Algebraic topological techniques for elliptic problems involving fractional Laplacian
aa r X i v : . [ m a t h . A P ] F e b Algebraic topological techniques for elliptic problems involving fractionalLaplacian
A. Panda , D. Choudhuri ∗ , A. Bahrouni Abstract
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the tech-niques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
Keywords : Fractional Laplacian, Fractional Sobolev Space, homology group.
AMS Classification : 35R11, 35J75, 35J60, 46E35.
1. Introduction
We propose to study the following singular problem with a mixed operator: a ( − ∆) s u + b ( − ∆) u = λ | u | − γ − u + µ | u | ∗ s − u, in Ω ,u = 0 , in R N \ Ω , (1.1)where Ω ⊂ R N ( N ≥
2) is a bounded domain, a, b ≥
0, 0 < s < < < ∗ s < ∗ , λ > µ ∈ R , γ ∈ (0 , − ∆) s u ( x ) = P.V. ˆ R N ( u ( x ) − u ( y )) | x − y | N +2 s dy, ∀ x ∈ Ω , is the fractional Laplacian. We refer the operator “ a ( − ∆) s + b ( − ∆)” as a mixed operator since itpossesses both local as well as nonlocal features, and thus we refer the problems of kind (1.1) as“nonlocal-local” elliptic problems. For a detailed study on this operator refer [6] and [14]. The mixedoperators with different order are nowadays gaining popularity in applied sciences, in theoreticalstudies and also in real world applications. The development of the literature includes viscositysolution methods [4], Cahn-Hilliard equations [9], Aubry-Mather theory [12], phase transitions[8], probability and stochastics [29], fractional damping effects [13], decay estimates for parabolicequations [15], population dynamics [14], Bernstein-type regularity results [7].We begin by considering the following two spaces X and Y which are the closures of C ∞ c (Ω) in H s (Ω) ∗ Corresponding author
Email addresses: [email protected] (A. Panda), [email protected] (D. Choudhuri), [email protected] (A. Bahrouni) Department of Mathematics, National Institute of Technology Rourkela University of Monastir
Preprint submitted to Elsevier February 25, 2021 nd H (Ω), respectively (refer Section 2 for these notations). Let Q = R N \ (( R N \ Ω) × ( R N \ Ω))and define the spaces as follows: X = (cid:26) u : Ω → R : u is measurable and ¨ Q | u ( x ) − u ( y ) | | x − y | N +2 s dxdy < ∞ (cid:27) Y = (cid:8) u ∈ L (Ω) : |∇ u | ∈ L (Ω) (cid:9) . The spaces
X, Y are Banach spaces with respect to the following norms: k u k X = k u k + (cid:18) ¨ Q | u ( x ) − u ( y ) | | x − y | N +2 s dydx (cid:19) = k u k + [ u ] s, , (1.2)and k u k Y = k u k + (cid:18) ˆ Ω |∇ u | dx (cid:19) = k u k + k u k , , (1.3)respectively. The norm [ · ] defined in (1.2) is the Gagliardo norm. Note that here k u k α = ( ´ Ω | u | α dx ) α for 1 < α < ∞ .The study of the nonlocal-local elliptic problem in (1.1), firstly directed our attention to fix a func-tion space in which the solution(s) will be seeked for. Define the space Z = { u : a [ u ] s, + b k u k , < ∞} equipped with the following norm: k u k = ( a [ u ] s, + b k u k , ) . (1.4)It is easy to see that the space Z is a Banach space and also reflexive with respect to the norm k · k ,defined in (1.4).The use of algebraic topological techniques to study problems having a singular nonlinearity is ararity in the literature. Thus, the problem discussed here is new as the consideration of a singularitywith a critical exponent with µ ∈ R handled with Morse theoretic approach is not found anywherein the literature to our knowledge. The question about the existence and multiplicity of positiveweak solutions to problem (1.1) with µ > a = 0 or b = 0 has been answered in[17, 