On the existence of three solutions for the Dirichlet problem on the Sierpinski gasket
aa r X i v : . [ m a t h . A P ] F e b ON THE EXISTENCE OF THREE SOLUTIONS FOR THEDIRICHLET PROBLEM ON THE SIERPINSKI GASKET
BRIGITTE E. BRECKNER, DUˇSAN REPOVˇS, AND CSABA VARGA
Abstract.
We apply a recently obtained three critical points theorem ofB. Ricceri to prove the existence of at least three solutions of certain two-parameters Dirichlet problems defined on the Sierpinski gasket. We also showthe existence of at least three nonzero solutions of certain perturbed two-parameters Dirichlet problems on the Sierpinski gasket, using both the moun-tain pass theorem of Ambrosetti-Rabinowitz and that of Pucci-Serrin. Introduction
The celebrated three critical points theorem obtained by Ricceri in [15] turnedout to be one of the most often applied abstract multiplicity results for the studyof different types of nonlinear problems of variational nature. In this sense we referto the references listed in [16]. Also, this three critical points theorem has beenextended to certain classes of non-smooth functions (see, for example, [2], [3], [12]).Ricceri has published both a revised form of his three critical points theorem ([16])and a refinement of it ([17]). A corollary of the latter, stated also in [17], is thefollowing result:
Theorem 1.1.
Let X be a separable and reflexive real Banach space, and Φ , J : X → R functionals satisfying the following conditions: (i) Φ is a coercive, sequentially weakly lower semicontinuous C -functional,bounded on each bounded subset of X , and whose derivative admits a con-tinuous inverse on X ∗ . (ii) If ( u n ) is a sequence in X converging weakly to u , and if lim inf n →∞ Φ( u n ) ≤ Φ( u ) ,then ( u n ) has a subsequence converging strongly to u . (iii) J is a C -functional with compact derivative. (iv) The functional Φ has a strict local minimum u with Φ( u ) = J ( u ) = 0 . (v) The inequality ρ < ρ holds, where ρ := max ( , lim sup || u ||→∞ J ( u )Φ( u ) , lim sup u → u J ( u )Φ( u ) ) and ρ := sup u ∈ Φ − (]0 , ∞ [) J ( u )Φ( u ) . Then, for each compact interval [ λ , λ ] ⊂ ] ρ , ρ [ ( where, by convention, := ∞ and ∞ = 0) , there exists a positive real number r with the following property: For Mathematics Subject Classification.
Primary 35J20; Secondary 28A80, 35J25, 35J60,47J30, 49J52.
Key words and phrases.
Sierpinski gasket, weak Laplacian, Dirichlet problem on the Sierpinskigasket, weak solution, critical point, minimax theorems, mountain pass theorems. every λ ∈ [ λ , λ ] and for every C -functional Ψ : X → R with compact derivativethere exists δ > such that, for every η ∈ [0 , δ ] , the equation Φ ′ ( u ) = λJ ′ ( u ) + η Ψ ′ ( u ) has at least three solutions in X whose norms are less than r . In the present paper we show with the aid of Theorem 1.1 that, under suit-able assumptions on the functions f, g : V × R → R , the following two-parametersDirichlet problem defined on the Sierpinski gasket V in R N − has at least threesolutions ( DP λ,η ) − ∆ u ( x ) = λf ( x, u ( x )) + ηg ( x, u ( x )) , ∀ x ∈ V \ V ,u | V = 0 . So far we know, this would be the first application of a Ricceri type three criticalpoints theorem to nonlinear partial differential equations on fractals. (Among thecontributions to the theory of nonlinear elliptic equations on fractals we mention[4], [5], [7], [8], [9], [19]).We also study, in a particular case, a perturbed version of problem ( DP λ,η ). Asimilar problem, but involving the p -Laplacian, has been recently investigated in[1]. Notations.
