POSITIVE SOLUTIONS AND MULTIPLE SOLUTIONS AT NON-RESONANCE, RESONANCE AND NEAR RESONANCE FOR HEMIVARIATIONAL INEQUALITIES WITH p-laplacian

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 5, May 2008, Pages 2527 2545 S 0002-9947(07)04449-2 Article electronically published on December 11, 2007 POSITIVE SOLUTIONS AND MULTIPLE SOLUTIONS AT NON-RESONANCE, RESONANCE AND NEAR RESONANCE FOR HEMIVARIATIONAL INEQUALITIES WITH p-laplacian D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU Abstract. In this paper we study eigenvalue problems for hemivariational inequalities driven by the p-laplacian differential operator. We prove the existence of positive smooth solutions for both non-resonant and resonant problems at the principal eigenvalue of the negative p-laplacian with homogeneous Dirichlet boundary condition. We also examine problems which are near resonance both from the left and from the right of the principal eigenvalue. For nearly resonant from the right problems we also prove a multiplicity result. 1. Introduction In this paper we study the following nonlinear elliptic eigenvalue problem: { div( Dx(z) (1.1) p 2 Dx(z)) λ x(z) p 2 x(z) j(z, x(z)) a.e. on, x =0, 1 <p<+. Here R N is a bounded domain with a C 1,α -boundary (0 <α<1), j(z, x) is a measurable function which is locally Lipschitz in the x R variable and j(z, x) stands for the generalized subdifferential of x j(z, x) in the sense of Clarke [5]. The solutions of (1.1) are sought in W 1,p 0 (). Problem (1.1) belongs to the class of hemivariational inequalities that are new types of variational expressions arising if one considers more realistic mechanical laws of multivalued and nonmonotone nature (see [17], [15]). The corresponding energy (Euler) functionals are nonsmooth and nonconvex. Eigenvalue problems as (1.1) enter in the stability analysis of structures whose equilibrium position is characterized by a hemivariational inequality. The mathematical theory of eigenvalue problems for hemivariational inequalities was studied in [3], [6], [9], [10], [13], [14] for semilinear problems (i.e., p = 2), whereas nonlinear eigenvalue problems driven by the p-laplacian were investigated in [7]. None of the aforementioned works addressed the problem of existence of positive solutions. Theorems on the existence of positive and multiple solutions for hemivariational inequalities involving the p-laplacian have recently been presented in [16]. In this paper we focus on the existence of positive solutions and nontrivial multiple solutions to the eigenvalue problem (1.1) under assumptions different than those of [16], including growth conditions of rate bigger than p for the nonsmooth potential j(z, x). Specifically, denoting by λ 1 the principal (or first) eigenvalue of the negative Received by the editors February 14, 2006. 2000 Mathematics Subject Classification. Primary 35J20, 35R70; Secondary 35J60, 35J85. Key words and phrases. Hemivariational inequality, eigenvalue problem, resonance. 2527 c 2007 American Mathematical Society

2528 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU p-laplacian ( p,w 1,p 0 ()) we are able to show the existence of positive solutions in the non-resonant case λ<λ 1 and in the resonant case λ = λ 1. First, under some natural assumptions set up around a weaker version of the Ambrosetti-Rabinowitz condition, it is shown that for λ<λ 1 problem (1.1) has a positive solution. If p =2, by slightly strengthening the assumptions a converse result is provided showing that generally in this setting we cannot relax the non-resonant condition λ<λ 1 to have positive solutions for problem (1.1). However, we show that keeping the generalized nonsmooth version of Ambrosetti-Rabinowitz type hypothesis but imposing a different assumption on the generalized gradient with respect to x R of the nonsmooth potential j(z, x) we still produce a result that demonstrates that in the resonant case λ = λ 1 a positive solution exists. Afterwards, dropping the assumption describing the generalized version of the Ambrosetti-Rabinowitz condition, we present another set of hypotheses on j(z, x) which still guarantee the existence of a positive solution for problem (1.1) in the case of near resonance from the left at λ 1. Knowing that in general we should not expect having positive solutions to (1.1) in the case of near resonance from the right, i.e., λ>λ 1 close to λ 1, in this situation