Multiple solutions for p-Laplacian type problems with asymptotically p-linear terms via a cohomological index theory
aa r X i v : . [ m a t h . A P ] O c t Multiple solutionsfor p -Laplacian type problemswith asymptotically p -linear termsvia a cohomological index theory A.M. Candela ∗ , , G. Palmieri ∗ , , K. Perera † ∗ Dipartimento di MatematicaUniversit`a degli Studi di Bari “Aldo Moro”Via E. Orabona 4, 70125 Bari, Italy [email protected], [email protected] † Department of Mathematical SciencesFlorida Institute of Technology150 W. University Blvd, Melbourne, FL 32901, USA kperera@fit.edu
Abstract
The aim of this paper is investigating the existence of weak solutionsof the quasilinear elliptic model problem − div( A ( x, u ) |∇ u | p − ∇ u ) + 1 p A t ( x, u ) |∇ u | p = f ( x, u ) in Ω, u = 0 on ∂ Ω,where Ω ⊂ R N is a bounded domain, N ≥ p > A is a given functionwhich admits partial derivative A t ( x, t ) = ∂A∂t ( x, t ) and f is asymptotically p -linear at infinity.Under suitable hypotheses both at the origin and at infinity, and if A ( x, · ) is even while f ( x, · ) is odd, by using variational tools, a cohomo-logical index theory and a related pseudo–index argument, we prove amultiplicity result if p > N in the non–resonant case. . . Key words . p -Laplacian type equation, asymptotically p -linear problem, Palais–Smale condition,cohomological index theory, pseudo–index theory . ∗ The authors acknowledge the support of Research Funds PRIN2009 and
Fondi d’Ateneo2010 . † This work was done while the third–named author was visiting the
Dipartimento diMatematica at the
Universit`a degli Studi di Bari , and he is grateful for the kind hospital-ity of the department. Introduction
Let us consider the p -Laplacian type equation( P ) − div( A ( x, u ) |∇ u | p − ∇ u ) + 1 p A t ( x, u ) |∇ u | p = f ( x, u ) in Ω, u = 0 on ∂ Ω,where Ω ⊂ R N is a bounded domain, N ≥ p > A , f : Ω × R → R aregiven functions such that the partial derivative A t ( x, t ) = ∂A∂t ( x, t ) exists for a.e. x ∈ Ω, all t ∈ R .If we set F ( x, t ) = R t f ( x, s ) ds , we can associate with problem ( P ) thefunctional J : D ⊂ W , p (Ω) → R defined by J ( u ) = 1 p Z Ω A ( x, u ) |∇ u | p dx − Z Ω F ( x, u ) dx. (1.1)In general, if no growth assumption is made on A with respect to t , thenatural domain D of J is contained in, but is not equal to, the Sobolev space W , p (Ω). Anyway, under the assumptions( H ) A , A t are Carath´eodory functions on Ω × R such thatsup | t |≤ r | A ( · , t ) | ∈ L ∞ (Ω) , sup | t |≤ r | A t ( · , t ) | ∈ L ∞ (Ω) for any r > h ) f is a Carath´eodory function on Ω × R such thatsup | t |≤ r | f ( · , t ) | ∈ L ∞ (Ω) for any r > J is surely well-defined on the Banach space X := W , p (Ω) ∩ L ∞ (Ω) , k u k X = k u k + | u | ∞ , (1.2)with k u k p = Z Ω |∇ u | p dx, | u | ∞ = ess sup x ∈ Ω | u ( x ) | , and, for any u, v ∈ X , its Gˆateaux derivative with respect to u in the direction v is given by h d J ( u ) , v i = Z Ω A ( x, u ) |∇ u | p − ∇ u · ∇ v dx + 1 p Z Ω A t ( x, u ) |∇ u | p v dx − Z Ω f ( x, u ) v dx. As our aim is investigating the existence of weak solutions of ( P ) when it isan asymptotically p -linear elliptic problem, we assume that A and f satisfy thefollowing hypotheses: 2 H ) there exists α > A ( x, t ) ≥ α a.e. in Ω , for all t ∈ R ;( H ) there exists A ∞ ∈ L ∞ (Ω) such thatlim | t |→ + ∞ A ( x, t ) = A ∞ ( x ) uniformly a.e. in Ω;( h ) there exist λ ∞ ∈ R and a (Carath´eodory) function g ∞ : Ω × R → R suchthat f ( x, t ) = λ ∞ | t | p − t + g ∞ ( x, t ) , where lim | t |→ + ∞ g ∞ ( x, t ) | t | p − = 0 uniformly a.e. in Ω . (1.3)As J is a C -functional on X under these hypotheses (see Proposition 3.1),we can seek weak solutions of ( P ) by means of variational tools.In the asymptotically linear case, i.e. under the hypotheses ( h ) and ( h ), avariational approach was first used for p = 2 and A ( x, t ) ≡ p = 2, but always for A ( x, t ) ≡ A ( x, t ) = A ( x ) independent of t (see [2, 4, 6, 16, 18, 21, 22, 23, 24, 25]). In fact, when p > − ∆ in H (Ω) is known,the full spectrum of − ∆ p is still unknown, even though various authors haveintroduced different characterizations of eigenvalues and definitions of quasi–eigenvalues.Clearly, the same problem arises in our setting when A ( x, t ) depends on t .Furthermore, we have difficulties with the Palais–Smale condition as well, andhave to consider the asymptotic behavior, both at the origin and at infinity, notonly of the term f ( x, t ), but also of the coefficient A ( x, t ).When ( h ) is replaced with different conditions at infinity, weaker versions ofthe Palais–Smale condition hold for arbitrary p >
1, and the existence of criticalpoints of J in X have been proved (see [10, 13]). However, these approaches donot distinguish between different critical points at the same critical level (see[11, 12]), and therefore, up to now, multiplicity results via a cohomological indextheory have been obtained only for p > N (see [9, 15]). In fact, in this case theSobolev Imbedding Theorem implies X = W ,p (Ω) and the classical Cerami’svariant of the Palais–Smale condition can be verified.In this paper, we will prove a multiplicity result for problem ( P ) when p > N and f ( x, t ) is asymptotically p -linear at infinity. To this aim, by consideringsome sequences of eigenvalues defined by means of the cohomological index, wewill prove the classical Palais–Smale condition and, by means of a cohomologicalindex theory and a related pseudo–index argument, we will extend the result3n [25] to our setting (see [14] for a result obtained by using the approach in[5]). In particular, let us point out that, if the coefficient A depends on t , theboundedness of each Palais–Smale sequence of J requires a careful proof alsoin the non–resonant assumption, unlike the t –independent case (see Proposition3.5). The aim of this section is to recall the abstract tools we need for the proof of ourmain result. Hence, let ( B , k · k B ) be a Banach space with dual space ( B ′ , k · k B ′ )and let J ∈ C ( B , R ).Furthermore, fixing a level β ∈ R , a point u ∈ B , a set C ⊂ B and a radius r >
