An abstract critical point theorem with applications to elliptic problems with combined nonlinearities
aa r X i v : . [ m a t h . A P ] F e b An abstract critical point theorem withapplications to elliptic problems withcombined nonlinearities ∗ Kanishka Perera
Department of Mathematical SciencesFlorida Institute of TechnologyMelbourne, FL 32901, USA kperera@fit.edu
Abstract
We prove an abstract critical point theorem based on a cohomological index theorythat produces pairs of nontrivial critical points with nontrivial higher critical groups.This theorem yields pairs of nontrivial solutions that are neither local minimizers nor ofmountain pass type for problems with combined nonlinearities. Applications are givento subcritical and critical p -Laplacian problems, Kirchhoff type nonlocal problems, andcritical fractional p -Laplacian problems. The purpose of this paper is to prove an abstract critical point theorem that can be used toobtain pairs of nontrivial solutions of problems of the type − ∆ p u = λ | u | p − u + µf ( x, u ) + | u | q − u in Ω u = 0 on ∂ Ω , (1.1)where Ω is a bounded domain in R N , N ≥ p >
1, ∆ p u = div( |∇ u | p − ∇ u ) is the p -Laplacianof u , p < q ≤ p ∗ = N p/ ( N − p ) if p < N and p < q < ∞ if p ≥ N , λ, µ > f is a Carath´eodory function on Ω × R satisfying f ( x, t ) = | t | σ − t + o( | t | σ − ) as t → , uniformly a.e. in Ω (1.2) ∗ MSC2010:
Primary 58E05, Secondary 35J92, 35R11
Key Words and Phrases:
Abstract critical point theory, pairs of critical points, critical groups, cohomo-logical index, local and nonlocal problems, combined nonlinearities, pairs of nontrivial solutions < σ < p , the sign condition F ( x, t ) = Z t f ( x, s ) ds > x ∈ Ω and all t ∈ R \ { } , (1.3)and the growth condition | f ( x, t ) | ≤ a ( | t | r − + 1) for a.a. x ∈ Ω and all t ∈ R (1.4)for some a > p < r < q . Denoting by λ > − ∆ p onΩ, the case where 0 < λ < λ and µ > λ ≥ λ , the associated variational functional E ( u ) = 1 p Z Ω |∇ u | p dx − λp Z Ω | u | p dx − µ Z Ω F ( x, u ) dx − q Z Ω | u | q dx, u ∈ W , p (Ω)no longer has the mountain pass geometry and no multiplicity results are available in theliterature. We will prove a linking theorem based on a cohomological index theory that cancapture the geometry of this functional to produce a pair of nontrivial critical points for all λ > µ >
0. These critical points are neither local minimizers nor ofmountain pass type in general. They are higher critical points in the sense that they eachhave a nontrivial higher critical group (see Definition 2.7).To state our main result, let W be a Banach space and let M be a bounded symmetricsubset of W \ { } radially homeomorphic to the unit sphere S = { u ∈ W : k u k = 1 } , i.e., therestriction to M of the radial projection π : W \ { } → S, u u/ k u k is a homeomorphism.Then the radial projection from W \ { } onto M is given by π M = ( π | M ) − ◦ π. For a symmetric set A ⊂ W \ { } , denote by i ( A ) its Z -cohomological index (see Definition2.1). We have the following theorem. Theorem 1.1.
Let E be a C -functional on W and let A and B be disjoint closed symmetricsubsets of M such that i ( A ) = i ( M \ B ) = k < ∞ . Assume that there exist w ∈ M \ A , ≤ r < ρ < R , and a < b such that, setting A = { π M ((1 − s ) v + sw ) : v ∈ A , ≤ s ≤ } , (1.5) A ∗ = { tu : u ∈ A , r ≤ t ≤ R } ,B ∗ = { tw : w ∈ B , ≤ t ≤ ρ } ,A = { ru : u ∈ A } ∪ { tv : v ∈ A , r ≤ t ≤ R } ∪ { Ru : u ∈ A } , (1.6) B = { ρw : w ∈ B } , (1.7)2 e have a < inf B ∗ E, sup A E < inf B E, sup A ∗ E < b. (1.8) If E satisfies the (PS) c condition for all c ∈ ( a, b ) , then E has a pair of critical points u , u with inf B ∗ E ≤ E ( u ) ≤ sup A E, inf B E ≤ E ( u ) ≤ sup A ∗ E. If, in addition, E has only a finite number of critical points with the corresponding criticalvalues in ( a, b ) , then u and u can be chosen to satisfy C k ( E, u ) = 0 , C k +1 ( E, u ) = 0 . We will prove this theorem in Section 2. The proof is based on the notion of a coho-mological linking. In the course of the proof, we will also establish a new linking result ofindependent interest (see Theorem 2.6).Next we prove a multiplicity result for a class of abstract operator equations that includesproblem (1.1) as a special case. Let ( W, k · k ) be a uniformly convex Banach space with dual( W ∗ , k · k ∗ ) and duality pairing ( · , · ). Recall that h ∈ C ( W, W ∗ ) is a potential operator ifthere is a functional H ∈ C ( W, R ), called a potential for h , such that H ′ = h . We considerthe nonlinear operator equation A p u = λB p u + µf ( u ) + g ( u ) (1.9)in W ∗ , where A p , B p , f, g ∈ C ( W, W ∗ ) are potential operators satisfying the following as-sumptions, and λ, µ > A ) A p is ( p − p ∈ (1 , ∞ ): A p ( tu ) = | t | p − t A p u for all u ∈ W and t ∈ R ,( A ) ( A p u, v ) ≤ k u k p − k v k for all u, v ∈ W , and equality holds if and only if αu = βv forsome α, β ≥
0, not both zero (in particular, ( A p u, u ) = k u k p for all u ∈ W ),( B ) B p is ( p − B p ( tu ) = | t | p − t B p u for all u ∈ W and t ∈ R ,( B ) ( B p u, u ) > u ∈ W \ { } , and ( B p u, v ) ≤ ( B p u, u ) ( p − /p ( B p v, v ) /p for all u, v ∈ W ,( B ) B p is a compact operator,( F ) the potential F of f with F (0) = 0 satisfies lim t → F ( tu ) | t | p = + ∞ uniformly on compactsubsets of W \ { } ,( F ) F ( u ) > u ∈ W \ { } ,( F ) F is bounded on bounded subsets of W ,3 G ) the potential G of g with G (0) = 0 satisfies G ( u ) = o( k u k p ) as u → G ) G ( u ) > u ∈ W \ { } ,( G ) G is bounded on bounded subsets of W ,( G ) lim t → + ∞ G ( tu ) t p = + ∞ uniformly on compact subsets of W \ { } .Solutions of equation (1.9) coincide with critical points of the C -functional E ( u ) = I p ( u ) − λJ p ( u ) − µF ( u ) − G ( u ) , u ∈ W, (1.10)where I p ( u ) = 1 p ( A p u, u ) = 1 p k u k p , J p ( u ) = 1 p ( B p u, u )are the potentials of A p and B p satisfying I p (0) = 0 = J p (0), respectively (see Proposition3.1). First we prove the following theorem, which assumes that E satisfies the (PS) condition. Theorem 1.2.