We denote by N the set of natural numbers { , , , . . . } , by N ∗ := N \ { } the set of positive naturals, and by | · | the Euclidian norm on the spaces R n , n ∈ N ∗ .If X is a topological space and M a subset of it, then M and ∂M denote theclosure, respectively, the boundary of M .If X is a normed space and r a positive real, then B r stands for the open ballwith radius r centered at the origin.2. The Sierpinski gasket
In its initial representation that goes back to the pioneering papers of the Polishmathematician Waclaw Sierpinski (1882–1969), the
Sierpinski gasket is the con-nected subset of the plane obtained from an equilateral triangle by removing theopen middle inscribed equilateral triangle of 4 − the area, removing the correspond-ing open triangle from each of the three constituent triangles, and continuing thisway. The gasket can also be obtained as the closure of the set of vertices arisingin this construction. Over the years, the Sierpinski gasket showed both to be ex-traordinarily useful in representing roughness in nature and man’s works. We referto [18] for an elementary introduction to this subject and to [20] for importantapplications to differential equations on fractals.We now rigorously describe the construction of the Sierpinski gasket in a generalsetting. Let N ≥ p , . . . , p N ∈ R N − be so that | p i − p j | = 1 for i = j . Define, for every i ∈ { , . . . , N } , the map S i : R N − → R N − by S i ( x ) = 12 x + 12 p i . HE DIRICHLET PROBLEM ON THE SIERPINSKI GASKET 3
Obviously every S i is a similarity with ratio . Let S := { S , . . . , S N } and denoteby F : P ( R N − ) → P ( R N − ) the map assigning to a subset A of R N − the set F ( A ) = N [ i =1 S i ( A ) . It is known (see, for example, Theorem 9.1 in [6]) that there is a unique nonemptycompact subset V of R N − , called the attractor of the family S , such that F ( V ) = V (that is, V is a fixed point of the map F ). The set V is called the Sierpinski gasket (SG for short) in R N − . It can be constructed inductively as follows: Put V := { p , . . . , p N } , V m := F ( V m − ), for m ≥
1, and V ∗ := ∪ m ≥ V m . Since p i = S i ( p i ) for i = 1 , N , we have V ⊆ V , hence F ( V ∗ ) = V ∗ . Taking into account that the maps S i , i = 1 , N , are homeomorphisms, we conclude that V ∗ is a fixed point of F . Onthe other hand, denoting by C the convex hull of the set { p , . . . , p N } , we observethat S i ( C ) ⊆ C for i = 1 , N . Thus V m ⊆ C for every m ∈ N , so V ∗ ⊆ C . It followsthat V ∗ is nonempty and compact, hence V = V ∗ . In the sequel V is consideredto be endowed with the relative topology induced from the Euclidean topology on R N − . The set V is called the intrinsic boundary of the SG.The family S of similarities satisfies the open set condition (see pg. 129 in [6])with the interior int C of C . (Note that int C = ∅ since the points p , . . . , p N areaffine independent.) Thus, by Theorem 9.3 of [6], the Hausdorff dimension d of V satisfies the equality N X i =1 (cid:18) (cid:19) d = 1 , hence d = ln N ln 2 , and 0 < H d ( V ) < ∞ , where H d is the d -dimensional Hausdorffmeasure on R N − . Let µ be the normalized restriction of H d to the subsets of V ,so µ ( V ) = 1. The following property of µ will be important for our investigations(2.1) µ ( B ) > , for every nonempty open subset B of V. In other words, the support of µ coincides with V . We refer, for example, to [4] forthe proof of (2.1). 3. The space H ( V )We retain the notations from the previous section and briefly recall from [7] thefollowing notions (see also [8] and [10] for the case N = 3). Denote by C ( V ) thespace of real-valued continuous functions on V and by C ( V ) := { u ∈ C ( V ) | u | V = 0 } . The spaces C ( V ) and C ( V ) are endowed with the usual supremum norm || · || sup .For a function u : V → R and for m ∈ N let(3.1) W m ( u ) = (cid:18) N + 2 N (cid:19) m X x,y ∈ V m | x − y | =2 − m ( u ( x ) − u ( y )) . We have W m ( u ) ≤ W m +1 ( u ) for every natural m , so we can put(3.2) W ( u ) = lim m →∞ W m ( u ) . Define now H ( V ) := { u ∈ C ( V ) | W ( u ) < ∞} . B. E. BRECKNER, D. REPOVˇS, AND CS. VARGA
It turns out that H ( V ) is a dense linear subset of L ( V, µ ) (equipped with theusual || · || norm). We now endow H ( V ) with the norm || u || = p W ( u ) . In fact, there is an inner product defining this norm: For u, v ∈ H ( V ) and m ∈ N let W m ( u, v ) = (cid:18) N + 2 N (cid:19) m X x,y ∈ V m | x − y | =2 − m ( u ( x ) − u ( y ))( v ( x ) − v ( y )) . Put W ( u, v ) = lim m →∞ W m ( u, v ) . Then W ( u, v ) ∈ R , and H ( V ), equipped with the inner product W (which obvi-ously induces the norm ||·|| ), becomes a real Hilbert space. Moreover, if c := 2 N +3,then(3.3) || u || sup ≤ c || u || , for every u ∈ H ( V ) , and the embedding(3.4) ( H ( V ) , || · || ) ֒ → ( C ( V ) , || · || sup )is compact.We now state a useful property of the space H ( V ) which shows, together withthe facts that ( H ( V ) , || · || ) is a Hilbert space and that H ( V ) is dense in L ( V, µ ),that W is a Dirichlet form on L ( V, µ ). (See, for example, Lemma 3.1 of [4] for thestraightforward proof.)