0, let us denote • K J = { u ∈ B : dJ ( u ) = 0 } the set of critical points of J in B ; • K Jβ = { u ∈ B : J ( u ) = β, dJ ( u ) = 0 } the set of critical points of J in B at the level β (clearly, K Jβ = ∅ if β is a regular value); • J β = { u ∈ B : J ( u ) ≤ β } the sublevel set of J associated with β ; • B B r ( u ) = { u ∈ B : k u − u k B ≤ r } the closed ball in B centered at u of radius r , with boundary ∂B B r ( u ); • dist B ( u, C ) = inf v ∈C k v − u k B the distance from C to u ∈ B .We say that a sequence ( u n ) n ⊂ B is a Palais–Smale sequence at the level β , brieftly a ( P S ) β –sequence , if J ( u n ) → β and k dJ ( u n ) k B ′ → n → + ∞ .The functional J satisfies the Palais–Smale condition at the level β in B , ( P S ) β condition for short, if every ( P S ) β –sequence admits a subsequence that con-verges in B .Now, we assume that J is even and J (0) = 0, and use the Z -cohomologicalindex of Fadell and Rabinowitz in [20] and the associated pseudo-index of Benciin [7] to obtain multiple critical points.Let us first recall the definition and some basic properties of the cohomolog-ical index.Let A be the class of symmetric subsets of B \ { } . For A ∈ A , we denoteby • A = A/ Z the quotient space of A with each u and − u identified, • f : A → R P ∞ the classifying map of A , • f ∗ : H ∗ ( R P ∞ ) → H ∗ ( A ) the induced homomorphism of the Alexander–Spanier cohomology rings. 4hen the cohomological index of A is defined by i ( A ) = ( sup (cid:8) m ≥ f ∗ ( ω m − ) = 0 (cid:9) if A = ∅ , A = ∅ , where ω ∈ H ( R P ∞ ) is the generator of the polynomial ring H ∗ ( R P ∞ ) = Z [ ω ].For example, if S n − is the unit sphere in R n , 1 ≤ n < + ∞ , then i ( S n − ) = n as the classifying map of S n − is the inclusion R P n − ⊂ R P ∞ , which inducesisomorphisms on H q for q ≤ n − Proposition 2.1 (Fadell–Rabinowitz [20]) . 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 partic-ular, if A ⊂ B ), then i ( A ) ≤ i ( B ) . Thus, equality holds when the map isan odd homeomorphism; ( i ) Dimension: i ( A ) ≤ dim B ; ( i ) Continuity: If A ∈ 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 bea δ -neighborhood N δ ( A ) = (cid:8) u ∈ B : dist B ( u, A ) ≤ δ (cid:9) ; ( i ) Subadditivity: If
A, B ∈ A are closed, then i ( A ∪ B ) ≤ i ( A ) + i ( B ) ; ( i ) Stability: If SA is the suspension of A = ∅ , obtained as the quotient spaceof A × [ − , with A × { } and A × {− } collapsed to different points,then i ( SA ) = i ( A ) + 1 ; ( i ) Piercing property: If
A, A , A are closed and ϕ : A × [0 , → A ∪ A is acontinuous mapping such that ϕ ( − u, t ) = − ϕ ( u, t ) for all ( u, t ) ∈ A × [0 , , ϕ ( A × [0 , is closed, 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 B . For any integer k ≥
1, let A k = (cid:8) A ∈ A : A is compact and i ( A ) ≥ k (cid:9) and set c k := inf A ∈A k max u ∈ A J ( u ) . Since A k +1 ⊂ A k , then c k ≤ c k +1 . Furthermore, for any k -dimensional subspace V of B and δ >
0, by ( i ) we have ∂B B δ (0) ∩ V ∈ A k , while by continuity itresults sup u ∈ ∂B B δ (0) J ( u ) → J (0) as δ → , c k ≤ J (0).The following theorem is standard (see, e.g., [24, Proposition 3.36]). Theorem 2.2.
Assume that J ∈ C ( B , R ) is even and J (0) = 0 . If −∞ < c k ≤ · · · ≤ c k + m − < and J satisfies ( P S ) c i for i = k, . . . , k + m − , then J has m distinct pairs ofnontrivial critical points. Now, let us recall the definition and some basic properties of a pseudo–indexrelated to the cohomological index i .Let A ∗ denote the class of symmetric subsets of B , let M ∈ A be closed,and let Γ denote the group of odd homeomorphisms γ of B such that γ | J is theidentity. Then the pseudo-index of A ∈ A ∗ related to i , M , and Γ is defined by i ∗ ( A ) = min γ ∈ Γ i ( γ ( A ) ∩ M ) . Proposition 2.3 (Benci [7]) . The pseudo-index i ∗ : A ∗ → N ∪ { , + ∞} has thefollowing properties: ( i ∗ ) If A ⊂ B , then i ∗ ( A ) ≤ i ∗ ( B ) ; ( i ∗ ) If η ∈ Γ , then i ∗ ( η ( A )) = i ∗ ( A ) ; ( i ∗ ) If A ∈ A ∗ and B ∈ A are closed, then i ∗ ( A ∪ B ) ≤ i ∗ ( A ) + i ( B ) . For any integer k ≥ k ≤ i ( M ), let A ∗ k = (cid:8) A ∈ A ∗ : A is compact and i ∗ ( A ) ≥ k (cid:9) and set c ∗ k := inf A ∈A ∗ k max u ∈ A J ( u ) . From A ∗ k +1 ⊂ A ∗ k , it follows c ∗ k ≤ c ∗ k +1 . The following theorem is standard (see,e.g., [24, Proposition 3.42]). Theorem 2.4.
Assume that J ∈ C ( B , R ) is even and J (0) = 0 . If < c ∗ k ≤ · · · ≤ c ∗ k + m − < + ∞ and J satisfies ( P S ) c ∗ i for i = k, . . . , k + m − , then J has m distinct pairs ofnontrivial critical points. The Palais–Smale condition
From here on, let X be the Banach space in (1.2) and let J : X → R be thefunctional in (1.1). Furthermore, we denote by • ( X ′ , k · k X ′ ) the dual space of ( X, k · k X ), • ( W − ,p ′ (Ω) , k · k W − ) the dual space of ( W ,p (Ω) , k · k ), • L q (Ω) the Lebesgue space equipped with the canonical norm | · | q for any q ≥ • meas( · ) the usual Lebesgue measure in R N .By definition, X ֒ → W ,p (Ω) and X ֒ → L ∞ (Ω) with continuous imbeddings;moreover, if p ∗ is the critical exponent, i.e. p ∗ = pNN − p if p ∈ [1 , N [, p ∗ = + ∞ otherwise, by the Sobolev Imbedding Theorem, for any 1 ≤ q < p ∗ , a constant γ q > | u | q ≤ γ q k u k for all u ∈ W ,p (Ω). (3.1)In particular, for 1 ≤ p < p ∗ , we have | u | p ≤ γ p k u k , | u | ≤ γ k u k for all u ∈ W ,p (Ω), (3.2)while, under the stronger assumption p > N , we have | u | ∞ ≤ γ ∞ k u k for all u ∈ W ,p (Ω). (3.3)Letting g ∞ be as in ( h ) and setting G ∞ ( x, t ) = R t g ∞ ( x, s ) ds , if ( h ) and( h ) hold, then g ∞ is a Carath´eodory function on Ω × R such thatsup | t |≤ r | g ∞ ( · , t ) | ∈ L ∞ (Ω) for any r >