Assume that ( A ) – ( G ) hold and E satisfies the (PS) c condition for all c ∈ R .If λ > , then ∃ µ > such that equation (1.9) has two nontrivial solutions u , u with E ( u ) < < E ( u ) for < µ < µ . We will prove Theorem 1.2 in Section 3. The proof will involve the nonlinear eigenvalueproblem A p u = λB p u. (1.11)Let M = { u ∈ W : I p ( u ) = 1 } . Then M ⊂ W \ { } is a bounded complete symmetric C -Finsler manifold radially homeomorphic to the unit sphere in W , and eigenvalues of problem(1.11) coincide with critical values of the C -functionalΨ( u ) = 1 J p ( u ) , u ∈ M . Denote by F the class of symmetric subsets of M and by i ( M ) the cohomological index of M ∈ F , let F k = { M ∈ F : i ( M ) ≥ k } , and set λ k := inf M ∈F k sup u ∈ M Ψ( u ) , k ∈ N . Then λ = inf u ∈M Ψ( u ) > λ ≤ λ ≤ · · · is an unbounded sequence of eigen-values. Moreover, denoting by Ψ a = { u ∈ M : Ψ( u ) ≤ a } (resp. Ψ a = { u ∈ M : Ψ( u ) ≥ a } )the sublevel (resp. superlevel) sets of Ψ, we have λ k < λ k +1 = ⇒ i (Ψ λ k ) = i ( M \ Ψ λ k +1 ) = k (1.12)(see Perera et al. [17, Theorem 4.6]). An intermediate step in the proof of Theorem 1.2 is thefollowing result of independent interest. 4 heorem 1.3. Assume that ( A ) – ( B ) hold. If λ k < λ k +1 , then the sublevel set Ψ λ k has acompact symmetric subset of index k .Remark . Eigenvalues based on the cohomological index were first introduced in Perera[16].Theorem 1.2 can be used to obtain a pair of nontrivial solutions of the p -Laplacian problem(1.1) for all λ > µ > Theorem 1.5.
Let < p < q , with q < p ∗ if p < N and q < ∞ if p ≥ N , let λ > , and let f be a Carath´eodory function on Ω × R satisfying (1.2) – (1.4) for some < σ < p < r < q .Then ∃ µ > such that problem (1.1) has two nontrivial solutions u , u with E ( u ) < < E ( u ) for < µ < µ . This theorem follows from Theorem 1.2 with W = W , p (Ω) and the operators A p , B p , f, g ∈ C ( W , p (Ω) , W − , p ′ (Ω)) given by( A p u, v ) = Z Ω |∇ u | p − ∇ u · ∇ v dx, ( B p u, v ) = Z Ω | u | p − uv dx, ( f ( u ) , v ) = Z Ω f ( x, u ) v dx, ( g ( u ) , v ) = Z Ω | u | q − uv dx, u, v ∈ W , p (Ω) . It is easily seen that ( A )–( G ) hold.As another application of Theorem 1.2, consider the Kirchhoff type nonlocal problem − (cid:18)Z Ω |∇ u | dx (cid:19) ∆ u = λu + µf ( x, u ) + | u | q − u in Ω u = 0 on ∂ Ω , (1.13)where Ω is a bounded domain in R N , N = 1 , , or 3, 4 < q < N = 3 and 4 < q < ∞ if N = 1 or 2, λ, µ > f is a Carath´eodory function on Ω × R satisfying(1.2)–(1.4) for some 1 < σ < < r < q . Solutions of this problem coincide with criticalpoints of the functional E ( u ) = 14 (cid:18)Z Ω |∇ u | dx (cid:19) − λ Z Ω u dx − µ Z Ω F ( x, u ) dx − q Z Ω | u | q dx, u ∈ H (Ω) , where F ( x, t ) = R t f ( x, s ) ds . Applying Theorem 1.2 with W = H (Ω) and the operators A p , B p , f, g ∈ C ( H (Ω) , H − (Ω)) given by( A p u, v ) = (cid:18)Z Ω |∇ u | dx (cid:19) Z Ω ∇ u · ∇ v dx, ( B p u, v ) = Z Ω u v dx, ( f ( u ) , v ) = Z Ω f ( x, u ) v dx, ( g ( u ) , v ) = Z Ω | u | q − uv dx, u, v ∈ H (Ω) , we have the following multiplicity result. 5 heorem 1.6. Let q > , with q < if N = 3 and q < ∞ if N = 1 or , let λ > , and let f be a Carath´eodory function on Ω × R satisfying (1.2) – (1.4) for some < σ < < r < q .Then ∃ µ > such that problem (1.13) has two nontrivial solutions u , u with E ( u ) < < E ( u ) for < µ < µ . Finally we prove a variant of Theorem 1.2 that only assumes a local (PS) condition andhence applicable to critical growth problems.
Theorem 1.7.
Assume that ( A ) – ( G ) hold and ∃ c µ > such that E satisfies the (PS) c condition for all c < c µ . If < λ < λ , assume that ∃ w ∈ M such that sup t ≥ E ( tw ) < c µ (1.14) for all sufficiently small µ > . If λ k ≤ λ < λ k +1 , assume that there exist a compact symmetricsubset C of Ψ λ with i ( C ) = k and w ∈ M \ C such that sup v ∈ C, s,t ≥ E ( sv + tw ) < c µ (1.15) for all sufficiently small µ > . Then ∃ µ > such that equation (1.9) has two nontrivialsolutions u , u with E ( u ) < < E ( u ) < c µ for < µ < µ . We will prove Theorem 1.7 in Section 3. As an application, consider the critical p -Laplacian problem − ∆ p u = λ | u | p − u + µf ( x, u ) + | u | p ∗ − u in Ω u = 0 on ∂ Ω , (1.16)where Ω is a bounded domain in R N , 1 < p < N , λ, µ > f is aCarath´eodory function on Ω × R satisfying (1.2)–(1.4) for some 1 < σ < p < r < p ∗ .Solutions of this problem coincide with critical points of the functional E ( u ) = 1 p Z Ω |∇ u | p dx − λp Z Ω | u | p dx − µ Z Ω F ( x, u ) dx − p ∗ Z Ω | u | p ∗ dx, u ∈ W , p (Ω) . Denote by σ ( − ∆ p ) the Dirichlet spectrum of − ∆ p in Ω, consisting of those λ ∈ R for whichthe problem − ∆ p u = λ | u | p − u in Ω u = 0 on ∂ Ω 6as a nontrivial solution. Let S = inf u ∈ W , p (Ω) \{ } Z Ω |∇ u | p dx (cid:18)Z Ω | u | p ∗ dx (cid:19) p/p ∗ (1.17)be the best Sobolev constant. We will prove the following multiplicity result in Section 4. Theorem 1.8.