Lemma 3.1.
Let h : R → R be a Lipschitz mapping with Lipschitz constant L ≥ and such that h (0) = 0 . Then, for every u ∈ H ( V ) , we have h ◦ u ∈ H ( V ) and || h ◦ u || ≤ L · || u || . The Dirichlet problem on the Sierpinski gasket
Keep the notations from the previous sections. We also recall from [7] (respec-tively, from [8] and [10] in the case N = 3) that one can define in a standard waya bijective, linear, and self-adjoint operator ∆ : D → L ( V, µ ), where D is a linearsubset of H ( V ) which is dense in L ( V, µ ) (and dense also in ( H ( V ) , || · || )), suchthat −W ( u, v ) = Z V ∆ u · vdµ, for every ( u, v ) ∈ D × H ( V ) . The operator ∆ is called the weak Laplacian on V . Remark 4.1.
Theorem 19.B of [22], applied to ∆ − : L ( V, µ ) → L ( V, µ ), yieldsin particular that H ( V ) is separable (see also sections 19.9 and 19.10 in [22]).Given a continuous function h : V × R → R , we can formulate now the following Dirichlet problem on the SG : Find appropriate functions u ∈ H ( V ) (in fact, u ∈ D )such that ( P ) − ∆ u ( x ) = h ( x, u ( x )) , ∀ x ∈ V \ V ,u | V = 0 . HE DIRICHLET PROBLEM ON THE SIERPINSKI GASKET 5
A function u ∈ H ( V ) is called a weak solution of ( P ) if W ( u, v ) − Z V h ( x, u ( x )) v ( x ) dµ = 0 , ∀ v ∈ H ( V ) . Remark 4.2.
Using the regularity result Lemma 2.12 of [7], it follows that everyweak solution of problem ( P ) is actually a strong solution (as defined in [7]). Forthis reason we will call in the sequel weak solutions of problem (P) simply solutionsof problem ( P ).Before defining the energy functional attached to problem ( P ) we recall a fewbasic notions. Definition 4.3.
Let E be a real Banach space and T : E → R a functional.(1) We say that T is Fr´echet differentiable at u ∈ E if there exists a continuouslinear map T ′ ( u ) : E → R , called the Fr´echet differential of T at u , such thatlim v → | T ( u + v ) − T ( u ) − T ′ ( u )( v ) ||| v || = 0 . The functional T is Fr´echet differentiable ( on E ) if T is Fr´echet differentiable atevery point u ∈ E . In this case the mapping T ′ : E → E ∗ assigning to each point u ∈ E the Fr´echet differential of T at u is called the Fr´echet derivative , or, shortly,the derivative of T on E . If T ′ : E → E ∗ is continuous, then T is called a C - functional .(2) If T is Fr´echet differentiable on E , then a point u ∈ E is a critical point of T if T ′ ( u ) = 0. The value of I at u is then called a critical value of I . Remark 4.4.
Note that if the Fr´echet differentiable functional T : E → R has in u ∈ E a local extremum, then u is a critical point of T . Proposition 4.5.
Let h : V × R → R be continuous and define H : V × R → R by H ( x, t ) = Z t h ( x, ξ ) dξ. Then the mapping J : H ( V ) → R given by J ( u ) = Z V H ( x, u ( x )) dµ satisfies the following properties: a) J is a C -functional. b) Its derivative J ′ : H ( V ) → ( H ( V )) ∗ is compact. c) J is sequentially weakly continuous.Proof. a) The proof of Proposition 2.19 in [7] implies that J is a C -functional andthat its derivative J ′ : H ( V ) → ( H ( V )) ∗ is given by J ′ ( u )( v ) = Z V h ( x, u ( x )) v ( x ) dµ, for all u, v ∈ H ( V ) . b) To show that J ′ is compact, pick a bounded sequence ( u n ) in H ( V ). Since H ( V ) is reflexive and since the embedding (3.4) is compact, there exists a subse-quence of ( u n ) which converges in ( C ( V ) , || · || sup ). Without any loss of generality B. E. BRECKNER, D. REPOVˇS, AND CS. VARGA we can assume that ( u n ) converges in ( C ( V ) , || · || sup ) to an element u ∈ C ( V ).Define T : H ( V ) → R by T ( v ) = Z V h ( x, u ( x )) v ( x ) dµ, for all v ∈ H ( V ) . According to (3.3), the functional T belongs to ( H ( V )) ∗ . We next show that thesequence ( J ′ ( u n )) converges to T in ( H ( V )) ∗ . By (3.3) the following inequalityholds for every index n || J ′ ( u n ) − T || ≤ c Z V | h ( x, u n ( x )) − h ( x, u ( x )) | dµ. Using the Lebesgue dominated convergence theorem, we conclude that ( J ′ ( u n ))converges to T in ( H ( V )) ∗ . Thus J ′ is compact.c) The assertion follows from b) and Corollary 41.9 of [21]. We also give a directproof: Clearly H is continuous. Let ( u n ) be a sequence which converges weakly to u in H ( V ). Since the embedding (3.4) is compact, ( u n ) converges to u in ( C ( V ) , || ·|| sup ). The Lebesgue dominated convergence theorem implies now that ( J ( u n ))converges to J ( u ). Thus J is sequentially weakly continuous. (cid:3) Proposition 4.6.