0; (3.4)furthermore, (3.4), respectively (1.3), implies thatsup | t |≤ r | G ∞ ( · , t ) | ∈ L ∞ (Ω) for any r >
0, (3.5)lim | t |→ + ∞ G ∞ ( x, t ) | t | p = 0 uniformly a.e. in Ω . (3.6)Hence (1.3) and (3.4), respectively (3.5) and (3.6), imply that for any ε > L ε > | g ∞ ( x, t ) | ≤ L ε + ε | t | p − for a.e. x ∈ Ω, all t ∈ R , (3.7) | G ∞ ( x, t ) | ≤ L ε + ε | t | p for a.e. x ∈ Ω, all t ∈ R . (3.8)Throughout this section, we consider the parametrized family of functionals J λ : X → R defined by J λ ( u ) = 1 p Z Ω ( A ( x, u ) |∇ u | p − λ | u | p ) dx − Z Ω G ∞ ( x, u ) dx. (3.9)7 roposition 3.1. Let p ≥ and assume that the conditions ( H ) , ( h ) and ( h ) hold. If ( u n ) n ⊂ X , u ∈ X are such that k u n − u k → as n → + ∞ (3.10) and k > exists so that | u n | ∞ ≤ k for all n ∈ N , (3.11) then for any λ ∈ R , we have J λ ( u n ) → J λ ( u ) and k d J λ ( u n ) − d J λ ( u ) k X ′ → as n → + ∞ .In particular, J λ ∈ C ( X, R ) with derivative d J λ : u ∈ X d J λ ( u ) ∈ X ′ defined by h d J λ ( u ) , ϕ i = Z Ω A ( x, u ) |∇ u | p − ∇ u · ∇ ϕ dx + 1 p Z Ω A t ( x, u ) ϕ |∇ u | p dx − λ Z Ω | u | p − uϕ dx − Z Ω g ∞ ( x, u ) ϕ dx, for any u , ϕ ∈ X .Proof. The proof is essentially a simpler version of [10, Proposition 3.1], but,for completeness, here we point out its main tools.First of all, consider the functional ¯ J : X → R which is defined as¯ J ( w ) = 1 p Z Ω A ( x, w ) |∇ w | p dx, w ∈ X, whose Gˆateaux differential in w along direction ϕ ( w , ϕ ∈ X ) is h d ¯ J ( w ) , ϕ i = Z Ω A ( x, w ) |∇ w | p − ∇ w · ∇ ϕ dx + 1 p Z Ω A t ( x, w ) ϕ |∇ w | p dx. Now, let ( u n ) n ⊂ X , u ∈ X be such that (3.10) and (3.11) hold. A directconsequence of (3.11) and ( H ) is the existence of a constant b > b dependingonly on k and | u | ∞ , such that for all n ∈ N and a.e. x ∈ Ω we have | A ( x, u ) | ≤ b, | A ( x, u n ) | ≤ b, | A t ( x, u ) | ≤ b, | A t ( x, u n ) | ≤ b. (3.12)On the other hand, by (3.10) it follows that( u n , ∇ u n ) → ( u, ∇ u ) in measure on Ω.Thus, being A and A t Carath´eodory functions, there results A ( x, u n ) |∇ u n | p → A ( x, u ) |∇ u | p ,A t ( x, u n ) |∇ u n | p → A t ( x, u ) |∇ u | p ,A ( x, u n ) |∇ u n | p − ∇ u n → A ( x, u ) |∇ u | p − ∇ u
8n measure on Ω, too, i.e., for all ε > n,ε ) → , meas(Ω tn,ε ) → , meas(Ω p − n,ε ) → , (3.13)whereΩ n,ε = { x ∈ Ω : (cid:12)(cid:12) A ( x, u n ) |∇ u n | p − A ( x, u ) |∇ u | p (cid:12)(cid:12) ≥ ε } , Ω tn,ε = { x ∈ Ω : (cid:12)(cid:12) A t ( x, u n ) |∇ u n | p − A t ( x, u ) |∇ u | p (cid:12)(cid:12) ≥ ε } , Ω p − n,ε = { x ∈ Ω : (cid:12)(cid:12) A ( x, u n ) |∇ u n | p − ∇ u n − A ( x, u ) |∇ u | p − ∇ u (cid:12)(cid:12) ≥ ε } . So, fixing ε >
0, by applying Vitali–Hahn–Saks Theorem and taking into ac-count the absolutely continuity of the Lebesgue integral, there exists δ ε > δ ε ≤ ε ), such that if E ⊂ Ω, meas( E ) < δ ε , then Z E |∇ u | p dx < ε, Z E |∇ u n | p dx < ε for all n ∈ N ; (3.14)moreover, by (3.13) an integer n ε exists such thatmeas(Ω n,ε ) < δ ε , meas(Ω tn,ε ) < δ ε , meas(Ω p − n,ε ) < δ ε for all n ≥ n ε . (3.15)Then, from (3.12), (3.14), (3.15) and direct computations it follows that | ¯ J ( u n ) − ¯ J ( u ) | ≤ p Z Ω n,ε ( | A ( x, u n ) ||∇ u n | p + | A ( x, u ) ||∇ u | p ) dx + 1 p Z Ω \ Ω n,ε (cid:12)(cid:12) A ( x, u n ) |∇ u n | p − A ( x, u ) |∇ u | p (cid:12)(cid:12) dx < b ε for all n ≥ n ε , where b > ε . Whence,¯ J ( u n ) → ¯ J ( u ).Now, fixing any ε > ϕ ∈ X , we have |h d ¯ J ( u n ) − d ¯ J ( u ) , ϕ i| ≤ Z Ω p − n,ε | A ( x, u n ) ||∇ u n | p − |∇ ϕ | dx + Z Ω p − n,ε | A ( x, u ) ||∇ u | p − |∇ ϕ | dx + Z Ω \ Ω p − n,ε (cid:12)(cid:12) A ( x, u n ) |∇ u n | p − ∇ u n − A ( x, u ) |∇ u | p − ∇ u (cid:12)(cid:12) |∇ ϕ | dx + 1 p Z Ω tn,ε ( | A t ( x, u n ) ||∇ u n | p + | A t ( x, u ) ||∇ u | p ) | ϕ | dx + 1 p Z Ω \ Ω tn,ε (cid:12)(cid:12) A t ( x, u n ) |∇ u n | p − A t ( x, u ) |∇ u | p (cid:12)(cid:12) | ϕ | dx. Thus, reasoning as above, from (1.2), (3.12), (3.14), (3.15) and direct computa-tions, a constant b > b independent of ε and ϕ , exists such that |h d ¯ J ( u n ) − d ¯ J ( u ) , ϕ i| ≤ (cid:0) bε − p + ε (meas(Ω)) − p (cid:1) k ϕ k + εp (cid:0) b + meas(Ω) (cid:1) | ϕ | ∞ ≤ b max { ε, ε − p }k ϕ k X n large enough. Hence, by the arbitrariness of ε and ϕ ∈ X , we have k d ¯ J ( u n ) − d ¯ J ( u ) k X ′ → . On the other hand, from (3.7) and standard arguments (see, e.g., [17, Subsection2.1]), it follows that the functional u ∈ W ,p (Ω) λp Z Ω | u | p dx + Z Ω G ∞ ( x, u ) dx ∈ R is C in ( W ,p (Ω) , k · k ), and so in ( X, k · k X ); hence, the tesis follows.Thus, if conditions ( H ), ( h ) and ( h ) hold, for each p ≥
1, problem ( P )has a variational structure and its bounded weak solutions are critical points of J = J λ ∞ in the Banach space X .As our aim is applying variational methods to the study of critical pointsof J in the asymptotically p -linear case, we introduce the following furtherconditions:( H ) we havelim | t |→ + ∞ A t ( x, t ) t = 0 uniformly a.e. in Ω;( H ) there exists α > α ≤
1) such that A ( x, t ) + 1 p A t ( x, t ) t ≥ α A ( x, t ) a.e. in Ω, for all t ∈ R . Remark 3.2.
Hypothesis ( H ) implies thatlim inf | t |→ + ∞ A t ( x, t ) t = 0 a.e. in Ω;hence, condition ( H ) is quite natural. Remark 3.3.
By ( H ) and ( H ), for each ε >
0, a radius R ε > | A ( x, t ) − A ∞ ( x ) | < ε for a.e. x ∈ Ω, if | t | ≥ R ε , (3.16) | A t ( x, t ) t | < ε for a.e. x ∈ Ω, if | t | ≥ R ε . (3.17)Since (3.16) implies | A ( x, t ) | ≤ | A ∞ | ∞ + ε for a.e. x ∈ Ω, if | t | ≥ R ε ,it follows from ( H ) and (3.17) that | A ( x, t ) | ≤ b, | A t ( x, t ) | ≤ b for a.e. x ∈ Ω, for all t ∈ R , (3.18)for a suitable b >
0. 10 emark 3.4.
