Flows and Critical Points

Nonlinear differ. equ. appl. 15 (2008), 495–509 c 2008 Birkh¨ auser Verlag Basel/Switzerland 1021-9722/040495-15 published online 26 November 2008 DOI 10.1007/s00030-008-7031-2

Nonlinear Differential Equations and Applications NoDEA

Flows and Critical Points Kanishka Perera and Martin Schechter Abstract. We use flows and the cohomological index to adapt the method of sandwich pairs to better suit quasilinear elliptic boundary value problems. Mathematics Subject Classification (2000). Primary 35J65, Secondary 47J10, 47J30. Keywords. p-Laplacian, boundary value problems, nonlinear eigenvalues, variational methods, sandwich pairs.

1. Introduction The notion of sandwich pairs was introduced by Schechter [12] based upon the sandwich theorem for complementing subspaces by Silva [13] and Schechter [9,10]. Definition 1. We say that a pair of subsets A, B of a Banach space E forms a sandwich pair if for any G ∈ C 1 (E, R), −∞ < b0 := inf G ≤ sup G =: a0 < +∞ B

(1)

A

implies that there is a sequence {uj } ⊂ E and a c ∈ [b0 , a0 ] such that G(uj ) → c ,

G (uj ) → 0 .

(2)

The sandwich pairs used in the literature so far have been formed using the eigenspaces of a semilinear operator and are therefore unsuitable for dealing with quasilinear problems where there are no eigenspaces. The purpose of the present paper is to show how this method can be modified to apply to p-Laplacian problems of the form −Δp u = f (x, u) in Ω (3) u=0 on ∂Ω where Ω is a bounded domain in Rn , n ≥ 1, Δp u = div |∇u|p−2 ∇u is the p-Laplacian of u, p ∈ (1, ∞), and f is a Carathéodory function on Ω × R with subcritical growth, i.e., |f (x, t)| ≤ C |t|r−1 + 1 ∀(x, t) ∈ Ω × R (4)

496

K. Perera and M. Schechter

for some r ∈ [1, p∗ ), where

NoDEA

np n−p

, p 0. Solutions of (3) coincide with the critical points of the C 1 functional |∇u|p − p F (x, u) , G(u) =

(6)

Ω

where

t

f (x, s) ds ,

F (x, t) =

(7)

0

defined on the Sobolev space W01, p (Ω). Sandwich pairs that produce Palais–Smale sequences for G were constructed in [7]. Here we modify the method to produce Cerami sequences, which give better results. The distinguishing feature of our current method is the use of flows to obtain critical values. This allows us more flexibility in dealing with minimax situations.

2. Flows Let E be a Banach space and let Σ be the set of all continuous maps σ = σ(t) from E × [0, 1] to E such that 1. σ(0) is the identity map, 2. for each t ∈ [0, 1], σ(t) is a homeomorphism of E onto E. Because of the special geometry of our problem, we shall require special types of flows. Let A1 , B1 be a pair of disjoint nonempty closed symmetric subsets of the unit sphere S in E and let A = π −1 (A1 ) ∪ {0} ,

B = π −1 (B1 ) ∪ {0}

(8)

where π : E \{0} → S, u → u/ u is the radial projection onto S. For each R > 1, let K = K(R) = e3 + R, and define K = λv : v ∈ B1 , 0 ≤ λ ≤ K . B Let R = Σ

K > 0, max σ(t)(λu) ≤ K, σ ∈ Σ : min d σ(t)(Ru), B t∈[0,1]

t∈[0,1]

u ∈ A1 , 0 ≤ λ ≤ R . (9)

R for each R > 1. This follows from the fact that Note that σ(t) u ≡ u is in Σ K > R and consequently that K ) , u ∈ A1 . d(Ru, B) = d(Ru, B

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We shall prove Theorem 2. Assume that d(RA1 , B) > e3 , K = ∅ , σ(1) A ∩ B

R , σ∈Σ

(10)

and −∞ < b0 := inf G ≤ sup G =: a0 < +∞ . B

(11)

A

c ≤ a0 and a sequence {uj } ⊂ E Then there are a number c satisfying b0 ≤ satisfying c, 1 + uj G (uj ) → 0 . (12) G(uj ) → For our applications, we shall need conditions which will imply (10). This will be done in the next section.