Let p > , let N ≥ p , let λ > with λ / ∈ σ ( − ∆ p ) , and let f be a Carath´eodoryfunction on Ω × R satisfying (1.2) – (1.4) for some < σ < p < r < p ∗ . Then ∃ µ , κ > suchthat problem (1.16) has two nontrivial solutions u , u with E ( u ) < < E ( u ) < N S
N/p − κµ for < µ < µ .Remark . When µ = 0, one nontrivial solution was obtained in Garc´ıa Azorero and PeralAlonso [10], Egnell [7], Guedda and V´eron [12], Arioli and Gazzola [2], and Degiovanni andLancelotti [5].As another application of Theorem 1.7, consider the critical fractional p -Laplacian prob-lem ( − ∆) sp u = λ | u | p − u + µf ( x, u ) + | u | p ∗ s − u in Ω u = 0 in R N \ Ω , (1.18)where Ω is a bounded domain in R N with Lipschitz boundary, s ∈ (0 , < p < N/s , ( − ∆) sp is the fractional p -Laplacian operator defined on smooth functions by( − ∆) sp u ( x ) = 2 lim ε ց Z R N \ B ε ( x ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) | x − y | N + sp dy, x ∈ R N ,p ∗ s = N p/ ( N − sp ) is the fractional critical Sobolev exponent, λ, µ > f is a Carath´eodory function on Ω × R satisfying (1.2)–(1.4) for some 1 < σ < p < r < p ∗ s . Let |·| p denote the norm in L p ( R N ), let[ u ] s, p = (cid:18)Z R N | u ( x ) − u ( y ) | p | x − y | N + sp dxdy (cid:19) /p be the Gagliardo seminorm of a measurable function u : R N → R , and let W s, p ( R N ) = (cid:8) u ∈ L p ( R N ) : [ u ] s, p < ∞ (cid:9) be the fractional Sobolev space endowed with the norm k u k s, p = (cid:16) | u | pp + [ u ] ps, p (cid:17) /p .
7e work in the closed linear subspace W s, p (Ω) = (cid:8) u ∈ W s, p ( R N ) : u = 0 a.e. in R N \ Ω (cid:9) , equivalently renormed by setting k·k = [ · ] s, p . Solutions of problem (1.18) coincide with criticalpoints of the functional E ( u ) = 1 p Z R N | u ( x ) − u ( y ) | p | x − y | N + sp dxdy − λp Z Ω | u | p dx − µ Z Ω F ( x, u ) dx − p ∗ s Z Ω | u | p ∗ s dx,u ∈ W s, p (Ω) . Denote by σ (( − ∆) sp ) the Dirichlet spectrum of ( − ∆) sp in Ω, consisting of those λ ∈ R forwhich the problem ( − ∆) sp u = λ | u | p − u in Ω u = 0 in R N \ Ωhas a nontrivial solution. Let˙ W s, p ( R N ) = (cid:8) u ∈ L p ∗ s ( R N ) : [ u ] s, p < ∞ (cid:9) endowed with the norm k·k , and let S = inf u ∈ ˙ W s, p ( R N ) \{ } Z R N | u ( x ) − u ( y ) | p | x − y | N + sp dxdy (cid:18)Z R N | u | p ∗ s dx (cid:19) p/p ∗ s (1.19)be the best fractional Sobolev constant. We will prove the following multiplicity result inSection 5. Theorem 1.10.
Let p > , let N > sp , let λ > with λ / ∈ σ (( − ∆) sp ) , and let f be aCarath´eodory function on Ω × R satisfying (1.2) – (1.4) for some < σ < p < r < p ∗ s . Then ∃ µ , κ > such that problem (1.18) has two nontrivial solutions u , u with E ( u ) < < E ( u ) < sN S N/sp − κµ for < µ < µ .Remark . When µ = 0, one nontrivial solution was obtained in Mosconi et al. [14]. In this section we prove Theorem 1.1. We begin by recalling the definition and some propertiesof the Z -cohomological index of Fadell and Rabinowitz [8].8 efinition 2.1. Let W be a Banach space and let A denote the class of symmetric subsets of W \ { } . For A ∈ A , let A = A/ Z be the quotient space of A with each u and − u identified,let f : A → R P ∞ be the classifying map of A , and let f ∗ : H ∗ ( R P ∞ ) → H ∗ ( A ) be the inducedhomomorphism of the Alexander-Spanier cohomology rings. The cohomological index of A isdefined by i ( A ) = A = ∅ sup { m ≥ f ∗ ( ω m − ) = 0 } if A = ∅ , where ω ∈ H ( R P ∞ ) is the generator of the polynomial ring H ∗ ( R P ∞ ) = Z [ ω ]. Example . The classifying map of the unit sphere S m − in R m , m ≥ R P m − ֒ → R P ∞ , which induces isomorphisms on the cohomology groups H q for q ≤ m − i ( S m − ) = m .The following proposition summarizes the basic properties of this index. Proposition 2.3 ([8]) . The index i : A → N ∪ { , ∞} has the following properties: ( i ) Definiteness: i ( A ) = 0 if and only if A = ∅ . ( i ) Monotonicity: If there is an odd continuous map from A to B (in particular, if A ⊂ B ),then i ( A ) ≤ i ( B ) . Thus, equality holds when the map is an odd homeomorphism. ( i ) Dimension: i ( A ) ≤ dim W . ( i ) Continuity: If A is closed, then there is a closed neighborhood N ∈ A of A such that i ( N ) = i ( A ) . When A is compact, N may be chosen to be a δ -neighborhood N δ ( A ) = { u ∈ W : dist ( u, A ) ≤ δ } . ( i ) Subadditivity: If A and B are closed, then i ( A ∪ B ) ≤ i ( A ) + i ( B ) . ( i ) Stability: If SA is the suspension of A = ∅ , obtained as the quotient space of A × [ − , with A × { } and A × {− } collapsed to different points, then i ( SA ) = i ( A ) + 1 . ( i ) Piercing property: If A , A and A are closed, and ϕ : A × [0 , → A ∪ A is acontinuous map such that ϕ ( − u, t ) = − ϕ ( u, t ) for all ( u, t ) ∈ A × [0 , , ϕ ( A × [0 , isclosed, ϕ ( A × { } ) ⊂ A and ϕ ( A × { } ) ⊂ A , then i ( ϕ ( A × [0 , ∩ A ∩ A ) ≥ i ( A ) . ( i ) Neighborhood of zero: If U is a bounded closed symmetric neighborhood of , then i ( ∂U ) = dim W . Next we recall the definition of a cohomological linking.