In the proof of Proposition 3.1, assumption (3.11) is required onlyfor the boundedness conditions (3.12), which are necessary for investigating thesmoothness of ¯ J . However, this uniform bound can be avoided in the hypotheses( H ) and ( H ) as (3.18) holds. Whence, in this set of hypotheses, for any p ≥ J is continuous in W ,p (Ω), and so is J λ for any λ ∈ R . However,in general, J λ is not C in W ,p (Ω) as it is Gˆateaux differentiable in u ∈ W ,p (Ω)only along bounded directions.Here and in the following, by σ ( A ∞ p ) we denote the spectrum of the operator A ∞ p : u ∈ W ,p (Ω)
7→ − div( A ∞ ( x ) |∇ u | p − ∇ u ) ∈ W − ,p ′ (Ω) , (3.19)which is the set of λ ∈ R such that the nonlinear eigenvalue problem (cid:26) − div( A ∞ ( x ) |∇ u | p − ∇ u ) = λ | u | p − u in Ω, u = 0 on ∂ Ωhas a nontrivial (weak) solution in W ,p (Ω), i.e. some u ∈ W ,p (Ω), u Z Ω A ∞ ( x ) |∇ u | p − ∇ u · ∇ ϕ dx = λ Z Ω | u | p − uϕ dx for all ϕ ∈ W ,p (Ω). Proposition 3.5. If p > , the hypotheses ( H ) – ( H ) , ( h ) and ( h ) hold, and λ σ ( A ∞ p ) , then for all β ∈ R , each ( P S ) β –sequence of J λ in X is bounded inthe W ,p –norm.Proof. Taking β >
0, let ( u n ) n ⊂ X be a ( P S ) β –sequence, i.e. J λ ( u n ) → β and k d J λ ( u n ) k X ′ → n → + ∞ . (3.20)Arguing by contradiction, we suppose that k u n k → + ∞ . (3.21)Hence, without loss of generality, for any n ∈ N we assume k u n k >
0, anddefine v n = u n k u n k , hence k v n k = 1 . (3.22)Then, there exists v ∈ W ,p (Ω) such that, up to subsequences, we have v n ⇀ v weakly in W ,p (Ω), (3.23) v n → v strongly in L q (Ω) for each 1 ≤ q < p ∗ , (3.24) v n → v a.e. in Ω. (3.25)In order to yield a contradiction, we organize the proof in some steps:1. v
0; 11. a constant b > µ > n µ ∈ N suchthat Z Ω \ Ω µn |∇ v n | p dx ≤ b max { µ, µ p } for all n ≥ n µ , (3.26)whereΩ µn = { x ∈ Ω : | v n ( x ) | ≥ µ } ; (3.27)3. taking Ω = { x ∈ Ω : v ( x ) = 0 } , if meas(Ω ) > Z Ω |∇ v | p dx = 0 , (3.28)which implies ∇ v = 0 a.e. in Ω and, clearly, Z Ω A ∞ ( x ) |∇ v | p − ∇ v ·∇ ϕ dx = Z Ω λ | v | p − vϕ dx for all ϕ ∈ W ,p (Ω);4. taking any ϕ ∈ X we have Z Ω A ∞ ( x ) |∇ v n | p − ∇ v n · ∇ ϕ dx − λ Z Ω | v n | p − v n ϕ dx →
0; (3.29)5. λ ∈ σ ( A ∞ p ), in contradiction with the hypotheses.For simplicity, here and in the following b i denotes any strictly positive constantindependent of n . Step 1.
Firstly, let us point out that for any ε > (cid:12)(cid:12) Z Ω G ∞ ( x, u n ) k u n k p dx (cid:12)(cid:12) ≤ L ε meas(Ω) k u n k p + εγ pp ;hence, (3.21) implies (cid:12)(cid:12) Z Ω G ∞ ( x, u n ) k u n k p dx (cid:12)(cid:12) ≤ ε (1 + γ pp ) for all n ≥ n ε for n ε large enough. Thus, we have Z Ω G ∞ ( x, u n ) k u n k p dx → . (3.30)Furthermore, (3.20) and (3.21) give J λ ( u n ) k u n k p → . (3.31)12ow, arguing by contradiction, assume v ≡
0. Then, from (3.24) it follows Z Ω | v n | p dx → , (3.32)but for any n ∈ N , condition ( H ), (3.9) and (3.22) imply0 < α p = α p k v n k p ≤ p Z Ω A ( x, u n ) |∇ v n | p dx = J λ ( u n ) k u n k p + λp Z Ω | v n | p dx + Z Ω G ∞ ( x, u n ) k u n k p dx in contradiction with (3.30)–(3.32). Step 2.
Taking any φ ∈ X , we have h d J λ ( u n ) , φ k u n k p − i = Z Ω A ( x, u n ) |∇ v n | p − ∇ v n · ∇ φ dx + 1 p Z Ω A t ( x, u n ) k u n k |∇ v n | p φ dx − λ Z Ω | v n | p − v n φ dx − Z Ω g ∞ ( x, u n ) k u n k p − φ dx. (3.33)Fix any µ > ε >
0. From one hand, (cid:12)(cid:12) h d J λ ( u n ) , φ k u n k p − i (cid:12)(cid:12) ≤ k d J λ ( u n ) k X ′ k u n k p − k φ k for all n ∈ N ; (3.34)while from (3.2), (3.7), (3.22) and the H¨older inequality it follows (cid:12)(cid:12) Z Ω g ∞ ( x, u n ) k u n k p − φ dx (cid:12)(cid:12) ≤ (cid:18) εγ pp + L ε γ k u n k p − (cid:19) k φ k for all n ∈ N . (3.35)On the other hand, by (3.22) and (3.27) we have | u n ( x ) | > µ k u n k for all x ∈ Ω µn , n ∈ N . (3.36)Then, from (3.21) an integer n µ,ε , independent of φ , exists such that (3.20) and(3.34) imply (cid:12)(cid:12) h d J λ ( u n ) , φ k u n k p − i (cid:12)(cid:12) ≤ ε k φ k for all n ≥ n µ,ε , (3.37)while inequality (3.35) becomes (cid:12)(cid:12) Z Ω g ∞ ( x, u n ) k u n k p − φ dx (cid:12)(cid:12) ≤ ε (cid:0) γ pp + 1 (cid:1) k φ k for all n ≥ n µ,ε , (3.38)and from (3.17) and (3.36) it follows | A t ( x, u n ( x )) u n ( x ) | < ε for a.e. x ∈ Ω µn , if n ≥ n µ,ε . (3.39)13ence, for all n ≥ n µ,ε by (3.22), (3.27) and (3.39), direct computations imply Z Ω µn (cid:12)(cid:12) A t ( x, u n ) u n (cid:12)(cid:12) | v n | |∇ v n | p dx ≤ εµ , (3.40)and then (cid:12)(cid:12) Z Ω µn A t ( x, u n ) k u n k |∇ v n | p φ dx (cid:12)(cid:12) ≤ Z Ω µn (cid:12)(cid:12) A t ( x, u n ) u n (cid:12)(cid:12) | v n | |∇ v n | p | φ | dx ≤ εµ | φ | ∞ . (3.41)Now, for any n ∈ N , let us consider the cut–off function T µ : R → R such that T µ ( t ) = (cid:26) t if | t | < µ , µ t | t | if | t | ≥ µ .As T µ ( v n ( x )) = ( v n ( x ) for a.e. x ∈ Ω \ Ω µn , µ v n ( x ) | v n ( x ) | if x ∈ Ω µn , (3.42) ∇ T µ ( v n ( x )) = (cid:26) ∇ v n ( x ) for a.e. x ∈ Ω \ Ω µn ,0 for a.e. x ∈ Ω µn , (3.43)then T µ ( v n ) ∈ X with k T µ ( v n ) k ≤ , | T µ ( v n ) | ∞ ≤ µ. (3.44)Thus, applying (3.33) on the test function φ = T µ ( v n ), we have Z Ω A ( x, u n ) |∇ v n | p − ∇ v n · ∇ T µ ( v n ) dx + 1 p Z Ω A t ( x, u n ) k u n k|∇ v n | p T µ ( v n ) dx = h d J λ ( u n ) , T µ ( v n ) k u n k p − i + λ Z Ω | v n | p − v n T µ ( v n ) dx + Z Ω g ∞ ( x, u n ) k u n k p − T µ ( v n ) dx, where (3.22), (3.42) and (3.43) imply Z Ω A ( x, u n ) |∇ v n | p − ∇ v n · ∇ T µ ( v n ) dx + 1 p Z Ω A t ( x, u n ) k u n k|∇ v n | p T µ ( v n ) dx = Z Ω \ Ω µn A ( x, u n ) |∇ v n | p dx + 1 p Z Ω \ Ω µn A t ( x, u n ) k u n k v n |∇ v n | p dx + µp Z Ω µn A t ( x, u n ) k u n k v n | v n | |∇ v n | p dx = Z Ω \ Ω µn (cid:0) A ( x, u n ) + 1 p A t ( x, u n ) u n (cid:1) |∇ v n | p dx + µp Z Ω µn A t ( x, u n ) u n | v n | |∇ v n | p dx, while (3.1) with q = p −
1, (3.22), (3.27) and (3.42) give (cid:12)(cid:12) Z Ω | v n | p − v n T µ ( v n ) dx (cid:12)(cid:12) ≤ µ p meas(Ω) + µγ p − p − . Z Ω \ Ω µn (cid:0) A ( x, u n ) + 1 p A t ( x, u n ) u n (cid:1) |∇ v n | p dx ≤ ε ( γ pp + 2 + 1 p )+ | λ | ( µ p meas(Ω) + µγ p − p − ) for all n ≥ n µ,ε . (3.45)As µ and ε are any and independent one from the other, we can fix ε = µ ;hence, n µ = n µ,µ and (3.45) becomes Z Ω \ Ω µn (cid:0) A ( x, u n ) + 1 p A t ( x, u n ) u n (cid:1) |∇ v n | p dx ≤ b max { µ, µ p } (3.46)for all n ≥ n µ , where b = γ pp + 2 + p + | λ | meas(Ω) + | λ | γ p − p − > H ) and ( H ) we have α α Z Ω \ Ω µn |∇ v n | p dx ≤ α Z Ω \ Ω µn A ( x, u n ) |∇ v n | p dx ≤ Z Ω \ Ω µn (cid:0) A ( x, u n ) + 1 p A t ( x, u n ) u n (cid:1) |∇ v n | p dx ;whence, summing up, (3.46) implies (3.26) with b = b α α . Step 3.