3. Cohomological index We recall the construction and some properties of the cohomological index of Fadell and Rabinowitz [5]. Writing the group Z2 multiplicatively as {1, −1}, a paracompact Z2 -space is a paracompact space X together with a mapping μ : Z2 × X → X, called a Z2 -action on X, such that μ(1, x) = x ,

−(−x) = x ∀x ∈ X

(13)

where −x := μ(−1, x). The action is fixed-point free if −x = x ∀x ∈ X . A subset A of X is invariant if −A :=

(14)

− x : x ∈ A = A,

(15)

and a map f : X → X between two paracompact Z2 -spaces is equivariant if f (−x) = −f (x) ∀x ∈ X .

(16)

Two spaces X and X are equivalent if there is an equivariant homeomorphism f : X → X . We denote by F the set of all paracompact free Z2 -spaces, identifying equivalent ones. A principal Z2 -bundle with paracompact base is a triple ξ = (E, p, B) consisting of an E ∈ F, called the total space, a paracompact space B, called the base space, and a map p : E → B, called the bundle projection, such that there are 1. an open covering {Uλ }λ∈Λ of B, 2. for each λ ∈ Λ, a homeomorphism ϕλ : Uλ × Z2 → p−1 (Uλ ) satisfying ϕλ (b, −1) = −ϕλ (b, 1) ,

p ϕλ (b, ±1) = b

∀b ∈ B .

(17)

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NoDEA

Then each p−1 (b), called a fiber, is some pair {e, −e} , e ∈ E. A bundle map f : ξ → ξ consists of an equivariant map f : E → E and a map f : B → B such that p f = f p, i.e., the diagram f

E −−−−→ E ⏐ ⏐ ⏐ ⏐ p p f

B −−−−→ B commutes. Two bundles ξ and ξ are equivalent if there are bundle maps f : ξ → ξ and f : ξ → ξ such that f f and f f are the identity bundle maps on ξ and ξ , respectively. We denote by PrinZ2 B the set of principal Z2 -bundles over B and Prin Z2 the set of all principal Z2 -bundles with paracompact base, identifying equivalent ones. Each X ∈ F can be identified with a ξ ∈ Prin Z2 as follows. Let X = X/Z2 be the quotient space of X ∈ F with each x and −x identified, called the orbit space of X, and π : X → X the quotient map. Then P : F → Prin Z2 ,

X → ξ := (X, π, X)

(18)

is a one-to-one correspondence. A map f : B → B induces a bundle f ∗ ξ = (f ∗ (E ), p, B) ∈ Prin Z2 , called the pullback, where (19) f ∗ (E ) = (b, e ) ∈ B × E : f (b) = p (e ) , −(b, e ) = (b, −e ) and

(20) p(b, e ) = b . Homotopic maps induce equivalent bundles, so for each B ∈ Prin Z2 , we have the mapping (21) T : [B, B ] → PrinZ2 B , [f ] → f ∗ ξ where [B, B ] is the set of homotopy classes of maps from B to B . For the bundle ξ = (S ∞ , π, RP∞ ), called the universal principal Z2 -bundle, where S ∞ is the unit sphere in R∞ , RP∞ is the infinite dimensional real projective space, and π identifies antipodal points ±x, T is a one-to-one correspondence (see Dold [3]). Thus for each X ∈ F, there is a map f : X → RP∞ , unique up to homotopy and called the classifying map, such that T ([f ]) = P(X) . ∗

∗

∞

(22)

∗

Let f : H (RP ) → H (X) be the induced homomorphism of the Alexander– Spanier cohomology rings. The cohomological index of X is defined by sup k ≥ 1 : f ∗ (ω k−1 ) = 0 , X = ∅ i(X) = (23) 0, X=∅ where ω ∈ H 1 (RP∞ ) is the generator of the polynomial ring H ∗ (RP∞ ) = Z2 [ω]. The index i : F → N ∪ {0, ∞} has the usual properties of an index theory:

