Flows and Critical Points

Transkript

1 Nonlinear differ. equ. appl. 15 (2008), c 2008 Birkhäuser Verlag Basel/Switzerland / published online 26 November 2008 DOI /s 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), <b 0 := inf B G sup A G =: a 0 < + (1) implies that there is a sequence u j } E and a c [b 0,a 0 ] such that G(u j ) c, G (u j ) 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 (3) u =0 on where is a bounded domain in R n,n 1, Δ p u = div ( u p 2 u ) is the p-laplacian of u, p (1, ), and f is a Carathéodory function on R with subcritical growth, i.e., f(x, t) C ( t r 1 +1 ) (x, t) R (4)

2 496 K. Perera and M. Schechter NoDEA for some r [1,p ), where np n p, p = p < n (5), p n is the critical Sobolev exponent, and C>0. Solutions of (3) coincide with the critical points of the C 1 functional G(u) = u p pf(x, u), (6) where F (x, t) = t 0 f(x, s) ds, (7) defined on the Sobolev space W 1,p 0 (). 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 A 1,B 1 be a pair of disjoint nonempty closed symmetric subsets of the unit sphere S in E and let A = π 1 (A 1 ) 0}, B = π 1 (B 1 ) 0} (8) where π : E \0} S, u u/ u is the radial projection onto S. ForeachR>1, let K = K(R) =e 3 + R, and define B K = λv : v B 1, 0 λ K }. Let Σ R = σ Σ : min σ(t)(ru), B ) K > 0, max σ(t)(λu) K, t [0,1] t [0,1] } u A 1, 0 λ R. (9) Note that σ(t) u u is in Σ R for each R>1. This follows from the fact that K>Rand consequently that d(ru, B) =d(ru, B K ), u A 1.

3 Vol. 15 (2008) Flows and Critical Points 497 We shall prove Theorem 2. Assume that d(ra 1,B) >e 3, and σ(1) A B K, σ Σ R, (10) <b 0 := inf B G sup A G =: a 0 < +. (11) Then there are a number c satisfying b 0 c a 0 andasequenceu j } E satisfying ( G(u j ) c, 1+ uj ) G (u j ) 0. (12) 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 Z 2 multiplicatively as 1, 1}, a paracompact Z 2 -space is a paracompact space X together with a mapping μ : Z 2 X X, called a Z 2 -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 A subset A of X is invariant if x x x X. (14) A := x : x A } = A, (15) and a map f : X X between two paracompact Z 2 -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 Z 2 -spaces, identifying equivalent ones. A principal Z 2 -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 U λ } λ Λ of B, 2. for each λ Λ, a homeomorphism ϕ λ : U λ Z 2 p 1 (U λ ) satisfying ϕ λ (b, 1) = ϕ λ (b, 1), pϕ λ (b, ±1) = b b B. (17)

4 498 K. Perera and M. Schechter 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 = fp, i.e., the diagram f E E p p B f 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 ff are the identity bundle maps on ξ and ξ, respectively. We denote by Prin Z2 B the set of principal Z 2 -bundles over B and Prin Z 2 the set of all principal Z 2 -bundles with paracompact base, identifying equivalent ones. Each X F can be identified with a ξ Prin Z 2 as follows. Let X = X/Z 2 be the quotient space of X Fwith each x and x identified, called the orbit space of X, andπ : X X the quotient map. Then P : F Prin Z 2, 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 Z 2, called the pullback, where f (E )= (b, e ) B E : f(b) =p (e ) }, (b, e )=(b, e ) (19) and p(b, e )=b. (20) Homotopic maps induce equivalent bundles, so for each B Prin Z 2,wehavethe mapping T :[B,B ] Prin Z2 B, [f] f ξ (21) 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 Z 2 -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 }, i(x) = X 0, X = where ω H 1 (RP ) is the generator of the polynomial ring H (RP )=Z 2 [ω]. The index i : F N 0, } has the usual properties of an index theory: (23)