Firstly, we claim that if meas(Ω ) > µ > n µ ∈ N such thatmeas(Ω ∩ Ω µn ) = 0 for all n ≥ n µ . (3.47)In fact, arguing by contradiction, we assume that ¯ µ > ∩ Ω ¯ µn ) > n ∈ N .From (3.25) a set ¯Ω ⊂ Ω exists such that meas( ¯Ω) = 0 and v n ( x ) → v ( x ) forall x ¯Ω; whence, for all n ∈ N it results meas((Ω ∩ Ω ¯ µn ) \ ¯Ω) > x ∈ (Ω ∩ Ω ¯ µn ) \ ¯Ω we have both | v n ( x ) | ≥ ¯ µ for all n ∈ N and v n ( x ) → n → + ∞ : a contradiction.Now, from Step 2 , (3.26) and (3.47) imply that Z Ω |∇ v n | p dx = Z Ω \ Ω µn |∇ v n | p dx ≤ Z Ω \ Ω µn |∇ v n | p dx ≤ b max { µ, µ p } for all n large enough, where from the weak lower semi–continuity of norms wehave Z Ω |∇ v | p dx ≤ lim inf n → + ∞ Z Ω |∇ v n | p dx ≤ b max { µ, µ p } . Hence, for the arbitrariness of µ >
0, (3.28) holds.15 tep 4.
Fixing any ρ >
0, we introduce another cut–off function χ ρ ∈ C ( R , R )which has to be even, nondecreasing in [0 , + ∞ [ and such that χ ρ ( t ) = (cid:26) | t | < ρ ,1 if | t | ≥ ρ , with | χ ′ ρ ( t ) | ≤ t ∈ R .Taking any ϕ ∈ X , n ∈ N , we denote ω ρ,n = χ ρ ( v n ) ϕ , hence, by definition, ω ρ,n ( x ) = (cid:26) | v n ( x ) | < ρ , ϕ ( x ) if | v n ( x ) | > ρ , (3.48) ∇ ω ρ,n ( x ) = (cid:26) | v n ( x ) | < ρ , ∇ ϕ ( x ) if | v n ( x ) | > ρ , (3.49)so direct computations imply ω ρ,n ∈ X with k ω ρ,n k ≤ k ϕ k + 2 | ϕ | ∞ ≤ k ϕ k X , | ω ρ,n | ∞ ≤ | ϕ | ∞ ≤ k ϕ k X . (3.50)Thus, we consider the test function φ = ω ρ,n in (3.33), and, from (3.27) with µ = ρ , we have Z Ω ρn A ( x, u n ) |∇ v n | p − ∇ v n · ∇ ω ρ,n dx − λ Z Ω ρn | v n | p − v n ω ρ,n dx = h d J λ ( u n ) , ω ρ,n k u n k p − i − p Z Ω ρn A t ( x, u n ) k u n k|∇ v n | p ω ρ,n dx + Z Ω g ∞ ( x, u n ) k u n k p − ω ρ,n dx. (3.51)Whence, by using (3.37) with φ = ω ρ,n and ε = ρ , (3.38) with φ = ω ρ,n and ε = ρ , (3.41) with φ = ω ρ,n , ε = ρ and µ = ρ , equation (3.51) with estimates(3.50) implies (cid:12)(cid:12) Z Ω ρn A ( x, u n ) |∇ v n | p − ∇ v n · ∇ ω ρ,n dx − λ Z Ω ρn | v n | p − v n ω ρ,n dx (cid:12)(cid:12) ≤ ρ k ω ρ,n k + ρp | ω ρ,n | ∞ + ρ ( γ pp + 1) k ω ρ,n k ≤ ρ b k ϕ k X (3.52)for all n ≥ n ρ , with n ρ large enough and b = 3 γ pp + 6 + p .On the other hand, by (3.16) with any ε >
0, (3.21) and (3.36) with µ = ε (andwith Ω εn as in (3.27)), an integer n ε exists such that | A ( x, u n ( x )) − A ∞ ( x ) | < ε for a.e. x ∈ Ω εn , if n ≥ n ε , (3.53)then, taking any φ ∈ X , for all n ≥ n ε by (3.22) and (3.53), the H¨older inequalityand direct computations imply (cid:12)(cid:12) Z Ω εn (cid:0) A ( x, u n ) − A ∞ ( x ) (cid:1) |∇ v n | p − ∇ v n · ∇ φ dx (cid:12)(cid:12) ≤ ε k φ k . (3.54)16n particular, if we take ε = ρ and φ = ω ρ,n in (3.54), an integer n ρ ≥ n ρ is suchthat from (3.50) and (3.52) it follows (cid:12)(cid:12) Z Ω ρn A ∞ ( x ) |∇ v n | p − ∇ v n · ∇ ω ρ,n dx − λ Z Ω ρn | v n | p − v n ω ρ,n dx (cid:12)(cid:12) ≤ ρb k ϕ k X (3.55)for all n ≥ n ρ , with b = 3 + b .Now, from definitions (3.27) with µ = 2 ρ , direct computations and (3.48), (3.49)imply (cid:12)(cid:12) Z Ω A ∞ ( x ) |∇ v n | p − ∇ v n · ∇ ϕdx − λ Z Ω | v n | p − v n ϕ dx (cid:12)(cid:12) ≤ (cid:12)(cid:12) Z Ω \ Ω ρn A ∞ ( x ) |∇ v n | p − ∇ v n · ∇ ((1 − χ ρ ( v n )) ϕ ) dx (cid:12)(cid:12) + | λ | (cid:12)(cid:12) Z Ω \ Ω ρn | v n | p − v n (1 − χ ρ ( v n )) ϕ dx (cid:12)(cid:12) + (cid:12)(cid:12) Z Ω ρn A ∞ ( x ) |∇ v n | p − ∇ v n · ∇ ω ρ,n dx − λ Z Ω ρn | v n | p − v n ω ρ,n dx (cid:12)(cid:12) , where (3.26) with µ = 2 ρ (in Step 2 ), (3.50), ( H ) and the H¨older inequalitygive (cid:12)(cid:12) Z Ω \ Ω ρn A ∞ ( x ) |∇ v n | p − ∇ v n · ∇ ((1 − χ ρ ( v n )) ϕ ) dx (cid:12)(cid:12) ≤ | A ∞ | ∞ Z Ω \ Ω ρn |∇ v n | p dx ! − p k ϕ − ω ρ,n k≤ b max { ρ p − , ρ − p } k ϕ k X for all n ≥ n ρ , with n ρ large enough and b > ρ and ϕ ,while (3.2) implies (cid:12)(cid:12) Z Ω \ Ω ρn | v n | p − v n (1 − χ ρ ( v n )) ϕ dx (cid:12)(cid:12) ≤ (2 ρ ) p − Z Ω \ Ω ρn | ϕ | dx ≤ ρ p − p − γ k ϕ k ≤ ρ p − p − γ k ϕ k X . Whence, taking n ρ ∈ N large enough, from (3.55) it follows (cid:12)(cid:12) Z Ω A ∞ ( x ) |∇ v n | p − ∇ v n · ∇ ϕdx − λ Z Ω | v n | p − v n ϕdx (cid:12)(cid:12) ≤ max { ρ − p , ρ, ρ p − } b k ϕ k X for all n ≥ n ρ , (3.56)with b > ρ and ϕ . Thus, from the arbitrariness of ρ ,(3.56) implies (3.29). 17 tep 5. Firstly, we apply (3.29) to ϕ = v n − v by taking into account (3.24),then by considering (3.23) we have Z Ω A ∞ ( x ) (cid:0) |∇ v n | p − ∇ v n − |∇ v | p − ∇ v (cid:1) · (cid:0) ∇ v n − ∇ v (cid:1) dx → . Whence, from the properties of A ∞ and the uniform convexity of ( W ,p (Ω) , k·k )(as p >
1) it follows that k v n − v k →
0. Thus, Z Ω A ∞ ( x ) |∇ v n | p − ∇ v n · ∇ ϕdx → Z Ω A ∞ ( x ) |∇ v | p − ∇ v · ∇ ϕdx for any ϕ ∈ W ,p (Ω), and by (3.24) and (3.29) it results Z Ω A ∞ ( x ) |∇ v | p − ∇ v · ∇ ϕdx = λ Z Ω | v | p − vϕdx, for any ϕ ∈ X , or better any ϕ ∈ W ,p (Ω).As pointed out in Remark 3.4, even if A and A t are bounded, we cannotsimply replace X with W ,p (Ω), so the classical Palais–Smale condition for J in X requires the convergence not only in the W ,p –norm, but also in the L ∞ –norm. This problem can be overcome if p > N since then X = W ,p (Ω) andthe two norms k · k and k · k X are equivalent. Proposition 3.6. If p > N and the hypotheses ( H ) – ( H ) , ( h ) and ( h ) hold,then for any λ σ ( A ∞ p ) , the functional J λ satisfies the ( P S ) β condition in W ,p (Ω) at each level β ∈ R .Proof. Taking β >