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1. Definiteness: i(X) = 0 if and only if X = ∅. 2. Monotonicity: If f : X → Y is an equivariant map, in particular, if X ⊂ Y , then i(X) ≤ i(Y ) . Thus, equality holds when f is an equivariant homeomorphism. 3. Subadditivity: If X ∈ F and A, B are closed invariant subsets of X such that X = A ∪ B, then i(A ∪ B) ≤ i(A) + i(B) . 4. Continuity: If X ∈ F and A is a closed invariant subset of X, then there is a closed invariant neighborhood N of A in X such that i(N ) = i(A) . 5. Neighborhood of zero: If U is a bounded symmetric neighborhood of 0 in a Banach space E, then i(∂U ) = dim E . Now let E be a Banach space and let A ⊂ F denote the class of symmetric subsets of E \ {0}. The suspension SA of a nonempty subset A of E is the quotient space of A × [−1, 1] with A × {1} and A × {−1} collapsed to different points, which can be realized in E ⊕ R as the union of all line segments joining the two points (0, ±1) ∈ E ⊕R to points of A ⊂ E. The cohomological index also has the following important stability property: If A ∈ A is closed, then i(SA) = i(A) + 1

(24)

(see Fadell and Rabinowitz [5]). Let S ∈ A be the unit sphere in E and let π be the radial projection onto S. Our main theorem for this section is Theorem 3. Let A1 , B1 be a pair of disjoint nonempty closed symmetric subsets of S such that i(A1 ) = i(S \ B1 ) < ∞ (25) and let A = π −1 (A1 ) ∪ {0} ,

B = π −1 (B1 ) ∪ {0} .

(26)

Assume that −∞ < b0 := inf G ≤ sup G =: a0 < +∞ . B

(27)

A

Then there are a number c satisfying b0 ≤ c ≤ a0 and a sequence {uj } ⊂ E satisfying G(uj ) → c, 1 + uj G (uj ) → 0 . (28) In proving Theorem 3, we shall make use of the following considerations. Theorem 4. If (25) holds, then for each R > 1, K = ∅ , σ(1) A ∩ B

R . σ∈Σ

(29)

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NoDEA

R such that Proof. Suppose there is a σ ∈ Σ K = ∅ . σ(1) A ∩ B

(30)

R , we have By the definition of Σ K = ∅ . σ(t)(RA1 ) : t ∈ [0, 1] ∩ B ⎧ ⎪ u ∈ A1 , 0 ≤ t ≤ 1/3 ⎨(1 − 3t + 3Rt) u , Γ(t) u = σ(3t − 1)(Ru) , u ∈ A1 , 1/3 < t ≤ 2/3 ⎪ ⎩ σ(1) (3(1 − t) Ru) , u ∈ A1 , 2/3 < t ≤ 1 .

Let

By (30) and (31),

K = ∅ . Γ(t) A1 : t ∈ [0, 1] ∩ B

(31)

(32)

(33)

Moreover, since max σ(t)(λu) ≤ K ,

t∈[0,1]

and u > K , we also have

u ∈ A1 ,

0≤λ≤R

K , u∈B\B

(34)

(35)

K = ∅ . (36) Γ(t) A1 : t ∈ [0, 1] ∩ B \ B So Γ = Γ(t) is a continuous map from A1 × [0, 1] to E \ B such that Γ(0) is the identity map on A1 and Γ(1) A1 is the single point σ(1)(0) ∈ E. Then π(Γ(t) u) , u ∈ A1 , t ∈ [0, 1] SA1 → S \ B1 , (u, t) → (37) −π(Γ(−t)(−u)) , u ∈ A1 , t ∈ [−1, 0)

is an odd map and hence i(S \ B1 ) ≥ i(SA1 ) = i(A1 ) + 1 by the monotonicity of the index and (24), contradicting (25).

(38)

Theorem 5. Assume that d(RA1 , B) > e3 and cR := inf sup G σ(1) u

(39)

is finite. Then there is a sequence {uj } ⊂ E satisfying K ) G (uj ) → 0 . G(uj ) → cR , 1 + d(uj , B

(40)

R u∈A σ∈Σ

Theorem 5 will be proved in Section 5. Proof of Theorem 2. Assumption (10) guarantees that cR is finite. In fact, we have cR ≤ a0 . b0 ≤ K ), Then by Theorem 5, there is a sequence satisfying (40). Since u ≤ K +d(u, B the result follows.

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Proof of Theorem 3. Take R so large that d(RA1 , B) > e3 . By (25) and Theorem 4, (10) holds, so the result follows from Theorem 2.