18, 19, 22, 23, 31] and the references therein. The authors of these works followed different toolssuch as variational method, concentration compactness method, Nehari manifold method etc. butnone of them used the techniques from algebraic topology to study (1.1) with a, b > µ ∈ R .With the help of algebraic topological techniques, the existence and multiplicity results for thefollowing problem have been established by many researchers under different growth conditions onthe reaction term f (either subcritical or critical growth conditions): L u = f ( x, u ) , in Ω ,u = 0 , on ∂ Ω . (1.5)Problems of type (1.5) were treated recently by Iannizzotto et al. in [24] (finite multiplicity with L being the fractional p -Laplacian), Ferrara et al. in [16] (at least one non-trivial solution with L L being the p ( x )-Laplacian) and the references therein. The paper by Papageorgiou & R˘adulescu[32], dealt with a nonlinear Robin problem and proved the multiplicity by producing three nontrivialsolutions. The techniques thus differed from problem-to-problem addressed.The double phase problems of type (1.5) with L being a ( p, q )-Laplacian or a fractional ( p, q )-Laplacian have been widely studied by many authors with different techniques. For instance, when L = ( − ∆ p − ∆ q ) with p, q >
1, Gongbao & Gao [21], Yin & Yang [36] used variational method, Lianget al. [27] used Morse theoretical technique for the case q = 2 = p , and Marano et al. [28], Mugani& Papageorgiou [30] used the variational method with Morse theory and truncation comparisontechniques. When L = (( − ∆ p ) s + ( − ∆ q ) s ) with p, q > s ∈ (0 , p − p − p, q )-Laplacian. For more details on doublephase problems one can refer the recent piece of works by A. Bahrouni, V. D. Rˇadulescu & D. D.Repovˇs [2, 3] and the bibliography therein.Motivated by the former works, in this article, we study the singular problem (1.1) using variationaltechniques and algebraic topological methods, specifically the Morse theory and the critical groups(refer Section 2). We establish the existence of infinitely many solutions to (1.1) in Section 3followed by two subsections. Subsection 3 . .
2. Mathematical preliminaries
A quintessential condition which the functional requires to satisfy is the the Palais-Smale condition(denoted by (
P S )-condition) which is as follows.
Definition 2.1.
Let X be a Banach space, and I : X → R be a C ( X, R ) functional. Given c ∈ R we say that the functional I satisfies the Palais-Smale condition (or the ( P S ) c -condition) at level c if any bounded sequence ( u n ) ⊂ X such that I ( u n ) → c , and I ′ ( u n ) → n → ∞ has a convergentsubsequence in X .Below are the Sobolev embedding results that will be used throughout the article. Theorem 2.2. [[26]] Let Ω ⊂ R N be a bounded domain, < s < , and s < N . Further, assumethat r ≤ ∗ s = NN − s . Then, there exists C = C ( r, s, N, Ω) > such that k u k L r (Ω) ≤ C k u k X , ∀ u ∈ X. Moreover, this embedding is continuous for any r ∈ [1 , ∗ s ] , and compact for any r ∈ [1 , ∗ s ) . Theabove embedding holds also for Z . Theorem 2.3 ([26]) . Let Ω ⊂ R N be a bounded domain, and N > . Then, for every ¯ r ≤ ∗ = NN − there exists ¯ C = ¯ C ( r, N, Ω) > such that k u k L ¯ r (Ω) ≤ ¯ C k u k Y , ∀ u ∈ Y. oreover, this embedding is continuous for any r ∈ [1 , ∗ ] , and compact for any r ∈ [1 , ∗ ) . Theabove embedding holds also for Z . We now present the fundamental tool that will be used to work with, namely the homology theory [33], which will be followed by the definition of deformation (see [33]).
Definition 2.4.