5 Vol. 15 (2008) Flows and Critical Points 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 Fand 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 Ais closed, then i(sa) = i(a) + 1 (24) (see Fadell and Rabinowitz [5]). Let S Abe the unit sphere in E and let π be the radial projection onto S. Our main theorem for this section is Theorem 3. Let A 1,B 1 be a pair of disjoint nonempty closed symmetric subsets of S such that i(a 1 )=i(s \ B 1 ) < (25) and let A = π 1 (A 1 ) 0}, B = π 1 (B 1 ) 0}. (26) Assume that <b 0 := inf G sup G =: a 0 < +. (27) B A Then there are a number c satisfying b 0 c a 0 andasequenceu j } E satisfying ( G(u j ) c, 1+ uj ) G (u j ) 0. (28) In proving Theorem 3, we shall make use of the following considerations. Theorem 4. If (25) holds, then for each R>1, σ(1) A B K, σ Σ R. (29)

6 500 K. Perera and M. Schechter NoDEA Proof. Suppose there is a σ Σ R such that σ(1) A B K =. (30) By the definition of Σ R,wehave σ(t)(ra1 ):t [0, 1] } B K =. (31) Let By (30) and (31), (1 3t +3Rt) u, u A 1, 0 t 1/3 Γ(t) u = σ(3t 1)(Ru), u A 1, 1/3 <t 2/3 σ(1) (3(1 t) Ru), u A 1, 2/3 <t 1. (32) Γ(t) A1 : t [0, 1] } B K =. (33) Moreover, since and we also have max σ(t)(λu) K, u A 1, 0 λ R (34) t [0,1] u >K, u B \ B K, (35) Γ(t) A1 : t [0, 1] } ( B \ B K ) =. (36) So Γ = Γ(t) is a continuous map from A 1 [0, 1] to E \ B such that Γ(0) is the identity map on A 1 and Γ(1) A the single point σ(1)(0) E. Then 1 is SA 1 S \ B 1, (u, t) π(γ(t) u), u A 1,t [0, 1] π(γ( t)( u)), u A 1,t [ 1, 0) (37) is an odd map and hence i(s \ B 1 ) i(sa 1 )=i(a 1 ) + 1 (38) by the monotonicity of the index and (24), contradicting (25). Theorem 5. Assume that d(ra 1,B) >e 3 and c R := inf sup G ( σ(1) u ) (39) σ Σ R u A is finite. Then there is a sequence u j } E satisfying ( G(u j ) c R, 1+d(u j, B ) K ) G (u j ) 0. (40) Theorem 5 will be proved in Section 5. Proof of Theorem 2. Assumption (10) guarantees that c R is finite. In fact, we have b 0 c R a 0. Then by Theorem 5, there is a sequence satisfying (40). Since u K +d(u, B K ), the result follows.

7 Vol. 15 (2008) Flows and Critical Points 501 Proof of Theorem 3. Take R so large that d(ra 1,B) >e 3. By (25) and Theorem 4, (10) holds, so the result follows from Theorem 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, ), withγ(t) nonnegative and ρ(t) positive. Assume that dτ T u 0 ρ(τ) > γ(s) ds, (41) where <T and u 0 are given positive numbers. Then there is a unique solution of u (t) =γ(t)ρ ( u(t) ), t [,T), u( )=u 0 (42) which is positive in [,T) and depends continuously on u 0. Proof. One can separate variables to obtain u dτ t W (u) = u 0 ρ(τ) = γ(s) ds. The function W (u) is differentiable and increasing in R, positive in [u 0, ), depends continuously on u 0 and satisfies dτ T W (u) L = u 0 ρ(τ) > γ(s) ds, as u. Thus, for each t [,T) there is a unique u [u 0, ) such that ( t ) u = W 1 γ(s) ds is the unique solution of (42), and it depends continuously on u 0. Lemma 7. Let γ(t) and ρ(t) be continuous functions on [0, ), withγ(t) nonnegative and ρ(t) positive. Assume that u0 dτ m ρ(τ) > γ(s) ds, (43) where <T and m<u 0 are given positive numbers. Then there is a unique solution of u (t) = γ(t)ρ ( u(t) ), t [,T), u( )=u 0 (44) which is m in [,T) and depends continuously on u 0. T