we supply existence and multiplicity results for nontrivial solutions. Our results extend in a nonsmooth quasilinear framework several classical properties of single- or multi-valued semilinear Dirichlet boundary value problems at non-resonance, resonance and near resonance. They allow one to cover a larger area of applicability in various nonsmooth and nonconvex problems arising in mechanics and engineering (see [15], [17]). The main tools in our approach are minimax theorems, nonsmooth analysis, spectrum of the negative p-laplacian, nonlinear regularity theory and nonlinear strong maximum principle. Examples illustrate all our results. The rest of the paper is organized as follows. Section 2 deals with some mathematical preliminaries. Section 3 contains our results on positive solutions for non-resonance and resonance. Section 4 focuses on the positive solutions in the case of near resonance from the left. Section 5 is devoted to multiple solutions with near resonance from the right. 2. Mathematical background Our approach is variational based on the nonsmooth critical point theory which uses the subdifferential theory of locally Lipschitz functions. For easy reference, first we recall some basic definitions which we will need in the sequel. Let X be a Banach space and X its topological dual. By, we denote the duality bracket for the pair (X, X ). For a locally Lipschitz function ϕ : X R, the generalized directional derivative ϕ 0 (x; h) ofϕ at x X in the direction h X is defined by ϕ 0 (x; h) = lim sup x x λ 0 ϕ(x + λh) ϕ(x ) λ whereas the generalized gradient ϕ(x) ofϕ at x X is introduced as ϕ(x) ={x X : x,h ϕ 0 (x; h), h X} (see Clarke [5]). We say that x X is a critical point of ϕ if 0 ϕ(x). A locally Lipschitz function ϕ : X R satisfies the nonsmooth Palais Smale condition at level c R (nonsmooth PS c -condition for short) if every sequence {x n } X such that ϕ(x n ) c and m(x n )=inf{ x : x ϕ(x n )} 0asn,

NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2529 has a strongly convergent subsequence. The locally Lipschitz function ϕ : X R satisfies the nonsmooth Palais Smale condition (nonsmooth PS-condition for short) if it satisfies the nonsmooth PS c -condition for every c R. Itisalsosaidthata locally Lipschitz function ϕ : X R satisfies the nonsmooth Cerami condition at level c R (nonsmooth C c -condition for short) if every sequence {x n } X such that ϕ(x n ) c and (1 + x n )m(x n ) 0asn has a strongly convergent subsequence. In this paper we use two minimax principles in the nonsmooth critical point theory. The first one is the nonsmooth version of the classical mountain pass theorem (see [4], [8], [15]). Theorem 2.1. Let X be a reflexive Banach space and let ϕ : X R be a locally Lipschitz function. Suppose that for some ρ>0 and x 1,x 2 X with x 1 x 2 >ρ one has max{ϕ(x 1 ),ϕ(x 2 )} <β:= inf{ϕ(x) : x x 1 = ρ}. If the function ϕ satisfies the nonsmooth C c -condition (in particular, the nonsmooth PS c -condition), with c =inf max ϕ(γ(t)) γ Γ t [0,1] where Γ={γ C([0, 1],X): γ(0) = x 1, γ(1) = x 2 }, then ϕ has a critical point x 0 X with ϕ(x 0 )=c β. The second minimax principle that we need is the nonsmooth variant of the Brezis-Nirenberg theorem with local linking (see [12], [19]). Theorem 2.2. If X is a reflexive Banach space admitting a direct sum decomposition X = Y V with dim Y < +, ϕ : X R is a locally Lipschitz function which is bounded below, satisfies the nonsmooth PS-condition, ϕ(0) = 0, inf ϕ<0 X and there exists ρ>0 such that { ϕ(y) 0 if y Y, y ρ, ϕ(v) 0 if v V, v ρ, then ϕ has at least two nontrivial critical points. Finally we recall some facts about the spectrum of the negative p-laplacian with Dirichlet boundary condition. Considering the nonlinear eigenvalue problem { div( Dx(z) (2.1) p 2 Dx(z)) = λ x(z) p 2 x(z) a.e. on, x =0, λ R, the least real number λ, denoted λ 1, for which problem (2.1) has a nontrivial solution in W 1,p 0 (), is called the principal eigenvalue of ( p,w 1,p 0 ()). We know that λ 1 is positive, isolated and simple. There is the following variational characterization of λ 1 > 0 using Rayleigh quotient: { Dx p } p (2.2) λ 1 =inf x p : x W 1,p 0 (), x 0. p This infimum is actually realized at the normalized eigenfunction u 1. We have that u 1 C 1,β () with0<β<1 and we may assume u 1 (z) > 0 for all z (see, e.g., [8, p. 117]). Throughout the rest of the paper we keep the notation for u 1.