A “homology theory” on a family of pairs of spaces (
X, A ) consists of:1. A sequence { H k ( X, A ) } k ∈ N of abelian groups known as “homology group” for the pair ( X, A )(note that for the pair (
X, φ ), we write H k ( X ) , k ∈ N ).2. To every map of pairs ϕ : ( X, A ) → ( Y, B ) is associated a homomorphism ϕ ∗ : H k ( X, A ) → H k ( Y, B ) for all k ∈ N .3. To every k ∈ N and every pair ( X, A ) is associated a homomorphism ∂ : H k ( X, A ) → H k − ( A )for all k ∈ N .Here, N = N ∪ { } . These objects satisfy the following axioms: ( A ) If ϕ = id X , then ϕ ∗ = id | H k ( X,A ) . ( A ) If ϕ : ( X, A ) → ( Y, B ), and ψ : ( Y, B ) → ( Z, C ) are maps of pairs, then ( ψ ◦ ϕ ) ∗ = ψ ∗ ◦ ϕ ∗ . ( A ) If ϕ : ( X, A ) → ( Y, B ) is a map of pairs, then ∂ ◦ ϕ ∗ = ( ϕ | A ) ∗ ◦ ∂ . ( A ) If i : A → X and j : ( X, φ ) → ( X, A ) are inclusion maps, then the following sequence is exact ... ∂ −→ H k ( A ) i ∗ −→ H k ( X ) j ∗ −→ H k ( X, A ) ∂ −→ H k − ( A ) → ... Recall that a chain ... ∂ K +1 −−−→ C k ( X ) ∂ k −→ C K − ( X ) ∂ k − −−→ C k − ( X ) ∂ k − −−→ ... is said to be exact if im ( ∂ k +1 ) = ker ( ∂ k ) for each k ∈ N . ( A ) If ϕ, ψ : ( X, A ) → ( Y, B ) are homotopic maps of pairs, then ϕ ∗ = ψ ∗ . ( A ) (Excision): If U ⊆ X is an open set with ¯ U ⊆ int( A ), and i : ( X \ U, A \ U ) → ( X, A ) is theinclusion map, then i ∗ : H k ( X \ U, A \ U ) → H k ( X, A ) is an isomorphism. ( A ) If X = {∗} , then H k ( ∗ ) = 0 for all k ∈ N . Definition 2.5.
A continuous map F : X × [0 , → X is a deformation retraction of a space X onto a subspace A if, for every x ∈ X and a ∈ A , F ( x,
0) = x , F ( x, ∈ A , and F ( a,
1) = a .An important result in Morse theory is stated below. Theorem 2.6.
Let I ∈ C ( X ) satisfy the Palais-Smale condition, and let ‘ a ’ be a regular value of I . Then, H ∗ ( X, I a ) = 0 , implies that K I ∩ I a = ∅ where K I = { u ∈ X : I ′ ( u ) = 0 } , and I a = { u ∈ X : I ( u ) ≤ a } . emark . Another notation which will be used in the article is K I,D = { u ∈ X : I ( u ) ∈ D } where D is a connected subset of R . Remark . Prior to applying the Morse lemma we recall that for a Morse function the followingholds:1. H ∗ ( I c , I c \ Crit(
I, c )) = ⊕ j H ∗ ( I c ∩ N j , ( I c \ { x j } ) ∩ N j ) . H k ( I c ∩ N, I c \ { x } ∩ N ) = ( R , k = m ( x )0 , otherwisewhere m ( x ) is a Morse index of x , and x is a critical point of f .3. Further H k ( I a , I b ) = ⊕ { i : m ( x i )= k } Z = Z m k ( a,b ) where m k ( a, b ) = n ( { i : m ( x i ) = k, x i ∈ K I, ( a,b ) } ). Here, n ( S ) is the number of elements inthe set S .4. Morse relation X u ∈ K J, [ a,b ] X k ≥ dimC k ( I, u ) t k = X k ≥ dimH k ( J a , J b ) t k + (1 + t ) Q t for all t ∈ R .In this paper, we use the notion of local ( m, n )-linking ( m, n ∈ N ) (see Definition 2.3, [34]) to provethe existence of solution. Definition 2.9.