8 502 K. Perera and M. Schechter NoDEA Proof. One can separate variables to obtain u0 dτ W (u) = u ρ(τ) = γ(s) ds. The function W (u) is differentiable and decreasing in R, positive in [m, u 0 ], depends continuously on u 0 and satisfies u0 dτ W (u) L = m ρ(τ) > γ(s) ds, as u m. Thus, for each t [,T) there is a unique u [m, u 0 ] such that ( t ) u = W 1 γ(s) ds T is the unique solution of (44), and it depends continuously on u 0. Theorem 8. Let g(t, x) be a continuous map from R H to H, whereh is a Banach space. Assume that for each point (,x 0 ) R H, there are constants K, b > 0 such that g(t, x) g(t, y) K x y, t <b, x x 0 <b, y x 0 <b. (45) Assume also that t g(t, x) γ(t)ρ( x ), x H, t [, ), (46) where γ(t), ρ(t) satisfy the hypotheses of Lemma 6 with ρ(t) nondecreasing. Then for each x 0 H and > 0 there is a unique solution x(t) of the equation dx(t) = g ( t, x(t) ), t [, ), x( )=x 0. (47) dt Moreover, x(t) depends continuously on x 0 and satisfies x(t) u(t), t [, ), (48) where u(t) is the solution of (42) in that interval satisfying u( )=u 0 x 0. We also have the following. Theorem 9. Let ρ, γ satisfy the hypotheses of Lemma 7, withρ locally Lipschitz continuous. Let u(t) be the solution of (44), andleth(t) be a continuous function satisfying Then h(t) h(s) t s γ(r)ρ ( h(r) ) dr, s<t<t, h( ) u 0. (49) u(t) h(t), t [,T). (50)

9 Vol. 15 (2008) Flows and Critical Points 503 Proof. Assume that there is a point t 1 in the interval such that h(t 1 ) <u(t 1 ). Let y(t) =u(t) h(t), t [,T). Then, y( ) 0andy(t 1 ) > 0. Let τ be the largest point <t 1 such that y(τ) =0. Then y(t) > 0, t (τ,t 1 ]. (51) Moreover, by (44) and (49) we have t [ y(t) γ(s) ρ ( u(s) ) ρ ( h(s) )] t ds L y(s) ds, (52) τ where L is the Lipschitz constant for ρ at u(τ) times the maximum of γ in the interval. Let t w(t) = y(s) ds. Then τ [ e Lt w(t) ] = e Lt [ y(t) Lw(t) ] 0, t [τ,t 1 ]. τ Consequently, e Lt w(t) e Lτ w(τ) =0, t [τ,t 1 ]. Hence, y(t) Lw(t) 0, t [τ,t 1 ], 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 ρ ( d(u, B K ) ) G (u) 1 (53) would hold for all u in the set Q = u E : c R 3δ G(u) c R +3δ }. (54) Let Q 0 = u Q : c R 2δ G(u) c R +2δ } (55) Q 1 = u Q : c R δ G(u) c R + δ } (56) and Q 2 = E \ Q 0, η(u) =d(u, Q 2 )/ [ d(u, Q 1 )+d(u, Q 2 ) ]. (57) It is easily checked that η(u) is locally Lipschitz continuous on E and satisfies η(u) =1, u Q 1, η(u) =0, u Q 2, (58) η(u) (0, 1), otherwise.

10 504 K. Perera and M. Schechter NoDEA For any θ<1 there is a locally Lipschitz continuous map Y (u) ofê = u E : G (u) 0} into itself such that Y (u) 1, θ G (u) ( G (u),y(u) ), u Ê (59) (cf., e.g., [11]). Let σ(t) be the flow generated by W (u) = η(u)y (u)ρ ( d(u, B K ) ). (60) Since W (u) ρ(d(u, B)) and W (u) is locally Lipschitz continuous, σ(t) exists for all t R + in view of Theorem 8. We also have dg ( σ(t)u ) /dt = ( G (σ),σ ) (61) = η(σ) ( G (σ),y(σ) ) ρ ( d(σ, B K ) ) θη(σ) G (σ) ρ ( d(σ, B K ) ) = θη(σ) in view of (53) and (59). Let T satisfy 2δ/θ < T < 3, and suppose u Q 1 is such that there is a t 1 [0,T] for which σ(t 1 )u/ Q 1. Then G ( σ(t 1 )u ) < c R δ, since we cannot have G(σ(t 1 )u) > c R + δ for u Q 1 by (61). But this implies G ( σ(t )u ) < c R δ. (62) On the other hand, if σ(t)u Q 1 for all t [0,T], then G ( σ(t )u ) G(u) θ T 0 dt c R + δ θt < c R δ by (61). Thus, (62) holds for u Q 1. We claim that σ 1 (t) =σ(tt ) Σ R. To see this, note that since we have If v B K, we have σ(t)u u = σ(t)u σ(s)u t 0 t s W ( σ(τ)u ) dτ, (63) ( ρ d ( σ(r)u, B ) ) K dr. h(s) =d ( σ(s)u, B K ) σ(s)u v σ(t)u v + t This implies, h(s) h(t)+ t s s ( ρ d ( σ(r)u, B )) dr. ρ ( h(r) ) dr. (64) 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,