Let h : V × R → R be continuous. Then the functional I : H ( V ) → R given by I ( u ) = 12 || u || − J ( u ) , where J : H ( V ) → R is defined in Proposition , is a C -functional and itsderivative I ′ : H ( V ) → ( H ( V )) ∗ is given by I ′ ( u )( v ) = W ( u, v ) − Z V h ( x, u ( x )) v ( x ) dµ, for all u, v ∈ H ( V ) . In particular, u ∈ H ( V ) is a solution of problem ( P ) if and only if u is a criticalpoint of I .Proof. See Proposition 2.19 in [7]. (cid:3)
Remark 4.7.
The functional I : H ( V ) → R defined in Proposition 4.6 is calledthe energy functional attached to problem ( P ).We now state for later use some fundamental properties of the energy functional I . Corollary 4.8.
Let h : V × R → R be continuous. Then the functional I : H ( V ) → R defined in Proposition is sequentially weakly lower semicontinuous.Proof. The function u ∈ H ( V )
7→ || u || ∈ R is continuous in the norm topology on H ( V ) and convex, thus it is sequentially weakly lower semicontinuous on H ( V ).The conclusion follows now from assertion c) of Proposition 4.5. (cid:3) Corollary 4.9.
Let h : V × R → R be continuous and consider the functional I : H ( V ) → R defined in Proposition . If ( u n ) is a bounded sequence in H ( V ) such that the sequence ( I ′ ( u n )) converges to , then ( u n ) contains a convergentsubsequence.Proof. Using Proposition 4.6, we know that for every index nI ′ ( u n ) = W ( u n , · ) − J ′ ( u n ) . Assertion b) of Proposition 4.5 yields now the conclusion. (cid:3)
HE DIRICHLET PROBLEM ON THE SIERPINSKI GASKET 7 A Dirichlet problem depending on two parameters
Let f, g : V × R → R be continuous, and define the functions F, G : V × R → R by F ( x, t ) = Z t f ( x, ξ ) dξ and G ( x, t ) = Z t g ( x, ξ ) dξ. For every λ, η ≥ DP λ,η ) − ∆ u ( x ) = λf ( x, u ( x )) + ηg ( x, u ( x )) , ∀ x ∈ V \ V ,u | V = 0 . By Proposition 4.6, the energy functional attached to the problem ( DP λ,η ) is themap I : H ( V ) → R defined by I ( u ) = 12 || u || − λ Z V F ( x, u ( x )) dµ − η Z V G ( x, u ( x )) dµ. The aim of this section is to apply Theorem 1.1 to show that, under suitable as-sumptions and for certain values of the parameters λ and η , problem ( DP λ,η ) hasat least three weak solutions. More precisely, we can state the following result. Theorem 5.1.
