with ρ i < r i < c 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 < c i r i and r i < s i such that (I ρ ,ρ ) , (I r ,r ) and (I s ,s ) hold.The system (2.1) has at least three positive solutions in K if one of the following conditionsholds. ( S ) For i = 1 , there exist ρ i , r i , s i , δ i ∈ (0 , ∞ ) with ρ i < r i < c i s i and s i < δ i such that (I ρ ,ρ ) [ or (I ρ ,ρ ) ⋆ ] , (I r ,r ) , (I s ,s ) and (I δ ,δ ) hold. ( S ) For i = 1 , there exist ρ i , r i , s i , δ i ∈ (0 , ∞ ) with ρ i < c i r i and r i < s i < c i δ i such that (I ρ ,ρ ) , (I r ,r ) , (I s ,s ) and (I δ ,δ ) hold. Non-existence results
We now provide some non-existence results for system (2.1).
Theorem 4.1.
Assume that one of the following conditions holds. (1)
For i = 1 , , (4.1) f i ( t, u , u ) < ϕ p i ( m i u i ) for every t ∈ [0 , and u i > . (2) For i = 1 , , (4.2) f i ( t, u , u ) > ϕ p i (cid:18) M i c i u i (cid:19) for every t ∈ [ a i , b i ] and u i > . (3) There exists k ∈ { , } such that (4.1) is verified for f k and for j = k condition (4.2) isverified for f j . Then there is no positive 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 , • Let be k u k ∞ = 0. Then we have u ( t ) = Z t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) < m Z t ϕ − p (cid:18)Z s g ( τ ) ϕ p ( u ( τ )) dτ (cid:19) ds + m h ϕ − p (cid:18)Z g ( τ ) ϕ p ( u ( τ )) dτ (cid:19) ≤ m k u k ∞ (cid:18)Z t ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) dτ (cid:19)(cid:19) . Taking t = 0 gives k u k ∞ = u (0) < m k u k ∞ (cid:18)Z ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) dτ (cid:19)(cid:19) = m k u k ∞ m , a contradiction. • Let be k v k ∞ = 0.Reasoning as in Lemma 3.2 we distinguish the cases σ u,v ≤ / σ u,v > / k v k ∞ = k T ( u, v ) k ∞ = T ( u, v )( σ u,v )= Z σ u,v ϕ − p (cid:18)Z σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z σ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) < m k v k ∞ (cid:18)Z σ u,v ϕ − p (cid:18)Z σ u,v s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z σ u,v g ( τ ) dτ (cid:19)(cid:19) ≤ m k v k ∞ Z ϕ − p Z s g ( τ ) dτ ! ds + h ϕ − p Z g ( τ ) dτ !! ≤ m k v k ∞ m , a contradiction.The proof is similar in the last case σ u,v > / u, v ) ∈ K such that ( u, v ) = T ( u, v ) and( u, v ) =(0 , p , p )-LAPLACIAN SYSTEM 13 • Let be k u k ∞ = 0. Then, for t ∈ [ a , b ] = [0 , b ], we have u ( t ) = Z t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) ≥ Z b t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + h ϕ − p (cid:18)Z g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ≥ Z b t ϕ − p (cid:18)Z s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + h ϕ − p (cid:18)Z b g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) > M c (cid:18)Z b t ϕ − p (cid:18)Z s g ( τ ) ϕ p ( u ( τ )) dτ (cid:19) ds + h ϕ − p (cid:18)Z b g ( τ ) ϕ p ( u ( τ )) dτ (cid:19)(cid:19) > M c (cid:18)Z b t ϕ − p (cid:18)Z s g ( τ ) ϕ p ( c k u k ∞ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z b g ( τ ) ϕ p ( c k u k ∞ ) dτ (cid:19)(cid:19) = M c c k u k ∞ (cid:18)Z b t ϕ − p (cid:18)Z s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z b g ( τ ) dτ (cid:19)(cid:19) . For t = 0 we obtain u (0) = k u k ∞ > M k u k ∞ M , a contradiction. • Let be k v k ∞ = 0. We examine the case σ u,v ≥ b . We have k v k ∞ = v ( σ u,v ) = T ( u, v )( σ u,v ) = Z σ u,v ϕ − p (cid:18)Z σ u,v s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + B (cid:18) ϕ − p (cid:18)Z σ u,v g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19)(cid:19) ≥ Z b a ϕ − p (cid:18)Z b s g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) ds + h ϕ − p (cid:18)Z b a g ( τ ) f ( τ, u ( τ ) , v ( τ )) dτ (cid:19) > M c c k v k ∞ Z b a ϕ − p (cid:18)Z b s g ( τ ) dτ (cid:19) ds + h ϕ − p (cid:18)Z b a g ( τ ) dτ (cid:19) ! ≥ M k v k ∞ M , a contradiction. By similar proofs, the cases 0 < σ u,v ≤ a and a < σ u,v < b can beexamined.(3) 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 the function f satisfies either (4.1) or (4.2) and the prooffollows as in the previous cases. If k v k ∞ = 0 then the function f satisfies either (4.1) or (4.2)and the proof follows as previous cases. (cid:3) An example
We illustrate in the following example that all the constants that occur in the Theorem 3.6can be computed.Consider the system(5.1) ( ϕ p ( u ′ )) ′ ( t ) + g ( t ) f ( t, u ( t ) , v ( t )) = 0 , t ∈ (0 , , ( ϕ p ( v ′ )) ′ ( t ) + g ( t ) f ( t, u ( t ) , v ( t )) = 0 , t ∈ (0 , , subject to boundary conditions(5.2) u ′ (0) = 0 , u (1) + B (cid:0) u ′ (1) (cid:1) = 0 , v (0) = B (cid:0) v ′ (0) (cid:1) , v (1) = 0 , where B and B are defined by: B ( w ) = w, w ≤ , w , ≤ w ≤ , w + , w ≥ , and B ( w ) = w , ≤ w ≤ , w + , w ≥ . Now we assume g = g ≡
1. Thus we have1 m = p − p + h , m = p − p (cid:18) (cid:19) p p − + h (cid:18) (cid:19) p − , M = 1 M [0 , b ] = p − p b p p − + h b p − and1 M = 1 M [ a , b ] = 12 min a ≤ ν ≤ b (cid:18) p − p (cid:16) ( ν − a ) p p − + ( b − ν ) p p − (cid:17) + h ( ν − a ) p − (cid:19) . The choice p = , p = 3, b = , a = , b = , h = 1 / h = 1 / h = 1 / h = 1 / c = 13 ; c = 14 ; m = 1 . M = 5 . m = 2 . M = 9 . . Let us now consider f ( t, u, v ) = 116 ( u + t v ) + 2750 , f ( t, u, v ) = ( tu ) + 10 v . Then, with the choice of ρ = ρ = 1 / r = 1, r = 2 / s = s = 9, we obtaininf n f ( t, u, v ) : ( t, u, v ) ∈ [0 ,
23 ] × [0 , ρ ] × [0 , ρ ] o = f (0 , ,
0) = 0 . > p M ρ = 0 . , sup n f ( t, u, v ) : ( t, u, v ) ∈ [0 , × [0 , r ] × [0 , r ] o = f (1 , r , r ) = 0 . < √ m r = 1 . , inf n f ( t, u, v ) : ( t, u, v ) ∈ [0 , / × [ c s , s ] × [0 , s ] o = f (0 , c s ,
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E-mail address : [email protected] Paolamaria Pietramala, Dipartimento di Matematica e Informatica, Universit`a della Calabria,87036 Arcavacata di Rende, Cosenza, Italy
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