11 Vol. 15 (2008) Flows and Critical Points 505 where m(s) is given by m(0) dτ m(s) ρ(τ) = s. Note that for u A 1, we have e 3 m(t ) 1. Thus min σ 1 (t)(ru), B ) K > 0 t [0,1] when u A 1. Moreover,byTheorem8,wehave x(t) = σ(t)(λu) λu e 3, u A 1, 0 λ R. This follows from the fact that x(0) = 0 and x(t) m(t), t [0,T], where m(t) satisfies m(t) ds = t, t [0,T]. 0 1+s Thus, m(t) e T e 3. Hence, σ(t)(λu) K, t [0,T], u A 1, 0 λ R. This tells us that σ 1 Σ R and G ( σ 1 (1)u ) < c R δ, 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 (66) u =0 on. Its eigenvalues coincide with the critical values of the C 1 functional 1 I(u) = (67) u p defined on the unit sphere S in W 1,p 0 (). Let F denote the class of symmetric subsets of S and set λ l = inf M F i(m) l sup I(u) (68) u M where i is the cohomological index. Then 0 <λ 1 <λ 2 (69) are eigenvalues of (66) (cf. Theorem of Perera et al. [8]).

12 506 K. Perera and M. Schechter NoDEA Setting H(x, t) =pf(x, t) tf(x, t), we shall prove Theorem 10. If λ l <λ l+1 and λ l t p W (x) pf(x, t) λ l+1 t p + W (x), t R (70) for some l>0 and W L 1 (), and H(x, t) C ( t α +1), H(x) := lim t H(x, t) t α < 0 a.e. (71) for some α satisfying 0 <α p, then (3) has a solution. Theorem 11. If λ l <λ l+1 and (70) holds for some l>0 and W L 1 (), and H(x, t) C ( t α +1), H(x) := lim t H(x, t) t α > 0 a.e. (72) for some α satisfying 0 <α p, then (3) has a solution. Theorem 12. If λ l <λ l+1 and (70) holds for some l>0 and W L 1 (), and H(x, t) W 1 (x) L 1 (), H(x, t) as t, (73) then (3) has a solution. Theorem 13. If λ l <λ l+1 and (70) holds for some l>0 and W L 1 (), and H(x, t) W 1 (x) L 1 (), 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 () as t ± and by Arcoya and Orsina [1], Bouchala and Drábek [2], and Drábek and Robinson [4] for the special case α ± = const. Proof of Theorem 10. Let A 1 = } u S : I(u) λ l, B1 = } u S : I(u) λ l+1. (75) Then i(a 1 )=i(s \ B 1 )=l (76) by Theorem of Perera et al. [8]. Let A, B be as in Theorem 3 and let G be given by (6). Since u p λ l+1 u p, u B (77) and u p λ l u p, u A, (78) (70) implies W inf G sup G W, (79) B A so there is a sequence u j } W 1,p 0 () satisfying (28) by Theorem 3.