2530 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU 3. Positive solutions with non-resonance and resonance First, we establish the existence of positive smooth solutions for problem (1.1) in the non-resonant case. The hypotheses on the nonsmooth potential j(z, x) are the following: H(j) 1 j : R R is a function such that j(z, 0) = 0 a.e. on and (i) for all x R, z j(z, x) is measurable; (ii) for almost all z, x j(z, x) is locally Lipschitz; (iii) for almost all z, allx R and all u j(z, x), we have u a(z)+c x r 1 with a L () +,c>0, p<r<p { Np N p where p if N>p, = + if N p ; (iv) there exist constants µ>pand M>0such that inf j(z, M) > 0 and µj(z, x) z j0 (z, x; x) a.a.z, allx M; j(z, x) (v) lim sup x 0 x p 0 uniformly for almost all z ; + (vi) for almost all z, allx>0andallu j(z, x), we have u ξx p 1 with a constant ξ 0. Assumption H(j) 1 (iv) is a nonsmooth variant of the Ambrosetti-Rabinowitz condition in [1]. Notice that here we do not require the sign condition j(z, x) > 0for almost all z and all x M. Theorem 3.1. If hypotheses H(j) 1 hold and λ<λ 1, then problem (1.1) has a solution x C 1 0() with x(z) > 0 for all z. Proof. Let τ : R R be the Lipschitz continuous truncation function { x if x 0, (3.1) τ(x) = 0 if x<0. Let j 1 (z, x) =j(z, τ(x)). For almost all z, x j 1 (z, x) is locally Lipschitz and j 1 (z, ) has the generalized gradient { {0} if x<0, (3.2) j 1 (z, x) = and j j(z, x) if x>0 1 (z, 0) conv{{0} j(z, 0)}. We consider the functional ϕ 1,λ : W 1,p 0 () R defined by (3.3) ϕ 1,λ (x) = 1 p Dx p p λ p x p p j 1 (z, x(z)) dz, x W 1,p 0 (). By H(j) 1 (i) (iii), ϕ 1,λ is Lipschitz continuous on bounded sets, so locally Lipschitz. Claim 1. ϕ 1,λ satisfies the nonsmooth PS-condition. Let {x n } W 1,p 0 () be such that ϕ 1,λ (x n ) M 1 for some constant M 1 > 0, all n 1andm(x n ) 0asn. Since the set ϕ 1,λ (x n ) W 1,p () =W 1,p 0 () (1/p +1/p =1) is w-compact and the norm functional on a Banach space is weakly lower semicontinuous, we can find x n ϕ 1,λ (x n ) such that m(x n )= x n, n 1.

NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2531 Let A : W 1,p 0 () W 1,p () be the negative p-laplacian that is the nonlinear operator defined by A(x),y = Dx(z) p 2 (Dx(z),Dy(z)) R N dz. It is known that A is monotone and demicontinuous, hence maximal monotone. For every n 1wehave x n = A(x n ) λ x n p 2 x n u n with u n L r (), where 1/r +1/r = 1, satisfying u n (z) j 1 (z, x n (z)) a.e. on. Thechoiceofthesequence{x n } W 1,p 0 () ensures x n,x n ε n x n with ε n 0, which implies that (3.4) Dx n p p + λ x n p p j1(z, 0 x n (z); x n (z)) dz ε n x n. We also have 1 (3.5) p Dx n p p λ p x n p p j 1 (z, x n (z)) dz M 1. Combining (3.4) and (3.5), we obtain ( ) ( ) µ µ p 1 Dx n p p λ p 1 x n p p (3.6) [ µj1 (z, x n (z)) + j1(z, 0 x n (z); x n (z)) ] dz µm 1 + ε n x n. Note that (3.2) yields j 1 (z, x n (z)) = 0 and j1(z, 0 x n (z); x n (z)) = 0 a.e. on {x n 0}. So, by hypotheses H(j) 1 (iii) and (iv), we have [ µj1 (z, x n (z)) + j1(z, 0 x n (z); x n (z)) ] dz [ = µj(z, xn (z)) + j 0 (z, x n (z); x n (z)) ] (3.7) dz {0<x n <M} [ µj(z, xn (z)) + j 0 (z, x n (z); x n (z)) ] dz β 1, n 1, {x n M} for some constant β 1 > 0. Returning to (3.6) and using (3.7), we derive ( ) ( ) µ µ p 1 Dx n p p λ p 1 x n p p µm 1 + ε n x n + β 1. In order to show the boundedness of {x n } in W 1,p 0 (), we may suppose without loss of generality that λ>0. Then by (2.2) it follows that ( ) µ (3.8) )(1 p 1 λλ1 Dx n p p µm 1 + ε n x n + β 1, n 1. Because λ < λ 1 and µ > p > 1, from (3.8) we infer that {x n } W 1,p 0 () is bounded. So by passing to a subsequence if necessary, we may assume w x n x in W 1,p 0 (), x n x in L r (), x n (z) x(z) a.e.on