Assume that the following hypotheses hold: (C1)
The function f : V × R → R is continuous. (C2) The function F : V × R → R satisfies the following conditions: (1) There exist α ∈ [0 , , a ∈ L ( V, µ ) , and m ≥ such that F ( x, t ) ≤ m ( a ( x ) + | t | α ) , for all ( x, t ) ∈ V × R . (2) There exist t > , M ≥ and β > such that F ( x, t ) ≤ M | t | β , for all ( x, t ) ∈ V × [ − t , t ] . (3) There exists t ∈ R \ { } such that for all x ∈ V and for all t between and t we have F ( x, t ) > and F ( x, t ) ≥ . Then there exists a real number Λ ≥ such that, for each compact interval [ λ , λ ] ⊂ ]Λ , ∞ [ , there exists a positive real number r with the following property: For every λ ∈ [ λ , λ ] and every continuous function g : V × R → R there exists δ > suchthat, for each η ∈ [0 , δ ] , the problem ( DP λ,η ) has at least three solutions whosenorms are less than r .Proof. Set X := H ( V ). Then X is separable (by Remark 4.1) and reflexive (as aHilbert space). Define the functions Φ , J : X → R for every u ∈ X byΦ( u ) = 12 || u || , J ( u ) = Z V F ( x, u ( x )) dµ. In order to apply Theorem 1.1, we show that the conditions (i)–(v) required in thistheorem are satisfied for the above defined functions.Clearly condition (i) of Theorem 1.1 is satisfied. (Note that Φ ′ : X → X ∗ isdefined by Φ ′ ( u )( v ) = W ( u, v ) for every u, v ∈ X .) Condition (ii) is a consequenceof the facts that X is uniformly convex and that Φ is sequentially weakly lowersemicontinuous. Condition (iii) follows from assertions a) and b) of Proposition 4.5.Obviously condition (iv) holds for u = 0. B. E. BRECKNER, D. REPOVˇS, AND CS. VARGA
To verify (v), observe first that assumption (1) of ( C
2) implies, together with(3.3), that for every u ∈ X \ { } the following inequality holds: J ( u )Φ( u ) ≤ m || u || Z V adµ + 2 mc α || u || α − . Since α <
2, we conclude that(5.1) lim sup || u ||→∞ J ( u )Φ( u ) ≤ . Note that if u ∈ X is so that || u || ≤ t c , then, by (3.3), || u || sup ≤ t . It followsthat u ( x ) ∈ [ − t , t ] for every x ∈ V . Using (2) of ( C x ∈ V F ( x, u ( x )) ≤ M | u ( x ) | β ≤ M c β || u || β . Hence the following inequality holds for every u ∈ X \ { } with || u || ≤ t c J ( u )Φ( u ) ≤ M c β || u || β − . Since β >
2, we obtain(5.2) lim sup u → J ( u )Φ( u ) ≤ . The inequalities (5.1) and (5.2) yield that(5.3) ρ := max ( , lim sup || u ||→∞ J ( u )Φ( u ) , lim sup u → u J ( u )Φ( u ) ) = 0 . Without loss of generality we may assume that the real number t in condition (3)of ( C
2) is positive. Lemma 3.1 implies that | u | ∈ H ( V ) whenever u ∈ H ( V ). Thuswe can pick a function u ∈ H ( V ) such that u ( x ) ≥ x ∈ V , and such thatthere is an element x ∈ V with u ( x ) > t . It follows that U := { x ∈ V | u ( x ) > t } is a nonempty open subset of V . Let h : R → R be defined by h ( t ) = min { t, t } ,for every t ∈ R . Then h (0) = 0 and h is a Lipschitz map with Lipschitz constant L = 1. Lemma 3.1 yields that u := h ◦ u ∈ H ( V ). Moreover, u ( x ) = t for every x ∈ U , and 0 ≤ u ( x ) ≤ t for every x ∈ V . Then, according to condition (3) of( C F ( x, u ( x )) > , for every x ∈ U, and F ( x, u ( x )) ≥ , for every x ∈ V. Together with (2.1) we then conclude that J ( u ) >
0. Thus(5.4) ρ := sup u ∈ Φ − (]0 , ∞ [) J ( u )Φ( u ) > . Relations (5.3) and (5.4) finally imply that assertion (v) of Theorem 1.1 is alsofulfilled. Put Λ := ρ (with the convention ∞ := 0). Note that if g : V × R → R iscontinuous, then the map Ψ : X → R , defined byΨ( u ) = Z V G ( x, u ( x )) dµ, is, by the assertions a) and b) of Proposition 4.5, a C -functional with compactderivative. So, applying Theorem 1.1 and Proposition 4.6, we obtain the assertedconclusion. (cid:3) HE DIRICHLET PROBLEM ON THE SIERPINSKI GASKET 9
Example 5.2.
Let 0 < α < < β and define f : R → R by f ( t ) = (cid:26) | t | β − t, if | t | ≤ | t | α − t, if | t | > . Then F : R → R , F ( t ) = R t f ( ξ ) dξ , is given by F ( t ) = (cid:26) β | t | β , if | t | ≤ β − α + α | t | α , if | t | > . Consider a continuous map a : V → R with a ( x ) >
0, for every x ∈ V , and define f : V × R → R by f ( x, t ) = a ( x ) f ( t ). Then F : V × R → R , F ( x, t ) = R t f ( x, ξ ) dξ ,is given by F ( x, t ) = a ( x ) F ( t ). Hence F satisfies condition ( C
2) of Theorem 5.1.6.