Definition 2.4.
Let A and B be disjoint nonempty subsets of a Banach space W . We saythat A links B cohomologically in dimension k ≥ ι : A ֒ → W \ B induces anontrivial homomorphism ι ∗ : e H k ( W \ B ) → e H k ( A ) between reduced cohomology groups indimension k . 9 xample . Let M be as in the introduction and let A and B be disjoint nonempty closedsymmetric subsets of M such that i ( A ) = i ( M \ B ) = k < ∞ . Then for any
R >
0, the set { Rv : v ∈ A } links the set { tw : w ∈ B , t ≥ } cohomologicallyin dimension k − Theorem 2.6.
Let A and B be disjoint closed symmetric subsets of M such that i ( A ) = i ( M \ B ) = k < ∞ . (2.1) Let w ∈ M \ A and ≤ r < ρ < R . Set A = { π M ((1 − s ) v + sw ) : v ∈ A , ≤ s ≤ } and A = { ru : u ∈ A } ∪ { tv : v ∈ A , r ≤ t ≤ R } ∪ { Ru : u ∈ A } , B = { ρw : w ∈ B } . Then A links B cohomologically in dimension k .Proof. First we note that A is contractible. Indeed, the mapping A × [0 , → A , ( u, t ) π M ((1 − t ) u + tw )is a contraction of A to w .Let A = { ru : u ∈ A } ∪ { tv : v ∈ A , r ≤ t ≤ R } , A = { Rv : v ∈ A } ,B = { tw : w ∈ B , t ≥ } , B = { tw : w ∈ B , t ≥ ρ } . The set A is contractible. Indeed, the mapping A × [0 , → A , ( u, t ) (1 − t ) u + tr π M ( u )is a strong deformation retraction of A onto { ru : u ∈ A } , which is homeomorphic to A and hence contractible. Consider the commutative diagram e H k − ( E \ B ) −−−→ e H k − ( E \ B ) −−−→ H k ( E \ B , E \ B ) −−−→ e H k ( E \ B ) y y ι ∗ y ι ∗ ye H k − ( A ) −−−→ e H k − ( A ) δ −−−→ H k ( A , A ) −−−→ e H k ( A )where the rows come from the exact sequences of the pairs ( A , A ) ֒ → ( E \ B , E \ B ). ByExample 2.5, A links B cohomologically in dimension k − ι ∗ = 0. Since A iscontractible, e H ∗ ( A ) = 0 and hence δ is an isomorphism by the exactness of the bottom row.So it follows from the commutativity of the middle square that ι ∗ = 0.10ext let A = { Ru : u ∈ A } , and consider the commutative diagram H k ( E \ B, E \ B ∗ ) ≈ −−−→ H k ( E \ B , E \ B ) y ι ∗ y ι ∗ H k ( A, A ) −−−→ H k ( A , A )induced by inclusions. The top arrow is an isomorphism by the excision property since { tw : w ∈ B , t > ρ } is a closed subset of E \ B contained in the open subset E \ B ∗ . Since ι ∗ = 0, it follows that ι ∗ = 0.Finally consider the commutative diagram e H k − ( E \ B ∗ ) −−−→ H k ( E \ B, E \ B ∗ ) −−−→ e H k ( E \ B ) −−−→ e H k ( E \ B ∗ ) y y ι ∗ y ι ∗ ye H k − ( A ) −−−→ H k ( A, A ) j ∗ −−−→ e H k ( A ) −−−→ e H k ( A )where the rows come from the exact sequences of the pairs ( A, A ) ֒ → ( E \ B, E \ B ∗ ). Since A is homeomorphic to A and hence contractible, e H ∗ ( A ) = 0. So j ∗ is an isomorphismby the exactness of the bottom row. Since ι ∗ = 0, it follows from the commutativity of themiddle square that ι ∗ = 0.Before proceeding to the proof of Theorem 1.1, we recall the definition of critical groups. Definition 2.7.
Let E be a C -functional on a Banach space W and let u be an isolatedcritical point of E . The critical groups of E at u are defined by C q ( E, u ) = H q ( E c , E c \ { u } ) , q ≥ , where c = E ( u ) is the corresponding critical value and E c = { u ∈ W : E ( u ) ≤ c } is thesublevel set.If E satisfies the (PS) c condition for all c ∈ [ a, b ] and H q ( E b , E a ) = 0 for some q ≥
0, then E has a critical point u with a ≤ E ( u ) ≤ b . If, in addition, a and b are regular values and E has only a finite number of critical points with the corresponding critical values in ( a, b ),then u can be chosen so that C q ( E, u ) = 0 (see, e.g., Perera et al. [17, Proposition 3.13]).We will make use of this fact in the proof of Theorem 1.1. Proof of Theorem 1.1.