13 Vol. 15 (2008) Flows and Critical Points 507 We claim that u j } is bounded and hence has a convergent subsequence by a standard argument. If ρ j = u j, a subsequence of ũ j = u j /ρ j converges to some ũ weakly in W 1,p 0 (), strongly in L p (), and a.e. in. Then by (12) and lim H(x, u j ) ρ α j H(x, u j ) ρ α j = G (u j ) u j /p G(u j ) ρ α j 0 (80) lim H(x, u j) u j α ũ j α H(x) ũ α 0 (81) by (71). Since H<0a.e., it follows that ũ = 0 a.e. Now passing to the limit in 1 G(u j) pf(x, u j ) ρ p = j ρ p λ l+1 ũ j p + W j ρ p (82) j gives 1 λ l+1 ũ p, contradicting the fact that ũ = 0 a.e. Proof of Theorem 11. As in the proof of Theorem 10, there is a sequence u j } W 1,p 0 () satisfying (28). We claim that u j } is bounded and hence a subsequence converges to a critical point of G. Ifρ j = u j, a subsequence of ũ j = u j /ρ j converges to some ũ weakly in W 1,p 0 (), strongly in L p (), and a.e. in. Then by (12) and lim H(x, u j ) ρ α j H(x, u j ) ρ α j = G (u j ) u j /p G(u j ) ρ α j 0 (83) lim H(x, u j) u j α ũ j α H(x) ũ α 0 (84) by (72). Since H > 0 a.e., it follows that ũ = 0 a.e. Now passing to the limit in 1 G(u j) pf(x, u j ) ρ p = j ρ p λ l+1 ũ j p + W j ρ p (85) j again gives 1 λ l+1 ũ p, contradicting the fact that ũ = 0 a.e. Proof of Theorem 12. We follow the proof of Theorem 10. We can conclude that there is a sequence u j } W 1,p 0 () satisfying (28). Again we claim that u j } is bounded, and hence a subsequence converges to a critical point of G. Ifρ j =

14 508 K. Perera and M. Schechter NoDEA u j, a subsequence of ũ j = u j /ρ j converges to some ũ weakly in W 1,p 0 (), strongly in L p (), and a.e. in. Now H(x, u j ) dx = G (u j ) u j /p G(u j ) } dx c. (86) This implies H(x, u j ) dx K. (87) As before, we show that ũ(x) 0. Let 0 be the subset of on which ũ 0. Then u j (x) = ρ j ũ j (x), x 0. (88) If 1 =\ 0, then we have H(x, u j ) dx = H(x, u j ) dx + 0 W 1 (x) dx. 1 (89) This contradicts (87), and we see that ρ j = u j is bounded. Proof of Theorem 13. In this case we have H(x, u j ) dx = H(x, u j ) dx 0 W 1 (x) dx, 1 (90) which contradicts (87) as well. References [1] D. Arcoya and L. Orsina. Landesman Lazer conditions and quasilinear elliptic equations. Nonlinear Anal., 28(10): , [2] J. Bouchala and P. Drábek. Strong resonance for some quasilinear elliptic equations. J. Math. Anal. Appl., 245(1):7 19, [3] A. Dold. Partitions of unity in the theory of fibrations. Ann. of Math. (2), 78: , [4] P. Drábek and S. B. Robinson. Resonance problems for the p-laplacian. J. Funct. Anal., 169(1): , [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): , [6] K. Perera. One-sided resonance for quasilinear problems with asymmetric nonlinearities. Abstr. Appl. Anal., 7(1):53 60, [7] K. Perera and M. Schechter. Sandwich pairs in p-laplacian problems. Topol. Methods Nonlinear Anal., 29(1):29 34, [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, Birkhäuser Boston Inc., Boston, MA, to appear. [9] M. Schechter. A generalization of the saddle point method with applications. Ann. Polon. Math., 57(3): , 1992.

15 Vol. 15 (2008) Flows and Critical Points 509 [10] M. Schechter. New saddle point theorems. In Generalized Functions and Their Applications (Varanasi, 1991), pages Plenum, New York, [11] M. Schechter. Linking Methods in Critical Point Theory, Birkhäuser Boston, [12] M. Schechter. Sandwich pairs in critical point theory. Trans. Amer. Math. Soc., 360(6): , [13] Silva, Elves A. B. Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal., 16(5): , Kanishka Perera Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL USA Martin Schechter Department of Mathematics University of California Irvine, CA USA Received: 17 July Accepted: 12 February 2008.