2532 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU and x n (z) k(z) a.e. on, for all n 1, with k L r (). The inequality x n,x n x ε n x n x reads A(x n),x n x λ x n p 2 x n (x n x) dz u n (x n x) dz ε n x n x. Since λ x n p 2 x n (x n x) dz 0 and u n (x n x) dz 0 as n, it results that lim n),x n x = 0. The operator A being maximal monotone, n it is generalized pseudomonotone and so A(x n ),x n A(x),x, orequivalently, w Dx n p Dx p. Recalling that Dx n Dx in L p (, R N )andl p (, R N )being uniformly convex we have Dx n Dx in L p (, R N ), which means x n x in W 1,p 0 (). This proves Claim 1. Claim 2. There exists ρ>0 such that β := inf{ϕ 1,λ (x) : x = ρ} > 0. By hypothesis H(j) 1 (v), given ε>0, we can find a number δ = δ(ε) > 0such that (3.9) j 1 (z, x) =j(z, x) ε p xp for a.a. z, x [0,δ]. On the other hand, due to hypothesis H(j) 1 (iii) and the mean value theorem for locally Lipschitz functions we can find a number c ε > 0 such that (3.10) j 1 (z, x) =j(z, x) c ε x r for a.a. z, x δ. From (3.9) and (3.10) we see that (3.11) j 1 (z, x) ε p x p + c ε x r for a.a. z, x R. It is clear that for obtaining the estimate in Claim 2 it suffices to admit λ>0. Therefore, by (3.3), (3.11) and (2.2), we have (3.12) ϕ 1,λ (x) 1 ( 1 λ + ε ) Dx p p c 1 Dx r p, x W 1,p 0 (), p λ 1 for some constant c 1 = c 1 (ε) > 0. Because λ<λ 1,choosingε>0such that λ + ε<λ 1 and taking into account that p<rand D p is an equivalent norm on W 1,p 0 (), from (3.12) we see that Claim 2 is satisfied for ρ>0 sufficiently small. Claim 3. There exists ˆx W 1,p 0 () with ˆx >ρsuch that ϕ 1,λ (ˆx) <ϕ 1,λ (0) = 0. For almost all z and all x R, the function s 1 s j(z, sx) is locally µ Lipschitz on (0, + ) and we have ( 1 s j(z, sx) s µ ) µ 1 j(z, sx)+ s µ+1 s µ xj(z, sx)x. Here by s and x we denote the generalized gradient with respect to s>0and x R, respectively. Using the mean value theorem for locally Lipschitz functions, for s>1 we can find θ (1,s) such that 1 j(z, sx) j(z, x) s µ ( µ 1 j(z, θx)+ θ µ+1 θ µ xj(z, θx)x = s 1 θ µ+1 ( µj(z, θx)+ xj(z, θx)θx). ) (s 1)

NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2533 By hypothesis H(j) 1 (iv), for almost all z and all x M, this implies 1 s µ j 1(z, sx) j 1 (z, x) s 1 ( µj(z, θx) j 0 θ µ+1 (z, θx; θx) ) 0. Then for almost all z and all x M we have j 1 (z, x) =j 1 (z, x ( x ) µ ( x ) µ M M) j(z, M) inf j(z, M). M M z From this inequality and the first part of hypothesis H(j) 1 (iv), it is seen that given η>0, we can find M η > 0 such that (3.13) j 1 (z, x) η p xp for a.a. z, x M η. Combining (3.13) and H(j) 1 (iii) shows that for a constant c η > 0 one has (3.14) j 1 (z, x) η p xp c η for a.a. z, x 0. By (3.3), (2.2), (3.14) it turns out that ) ϕ 1,λ (tu 1 ) (1 tp λλ1 ηλ1 Du 1 p p + β 3, t >0, p with a constant β 3 > 0. Now choose η>λ 1 λ>0. Claim 3 follows because