For the sake of simplicity we only consider the case where W is infinitedimensional. Since B ∗ ∩ A and B ∩ A ∗ are nonempty, inf E ( B ∗ ) ≤ sup E ( A ) and inf E ( B ) ≤ sup E ( A ∗ ). We will show that if α < β < γ satisfy a < α < inf B ∗ E, sup A E < β < inf B E, sup A ∗ E < γ < b, H k ( E β , E α ) = 0 , H k +1 ( E γ , E β ) = 0 . Then E has a pair of critical points u , u with α ≤ E ( u ) ≤ β, β ≤ E ( u ) ≤ γ. If inf E ( B ∗ ) or sup E ( A ) is a critical value of E , we can take u to be at one of thoselevels. Otherwise, α and β can be chosen so that E has no critical values in [ α, inf E ( B ∗ )] ∪ [sup E ( A ) , β ] since E satisfies the (PS) condition at those levels, and theninf B ∗ E < E ( u ) < sup A E. Similarly, u can be chosen to satisfyinf B E ≤ E ( u ) ≤ sup A ∗ E. If E has only a finite number of critical points with the corresponding critical values in ( a, b ),then α and β can be chosen so that E has no critical values in [ α, inf E ( B ∗ )) ∪ (sup E ( A ) , β ]and C k ( E, u ) = 0 , C k +1 ( E, u ) = 0 . By Theorem 2.6, A links B cohomologically in dimension k and hence the inclusion ι : A ֒ → W \ B induces a nontrivial homomorphism ι ∗ : e H k ( W \ B ) → e H k ( A ). Since E < β on A and E > β on B , we also have the inclusions ι : A ֒ → E β and ι : E β ֒ → W \ B , whichinduce the commutative diagram e H k ( W \ B ) ι ∗ (cid:15) (cid:15) ι ∗ & & ▼▼▼▼▼▼▼▼▼▼ e H k ( E β ) ι ∗ / / e H k ( A ) . This gives ι ∗ ι ∗ = ι ∗ = 0 , so both ι ∗ and ι ∗ are nontrivial homomorphisms.First we show that H k ( E β , E α ) = 0. Since E > α on B ∗ and α < β , we have the inclusions E α ֒ → W \ B ∗ ֒ → W \ B and E α ֒ → E β ֒ → W \ B , yielding the commutative diagram e H k ( W \ B ) −−−→ e H k ( W \ B ∗ ) y ι ∗ ye H k ( E β ) i ∗ −−−→ e H k ( E α ) . W \ B ∗ is contractible. Indeed, for any ρ ′ > ρ , the mapping( W \ B ∗ ) × [0 , → W \ B ∗ , ( u, t ) (1 − t ) u + tρ ′ π M ( u )is a strong deformation retraction of W \ B ∗ onto { ρ ′ u : u ∈ M} , which is homeomorphic to S and hence contractible since W is infinite dimensional. So e H ∗ ( W \ B ∗ ) = 0 and hence i ∗ ι ∗ = 0 . Since ι ∗ is nontrivial, this implies that i ∗ is not injective. Now consider the exact sequence ofthe pair ( E β , E α ): · · · δ −−−→ H k ( E β , E α ) j ∗ −−−→ e H k ( E β ) i ∗ −−−→ e H k ( E α ) δ −−−→ · · · . By exactness,im j ∗ = ker i ∗ = 0 , so H k ( E β , E α ) = 0.Finally we show that H k +1 ( E γ , E β ) = 0. Since E < γ on A ∗ and β < γ , we have theinclusions A ֒ → A ∗ ֒ → E γ and A ֒ → E β ֒ → E γ , yielding the commutative diagram e H k ( E γ ) −−−→ e H k ( A ∗ ) y i ∗ ye H k ( E β ) ι ∗ −−−→ e H k ( A ) . The set A ∗ is contractible. Indeed, the mapping A ∗ × [0 , → A ∗ , ( u, t ) (1 − t ) u + tr π M ( u )is a strong deformation retraction of A ∗ onto { ru : u ∈ A } , which is homeomorphic to A and hence contractible as in the proof of Theorem 2.6. So e H ∗ ( A ∗ ) = 0 and hence ι ∗ i ∗ = 0 . Since ι ∗ is nontrivial, this implies that i ∗ is not surjective. Now consider the exact sequenceof the pair ( E γ , E β ): · · · j ∗ −−−→ e H k ( E γ ) i ∗ −−−→ e H k ( E β ) δ −−−→ H k +1 ( E γ , E β ) j ∗ −−−→ · · · . By exactness,ker δ = im i ∗ = e H k ( E β ) , so H k +1 ( E γ , E β ) = 0. 13 Proofs of Theorems 1.2, 1.3, and 1.7
In this section we prove Theorems 1.2, 1.3, and 1.7. We begin by proving the followingproposition, which shows that E ′ ( u ) = A p u − λB p u − µf ( u ) − g ( u ) , u ∈ W and hence solutions of equation (1.9) coincide with critical points of E . Proposition 3.1. If h ∈ C ( W, W ∗ ) is a potential operator and H ∈ C ( W, R ) is its potentialsatisfying H (0) = 0 , then H ( u ) = Z ( h ( tu ) , u ) dt ∀ u ∈ W. In particular, H is even if h is odd. If h is ( p − -homogeneous, then H ( u ) = 1 p ( h ( u ) , u ) ∀ u ∈ W and is p -homogeneous.Proof. We have H ( u ) = H (0) + Z ddt (cid:0) H ( tu ) (cid:1) dt = Z ( H ′ ( tu ) , u ) dt = Z ( h ( tu ) , u ) dt. The last integral equals Z t p − ( h ( u ) , u ) dt = 1 p ( h ( u ) , u )if h is ( p − A p that follows from the assumption( A ). Proposition 3.2. If ( A ) holds, then ( i ) A p is strictly monotone: ( A p u − A p v, u − v ) > for all u = v in W , ( ii ) A p is of type ( S ) : every sequence ( u j ) ⊂ W such that u j ⇀ u and ( A p u j , u j − u ) → has a subsequence that converges strongly to u .Proof. ( i ) By ( A ),( A p u − A p v, u − v ) = ( A p u, u ) − ( A p u, v ) − ( A p v, u ) + ( A p v, v ) ≥ k u k p − k u k p − k v k − k v k p − k u k + k v k p = (cid:0) k u k p − − k v k p − (cid:1)(cid:0) k u k − k v k (cid:1) ≥ u, v ∈ W . If ( A p u − A p v, u − v ) = 0, then equality holds throughout and hence( A p u, v ) = k u k p − k v k and k u k = k v k . The first equality implies that αu = βv for some α, β ≥
0, not both zero. The second equality then implies that either u = v = 0, or α = β >
0. In the latter case, u = v since αu = βv .( ii ) As in the proof of ( i ),( A p u j − A p u, u j − u ) ≥ (cid:0) k u j k p − − k u k p − (cid:1)(cid:0) k u j k − k u k (cid:1) ≥ . (3.1)Since ( A p u j , u j − u ) → u j ⇀ u ,( A p u j − A p u, u j − u ) = ( A p u j , u j − u ) − ( A p u, u j − u ) → , so (3.1) implies that k u j k → k u k . Then u j → u since W is uniformly convex.Next we prove Theorem 1.3. Proof of Theorem 1.3.