A perturbed two-parameters Dirichlet problem
Now we study, in a particular case, a perturbed version of the two-parametersproblem ( DP λ,η ) of the previous section. More exactly, for fixed reals r, s, q with1 < r < s < < q and for the parameters λ, η ≥
0, consider the following Dirichletproblem on the SG:( P λ,η ) − ∆ u ( x ) = λ | u ( x ) | s − u ( x ) − η | u ( x ) | r − u ( x ) + | u ( x ) | q − u ( x ) , ∀ x ∈ V \ V ,u | V = 0 , where we put, by definition, | | ℓ · ℓ <
0. By Proposition 4.6 andRemark 4.7, the map I λ,η : H ( V ) → R , defined by(6.1) I λ,η ( u ) = 12 || u || − λs Z V | u | s dµ + ηr Z V | u | r dµ − q Z V | u | q dµ, is the energy functional attached to problem ( P λ,η ). The derivative of this map isgiven, for every u, v ∈ H ( V ), by(6.2) I ′ λ,η ( u )( v ) = W ( u, v ) − λ Z V | u | s − uvdµ + η Z V | u | r − uvdµ − Z V | u | q − uvdµ. For the sake of completeness we recall the two mountain pass theorems that willbe used to prove the main result of this section. The first one is the celebratedmountain pass theorem due to Ambrosetti and Rabinowitz (e.g., Theorem 2.2 in[14]):
Theorem 6.1.
Let X be a real Banach space and let I : X → R be a C -functionalsatisfying the Palais-Smale condition. Furthermore assume that I (0) = 0 and thatthe following conditions hold: (i) There are reals ρ, α > such that I | ∂B ρ ≥ α . (ii) There is an element e ∈ E \ B ρ such that I ( e ) ≤ .Then the real number κ , characterized as (6.3) κ := inf g ∈ Γ max t ∈ [0 , I ( g ( t )) , where Γ := { g : [0 , → X | g continuous , g (0) = 0 , g (1) = e } , is a critical value of I with κ ≥ α . The next result generalizes the above Theorem by weakening condition (i). Itgoes back to P. Pucci and J. Serrin, and can be found in [13].
Theorem 6.2.
Let X be a real Banach space and let I : X → R be a C -functionalsatisfying the Palais-Smale condition. Furthermore assume that I (0) = 0 and thatthe following conditions hold: (i) There exists a real number ρ > such that I | ∂B ρ ≥ . (ii) There is an element e ∈ E \ B ρ with I ( e ) ≤ .Then the real number κ defined in (6.3) is a critical value of I with κ ≥ . If κ = 0 ,there exists a critical point of I on ∂B ρ corresponding to the critical value . We also recall two standard results concerning the existence of minimum pointsof sequentially weakly lower semicontinuous functionals.
Proposition 6.3.
Let X be a reflexive real Banach space, M a bounded and se-quentially weakly closed subset of X , and f : M → R a sequentially weakly lowersemicontinuous functional. Then f possesses at least one minimum point. Proposition 6.4.
Let X be a reflexive real Banach space, M a sequentially weaklyclosed subset of X , and f : M → R a sequentially weakly lower semicontinuous andcoercive functional. Then f possesses at least one minimum point. We next establish some important properties of the energy functional I λ,η : H ( V ) → R attached to problem ( P λ,η ). Lemma 6.5.
Let λ, η ≥ . If ( u n ) is a sequence in H ( V ) such that both of thesequences ( I λ,η ( u n )) and ( I ′ λ,η ( u n )) are bounded, then ( u n ) is bounded, too.Proof. Let d be a real number such that I λ,η ( u n ) ≤ d and || I ′ λ,η ( u n ) || ≤ d for everyindex n . Relations (6.1) and (6.2) yield for every index nI λ,η ( u n ) = 12 || u n || − λs Z V | u n | s dµ + ηr Z V | u n | r dµ − q Z V | u n | q dµ = (cid:18) − q (cid:19) || u n || − λ (cid:18) s − q (cid:19) Z V | u n | s dµ + η (cid:18) r − q (cid:19) Z V | u n | r dµ + 1 q I ′ λ,η ( u n )( u n ) . Using (3.3), we get d ≥ I λ,η ( u n ) ≥ (cid:18) − q (cid:19) || u n || − λ (cid:18) s − q (cid:19) c s || u n || s − dq || u n || . Since lim t →∞ (cid:18)(cid:18) − q (cid:19) t − λ (cid:18) s − q (cid:19) c s t s − dq t (cid:19) = ∞ , we conclude that the sequence ( u n ) has to be bounded. (cid:3) Proposition 6.6.