ϕ 1,λ (tu 1 ) as t +. Claims 1, 2 and 3 permit the use of Theorem 2.1. We obtain x W 1,p 0 () such that ϕ 1,λ (x) β>0=ϕ 1,λ (0), thereby x 0,and0 ϕ 1,λ (x). The last inclusion ensures (3.15) A(x) λ x p 2 x = u with u L r (), u(z) j 1 (z, x(z)) a.e. on. From nonlinear regularity theory (see, e.g., [8, p. 115]) we deduce x C0 1(). Let x =max{ x, 0} W 1,p 0 () and suppose x 0. Substituting { Dx Dx(z) a.e. on {x <0}, (z) = 0 a.e. on {x 0} in A(x), x λ x p 2 x, x = u, x (see (3.15)), we obtain on the basis of (3.2) that Dx p p λ x p p = ux dz =0. Recalling λ<λ 1 and x 0, this implies that Dx p p <λ 1 x p p, which contradicts (2.2). Hence x =0andsox 0, x 0. Then by H(j) 1 (vi), we have A(x) λx p 1 = u ξx p 1 a.e. on {x >0}, thus div ( Dx(z) p 2 Dx(z)) ( λ + ξ)x(z) p 1 a.e. on. Through the nonlinear strong maximum principle due to Vázquez [18] (see also Gasiński Papageorgiou [8, p. 116]) applied with the function therein β(u) = ( λ + ξ)u p 1 for u 0, we conclude that x(z) > 0 for all z. In view of (3.15) and (3.2) we note that x solves (1.1) which completes the proof. Example 3.2. Consider problem (1.1) with the following potential (for simplicity we drop the z-dependence): { 1 1 j(x) =max η x η, r x r c } p x p,

2534 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU for c 0, p<η<p, p r<p and c 1ifp = r. It is readily seen that hypothesis H(j) 1 is fulfilled, so Theorem 3.1 applies to the corresponding problem (1.1). If p = 2, by strengthening assumption H(j) 1 we obtain a converse result. Namely, we state H(j) 2 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i) (v) with p =2,and (vi) for almost all z, allx>0andallu j(z, x), one has u>0. Theorem 3.3. If hypotheses H(j) 2 hold, then problem (1.1) has a solution x C 1 0() with x(z) > 0 for all z if and only if λ<λ 1. Proof. The sufficiency part follows from Theorem 3.1 because assumption H(j) 2 is stronger than H(j) 1. For the necessity part let x C0() 1 be a positive solution of (1.1), thus x(z) λx(z) =u(z) a.e. on with u L r (), u(z) j(z, x(z)) a.e. on. It yields x(z)u 1 (z) dz λ x(z)u 1 (z) dz = u(z)u 1 (z) dz > 0 accordingtohypothesish(j) 2 (vi). This implies x(z) u 1 (z) dz = λ 1 x(z)u 1 (z) dz > λ which is equivalent to λ 1 >λ. x(z)u 1 (z) dz, In the following we focus on the resonant case λ = λ 1. To this end we need an auxiliary result. Lemma 3.4. If ϑ 1 L () +, ϑ 1 (z) λ 1 a.e. on with strict inequality on a set of positive measure, then there exists a constant ξ 0 > 0 such that ψ(x) := Dx p p ϑ 1 (z) x(z) p dz ξ 0 Dx p p, x W 1,p 0 (). Proof. From (2.2) and the hypothesis on ϑ 1,weseethatψ 0. Suppose that the conclusion of the lemma is not true. Exploiting the p-homogeneity of ψ, we find {x n } W 1,p 0 () such that ψ(x n ) 0asn and Dx n p =1foralln 1. By passing to a subsequence if necessary, we may assume that w x n x in W 1,p 0 (), x n x in L p (), x n (z) x(z) a.e.on and x n (z) k(z) a.e.on, fork L p (). Since the norm on a Banach space is weakly lower semicontinuous, in the limit as n we obtain (3.16) Dx p p ϑ 1 (z) x(z) p dz λ 1 x p p. By (2.2), it follows that x = tu 1, t R. If x = 0, it turns out that Dx n p 0 because ψ(x n ) 0, which contradicts Dx n p =1. Thusx = tu 1, t 0. From (3.16) we have Dx p p <λ 1 x p p contradicting (2.2).

NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2535 Restricting H(j) 1 (with 1 <p<+ ) in a different way than was done in H(j) 2 for p = 2, in particular allowing the elements of j(z, x) to be not everywhere positive whenever x>0, we incorporate in our considerations resonant problems, that is, for λ = λ 1. We formulate the new hypotheses: H(j) 3 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i) (iv), (vi), and (v) there exists ϑ L () such that ϑ(z) 0a.e.on with strict inequality on a set of positive measure and pj(z, x) lim sup x 0 x p ϑ(z) uniformly for almost all z ; + (vii) for almost all z, allx, y > 0andu j(z, x), v j(z, y) one has ( u x p 1 v ) y p 1 (x y) 0. Hypothesis H(j) 3 (vii) extends a condition used in [11] in the context of asymptotically linear problems. In order to deal with the resonant case λ = λ 1 we consider the locally Lipschitz function ϕ 1 : W 1,p 0 () R defined by (3.17) ϕ 1 (x) = 1 p Dx p p λ 1 p x p p j 1 (z, x(z)) dz, x W 1,p 0 (). Here enters the modified potential function j 1 (z, x) =j(z, τ(x)) with the truncation τ : R R in (3.1). The following auxiliary result on ϕ 1 in (3.17) extends [11, Lemma 2.4]. Lemma 3.5. Under hypotheses H(j) 3, any sequence {x n } W 1,p 0 (Ω) such that x n 0 a.e. in Ω and for which there exists x n ϕ 1 (x n ) provided x n,x n 0 as n contains a relabelled subsequence satisfying ϕ 1 (tx n ) 1+tp pn + ϕ 1(x n ), t >0, n 1. Proof. By passing to a subsequence if necessary, we may assume (3.18) 1 n x n,x n = Dx n p p λ 1 x n p p u n x n dz 1, n 1, n with u n L r () satisfying u n (z) j 1 (z, x n (z)) for a.a. z. Denoting by 1 the Lebesgue measure on R, we know that j 1 (z, ) is locally Lipschitz for all z \ D 0,where D 0 1 =0. Withfixedz \ D 0 and n 1, we introduce ζ(t) = tp p u n(z)x n (z) j 1 (z, tx n (z)), t >0. The function ζ is differentiable for almost all t>0, and from assumption H(j) 3 (vii), for a.a. t>0wehave ( d ζ (t) =t p 1 ds x n (z) u n (z) j ) { 1(z, tx n (z)) 0 if t 1, t p 1 0 if t (0, 1] because d ds j 1(z, tx n (z)) j 1 (z, tx n (z)), where d ds j 1(z, s) stands for the derivative of j 1 (z, s) with respect to s. This implies that (3.19) ζ(t) ζ(1), t >0.

2536 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU Then, by (3.17) (3.19), for t>0andn 1wederive ( ) (3.20) ϕ 1 (tx n ) tp 1 pn + p u n(z)x n (z) j 1 (z, x n (z)) Also by (3.17) and (3.18) for every n 1wehave (3.21) ϕ 1 (x n ) 1 ( ) 1 pn + p u n(z)x n (z) j 1 (z, x n (z)) Using (3.21) and (3.20) we achieve the desired conclusion. We can now handle the resonant case. Theorem 3.6. If hypotheses H(j) 3 hold and λ = λ 1, then problem (1.1) has a solution x C 1 0() with x(z) > 0 for all z. Proof. First we show that ϕ 1 in (3.17) satisfies the nonsmooth C c -condition for any number c>0. Consider a sequence {x n } W 1,p 0 () such that ϕ 1 (x n ) c and (1 + x n )m(x n ) 0 as n. We f ind x n ϕ 1 (x n )withm(x n )= x n and x n = A(x n ) λ 1 x n p 2 x n u n for u n L r (), u n (z) j 1 (z, x n (z)) a.e. on. Weinfer (1 + x n ) x n,v ε n v, v W 1,p 0 (), where ε n 0. Writing x n = x + n x n with x + n =max{x n, 0} and x n =max{ x n, 0}, we set v = x n W 1,p 0 () to obtain via (3.2) that (3.22) Dx n p p λ 1 x n p p ε n This ensures that x n 1+ x n <ε n. ϕ 1 ( x n )= 1 p Dx n p p λ 1 p x n p p 0 as n. Since ϕ 1 (x n )=ϕ 1 (x + n )+ϕ 1 ( x n ), we obtain (3.23) ϕ 1 (x + n ) c as n. Suppose that x n along a relabelled subsequence. Having x n p = x n p + x + n p and admitting without loss of generality that x + n = Dx + n p, first we assume that x + n.let t n = (2pc) 1 p x + and y n = t n x + n. n We see that x n = vn A(x n )+λ 1 (x n ) p 1, where, according to (3.2), vn = A(x + n ) λ 1 (x + n ) p 1 u n ϕ 1 (x + n ). We deduce vn,x + n = x n,x + n 0asn, which enables us to apply Lemma 3.5 and (3.23) for obtaining along a relabelled subsequence that (3.24) ϕ 1 (y n )=ϕ 1 (t n x + n ) 1+tp n pn dz. dz. + ϕ 1(x + n ) c as n.

NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2537 Eventually passing to a subsequence, we may suppose w y n y in W 1,p 0 (), y n y in L r (), y n (z) y(z) a.e.on and y n (z) k(z) a.e.on, for all n 1, with k L r (). If y =0,then (3.25) ϕ 1 (y n )=2c λ 1 p y n p p j 1 (z, y n (z)) dz 2c. Comparing (3.25) and (3.24), we achieve a contradiction since c>0. This proves that y 0 and clearly y 0. Thus the set C = {y >0} R N has the Lebesgue measure C N > 0. So x + n (z) + a.e. on C. On the other hand, (3.23) shows (3.26) pϕ 1 (x + n ) vn,x + n pc as n. As in Claim 3 of the proof of Theorem 3.1, using H(j) 1 (iii)-(iv) we derive j 1 (z, x) a 0 x µ ˆβ, for a.a. z, x 0, with constants a 0 > 0and ˆβ >0. Due to H(j) 1 (iv), this gives ux pj(z, x) (µ p)j(z, x) (µ p)a 0 x µ (µ p) ˆβ, for almost all z, allx M and all u j 1 (z, x). We conclude that (3.27) ux pj 1 (z, x) + as x + uniformly for a.a. z and all u j 1 (z, x). Thus we can find ˆη >0 such that ux pj 1 (z, x) 1 for a.a. z, x ˆη, u j 1 (z, x). In addition, from H(j) 1 (iii) there exists ˆη 1 > 0 satisfying ux pj 1 (z, x) ˆη 1 for a.a. z, x [0, ˆη], u j 1 (z, x). Consequently, a constant η 2 R can be found such that (3.28) ux pj 1 (z, x) η 2 for a.a. z, x 0, u j 1 (z, x). Then from (3.28) and (3.2) we have pϕ 1 (x + n ) vn,x + n = (u n (z)x + n (z) pj 1 (z, x + n (z))) dz (u n (z)x + n (z) pj 1 (z, x + n (z))) dz + η 2 \ C N. C By (3.27) and x + n (z) + a.e. on C it follows that (u n (z)x + n (z) pj 1 (z, x + n (z))) dz +. C Since this contradicts (3.26), we infer {x + n } W 1,p 0 () is bounded. Therefore we must have that x n. Set ŷ n = x n x n. We may assume w ŷ n ŷ in W 1,p 0 (), ŷ n ŷ in L p (), ŷ n (z) ŷ(z) a.e.on and ŷ n (z) ˆk(z) a.e. on, for all n 1, with ˆk L p (). Because {x + n } W 1,p 0 () is bounded, we have ŷ 0. By (3.22) we obtain (3.29) Dŷn p p λ 1 ŷn p p <ε n provided n is sufficiently large to have x n 1. Letting n gives Dŷ p p λ 1 ŷ p p. Recalling that ŷ 0, this implies ŷ = tu 1 with t 0. If ŷ =0,thenfrom

2538 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU (3.29) we have ŷ n 0inW 1,p 0 (), which contradicts that ŷ n =1. Itremains that ŷ = tu 1 with t>0, thus x n (z) a.e. on. The choice of the sequence {x n } W 1,p 0 () shows (3.30) pϕ 1 (x n ) x n,x n = (u n (z)x n (z) pj 1 (z, x n (z))) dz pc as n. By (3.2) and the definition of the truncated potential j 1 (z, x), we obtain (u n (z)x n (z) pj 1 (z, x n (z))) dz =0, n 1. {x n <0} Since {x + n } W 1,p 0 () is bounded and {x n 0} N 0asn because x n (z) a.e. on, wehave (u n (z)x n (z) pj 1 (z, x n (z))) dz 0 as n. {x n 0} So finally we get (3.31) (u n (z)x n (z) pj 1 (z, x n (z))) dz 0 as n. Comparing (3.30) and (3.31) a contradiction is reached because c>0. This proves the boundedness of {x n } W 1,p 0 (). From now on we can proceed as in Claim 1 of the proof of Theorem 3.1 to check that ϕ 1 satisfies the nonsmooth C c -condition for c>0. From the hypotheses H(j) 1 (iii), H(j) 3 (v) and the mean value theorem for locally Lipschitz functions, given ε>0, there is c ε > 0 such that j 1 (z, x) 1 p (ϑ(z)+ε) x p + c ε x