For each w ∈ W , the equation A p u = B p w has a unique solution u ∈ W . Indeed, a solution can be obtained by minimizing the functionalΦ( u ) = 1 p ( A p u, u ) − ( B p w, u ) , u ∈ W, which is coercive and weakly lower semicontinuous since ( A p u, u ) = k u k p by ( A ) and p > A p is strictly monotone (see Proposition 3.2 ( i )).Since both A p and B p are ( p − K : W → W, w u is linear. Moreover, K is compact. To see this, let ( w j ) be a bounded sequence in W and let u j = Kw j . Since B p is a compact operator, A p u j = B p w j → l for a renamed subsequence and some l ∈ W ∗ . By ( A ), k u j k p = ( A p u j , u j ) = ( B p w j , u j ) ≤ k B p w j k ∗ k u j k , which implies that ( u j ) is bounded since p > B p w j ) is bounded. Since W is reflexive,then a further subsequence of ( u j ) converges weakly to some u ∈ W . Then( A p u j , u j − u ) = ( A p u j − l, u j − u ) + ( l, u j − u ) → . Since A p is of type ( S ) (see Proposition 3.2 ( ii )), then ( u j ) has a subsequence that convergesstrongly to u .Let w ∈ W \ { } and let u = Kw . Then u = 0 since ( A p u, w ) = ( B p w, w ) > B ).Since I p is p -homogeneous, the radial projection of u on M is given by π M ( u ) = uI p ( u ) /p . J p is p -homogeneous, this givesΨ( π M ( u )) = 1 J p ( π M ( u )) = I p ( u ) J p ( u ) . (3.2)We have I p ( u ) = 1 p ( A p u, u ) = 1 p ( B p w, u ) ≤ p ( B p w, w ) ( p − /p ( B p u, u ) /p = J p ( w ) ( p − /p J p ( u ) /p by ( B ) and J p ( w ) = 1 p ( B p w, w ) = 1 p ( A p u, w ) ≤ p k u k p − k w k = I p ( u ) ( p − /p I p ( w ) /p (3.3)by ( A ), so I p ( u ) J p ( u ) ≤ I p ( w ) J p ( w ) . For w ∈ M , I p ( w ) = 1 and 1 /J p ( w ) = Ψ( w ), so combining this with (3.2) givesΨ( π M ( Kw )) ≤ Ψ( w ) . (3.4)Let C = π M ( K (Ψ λ k )) . Since π M ( K (Ψ λ k )) is contained in the closed set Ψ λ k by (3.4), we have π M ( K (Ψ λ k )) ⊂ C ⊂ Ψ λ k . (3.5)Since π M ◦ K is an odd continuous map on Ψ λ k , then i (Ψ λ k ) ≤ i ( π M ( K (Ψ λ k ))) ≤ i ( C ) ≤ i (Ψ λ k )by the monotonicity of the index (see Proposition 2.3 ( i )). Since λ k < λ k +1 , this togetherwith (1.12) gives i ( C ) = k . Since I p ( w ) = 1 and J p ( w ) ≥ /λ k for w ∈ Ψ λ k , (3.3) implies that K (Ψ λ k ) ⊂ W \ { } . The set π M ( K (Ψ λ k )) is compact since Ψ λ k is bounded, K is compact,and π M is continuous on W \ { } . Then so is the closed subset C .We are now ready to prove Theorem 1.2. Proof of Theorem 1.2.
We apply Theorem 1.1 to the functional E in (1.10). If 0 < λ < λ ,take A = ∅ and B = M . If λ ≥ λ , then λ k ≤ λ < λ k +1 for some k ≥
1. Let A be acompact symmetric subset of Ψ λ k of index k (see Theorem 1.3) and let B = Ψ λ k +1 . Then i ( A ) = i ( M \ B ) = k by (1.12). 16or u ∈ M and t > E ( tu ) = t p (cid:18) − λ Ψ( u ) (cid:19) − µF ( tu ) − G ( tu ) . (3.6)For w ∈ B , this together with ( G ) gives E ( tw ) ≥ t p (cid:18) − λλ k +1 + o(1) (cid:19) − µF ( tw ) as t → . Since λ < λ k +1 , it follows from this and ( F ) that ∃ ρ, µ > B E > < µ < µ , where B is as in (1.7). Fix 0 < µ < µ , let w ∈ M \ A , and let A be asin (1.5). Since A is compact, so is A . For u ∈ A , (3.6) together with ( G ) gives E ( tu ) ≤ t p (cid:18) − µ F ( tu ) t p (cid:19) . Since A is compact, it follows from this and ( F ) that ∃ < r < ρ such thatsup { E ( ru ) : u ∈ A } < . (3.7)Similarly, (3.6) together with ( F ) gives E ( tu ) ≤ t p (cid:18) − G ( tu ) t p (cid:19) for u ∈ A , and it follows from this and ( G ) that ∃ R > ρ such thatsup { E ( Ru ) : u ∈ A } < . (3.8)For v ∈ A , E ( tv ) < − t p (cid:18) λ Ψ( v ) − (cid:19) ≤ v ) ≤ λ k ≤ λ . Since A is compact, it follows from this thatsup { E ( tv ) : v ∈ A , r ≤ t ≤ R } < . (3.9)Combining (3.7)–(3.9) givessup A E < , where A is as in (1.6). So the second inequality in (1.8) holds. The first and the thirdinequalities hold for some a < b since A ∗ and B ∗ are bounded sets and E is bounded onbounded sets by ( A ), ( B ), ( F ), and ( G ). So E has two nontrivial critical points u , u with E ( u ) ≤ sup A E < < inf B E ≤ E ( u ) . Proof of Theorem 1.7.
We proceed as in the proof of Theorem 1.2. If 0 < λ < λ , take A = ∅ and B = M . Then A ∗ ⊂ { tw : t ≥ } and hence sup E ( A ∗ ) < c µ for all sufficiently small µ > λ k ≤ λ < λ k +1 , let A = C and B = Ψ λ k +1 . Then A ∗ ⊂ { sv + tw : v ∈ C, s, t ≥ } and hence sup E ( A ∗ ) < c µ for all sufficiently small µ > a < inf E ( B ∗ ) and b = c µ to conclude the proof. In this section we prove Theorem 1.8 by applying Theorem 1.7 with W = W , p (Ω) and theoperators A p , B p , f, g ∈ C ( W , p (Ω) , W − , p ′ (Ω)) given by( A p u, v ) = Z Ω |∇ u | p − ∇ u · ∇ v dx, ( B p u, v ) = Z Ω | u | p − uv dx, ( f ( u ) , v ) = Z Ω f ( x, u ) v dx, ( g ( u ) , v ) = Z Ω | u | p ∗ − uv dx, u, v ∈ W , p (Ω) . We begin by determining a threshold level below which the functional E satisfies the (PS)condition. Lemma 4.1.
Let < µ ≤ . Then ∃ κ > such that E satisfies the (PS) c condition for all c < N S
N/p − κµ. (4.1) Proof.