Let W be a Banach space, I ∈ C ( W, R ), and 0 an isolated critical point of I with I (0) = 0. Further, assume that m, n ∈ N . We say that I has a “local ( m, n )-linking” near theorigin if there exist a neighborhood U of 0, E = ∅ , E ⊆ U , and D ⊆ W such that 0 / ∈ E ⊆ E , E ∩ D = ∅ , and1. 0 is the only critical point of I in I ∩ U , where I = { u ∈ W : I ( u ) ≤ } ,2. Dim im ( i ∗ ) − Dim im ( j ∗ ) ≥ n , where i ∗ : H m − ( E ) → H m − ( W \ D ) and j ∗ : H m − ( E ) → H m − ( E )are the homomorphisms induced by the inclusion maps i : E → W \ D and j : E → E ,3. I | E ≤ ≤ I | U ∩ D \{ } . 5 . Proof of the main results Let F ( x, t ) = ´ t f ( s, x ) ds be the primitive of f ( x, t ) = λ | t | − γ − t + µ | t | ∗ s − t .We say u ∈ Z to be a weak solution of (1.1) if for every ϕ ∈ Z , we have a ¨ Q ( u ( x ) − u ( y )) | x − y | N +2 s ( ϕ ( x ) − ϕ ( y )) dxdy + b ˆ Ω ∇ u · ∇ ϕdx = ˆ Ω λ | u | − γ − uϕ ( x ) dx + ˆ Ω µ | u | ∗ s − uϕ ( x ) dx. Clearly, a weak solution to problem (1.1) is a critical point of the corresponding energy functional I ( u ) = a ¨ Q | u ( x ) − u ( y ) | | x − y | N +2 s dxdy + b ˆ Ω |∇ u | dx − ˆ Ω F ( x, u ) dx. (3.1)However, it is easy to see that the functional I is not C ( Z ) due to the presence of the singularterm. Therefore, instead of working with the original functional I , we will use a cut-off functional,¯ I . (1 , linking at 0 We first define the functional ¯ I . Let us consider the following cut-off function. ξ ( t ) = , if | t | ≤ lξ is decreassing, if l ≤ t ≤ l , if | t | ≥ l. Let us now consider the following cut-off problem: a ( − ∆) s u − b ∆ u = λ u | u | γ +1 + ˜ g ( u ) in Ω ,u = 0 in R N \ Ω , (3.2)where, ˜ g ( u ) = µ | u | ∗ s − uξ ( k u k ) . Let ˜ F ( x, t ) = ´ t ˜ f ( s, x ) ds be the primitive of ˜ f ( x, t ) = λ | t | − γ − t + ˜ g ( t ). We say ˜ u ∈ Z to be a weaksolution of (3.2) if for every ϕ ∈ Z , we have a ¨ Q (˜ u ( x ) − ˜ u ( y )) | x − y | N +2 s ( ϕ ( x ) − ϕ ( y )) dxdy + b ˆ Ω ∇ ˜ u · ∇ ϕdx = ˆ Ω λ | ˜ u | − γ − ˜ uϕ ( x ) dx + ˆ Ω µξ ( k ˜ u k ) | ˜ u | ∗ s − ˜ uϕ ( x ) dx. Consequently, a weak solution to problem (3.2) is a critical point of the corresponding energyfunctional ¯ I ( u ) = a ¨ Q | u ( x ) − u ( y ) | | x − y | N +2 s dxdy + b ˆ Ω |∇ u | dx − ˆ Ω ˜ F ( x, u ) dx. (3.3)6 emark . One can easily see that if u is a weak solution to (3.2) with k u k ≤ l , then u is also aweak solution to (1.1). Remark . It is easy to observe that I ( u ) = I ( | u | ). Hence, if a solution exists it has to benonnegative. Furthermore, due to the presence of the singular term the solution is forced to bepositive a.e. in Ω.We first verify that that functional ¯ I satisfies the (PS)- condition. Let us denote S = inf u ∈ X \{ } k u k X k u k L ∗ s (Ω) (3.4)which is the best Sobolev constant in the Sobolev embedding (Theorem 2 . Theorem 3.3.