0, let ( u n ) n ⊂ W ,p (Ω) be a ( P S ) β –sequence, i.e. (3.20)holds. From Proposition 3.5 and (3.3) a constant L > k u n k ≤ L and | u n | ∞ ≤ γ ∞ L for all n ∈ N . (3.57)Hence, up to subsequences, there exists u ∈ W ,p (Ω) such that u n ⇀ u weakly in W ,p (Ω), (3.58) u n → u strongly in L q (Ω) for each q ≥
1, (3.59) u n → u a.e. in Ω, (3.60)and h ∈ L p (Ω) exists such that | u n ( x ) | ≤ h ( x ) a.e. in Ω, for all n ∈ N . (3.61)We claim that u n → u strongly in W ,p (Ω).This proof is essentially as in Step 4. of the proof of [10, Proposition 4.6] andfollows some arguments in [3] according to an idea introduced in [8]. Anyway,for completeness, here we prove it. 18et us consider the real map ψ ( t ) = t e ηt , where η > ( β β ) will be fixed once β , β > β ψ ′ ( t ) − β | ψ ( t ) | > β t ∈ R . (3.62)Taking w n = u n − u , from (3.57) it follows | w n | ∞ ≤ γ ∞ L + | u | ∞ ;moreover, (3.58) – (3.61) imply w n ⇀ W ,p (Ω), (3.63) w n → L q (Ω) for all q ≥ w n → | w n ( x ) | ≤ h ( x ) + | u ( x ) | a.e. in Ω, for all n ∈ N , with h + | u | ∈ L p (Ω).Hence, σ > | ψ ( w n ) | ≤ σ , < ψ ′ ( w n ) ≤ σ a.e. in Ω, for all n ∈ N , (3.64) ψ ( w n ) → , ψ ′ ( w n ) → n → + ∞ . (3.65)Thus, ( ψ ( w n )) n is bounded in W ,p (Ω), and (3.20) implies h d J λ ( u n ) , ψ ( w n ) i → n → + ∞ , (3.66)where it is h d J λ ( u n ) , ψ ( w n ) i = Z Ω ψ ′ ( w n ) A ( x, u n ) |∇ u n | p − ∇ u n · ∇ w n dx + 1 p Z Ω A t ( x, u n ) ψ ( w n ) |∇ u n | p dx − λ Z Ω | u n | p − u n ψ ( w n ) dx − Z Ω g ∞ ( x, u n ) ψ ( v k,n ) dx. (3.67)By (3.7), (3.60), (3.61), (3.64) and (3.65), the Lebesgue Dominated ConvergenceTheorem implies Z Ω | u n | p − u n ψ ( w n ) dx → , Z Ω g ∞ ( x, u n ) ψ ( w n ) dx → Z Ω ψ ′ ( w n ) A ( x, u n ) |∇ u n | p − ∇ u n · ∇ w n dx + 1 p Z Ω A t ( x, u n ) ψ ( w n ) |∇ u n | p dx = ε ,n , (3.68)19ith ε ,n →
0. On the other hand, from ( H ) and (3.18) it follows (cid:12)(cid:12) Z Ω A t ( x, u n ) ψ ( w n ) |∇ u n | p dx (cid:12)(cid:12) ≤ bα Z Ω A ( x, u n ) | ψ ( w n ) | |∇ u n | p dx = bα Z Ω A ( x, u n ) | ψ ( w n ) | |∇ u n | p − ∇ u n · ∇ w n dx + bα Z Ω A ( x, u n ) | ψ ( w n ) | |∇ u n | p − ∇ u n · ∇ u dx, where (3.18), H¨older inequality, (3.57), (3.64), (3.65), and the Lebesgue Domi-nated Convergence Theorem give Z Ω A ( x, u n ) | ψ ( w n ) | |∇ u n | p − ∇ u n · ∇ u dx → . Whence, a sequence ε ,n → ε ,n ≥ Z Ω (cid:0) ψ ′ ( w n ) − bpα | ψ ( w n ) | (cid:1) A ( x, u n ) |∇ u n | p − ∇ u n · ∇ w n dx (3.69)for all n ∈ N . Now, taking β = 1 and β = bpα in the definition of ψ , anddenoting h n = β ψ ′ ( w n ) − β | ψ ( w n ) | , from (3.62) and (3.64) it follows12 ≤ h n ( x ) ≤ σ (1 + β ) a.e. in Ω, for all n ∈ N ; (3.70)while from (3.65) it is h n ( x ) → n → + ∞ . (3.71)Moreover, it is Z Ω h n A ( x, u n ) |∇ u n | p − ∇ u n · ∇ w n dx = Z Ω A ( x, u ) |∇ u | p − ∇ u · ∇ w n dx + Z Ω (cid:0) h n A ( x, u n ) − A ( x, u ) (cid:1) |∇ u | p − ∇ u · ∇ w n dx + Z Ω h n A ( x, u n ) (cid:0) |∇ u n | p − ∇ u n − |∇ u | p − ∇ u (cid:1) · ∇ w n dx, where (3.63) implies Z Ω A ( x, u ) |∇ u | p − ∇ u · ∇ w n dx → , while H¨older inequality, (3.57), and also (3.18), (3.60), (3.70), (3.71) and theLebesgue Dominated Convergence Theorem, imply Z Ω (cid:0) h n A ( x, u n ) − A ( x, u ) (cid:1) |∇ u | p − ∇ u · ∇ w n dx → . (cid:0) |∇ u n | p − ∇ u n − |∇ u | p − ∇ u (cid:1) · ∇ w n ≥ H ), (3.69) and (3.70) give ε ,n ≥ Z Ω h n A ( x, u n ) (cid:0) |∇ u n | p − ∇ u n − |∇ u | p − ∇ u (cid:1) · ∇ w n dx ≥ α Z Ω (cid:0) |∇ u n | p − ∇ u n − |∇ u | p − ∇ u (cid:1) · ∇ w n dx ≥ . for a suitable ε ,n →
0. Whence, k u n − u k → In addition to the hypotheses ( H )–( H ), ( h ) and ( h ), we assume( h ) there exist λ ∈ R and a (Carath´eodory) function g : Ω × R → R suchthat f ( x, t ) = λ | t | p − t + g ( x, t )and lim t → g ( x, t ) | t | p − = 0 uniformly a.e. in Ω . From ( h ) it followslim t → G ( x, t ) | t | p = 0 uniformly a.e. in Ω , (4.1)where G ( x, t ) = R Ω g ( x, s ) ds .Moreover, if we write A ( x ) = A ( x, H ) implies A ∈ L ∞ (Ω), whilefrom ( H ) it follows A ( x ) ≥ α > H ) and ( H )imply (3.18); whence,lim t → A ( x, t ) = A ( x ) uniformly a.e. in Ω . (4.2)For simplicity, as in (3.19), we introduce the operator A p : u ∈ W ,p (Ω)
7→ − div( A ( x ) |∇ u | p − ∇ u ) ∈ W − ,p ′ (Ω)and denote its spectrum by σ ( A p ).For ♯ = 0 , ∞ , let I ♯ ( u ) = Z Ω A ♯ ( x ) |∇ u | p dx, u ∈ W ,p (Ω) , M ♯ = (cid:8) u ∈ W ,p (Ω) : I ♯ ( u ) = 1 (cid:9) . Since the hypotheses imply that A ♯ ∈ L ∞ (Ω) and A ♯ ( x ) ≥ α > x ∈ Ω, (4.3)then M ♯ ⊂ W ,p (Ω) \{ } is a bounded symmetric complete C -Finsler manifoldradially homeomorphic to the unit sphere in W ,p (Ω). LetΨ( u ) = 1 Z Ω | u | p dx , u ∈ W ,p (Ω) \ { } . Then λ ∈ σ ( A ♯p ) if and only if λ is a critical value of Ψ | M ♯ by the Lagrangemultiplier rule.Now, let F ♯ denote the class of compact symmetric subsets of M ♯ and set λ ♯k := inf M ∈F ♯k max u ∈ M Ψ( u ) , k ≥ , where F ♯k = (cid:8) M ∈ F ♯ : i ( M ) ≥ k (cid:9) , and i is the cohomological index. Then λ ♯k ∈ σ ( A ♯p ) and 0 < λ ♯k ր + ∞ (see [24, Proposition 3.52]). In particular, λ ♯ Z Ω | u | p dx ≤ Z Ω A ♯ ( x ) |∇ u | p dx for all u ∈ W ,p (Ω). (4.4)Our main result is the following. Theorem 4.1.