Let c ∈ R and let ( u j ) be a sequence in W , p (Ω) such that E ( u j ) = Z Ω (cid:18) p |∇ u j | p − λp | u j | p − µF ( x, u j ) − p ∗ | u j | p ∗ (cid:19) dx = c + o(1) (4.2)and ( E ′ ( u j ) , v ) = Z Ω (cid:0) |∇ u j | p − ∇ u j · ∇ v − λ | u j | p − u j v − µf ( x, u j ) v − | u j | p ∗ − u j v (cid:1) dx = o( k v k ) ∀ v ∈ W , p (Ω) . (4.3)Taking v = u j in (4.3) gives Z Ω (cid:0) |∇ u j | p − λ | u j | p − µf ( x, u j ) u j − | u j | p ∗ (cid:1) dx = o( k u j k ) . (4.4)18ince r < p ∗ , (4.2) and (4.4) imply that ( u j ) is bounded, so a renamed subsequence convergesto some u weakly in W , p (Ω), strongly in L s (Ω) for all s ∈ [1 , p ∗ ), and a.e. in Ω. Setting e u j = u j − u , we will show that e u j → W , p (Ω).Equation (4.4) implies k u j k p = | u j | p ∗ p ∗ + Z Ω ( λ | u | p + µf ( x, u ) u ) dx + o(1) , (4.5)where |·| p ∗ denotes the L p ∗ (Ω)-norm. Taking v = u in (4.3) and passing to the limit gives k u k p = | u | p ∗ p ∗ + Z Ω ( λ | u | p + µf ( x, u ) u ) dx. (4.6)Since k e u j k p = k u j k p − k u k p + o(1) (4.7)and | e u j | p ∗ p ∗ = | u j | p ∗ p ∗ − | u | p ∗ p ∗ + o(1)by the Br´ezis-Lieb lemma [4, Theorem 1], (4.5) and (4.6) imply k e u j k p = | e u j | p ∗ p ∗ + o(1) ≤ k e u j k p ∗ S p ∗ /p + o(1) , so k e u j k p (cid:16) S N/ ( N − p ) − k e u j k p / ( N − p ) (cid:17) ≤ o(1) . (4.8)On the other hand, (4.2) implies c = 1 p k u j k p − p ∗ | u j | p ∗ p ∗ − Z Ω (cid:18) λp | u | p + µF ( x, u ) (cid:19) dx + o(1) , and a straightforward calculation combining this with (4.5)–(4.7) gives c = 1 N k e u j k p + Z Ω K ( x, u ) dx + o(1) , (4.9)where K ( x, t ) = 1 N | t | p ∗ + µ (cid:18) p f ( x, t ) t − F ( x, t ) (cid:19) . By (1.4),1 p f ( x, t ) t − F ( x, t ) ≥ − a ( | t | r + 1) for a.a. x ∈ Ω and all t ∈ R for some a >
0. Since r < p ∗ and µ ≤
1, this together with the Young’s inequality gives K ( x, t ) ≥ − a µ for a.a. x ∈ Ω and all t ∈ R for some a >
0. Then (4.9) gives k e u j k p ≤ N ( c + κµ ) + o(1)for some κ >
0. Combining this with (4.8) shows that e u j → c µ = 1 N S
N/p − κµ, where κ > λ k < λ < λ k +1 forsome k ≥ < λ < λ is similar and simpler. We have M = (cid:8) u ∈ W , p (Ω) : k u k p = p (cid:9) and Ψ( u ) = p/ R Ω | u | p dx for u ∈ M . We need to show that thereexist a compact symmetric subset C of Ψ λ with i ( C ) = k and w ∈ M \ C such thatsup v ∈ C, s,t ≥ E ( sv + tw ) < N S
N/p − κµ (4.10)for all sufficiently small µ > K used in the proof of Theorem 1.3, wemay assume that the compact symmetric subset C of Ψ λ k with i ( C ) = k constructed in thattheorem is bounded in L ∞ (Ω) ∩ C , α loc (Ω) (see Degiovanni and Lancelotti [5, Theorem 2.3]). Wemay assume without loss of generality that 0 ∈ Ω. Let δ = dist (0 , ∂ Ω), let η : [0 , ∞ ) → [0 , η ( s ) = 0 for s ≤ / η ( s ) = 1 for s ≥
1, and set u δ ( x ) = η (cid:18) | x | δ (cid:19) u ( x ) , < δ ≤ δ / u ∈ C . Then set v = π M ( u δ ) , where π M : W , p (Ω) \ { } → M , u p /p u/ k u k is the radial projection onto M , and let C = { v : u ∈ C } . Lemma 4.2. If δ > is sufficiently small, then C is a compact symmetric subset of Ψ λ with i ( C ) = k .Proof. First we show that C ⊂ Ψ λ if δ > u ∈ C . Since functionsin C are bounded in C ( B δ / (0)) and belong to Ψ λ k , Z Ω |∇ u δ | p dx ≤ Z Ω \ B δ (0) |∇ u | p dx + Z B δ (0) (cid:0) |∇ u | p + a δ − p | u | p (cid:1) dx ≤ p + a δ N − p for some a , a > Z Ω | u δ | p dx ≥ Z Ω \ B δ (0) | u | p dx = Z Ω | u | p dx − Z B δ (0) | u | p dx ≥ pλ k − a δ N for some a >
0. SoΨ( v ) = Z Ω |∇ u δ | p dx Z Ω | u δ | p dx ≤ λ k + a δ N − p a >
0. Since λ k < λ , the last expression is less than or equal to λ , and hence v ∈ Ψ λ , for all sufficiently small δ > C is a compact symmetric set and u v is an odd continuous map of C onto C , C is also a compact symmetric set and i ( C ) ≥ i ( C ) = k by the monotonicity of the index (see Proposition 2.3 ( i )). On the other hand, since C ⊂ Ψ λ ⊂ M \ Ψ λ k +1 , i ( C ) ≤ i ( M \ Ψ λ k +1 ) = k by (1.12). So i ( C ) = k .We are now ready to prove Theorem 1.8. Proof of Theorem 1.8.