4. Ordinary differential equations In proving Theorem 5 we shall make use of various extensions of Picard’s theorem in a Banach space. Some are well known. Lemma 6. Let γ(t) and ρ(t) be continuous functions on [0, ∞), with γ(t) nonnegative and ρ(t) positive. Assume that T ∞ dτ > γ(s) ds , (41) u0 ρ(τ ) t0 where t0 < T and u0 are given positive numbers. Then there is a unique solution of (42) u (t) = γ(t)ρ u(t) , t ∈ [t0 , T ) , u(t0 ) = u0 which is positive in [t0 , T ) and depends continuously on u0 . Proof. One can separate variables to obtain u t dτ W (u) = = γ(s) ds . u0 ρ(τ ) t0 The function W (u) is differentiable and increasing in R, positive in [u0 , ∞), depends continuously on u0 and satisfies ∞ T dτ W (u) → L = > γ(s) ds , as u → ∞ . u0 ρ(τ ) t0 Thus, for each t ∈ [t0 , T ) there is a unique u ∈ [u0 , ∞) such that t −1 γ(s) ds u=W t0

is the unique solution of (42), and it depends continuously on u0 .

Lemma 7. Let γ(t) and ρ(t) be continuous functions on [0, ∞), with γ(t) nonnegative and ρ(t) positive. Assume that T u0 dτ > γ(s) ds , (43) t0 m ρ(τ ) where t0 < T and m < u0 are given positive numbers. Then there is a unique solution of (44) u (t) = −γ(t)ρ u(t) , t ∈ [t0 , T ) , u(t0 ) = u0 which is ≥ m in [t0 , T ) and depends continuously on u0 .

502


NoDEA

Proof. One can separate variables to obtain t u0 dτ = γ(s) ds . W (u) = ρ(τ ) t0 u The function W (u) is differentiable and decreasing in R, positive in [m, u0 ], depends continuously on u0 and satisfies u0 T dτ W (u) → L = > γ(s) ds , as u → m . t0 m ρ(τ ) Thus, for each t ∈ [t0 , T ) there is a unique u ∈ [m, u0 ] such that t γ(s) ds u = W −1 t0

is the unique solution of (44), and it depends continuously on u0 .

Theorem 8. Let g(t, x) be a continuous map from R × H to H, where H is a Banach space. Assume that for each point (t0 , x0 ) ∈ R × H, there are constants K, b > 0 such that g(t, x)−g(t, y) ≤ K x−y ,

|t−t0 | < b ,

x−x0 < b ,

y −x0 < b . (45)

Assume also that g(t, x) ≤ γ(t)ρ( x ) ,

x∈H,

t ∈ [t0 , ∞) ,

(46)

where γ(t), ρ(t) satisfy the hypotheses of Lemma 6 with ρ(t) nondecreasing. Then for each x0 ∈ H and t0 > 0 there is a unique solution x(t) of the equation dx(t) = g t, x(t) , dt

t ∈ [t0 , ∞) ,

x(t0 ) = x0 .

(47)

Moreover, x(t) depends continuously on x0 and satisfies x(t) ≤ u(t) ,

t ∈ [t0 , ∞) ,

(48)

where u(t) is the solution of (42) in that interval satisfying u(t0 ) = u0 ≥ x0 . We also have the following. Theorem 9. Let ρ, γ satisfy the hypotheses of Lemma 7, with ρ locally Lipschitz continuous. Let u(t) be the solution of (44), and let h(t) be a continuous function satisfying t γ(r)ρ h(r) dr , t0 ≤ s < t < T , h(t0 ) ≥ u0 . (49) h(t) ≥ h(s) − s

Then u(t) ≤ h(t) ,

t ∈ [t0 , T ) .

(50)

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503

Proof. Assume that there is a point t1 in the interval such that h(t1 ) < u(t1 ) . Let y(t) = u(t) − h(t) , t ∈ [t0 , T ) . Then, y(t0 ) ≤ 0 and y(t1 ) > 0. Let τ be the largest point < t1 such that y(τ ) = 0. Then y(t) > 0 , t ∈ (τ, t1 ] . (51) Moreover, by (44) and (49) we have t t y(t) ≤ − γ(s) ρ u(s) − ρ h(s) ds ≤ L y(s) ds , (52) τ

τ

where L is the Lipschitz constant for ρ at u(τ ) times the maximum of γ in the interval. Let t y(s) ds . w(t) = τ

Then Consequently,

e−Lt w(t) = e−Lt y(t) − Lw(t) ≤ 0 , e−Lt w(t) ≤ e−Lτ w(τ ) = 0 ,

t ∈ [τ, t1 ] .

t ∈ [τ, t1 ] .