Assume that p > N , ( H ) – ( H ) and ( h ) – ( h ) hold, and • A ( x, · ) is an even function for a.a. x ∈ Ω and f ( x, · ) is an odd functionfor a.a. x ∈ Ω , • λ ∞ σ ( A ∞ p ) .If m, l ∈ N , l = m , exist such that one of the two following conditions hold: (i) l > m and λ l < λ , λ ∞ < λ ∞ m +1 ; (ii) l < m and λ < λ l +1 , λ ∞ m < λ ∞ ;then problem ( P ) has at least | l − m | distinct pairs of nontrivial solutions. From here on, let p > N and assume that the hypotheses of Theorem 4.1hold. Thus, X = W ,p (Ω) and, from (1.1) and condition ( h ), J ( u ) = 1 p Z Ω ( A ( x, u ) |∇ u | p − λ ∞ | u | p ) dx − Z Ω G ∞ ( x, u ) dx, u ∈ W ,p (Ω) , h ), J ( u ) = 1 p Z Ω (cid:0) A ( x, u ) |∇ u | p − λ | u | p (cid:1) dx − Z Ω G ( x, u ) dx, u ∈ W ,p (Ω) . Furthermore, for ♯ = 0 , ∞ , we write J ♯ ( u ) = 1 p Z Ω (cid:0) A ♯ ( x ) |∇ u | p − λ ♯ | u | p (cid:1) dx = 1 p (cid:0) I ♯ ( u ) − λ ♯ | u | pp (cid:1) ; (4.5)whence, J ( u ) − J ♯ ( u ) = 1 p Z Ω (cid:0) A ( x, u ) − A ♯ ( x ) (cid:1) |∇ u | p dx − Z Ω G ♯ ( x, u ) dx, (4.6)for u ∈ W ,p (Ω).In order to prove our main result, we need the following lemmas. Lemma 4.2.
For any ε > , a suitable r ε > exists such that u ∈ W ,p (Ω) , | u | ∞ ≤ r ε = ⇒ (cid:12)(cid:12) J ( u ) − J ( u ) (cid:12)(cid:12) ≤ εp I ( u ) . (4.7) Proof.
Fixing any ε >
0, by (4.2), respectively (4.1), there is a r ε > | t | ≤ r ε = ⇒ (cid:12)(cid:12) A ( x, t ) − A ( x ) (cid:12)(cid:12) ≤ εα , (cid:12)(cid:12) G ( x, t ) (cid:12)(cid:12) ≤ ελ p | t | p a.e. in Ω , where α is as in ( H ). Then, if | u | ∞ ≤ r ε from (4.6), the estimates (4.3) and(4.4) imply (cid:12)(cid:12) J ( u ) − J ( u ) (cid:12)(cid:12) ≤ ε p α Z Ω |∇ u | p dx + ε p λ Z Ω | u | p dx ≤ εp I ( u ) . Lemma 4.3.
Let K ∞ be a compact subset of M ∞ . Then, for any ε > thereexists a constant C ε = C ( K ∞ , ε ) > such that |J ( Ru ) − J ∞ ( Ru ) | < εp I ∞ ( Ru ) + C ε for all R ≥ , u ∈ K ∞ . (4.8) Proof.
We organize the proof in different steps: (a) if K is a compact subset of W ,p (Ω), taking any ε > ρ ε = ρ ( K , ε ) > Z Ω uρε |∇ u | p dx < ε for all u ∈ K , with Ω uρ ε = { x ∈ Ω : | u ( x ) | < ρ ε } ; (b) if K is a compact subset of W ,p (Ω), taking any ε > R ∗ ε = R ∗ ( K , ε ) > (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( A ( x, Ru ) − A ∞ ( x )) |∇ u | p dx (cid:12)(cid:12)(cid:12)(cid:12) < ε for all R ≥ R ∗ ε , u ∈ K ;23 c) if K ∞ is a compact subset of M ∞ , taking any ε > C ε = C ( K ∞ , ε ) > Step (a)
Firstly, we claim that for any u ∈ W ,p (Ω) and ε > r ε > Z Ω urε |∇ u | p dx < ε. (4.9)In fact, the monotonicity property of the Lebesgue integral implieslim r → Z Ω ur |∇ u | p dx = Z Ω u |∇ u | p dx, with Ω u = { x ∈ Ω : u ( x ) = 0 } , (4.10)where Z Ω u |∇ u | p dx = 0 (4.11)not only if meas(Ω u ) = 0 but also if meas(Ω u ) > { x ∈ Ω : u ( x ) = 0 , ∇ u ( x ) = 0 } ) = 0(see, e.g., [19, Ex. 17, pp. 292]). Whence, (4.9) follows from (4.10) and (4.11).Now, arguing by contradiction, assume that for the compact K the thesis in Step(a) does not hold; hence, there exist a constant ¯ ε > u n ) n ⊂ K such that Z Ω n |∇ u n | p dx ≥ ¯ ε for all n ≥
1, with Ω n = { x ∈ Ω : | u n ( x ) | < n } . (4.12)As K is compact, then ¯ u ∈ K exists such that, up to subsequences, k u n − ¯ u k → , and so u n → ¯ u a.e. in Ω. (4.13)Now, taking ε < ¯ ε , from (4.9) applied to ¯ u , there exists ¯ r > Z Ω ¯ u ¯ r |∇ ¯ u | p dx < ε . Then, taking a ρ < ¯ r , if n is large enough, not only we have Ω n ⊂ Ω u n ρ but alsofrom (4.13) it follows that Z Ω unρ |∇ u n | p dx < ε in contradiction with (4.12). Step (b)
For the compacteness of K , a constant γ K > k u k p ≤ γ K for all u ∈ K . (4.14)24urthermore, taking ε >
0, let ρ ε > Step (a) so that Z Ω uρε |∇ u | p dx < ε b + | A ∞ | ∞ ) for all u ∈ K , (4.15)where b > H ), a constant σ ε > | A ( x, t ) − A ∞ ( x ) | < ε γ K for a.e. x ∈ Ω, if | t | ≥ σ ε , (4.16)then, taking R ∗ ε = σ ε ρ ε , for all u ∈ K , R ≥ R ∗ ε , from (3.18), (4.14) – (4.16) wehave (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( A ( x, Ru ) − A ∞ ( x )) |∇ u | p dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω uρε ( | A ( x, Ru ) | + | A ∞ ( x ) | ) |∇ u | p dx + Z Ω \ Ω uρε | A ( x, Ru ) − A ∞ ( x ) | |∇ u | p dx < ε. Step (c)
Consider K ∞ , compact subset of M ∞ , and take any ε > L ε > | G ∞ ( x, t ) | ≤ ελ ∞ p | t | p + L ε for a.a. x ∈ Ω and all t ∈ R . Hence, (4.4) implies that (cid:12)(cid:12)(cid:12)(cid:12)Z Ω G ∞ ( x, u ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε p I ∞ ( u ) + L ε meas(Ω) for all u ∈ W ,p (Ω). (4.17)Now, taking u ∈ K ∞ , from Step (b) applied to K ∞ and ε , a constant R ∗ > R ≤ R ∗ , then (3.18), (4.3), (4.6) and (4.17) imply that |J ( Ru ) − J ∞ ( Ru ) | ≤ ( R ∗ ) p p ( b + | A ∞ | ∞ ) γ K ∞ + ε p I ∞ ( Ru ) + L ε meas(Ω) . (4.18)On the contrary, if R > R ∗ , then by (4.6), (4.17) and Step (b) , as K ∞ ⊂ M ∞ ,it follows |J ( Ru ) − J ∞ ( Ru ) | ≤ R p ε p + ε p I ∞ ( Ru ) + L ε meas(Ω)= R p ε p I ∞ ( u ) + ε p I ∞ ( Ru ) + L ε meas(Ω)= εp I ∞ ( Ru ) + L ε meas(Ω) . (4.19)Thus, (4.8) follows from (4.18) and (4.19) if we choose C ε > Proof of Theorem 4.1.