Recall that the infimum in (1.17) is attained by the Aubin-Talentifunctions u ∗ ε ( x ) = c N,p ε ( N − p ) /p (cid:0) ε + | x | p/ ( p − (cid:1) ( N − p ) /p , ε > R N , where the constant c N,p > Z R N |∇ u ∗ ε | p dx = Z R N | u ∗ ε | p ∗ dx = S N/p . Fix δ > C is a compact symmetric subset of Ψ λ with i ( C ) = k (see Lemma4.2), let θ : [0 , ∞ ) → [0 ,
1] be a smooth function such that θ ( s ) = 1 for s ≤ / θ ( s ) = 0for s ≥ /
2, and set u ε ( x ) = θ (cid:18) | x | δ (cid:19) u ∗ ε ( x ) , e u ε ( x ) = u ε ( x ) (cid:18)Z R N | u ε | p ∗ dx (cid:19) /p ∗ , ε > . We have the well-known estimates Z R N |∇ e u ε | p dx ≤ S + a ε ( N − p ) /p , (4.11) Z R N | e u ε | p ∗ dx = 1 , (4.12) Z R N | e u ε | p dx ≥ a ε p − if N > p a ε p − | log ε | if N = p (4.13)for some a , a > w = π M ( e u ε ) . C have their supports in Ω \ B δ/ (0), while the support of w is in B δ/ (0), w ∈ M \ C . We will show that (4.10) holds for all sufficiently small ε, µ > v ∈ C and s, t ≥
0. Since v and w have disjoint supports, E ( sv + tw ) = E ( sv ) + E ( tw ) . (4.14)By (1.3), E ( sv ) ≤ s p p (cid:18)Z Ω |∇ v | p dx − λ Z Ω | v | p dx (cid:19) = − s p (cid:18) λ Ψ( v ) − (cid:19) ≤ v ∈ Ψ λ . Moreover, E ( tw ) ≤ t p p (cid:18)Z Ω |∇ w | p dx − λ Z Ω | w | p dx (cid:19) − t p ∗ p ∗ Z Ω | w | p ∗ dx, and maximizing the right-hand side over all t ≥ E ( tw ) ≤ N (cid:18)Z Ω |∇ w | p dx − λ Z Ω | w | p dx (cid:19) p ∗ / ( p ∗ − p ) (cid:18)Z Ω | w | p ∗ dx (cid:19) p/ ( p ∗ − p ) = 1 N (cid:18)Z Ω |∇ e u ε | p dx − λ Z Ω | e u ε | p dx (cid:19) p ∗ / ( p ∗ − p ) (cid:18)Z Ω | e u ε | p ∗ dx (cid:19) p/ ( p ∗ − p ) ≤ N (cid:0) S + a ε ( N − p ) /p − λa ε p − (cid:1) N/p if N > p ( S + a ε p − − λa ε p − | log ε | ) p if N = p (4.16)by (4.11)–(4.13). It follows from (4.14)–(4.16) thatsup v ∈ C, s,t ≥ E ( sv + tw ) < N S
N/p if ε > µ > In this section we prove Theorem 1.10 by applying Theorem 1.7 with W = W s, p (Ω) and theoperators A p , B p , f , and g on W s, p (Ω) given by( A p u, v ) = Z R N | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) ( v ( x ) − v ( y )) | x − y | N + sp dxdy, ( B p u, v ) = Z Ω | u | p − uv dx, ( f ( u ) , v ) = Z Ω f ( x, u ) v dx, ( g ( u ) , v ) = Z Ω | u | p ∗ − uv dx,u, v ∈ W s, p (Ω) . Proof of Theorem 1.10.
An argument similar to that in the proof of Lemma 4.1 shows thatif 0 < µ ≤
1, then ∃ κ > E satisfies the (PS) c condition for all c < sN S N/sp − κµ. We will apply Theorem 1.7 with c µ = sN S N/sp − κµ. We only consider the case where λ k < λ < λ k +1 for some k ≥ < λ < λ is similar and simpler. We have M = { u ∈ W s, p (Ω) : k u k p = p } and Ψ( u ) = p/ R Ω | u | p dx for u ∈ M . We need to show that there exist a compact symmetric subset C of Ψ λ with i ( C ) = k and w ∈ M \ C such thatsup v ∈ C, s,t ≥ E ( sv + tw ) < sN S N/sp − κµ (5.1)for all sufficiently small µ > S in (1.19), we will use certainasymptotic estimates for minimizers obtained in Brasco et al. [3]. It was shown there thatthere exists a nonnegative, radially symmetric, and decreasing minimizer U = U ( r ) satisfying Z R N | U ( x ) − U ( y ) | p | x − y | N + sp dxdy = Z R N | U | p ∗ s dx = S N/sp (5.2)and c r − ( N − sp ) / ( p − ≤ U ( r ) ≤ c r − ( N − sp ) / ( p − ∀ r ≥ c , c >
0. By (5.3), U ( θr ) U ( r ) ≤ c c θ − ( N − sp ) / ( p − ≤ ∀ r ≥ θ > U ε ( x ) = ε − ( N − sp ) /p U (cid:18) | x | ε (cid:19) is also a minimizer for S satisfying (5.2) for any ε >
0. For ε, δ >
0, let m ε,δ = U ε ( δ ) U ε ( δ ) − U ε ( θδ ) , let g ε,δ ( t ) = ≤ t ≤ U ε ( θδ ) m pε,δ ( t − U ε ( θδ )) if U ε ( θδ ) ≤ t ≤ U ε ( δ ) t + U ε ( δ ) ( m p − ε,δ −
1) if t ≥ U ε ( δ ) , G ε,δ ( t ) = Z t g ′ ε,δ ( τ ) /p dτ = ≤ t ≤ U ε ( θδ ) m ε,δ ( t − U ε ( θδ )) if U ε ( θδ ) ≤ t ≤ U ε ( δ ) t if t ≥ U ε ( δ ) , and set u ε,δ ( r ) = G ε,δ ( U ε ( r )) . For ε ≤ δ/
2, we have the estimates Z R N | u ε,δ ( x ) − u ε,δ ( y ) | p | x − y | N + sp dxdy ≤ S N/sp + a (cid:16) εδ (cid:17) ( N − sp ) / ( p − , (5.4) Z R N | u ε,δ | p ∗ s dx ≥ S N/sp − a (cid:16) εδ (cid:17) N/ ( p − , (5.5) Z R N | u ε,δ | p dx ≥ a ε sp if N > sp a ε sp (cid:12)(cid:12)(cid:12) log (cid:16) εδ (cid:17)(cid:12)(cid:12)(cid:12) if N = sp (5.6)for some a , a > w = π M ( u ε,δ ) , where π M : W s, p (Ω) \ { } → M , u p /p u/ k u k is the radial projection onto M .By iterating a sufficient number of times the map K used in the proof of Theorem 1.3,we may assume that the compact symmetric subset C of Ψ λ k with i ( C ) = k constructed inthat theorem consists of functions u such that ( − ∆) sp u is bounded in L ∞ (Ω) (see Mosconiet al. [14, Proposition 3.1]). We may assume without loss of generality that 0 ∈ Ω. Let η : R N → [0 ,
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