The functional ¯ I satisfies the ( P S ) c -condition for c < c ∗ = (cid:18) − ∗ s (cid:19) µ − ∗ s − S ∗ s ∗ s − − (cid:18) − ∗ s (cid:19) − γ − γ +1 (cid:20) | Ω | ∗ s − γ ∗ s S − − γ λ − γ (cid:21) γ , where c ∗ > for sufficiently small λ, | µ | > .Proof. We see that although the functional has been modified with a cut-off, yet the functional isnot C which is not in the premise of the Palais-Smale condition. However, we will devise a schemeto tackle this situation by picking the sequence in a way that the singularity at 0 gets avoided sincethe functional is C in Z \ { } . Suppose ( u n ) ⊂ Z is an eventually zero sequence, then apparentlyit converges to 0, and we discard the sequence immediately. Suppose, ( u n ) ⊂ Z is a sequence withinfinitely many terms of the sequence equal to 0, then we will hand-pick a subsequence of ( u n ) withall non-zero terms. Thus, without loss of generality we will let ( u n ) such that u n = 0 for every n ∈ N . Let this sequence ( u n ) be such that¯ I ( u n ) → c, and ¯ I ′ ( u n ) → n → ∞ . Therefore, let us consider a sequence ( u n ) ⊂ Z such that ¯ I ( u n ) → c for some c ∈ R ,and ¯ I ′ ( u n ) → n → ∞ . It is not difficult to see that the subsequence is bounded in Z . This hasthe following consequences: u n ⇀ u in Z ; k u n k → M,u n → u in L r (Ω) for any 1 ≤ r < ∗ s ,u n → u = 0 a.e. in Ω , (3.6)as n → ∞ . Now, consider o (1) = h ¯ I ′ ( u n ) , u n − u i =( k u n k − k u k ) − λ ˆ Ω | u n | − γ − u n ( u n − u ) dx − µ ˆ Ω ξ ( k u n k ) | u n | ∗ s − u n ( u n − u ) dx =( k u n k − k u k ) − µξ ( M ) ˆ Ω | u n | ∗ s dx + µξ ( M ) ˆ Ω | u | ∗ s dx + o (1)= k u n − u k − µξ ( M ) ˆ Ω | u n − u | ∗ s dx + o (1) . (3.7)7e obtain lim n →∞ k u n − u k = µξ ( M ) lim n →∞ ˆ Ω | u n − u | ∗ s dx = µξ ( M ) N ∗ s . (3.8)If µ ≤
0, we produce a contradiction from (3.8), and we guarantee that u n → u strongly in Z .Therefore, we now proceed for µ > N = 0, then we obtain u n → u in Z as n → ∞ , since M >
0. Therefore, we will showthat N = 0. On the contrary let us assume that N >
0. Thus, we get0 ≤ lim n →∞ k u n − u k = µξ ( M ) N ∗ s . (3.9)Next, from (3.6), (3.9), and (3.4) we have SN ≤ µξ ( M ) N ∗ s ≤ µN ∗ s M − k u k = µξ ( M ) N ∗ s . (3.10)It also follows from (3.10) that N ≥ (cid:18) Sµ (cid:19) ∗ s − , (3.11)and M ≥ SN ≥ µ − ∗ s − S ( S ∗ s − )= µ − ∗ s − S ∗ s ∗ s − . (3.12)Consider c + o (1) = ¯ I ( u n ) − ∗ s h ¯ I ′ ( u n ) , u n i = a (cid:18) − ∗ s (cid:19) [ u n ] s, + b (cid:18) − ∗ s (cid:19) k∇ u n k + λ (cid:18) ∗ s − − γ (cid:19) ˆ Ω | u n | − γ dx ≥ (cid:18) − ∗ s (cid:19) k u n k − λ − γ ˆ Ω | u n | − γ dx. (3.13)Therefore, by the Brezis-Lieb theorem, the Young’s inequality, and the results derived in this8heorem, we pass the limit n → ∞ in (3.13) to obtain the following: c ≥ (cid:18) − ∗ s (cid:19) ( M + k u k ) − λ − γ ˆ Ω | u | − γ dx ≥ (cid:18) − ∗ s (cid:19) ( M + k u k ) − | Ω | ∗ s − γ ∗ s S − − γ λ − γ k u k − γ ≥ (cid:18) − ∗ s (cid:19) ( M + k u k ) − (cid:18) − ∗ s (cid:19) k u k − (cid:18) − ∗ s (cid:19) − γ − γ +1 (cid:20) | Ω | ∗ s − γ ∗ s S − − γ λ − γ (cid:21) γ ≥ (cid:18) − ∗ s (cid:19) M − (cid:18) − ∗ s (cid:19) − γ − γ +1 (cid:20) | Ω | ∗ s − γ ∗ s S − − γ λ − γ (cid:21) γ ≥ (cid:18) − ∗ s (cid:19) µ − ∗ s − S ∗ s ∗ s − − (cid:18) − ∗ s (cid:19) − γ − γ +1 (cid:20) | Ω | ∗ s − γ ∗ s S − − γ λ − γ (cid:21) γ = c ∗ (3.14)for sufficiently small λ > | µ | >
0. This is a contradiction.
Theorem 3.4.