Firstly, we note that by Proposition 3.6 J satisfies the( P S ) β condition for all β ∈ R .Then, we split the proof in two steps. (i) Case l > m . Let A denote the class of symmetric subsets of W ,p (Ω) \ { } and A k = (cid:8) M ∈ A : M is compact and i ( M ) ≥ k (cid:9) . Set c k := inf M ∈A k max u ∈ M J ( u ) , m + 1 ≤ k ≤ l. We will show that −∞ < c m +1 ≤ · · · ≤ c l <
0, so we can apply Theorem 2.2.In order to see that c l <
0, let ε > ε )( λ l + ε ) < λ .Then, there is M ∈ F l such that Ψ( u ) ≤ λ l + ε for all u ∈ M . Let r = r ε γ ∞ with r ε as in Lemma 4.2 and γ ∞ as in (3.3), and let˜ M = (cid:8) v = r u k u k : u ∈ M (cid:9) . As the map u ∈ M v ∈ ˜ M is an odd homeomorphism, then ˜ M is compactand i ( ˜ M ) = i ( M ) ≥ l by ( i ), so ˜ M ∈ A l . By (4.5) and (4.7), for any v ∈ ˜ M we have J ( v ) ≤ J ( v ) + εp I ( v ) = 1 p (cid:0) (1 + ε ) I ( v ) − λ | v | pp (cid:1) = r p p k u k p (cid:0) (1 + ε ) I ( u ) − λ | u | pp (cid:1) ≤ r p p k u k p (cid:18) ε − λ λ l + ε (cid:19) < c l < c m +1 > −∞ , take any M ∞ ∈ A m +1 and let ε > − ε ) λ ∞ m +1 ≥ λ ∞ . Then, consider˜ M ∞ = (cid:8) u = v/ [ I ∞ ( v )] /p : v ∈ M ∞ (cid:9) ⊂ M ∞ . As the map v ∈ M ∞ u ∈ ˜ M ∞ is an odd homeomorphism, then ˜ M ∞ iscompact and i ( ˜ M ∞ ) = i ( M ∞ ) ≥ m + 1 by ( i ). So, ˜ M ∞ ∈ F ∞ m +1 ; hence,max u ∈ ˜ M ∞ Ψ( u ) ≥ λ ∞ m +1 . Now, let C ε be as in Lemma 4.3 with K ∞ = M ∞ . By (4.5) and (4.8), for any v ∈ M ∞ , it results J ( v ) ≥ J ∞ ( v ) − εp I ∞ ( v ) − C ε = 1 p (cid:0) (1 − ε ) I ∞ ( v ) − λ ∞ | v | pp (cid:1) − C ε = I ∞ ( v ) p (cid:0) − ε − λ ∞ | u | pp (cid:1) − C ε , I ∞ ( v ) ≥
0. Whence,max v ∈ M ∞ J ( v ) ≥ − C ε ;thus c m +1 ≥ − C ε . (ii) Case l < m . Let A ∗ denote the class of symmetric subsets of W ,p (Ω), Γ thegroup of odd homeomorphisms γ of W ,p (Ω) such that γ | {J ≤ } is the identity,and i ∗ the pseudo-index related to i , ∂B Wr (0), and Γ, where W = W ,p (Ω).Then, let A ∗ k = (cid:8) M ∈ A ∗ : M is compact and i ∗ ( M ) ≥ k (cid:9) and set c ∗ k := inf M ∈A ∗ k max u ∈ M J ( u ) , l + 1 ≤ k ≤ m. We will show that 0 < c ∗ l +1 ≤ · · · ≤ c ∗ m < + ∞ if r > c ∗ l +1 >
0, fix ε > − ε ) λ l +1 > λ , define r = r ε γ ∞ with r ε as in Lemma 4.2 and γ ∞ as in (3.3), take any M ∗ ∈ A ∗ l +1 , andconsider˜ M ∗ = (cid:8) u = v [ I ( v )] /p : v ∈ M ∗ ∩ ∂B Wr (0) (cid:9) ⊂ M . The map v ∈ M ∗ ∩ ∂B Wr (0) u ∈ ˜ M ∗ is an odd homeomorphism; hence, ˜ M ∗ is compact and i ( ˜ M ∗ ) = i ( M ∗ ∩ ∂B Wr (0)) ≥ i ∗ ( M ∗ ) ≥ l + 1by ( i ). So ˜ M ∗ ∈ F l +1 and hencemax u ∈ ˜ M ∗ Ψ( u ) ≥ λ l +1 . By (4.5) and (4.7), for any v ∈ M ∗ ∩ ∂B Wr (0) we have J ( v ) ≥ J ( v ) − εp I ( v ) = 1 p (cid:0) (1 − ε ) I ( v ) − λ | v | pp (cid:1) = I ( v ) p (cid:0) − ε − λ | u | pp (cid:1) . Since I ( v ) ≥ α k v k p , it results δ := inf v ∈ ∂B Wr (0) I ( v ) ≥ α r p > . Whence, it follows thatmax v ∈ M ∗ J ( v ) ≥ max v ∈ M ∗ ∩ ∂B Wr (0) J ( v ) ≥ δp (cid:18) − ε − λ λ l +1 (cid:19) > c ∗ l +1 > c ∗ m < + ∞ , let ε > ε )( λ ∞ m + ε ) < λ ∞ .There is a M ∗∞ ∈ F ∞ m such that Ψ( u ) ≤ λ ∞ m + ε for all u ∈ M ∗∞ . Let C ε be asin Lemma 4.3 with K ∞ = M ∗∞ and consider˜ M ∗ R = (cid:8) v = Ru : u ∈ M ∗∞ (cid:9) , R > . The map u ∈ M ∗∞ v ∈ ˜ M ∗ R is an odd homeomorphism; hence, ˜ M ∗ R is compactand i ( ˜ M ∗ R ) = i ( M ∗∞ ) ≥ m by ( i ). By (4.5) and (4.8), for any v ∈ ˜ M ∗ R we have J ( v ) ≤ J ∞ ( v ) + εp I ∞ ( v ) + C ε = 1 p (cid:0) (1 + ε ) I ∞ ( v ) − λ ∞ | v | pp (cid:1) + C ε = R p p (cid:0) (1 + ε ) I ∞ ( u ) − λ ∞ | u | pp (cid:1) + C ε ≤ R p p (cid:18) ε − λ ∞ λ ∞ m + ε (cid:19) + C ε . Fixing R so large that the last term of the previous estimates is ≤
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