Let λ, η ≥ . The energy functional I λ,η : H ( V ) → R attachedto problem ( P λ,η ) has the following properties: a) I λ,η is a C -functional. b) u ∈ H ( V ) is a solution of problem ( P λ,η ) if and only if u is a critical pointof I λ,η . c) I λ,η is sequentially weakly lower semicontinuous. d) I λ,η satisfies the Palais-Smale condition. e) 0 is a local minimum of I λ,η . HE DIRICHLET PROBLEM ON THE SIERPINSKI GASKET 11
Proof.
The assertions a) and b) follow from Proposition 4.6, while c) is a conse-quence of Corollary 4.8.d) Consider a sequence ( u n ) in H ( V ) such that ( I λ,η ( u n )) is bounded and suchthat ( I ′ ( u n )) converges to 0. By Lemma 6.5 we know that ( u n ) is bounded, thus, inview of Corollary 4.9, the sequence ( u n ) contains a convergent subsequence. Hence I λ,η satisfies the Palais-Smale condition.e) We know from (6.1) that for every u ∈ H ( V ) I λ,η ( u ) = 12 || u || + Z V (cid:18) ηr − λs | u | s − r − q | u | q − r (cid:19) | u | r dµ. Let h : R → R be defined by h ( t ) = ηr − λs | t | s − r − q | t | q − r . Since h (0) = ηr > h is continuous, there exists δ > h ( t ) > t ∈ ] − δ, δ [.Put r := δc . If u ∈ B r , then (3.3) implies || u || sup ≤ c || u || < δ , hence u ( x ) ∈ ] − δ, δ [for every x ∈ V . It follows that I λ,η ≥ I λ,η (0) , for every u ∈ B r , thus 0 is alocal minimum of I λ,η . (cid:3) Lemma 6.7. If λ > , then there exists a nonzero element u λ ∈ H ( V ) satisfyingthe following equality (6.4) || u λ || = λ Z V | u λ | s dµ. In particular, the inequality (6.5) || u λ || ≤ ( λc s ) − s holds.Proof. Let ψ λ : H ( V ) → R be defined by ψ λ ( u ) = || u || − λs R V | u | s dµ . By (3.3),the following relations hold for every u ∈ H ( V ) ψ λ ( u ) ≥ || u || − λs c s || u || s = || u || (cid:18) − λs c s || u || s − (cid:19) . Since s <
2, we have that lim || u ||→∞ ψ λ ( u ) = ∞ , i.e., ψ λ is coercive. From Corollary 4.8we know that ψ λ is sequentially weakly lower semicontinuous, thus, by Proposition6.4, ψ λ admits at least one global minimum which we denote by u λ . Since u λ is acritical point of ψ λ , we have that ψ ′ λ ( u λ )( v ) = 0, for every v ∈ H ( V ), i.e., in viewof Proposition 4.6, W ( u λ , v ) − λ Z V | u λ | s − u λ vdµ = 0 , for every v ∈ H ( V ) . If we take v = u λ in the above equality, we get (6.4). Inequality (3.3) then yields(6.5).We finally show that u λ is nonzero. For this fix an arbitrary nonzero element v ∈ H ( V ). According to (2.1), we have that R V | v | s dµ >
0. On the other hand, thefollowing equality holds for every t > ψ t ( tv ) = t s (cid:18) t − s || v || − λs Z V | v | s dµ (cid:19) . For t sufficiently small (more exactly, for 0 < t − s < λs || v || R V | v | s dµ ) we thus havethat ψ λ ( u λ ) ≤ ψ λ ( tv ) < ψ λ (0) , hence u λ is nonzero. (cid:3) Now we can state the main result of this section.
Theorem 6.8.