The functional ¯ I has a local (1 , - linking at the origin.Proof. We define V = R . Clearly V is a one dimensional vector subspace of Z . We now choose ν ∈ (0 ,
1) small enough so that K ¯ I ∩ B ν (0) = { } where B ν (0) = { u ∈ Z : k u k < ν } , and K ¯ I = { u ∈ Z : ¯ I ′ ( u ) = 0 } . Define E = V ∩ B ν (0)for small enough ν ∈ (0 , E . Therefore, using thisto our advantage, for a sufficiently small ν > δ >
0, we have k u k ≤ ν ⇒ k u k ∞ ≤ δ for all u ∈ E. Therefore,¯ I ( u ) = a ¨ Q | u ( x ) − u ( y ) | | x − y | N +2 s dxdy + b ˆ Ω |∇ u | dx − λ − γ ˆ Ω | u | − γ dx − µ ∗ s ˆ Ω ξ ( k u k ) | u | ∗ s dx = 12 k u k − λ − γ ˆ Ω | u | − γ dx − µ ∗ s ˆ Ω ξ ( k u k ) | u | ∗ s dx ≤ k u k − λ − γ C k u k − γ − µ ∗ s C ′ k u k ∗ s ≤ E being in a finite dimensional space V . Further,define D ′ = (cid:26) u ∈ Z : k u k ≥ λ − γ k u k − γ − γ (cid:27) . u ∈ D ′ ,¯ I ( u ) = a ¨ Q | u ( x ) − u ( y ) | | x − y | N +2 s dxdy + b ˆ Ω |∇ u | dx − λ − γ ˆ Ω | u | − γ dx − µ ∗ s ˆ Ω ξ ( k u k ) | u | ∗ s dx = 12 k u k − λ − γ ˆ Ω | u | − γ dx − µ ∗ s ˆ Ω | u | ∗ s dx ≥ k u k − | µ | ∗ s C k u k ∗ s > k u k < ν . This holds for all D = D ′ ∩ ( B ν (0) \ { } ). Define E = V ∩ ∂B ν (0) . Clearly E ∩ D = ∅ . Therefore, we arrive at the following I | E ≤ < I | D . Further, define h : [0 , × Z \ D ′ → Z \ D ′ as h ( t, u ) = (1 − t ) u + t · ν v k v k where v = α ∈ V . Clearly 0 ∈ D ′ which makes the definition of h ( · , · ) valid. Observe that h (0 , u ) = u, in Z \ D ′ h (1 , u ) = ν v k v k , in V ∩ ∂B ν (0) = E . (3.17)Thus, E is a retract of Z \ D ′ . Therefore, i ∗ : H ( E ) → H ( Z \ D )is an isomorphism. Note that E = { v, − v } for some v = 0. Therefore, from DimH ( E ) = 2 since H ( E ) = R ⊕ R . Thus, Dim im ( i ∗ ) = 2.Further, E is an interval [ − α, α ] which is contractible to a point. Thus, H ( E ) = R . Hence, if j ∗ : H ( E ) → H ( E ) , then Dim im ( i ∗ ) − Dim im ( j ∗ ) = 2 − ,
1) linking at 0.
Proposition 3.5. C k ( ¯ I, ∞ ) = 0 for all k ∈ N . roof. Let t >
0, and consider ddt ¯ I ( tu ) = h ¯ I ′ ( tu ) , u i = 1 t h ¯ I ′ ( tu ) , tu i = 1 t (cid:20) a ¨ Q | tu ( x ) − tu ( y ) | | x − y | N +2 s dxdy + b ˆ Ω |∇ tu | dx − ˆ Ω ˜ f ( x, tu ) tudx (cid:21) = 1 t (cid:20) a ¨ Q | tu ( x ) − tu ( y ) | | x − y | N +2 s dxdy + b ˆ Ω |∇ tu | dx − λ ˆ Ω | tu | − γ dx − µ ˆ Ω ξ ( k tu k ) | tu | ∗ s dx (cid:21) (3.18)Observe that for large t > I ( tu ) ≥ J > c , for some c >
0, since ¯ I ( tu ) → ∞ as t → ∞ .Therefore, ddt ¯ I ( tu ) > t >
0. Thus, there exists a unique h ( u ) > I ( h ( u ) u ) = J . Thisactually implies that h ∈ C ( ∂B ). On extending h ( · ) to Z \ { } by defining˜ h ( u ) = 1 k u k h (cid:18) u k u k (cid:19) for all u ∈ Z \ { } , we obtain ˜ h ∈ Z \ { } , and ¯ I (˜ h ( u ) u ) = J . Also if ¯ I ( u ) = J then ˜ h ( u ) = 1.Therefore, we define ˆ h ( u ) = (cid:26) , if ¯ I ( u ) ≥ J ˜ h ( u ) , if ¯ I ( u ) ≤ J which makes ˆ h continuous. Further define g ( t, u ) = (1 − s ) u + s ˆ h ( u ) u for all ( t, u ) ∈ [0 , × ( Z \ { } ). Thus, we have g (0 , u ) = u, g (1 , u ) = ˜ h ( u ) u ∈ ¯ I J , and g ( t, . ) | ¯ I J = id | ¯ I J for all t ∈ [0 , I J is a deformation of Z \ { } . By standarddefinition of a homotopy, it is easy to see that ∂B = { u ∈ ¯ Z : k u k = 1 } is a deformation of Z \ { } .Thus, on choosing J sufficiently negative we get C k ( ¯ I, ∞ ) = H k ( ¯ Z, ∂B ) = 0for all k ∈ N . 11 emark . We use a variant of the result - If X is a Banach space, ¯ I ∈ C ( X ) , ∈ K I is isolated,and ¯ I has a local ( m, n ) -linking near the origin, then rank C m ( ¯ I, ≥ n . Since we obtained a (1 , C ( ¯ I, ≥
1. In our case 0 is not a critical point of ¯ I , however onecan still construct C ( ¯ I,
0) owing to not only the fact that ¯ I is well-defined at 0 but also having a‘ sharp ’ isolated non regularity there. Theorem 3.7.