Hence, y(t) ≤ Lw(t) ≤ 0 , t ∈ [τ, t1 ] , contradicting (51). This completes the proof.

5. Proof of Theorem 5 Let ρ(r) = 1 + r. If the theorem were not true, there would be a δ > 0 such that K ) G (u) ≥ 1 (53) ρ d(u, B would hold for all u in the set Q= u∈E: cR − 3δ ≤ G(u) ≤ cR + 3δ . Let

and

cR − 2δ ≤ G(u) ≤ cR + 2δ Q0 = u ∈ Q : Q1 = u ∈ Q : cR − δ ≤ G(u) ≤ cR + δ

(54) (55) (56)

Q2 = E \ Q0 , η(u) = d(u, Q2 )/ d(u, Q1 ) + d(u, Q2 ) . (57) It is easily checked that η(u) is locally Lipschitz continuous on E and satisfies ⎧ ⎪ u ∈ Q1 , ⎨η(u) = 1 , (58) η(u) = 0 , u ∈ Q2 , ⎪ ⎩ η(u) ∈ (0, 1) , otherwise .

504


NoDEA

ˆ = {u ∈ E : For any θ < 1 there is a locally Lipschitz continuous map Y (u) of E G (u) = 0} into itself such that ˆ (59) Y (u) ≤ 1 , θ G (u) ≤ G (u), Y (u) , u ∈ E (cf., e.g., [11]). Let σ(t) be the flow generated by K ) . W (u) = −η(u)Y (u)ρ d(u, B

(60)

and W (u) is locally Lipschitz continuous, σ(t) exists Since W (u) ≤ ρ(d(u, B)) for all t ∈ R+ in view of Theorem 8. We also have (61) dG σ(t)u /dt = G (σ), σ K ) = −η(σ) G (σ), Y (σ) ρ d(σ, B K ) ≤ −θη(σ) G (σ) ρ d(σ, B = −θη(σ) in view of (53) and (59). Let T satisfy 2δ/θ < T < 3, and suppose u ∈ Q1 is such / Q1 . Then that there is a t1 ∈ [0, T ] for which σ(t1 )u ∈ G σ(t1 )u < cR − δ , since we cannot have G(σ(t1 )u) > cR + δ for u ∈ Q1 by (61). But this implies G σ(T )u < cR − δ . (62) On the other hand, if σ(t)u ∈ Q1 for all t ∈ [0, T ], then T dt ≤ cR + δ − θT < cR − δ G σ(T )u ≤ G(u) − θ 0

by (61). Thus, (62) holds for u ∈ Q1 . R . To see this, note that since We claim that σ1 (t) = σ(tT ) ∈ Σ t σ(t)u − u = W σ(τ )u dτ ,

(63)

0

we have

σ(t)u − σ(s)u ≤

t

K dr . ρ d σ(r)u, B

s

K , we have If v ∈ B K ≤ σ(s)u − v ≤ σ(t)u − v + h(s) = d σ(s)u, B

t

dr . ρ d σ(r)u, B

s

This implies,

h(s) ≤ h(t) +

t

ρ h(r) dr .

(64)

s

Moreover, if we replace u by Ru, we see by Lemma 7 and Theorem 9, that h(s) satisfies m(s) ≤ h(s) , 0 ≤ s ≤ T ,

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where m(s) is given by

m(0)

m(s)

505

dτ = s. ρ(τ )

Note that for u ∈ A1 , we have e3 m(T ) ≥ 1. Thus K > 0 min d σ1 (t)(Ru), B t∈[0,1]

when u ∈ A1 . Moreover, by Theorem 8, we have u ∈ A1 ,

x(t) = σ(t)(λu) − λu ≤ e3 ,

0 ≤ λ ≤ R.

This follows from the fact that x(0) = 0 and x(t) ≤ m(t) , where m(t) satisfies

m(t)

0

t ∈ [0, T ] ,

ds = t, 1+s

t ∈ [0, T ] .

t ∈ [0, T ] ,

u ∈ A1 ,

Thus, m(t) ≤ eT ≤ e3 . Hence, σ(t)(λu) ≤ K ,

R and This tells us that σ1 ∈ Σ cR − δ , G σ1 (1)u <

0 ≤ λ ≤ R.

u ∈ A.