There exists a real number Λ > with the following property: Forevery λ ∈ ]0 , Λ[ there exists η λ > such that for each η ∈ [0 , η λ [ the problem ( P λ,η ) has at least three nonzero solutions.Proof. Let R := c − qq − , so c q R q = R . Put m := ( − q ) R . Obviously m > λ, η ≥ u ∈ H ( V ) with || u || = R we then have, according to(3.3) and (6.1),(6.6) I λ,η ( u ) ≥ R − λs c s R s − q c q R q = (cid:18) − q (cid:19) R − λs c s R s = 2 m − λs c s R s . Consider Λ := min (cid:26) msc s R s , R − s c s (cid:27) . Fix now an arbitrary λ ∈ ]0 , Λ[. From (6.6) we then get that(6.7) inf || u || = R I λ,η ( u ) > m, for every η ≥ . Also, by (6.5) and the choice of Λ, we have that(6.8) || u λ || < R. By (6.1) and (6.4), the following equality holds for every η ≥ I λ,η ( u λ ) = || u λ || (cid:18) − s (cid:19) + ηr Z V | u λ | r dµ − q Z v | u λ | q dµ. Since u λ is nonzero, (2.1) implies that R V | u λ | r dµ >
0. Put η ,λ := r (cid:16) || u λ || (cid:0) s − (cid:1) + q R v | u λ | q dµ (cid:17)R V | u λ | r dµ . Then η ,λ > I λ,η ( u λ ) < , for every η ∈ [0 , η ,λ [ . For η ≥ I λ,η ( tu λ ) = 12 t || u λ || − t s s || u λ || + ηt r r Z V | u λ | r dµ − t q q Z V | u λ | q dµ, for t ≥ . Since s < t ≤ t s s , for every t ∈ [0 , I λ,η ( tu λ ) ≤ ηt r r Z V | u λ | r dµ, for η ≥ , and t ∈ [0 , . Let η ,λ := mr R V | u λ | r dµ . Then η ,λ > I λ,η ( tu λ ) < m, for every η ∈ [0 , η ,λ [ and every t ∈ [0 , . Define η λ := min { η ,λ , η ,λ } and pick an arbitrary η ∈ [0 , η λ [. Relation (6.9) implies(6.12) I λ,η ( u λ ) < , HE DIRICHLET PROBLEM ON THE SIERPINSKI GASKET 13 while relation (6.11) yields(6.13) I λ,η ( tu λ ) < m, for every t ∈ [0 , . We next proceed in three steps to get three nonzero solutions of problem ( P λ,η ).From the assertions a) and d) of Proposition 6.6 we know that I λ,η is a C -functionalwhich satisfies the Palais-Smale condition. The first step : The closed ball B R is weakly closed (being convex and closed inthe strong topology), hence it is also sequentially weakly closed. By assertion c) ofProposition 6.6 the restriction I λ,η | B R is sequentially weakly lower semicontinuous.Proposition 6.3 implies then that I λ,η | B R has at least one minimum point e u . By(6.8) we know that u λ ∈ B R , thus I λ,η ( e u ) ≤ I λ,η ( u λ ). Since I λ,η ( u λ ) < I λ,η (0)(by inequality 6.12), we conclude that e u is nonzero. Also, in view of (6.7), we havethat e u ∈ B R . This shows that e u is a local minimum point, hence a critical point,of I λ,η . The second step : From assertion e) of Proposition 6.6 we get a positive real r < || u λ || such that 0 = I λ,η (0) ≤ I λ,η ( u ), for every u ∈ B r . From (6.12) we knowthat I λ,η ( u λ ) <
0, so Theorem 6.2 guarantees the existence of a nonzero criticalpoint e u of I λ,η such that I λ,η ( e u ) ≥ I λ,η ( e u ) = inf g ∈ Γ max t ∈ [0 , I λ,η ( g ( t )) , where Γ := { g : [0 , → H ( V ) | g continuous , g (0) = 0 , g (1) = u λ } . Choosing e g : [0 , → H ( V ), e g ( t ) = tu λ , we get in virtue of (6.13) I λ,η ( e u ) ≤ max t ∈ [0 , I λ,η ( tu λ ) < m. The third step : Since R V | u λ | q dµ >
0, relation (6.10) yields lim t →∞ I λ,η ( tu λ ) = −∞ .Thus there is a positive real t such that I λ,η ( tu λ ) < t || u λ || > R . Taking intoaccount (6.7), Theorem 6.1 implies the existence of a critical point e u of I λ,η suchthat I λ,η ( e u ) ≥ m .Thus e u , e u , and e u are nonzero critical points of I λ,η satisfying the inequalities I λ,η ( e u ) < ≤ I λ,η ( e u ) < m ≤ I λ,η ( e u ) . Hence e u , e u , e u are pairwise distinct. So assertion b) of Proposition 6.6 finallyimplies that problem ( P λ,η ) has at least three nonzero solutions. (cid:3) Remark 6.9.
As already mentioned in the introduction, in [1] there is investigatedthe analogous of problem ( P λ,η ) in case of the p -Laplacian. Since H ( V ) satisfies(3.4), our situation differs from that one in [1], where the corresponding result toTheorem 6.8 has been obtained only for subcritical values of q , while for the criticalvalue or for supercritical values of q one can guarantee only the existence of at leasttwo nonzero solutions (see Theorem 1, respectively, Theorem 2 in [1]). Note thatour Theorem 6.8 holds for every q > Acknowledgements.
Varga was fully supported by the grant CNCSIS PCCE-55/2008 “Sisteme diferent¸iale ˆın analiza neliniar˘a ¸si aplicat¸ii. Breckner was sup-ported by the grant CNMP PN-II-P4-11-020/2007. Repovˇs was supported by SRAgrants P1-0292-0101, J1-2057-0101 and J1-4144-0101.
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