There exists a solution, say u , to the problem (3.2) .Proof. As seen in the Remark 3 . C ( ¯ I, ≥
1. In tandem with this andProposition 3 .
5, we are in a position to apply the Proposition 6 . .
42 (refer Theorem 4 . u ∈ Z such that u ∈ K ¯ I \ { } , which further implies that u ∈ Z ∩ L ∞ (Ω) is a solution of (3.2). For the boundedness of u , pleaserefer the Theorem 4 . Theorem 3.8.
The problem (3.2) has at least two nontrivial solutions in Z ∩ L ∞ (Ω) . Proof.
From the Theorem 3 . H k ( Z, ¯ I − a ) for all k ≥
0. Pick a u ∈ { v ∈ Z : k v k = 1 } = ∂B ∞ , where B ∞ = { v ∈ Z : k v k ≤ } .We make use of the equation (3.16) which explains that for small enough t > I ( tu ) > I being C in Z \ { } indicates that ¯ I ′ ( tu ) > t >
0. Also from(3.19) and in combination with the fact that the functional is superlinear, we have that for largeenough t >
0, ¯ I ′ ( tu ) <
0. Thus, there exists a unique t ( u ) such that ¯ I ′ ( t ( u ) u ) = 0 since due to ourassumption that there exists exactly one nontrivial solution. We can thus say that there exists a C -function T : Z \ { } → R + defined by u t ( u ). We now define a standard deformation retract η of Z \ B r (0) into ¯ I − a as follows (refer Definition 2.5). η ( s, u ) = ( (1 − s ) u + sT (cid:16) u k u k (cid:17) u k u k , k u k ≥ ν, ¯ I ( u ) ≥ − au, ¯ I ( u ) ≤ − a. It is not difficult to see that η is a C function over [0 , × Z \ B r (0). On using the map δ ( s, u ) = u k u k ,for u ∈ Z \ B r (0) we claim that H k ( Z, Z \ B r (0)) = H k ( B ∞ , S ∞ ) for all k ≥
0. This is because, H k ( B ∞ , S ∞ ) ∼ = H k ( ∗ , H k ( ∗ ,
0) = { } for each k ≥
0. A result in [35] tells us that C n ( ¯ I,
0) = ( R , if m (0) = N , otherwise . Clearly, m (0) ≥
2. Therefore, from the Morse relation in the Remark 2 . − a > X u ∈ K ¯ I, [ − a, ∞ ) X k ≥ dimC k ( ¯ I, u ) t k = t m (0) + p ( t ) (3.20)12here m (0) is the Morse index of 0, and p ( t ) contains the rest of the powers of t corresponding tothe other critical points, if any. On further using the the Morse relation we obtain t m (0) + p ( t ) = (1 + t ) Q t . (3.21)This is because the H k s are all trivial groups. Hence, Q t either contains t m (0) or t m (0) − or both.Thus, there exists at least two nontrivial u ∈ K ¯ I, [ − a, ∞ ) with 2 ≤ m (0) ≤ N + 1. Theorem 3.9.
Let Ω be as above.Then, ¯ I has infinitely many critical points in Z .Proof. We appeal to the Morse theory again from which we obtain the following. X u ∈ K ¯ I, [ − a, ∞ ) X k ≥ dimC k ( ¯ I, u ) t k = t m (0) + 2 X k ≥ α k t k . (3.22)The factor 2 is due to the fact that if u is a critical point then ( − u ) is also a critical point. α k s arenonnegative integers. As in the proof of the Theorem 3 .