(65)

But this contradicts (11). Hence (53) cannot hold for u satisfying (54). The proof is complete.

6. Applications to the p-Laplacian We shall now study problem (3). Consider the nonlinear eigenvalue problem − Δp u = λ |u|p−2 u in Ω u=0

on ∂Ω .

Its eigenvalues coincide with the critical values of the C 1 functional 1 I(u) = |u|p Ω

(66)

(67)

defined on the unit sphere S in W01, p (Ω). Let F denote the class of symmetric subsets of S and set λl = inf sup I(u) (68) u∈M M ∈F i(M )≥l

where i is the cohomological index. Then 0 < λ 1 < λ2 ≤ · · · → ∞ are eigenvalues of (66) (cf. Theorem 4.2.1 of Perera et al. [8]).

(69)

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Setting H(x, t) = p F (x, t) − tf (x, t), we shall prove Theorem 10. If λl < λl+1 and λl |t|p − W (x) ≤ p F (x, t) ≤ λl+1 |t|p + W (x) ,

t∈R

(70)

for some l > 0 and W ∈ L (Ω), and 1

H(x, t) ≤ C (|t|α + 1) ,

H(x) := lim

|t|→∞

H(x, t) < 0 a.e. |t|α

(71)

for some α satisfying 0 < α ≤ p, then (3) has a solution. Theorem 11. If λl < λl+1 and (70) holds for some l > 0 and W ∈ L1 (Ω), and H(x, t) ≥ −C (|t|α + 1) ,

H(x, t) > 0 a.e. |t|α

H(x) := lim |t|→∞

(72)

for some α satisfying 0 < α ≤ p, then (3) has a solution. Theorem 12. If λl < λl+1 and (70) holds for some l > 0 and W ∈ L1 (Ω), and H(x, t) ≤ W1 (x) ∈ L1 (Ω) ,

H(x, t) → −∞

as

|t| → ∞ ,

(73)

then (3) has a solution. Theorem 13. If λl < λl+1 and (70) holds for some l > 0 and W ∈ L1 (Ω), and H(x, t) ≥ −W1 (x) ∈ L1 (Ω) ,

H(x, t) → +∞

as

|t| → ∞ ,

(74)

then (3) has a solution. Similar resonance problems have been studied by Perera [6] when f (x, t)/|t|p−2 t → α± (x) ∈ L∞ (Ω) as t → ±∞ and by Arcoya and Orsina [1], Bouchala and Dr´ abek [2], and Dr´ abek and Robinson [4] for the special case α± = const. Proof of Theorem 10. Let A1 = u ∈ S : I(u) ≤ λl ,

B1 = u ∈ S : I(u) ≥ λl+1 .

(75)

Then i(A1 ) = i(S \ B1 ) = l (76) by Theorem 4.2.2 of Perera et al. [8]. Let A, B be as in Theorem 3 and let G be given by (6). Since p |∇u| ≥ λl+1 |u|p , u ∈ B (77) Ω

and

Ω

|∇u|p ≤ λl

Ω

(70) implies

|u|p ,

u ∈ A,

W ≤ inf G ≤ sup G ≤

− Ω

(78)

Ω

B

A

W, Ω

so there is a sequence {uj } ⊂ W01, p (Ω) satisfying (28) by Theorem 3.

(79)

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We claim that {uj } is bounded and hence has a convergent subsequence by j = uj /ρj converges a standard argument. If ρj = uj → ∞, a subsequence of u 1, p p to some u weakly in W0 (Ω), strongly in L (Ω), and a.e. in Ω. Then H(x, uj ) G (uj ) uj /p − G(uj ) = →0 (80) ρα ρα Ω j j by (12) and

lim Ω

H(x, uj ) ≤ ρα j

lim Ω

H(x, uj ) | uj |α ≤ |uj |α

H(x) | u|α ≤ 0

(81)

Ω

by (71). Since H < 0 a.e., it follows that u = 0 a.e. Now passing to the limit in p F (x, uj ) W G(uj ) = ≤ λl+1 | uj |p + p (82) 1− p ρpj ρ ρj Ω Ω j gives

1≤

λl+1 | u|p , Ω

contradicting the fact that u = 0 a.e.