8, we have H k ( Z, ¯ I ∞− a ) = 0. Therefore, wehave the following identity over R . t m (0) + 2 X k ≥ α k t k = (1 + t ) Q t . (3.23)In particular for t = 1 we have 1 + 2 A = 2 B , where the series on the left of the identity in (3.23)for t = 1 is denoted by A , and the term on the right of the same for t = 1 is denoted by B . Thisis possible only when the sum is infinite as finite sum leads to a contradiction that there exists anodd and an even number that agree. Thus, there exists infinitely many solutions.The following is the main existence result for the problem (1.1). Theorem 3.10.
The problem (1.1) admits infinitely many nontrivial solutions in Z ∩ L ∞ (Ω) .Proof. According to Theorem 3 . − .
9, the problem (3.2) has infinitely many solutions in Z ∩ L ∞ (Ω).With a suitable choice of l and by Remark 3 .
2, we conclude our proof.
4. AppendixTheorem 4.1. (Theorem . . of Papagiorgiou et al. [33]) If X is a Banach space, I ∈ C ( X ) , I satisfies the Ce -condition, K I is finite with ∈ K I , and for some k ∈ N we have C k ( I, = 0 , C k ( I, ∞ ) = 0 , then there exists a u ∈ K I such that ¯( u ) < , C k − ( I, u ) = 0 or I ( u ) > , and C k +1 ( I, u ) = 0 . Theorem 4.2.
Any solution to (3.2) is in L ∞ (Ω) .Proof. The argument sketched here is a standard one, and hence we shall only show that an im-provement in integrability is possible up to L ∞ assuming an integrability of certain order, say p > bootstrap argument. Without loss of generality we can considerthe set Ω ′ = { x ∈ Ω : u ( x ) > } , and thus by the positivity of a fixed solution (refer Remark 3 . u we have u = u + > u ∈ L p (Ω) for p >
1. Let a = max { λ, | µ |} . On testingwith u p to obtain the following: C k u p +12 k ∗ s ≤ k u p +12 k≤ (cid:18) λ ˆ Ω ′ | u | p − γ dx + µ ˆ Ω ′ | u | ∗ s − p dx (cid:19) ( p + 1) p ≤ a (cid:18) ˆ Ω ′ | u | p (1 + | u | ∗ s − ) dx (cid:19) ( p + 1) p ; since in Ω ′ we have u > ≤ a (cid:18) ˆ Ω ′ | u | p | u | ∗ s dx (cid:19) ( p + 1) p ≤ aC ′′ k u k ∗ s β ∗ k u p k t ; since by using H¨older’s inequality . (4.1)Here, t = β ∗ β ∗ − ∗ s for some β ∗ > t ∗ = tNN − ts < ∗ s . Thus, we also have C ′ k u p k β ∗ ≤ C ′ k u p +12 k β ∗ ≤ k u p +12 k ∗ s . (4.2)From the story so far, we know the following C ′ k u p k β ∗ ≤ aC ′′ k u k ∗ s β ∗ k u p k t . (4.3)For the fixed β ∗ >
1, we set η = β ∗ t > t , and τ = t p to get k u k ητ ≤ C t τ k u k τ ; where C = 2 aC ′′ k u k α + β ∗ is a fixed quantity for a fixed solution u. (4.4)(4.5)Let us now iterate with τ = t , τ n +1 = ητ n = η n +1 t . After n iterations, the inequality (4.4) yields k u k τ n +1 ≤ C n P i =0 t τi n Y i =0 (cid:16) τ i t (cid:17) t τi k u k t . (4.6)By using η > τ = t , τ n +1 = ητ n = η n +1 t , we have ∞ X i =0 t τ i = ∞ X i =0 η i = ηη − , and ∞ Y i =0 (cid:16) τ i t (cid:17) t τi = η η η − . Hence, on passing the limit n → ∞ in (4.6), we end up getting k u k ∞ ≤ C ηη − η η η − k u k t . (4.7)Thus, u ∈ L ∞ (Ω). 14 . ReferencesReferences [1] V. Ambrosio and T. Isernia. On a fractional p & q Laplacian problem with critical Sobolev-Hardyexponents,
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