Proof of Theorem 11. As in the proof of Theorem 10, there is a sequence {uj } ⊂ W01, p (Ω) satisfying (28). We claim that {uj } is bounded and hence a subsequence j = uj /ρj converges to a critical point of G. If ρj = uj → ∞, a subsequence of u 1, p p converges to some u weakly in W0 (Ω), strongly in L (Ω), and a.e. in Ω. Then H(x, uj ) G (uj ) uj /p − G(uj ) = →0 (83) ρα ρα Ω j j by (12) and

lim Ω

H(x, uj ) ≥ ρα j

lim Ω

H(x, uj ) | uj |α ≥ |uj |α

H(x) | u|α ≥ 0

(84)

Ω

by (72). Since H > 0 a.e., it follows that u = 0 a.e. Now passing to the limit in p F (x, uj ) W G(uj ) = ≤ λl+1 | uj |p + p (85) 1− p p ρj ρj ρj Ω Ω again gives

1≤

λl+1 | u|p , Ω

contradicting the fact that u = 0 a.e.

Proof of Theorem 12. We follow the proof of Theorem 10. We can conclude that there is a sequence {uj } ⊂ W01, p (Ω) satisfying (28). Again we claim that {uj } is bounded, and hence a subsequence converges to a critical point of G. If ρj =

508


NoDEA

uj → ∞, a subsequence of u j = uj /ρj converges to some u weakly in W01, p (Ω), p strongly in L (Ω), and a.e. in Ω. Now G (uj ) uj /p − G(uj ) dx → − H(x, uj ) dx = c. (86) Ω

Ω

This implies

H(x, uj ) dx ≤ K .

(87)

Ω

As before, we show that u (x) ≡ 0. Let Ω0 be the subset of Ω on which u ˜ = 0. Then |uj (x)| = ρj |˜ uj (x)| → ∞, If Ω1 = Ω \ Ω0 , then we have H(x, uj ) dx = + Ω

Ω0

x ∈ Ω0 .

≤

Ω1

W1 (x) dx → −∞ .

H(x, uj ) dx + Ω0

Proof of Theorem 13. In this case we have H(x, uj ) dx = + ≥ H(x, uj ) dx − Ω0

Ω1

which contradicts (87) as well.

Ω0

(89)

Ω1

This contradicts (87), and we see that ρj = uj is bounded.

Ω

(88)

W1 (x) dx → ∞ ,

(90)

Ω1

References [1] D. Arcoya and L. Orsina. Landesman–Lazer conditions and quasilinear elliptic equations. Nonlinear Anal., 28(10):1623–1632, 1997. [2] J. Bouchala and P. Dr´ abek. Strong resonance for some quasilinear elliptic equations. J. Math. Anal. Appl., 245(1):7–19, 2000. [3] A. Dold. Partitions of unity in the theory of fibrations. Ann. of Math. (2), 78:223–255, 1963. [4] P. Dr´ abek and S. B. Robinson. Resonance problems for the p-Laplacian. J. Funct. Anal., 169(1):189–200, 1999. [5] E. R. Fadell and P. H. Rabinowitz. Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math., 45(2):139–174, 1978. [6] K. Perera. One-sided resonance for quasilinear problems with asymmetric nonlinearities. Abstr. Appl. Anal., 7(1):53–60, 2002. [7] K. Perera and M. Schechter. Sandwich pairs in p-Laplacian problems. Topol. Methods Nonlinear Anal., 29(1):29–34, 2007. [8] K. Perera, R. P. Agarwal, and D. O’Regan. Morse-theoretic aspects of p-Laplacian type operators. Progress in Nonlinear Differential Equations and their Applications, Birkh¨ auser Boston Inc., Boston, MA, to appear. [9] M. Schechter. A generalization of the saddle point method with applications. Ann. Polon. Math., 57(3):269–281, 1992.

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[10] M. Schechter. New saddle point theorems. In Generalized Functions and Their Applications (Varanasi, 1991), pages 213–219. Plenum, New York, 1993. [11] M. Schechter. Linking Methods in Critical Point Theory, Birkh¨ auser Boston, 1999. [12] M. Schechter. Sandwich pairs in critical point theory. Trans. Amer. Math. Soc., 360(6):2811–2823, 2008. [13] Silva, Elves A. B. Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal., 16(5):455–477, 1991. Kanishka Perera Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901-6975 USA e-mail: [email protected] Martin Schechter Department of Mathematics University of California Irvine, CA 92697-3875 USA e-mail: [email protected] Received: 17 July 2007. Accepted: 12 February 2008.