r for a.a. z, x R. Consequently, for all x W 1,p 0 (), by (3.17) it is seen that ϕ 1 (x) 1 p Dx p p 1 (λ 1 + ϑ(z)) x(z) p dz ε p p x p p c 2 Dx r p, with a constant c 2 > 0. We set ϑ 1 (z) =ϑ(z) +λ 1 λ 1 a.e. on with strict inequality on a set of positive measure in view of H(j) 3 (v). By Lemma 3.4 we find ξ 0 > 0 with the property ϕ 1 (x) 1 p (ξ 0 ε λ 1 ) Dx p p c 2 Dx r p, x W 1,p 0 (). Choosing ε<ξ 0 λ 1, because r>pthere is ρ>0 satisfying inf{ϕ 1 (x) : x = ρ} = β>0. Furthermore, proceeding as for Claim 3 in the proof of Theorem 3.1 we achieve the assertion therein. Then using the inequality inf γ Γ max ϕ 1(γ(t)) β>0, t [0,1] with Γ as in Theorem 2.1, it follows that the nonsmooth C c -condition suffices for c>0 to be applied. The rest of the proof proceeds as that of Theorem 3.1. We obtain x =0sinceifx = tu 1 with t>0, then x = tu 1 and ϕ 1 (x) =0,a contradiction.

NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2539 Example 3.7. Consider problem (1.1) at resonance λ = λ 1 with the locally Lipschitz potential given below: ϑ(z) p x p if x 1, j(z, x) = 1 µ xµ + ϑ(z) p xp 1 if x 1, µ for p<µ<p and ϑ L () whereϑ(z) 0a.e. on with strict inequality on a set of positive measure. It is straightforward to verify that assumptions H(j) 3 are fulfilled, so Theorem 3.6 can be applied yielding a positive solution to problem (1.1) in the resonant case λ = λ 1 with j(z, x) asabove. 4. Non-negative solutions with near resonance from the left In this section the parameter λ approaches λ 1 > 0 from the left. Our hypotheses on the potential j(z, x) are the following: H(j) 4 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i), (ii), (v), (vi), and (iii) for almost all z, allx R and all u j(z, x), we have u a(z)+c x p 1 with a L () +,c>0; (iv) there exists a constant ε 0 > 0 such that u lim inf x + x p 1 ε 0 uniformly for almost all z and all u j(z, x). Theorem 4.1. If hypotheses H(j) 4 hold, then for all λ (λ 1 ε 0,λ 1 ) problem (1.1) has a solution x C0() 1 with x(z) > 0 for all z. Proof. We again consider the locally Lipschitz functional ϕ 1,λ : W 1,p 0 () R defined by (3.3). Claim 1. For every λ (λ 1 ε 0,λ 1 ), ϕ 1,λ satisfies the nonsmooth PS-condition. Suppose that {x n } W 1,p 0 () is a sequence such that ϕ 1,λ (x n ) M 2 for some constant M 2 > 0, all n 1 and m(x n ) 0. We can find x n ϕ 1,λ (x n ) such that m(x n )= x n for all n 1. We see x n = A(x n ) λ x n p 2 x n u n with u n L p (), u n (z) j 1 (z, x n (z)) a.e. on. Let us show that {x n } W 1,p 0 () is bounded. Arguing indirectly, we assume that x n along a relabelled subsequence. Set y n = x n x n, n 1. Then at least for a subsequence, we have w y n y in W 1,p 0 (), y n y in L p (), y n (z) y(z) a.e.on and (4.1) y n (z) k(z) a.a.z, for all n 1andwithk L p (). Since x n W 1,p 0 (), we have x n, x n ε n x n with ε n 0, which implies by (3.2) that Dx n p p λ x n p p ε n x n. Ifλ 0, it is clear that {x n } W 1,p 0 () is bounded. If 0 <λ<λ 1,wenote ) (1 λλ1 Dx n p p ε n x n,

2540 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU so again {x n } W 1,p 0 () is bounded. Writing y n =(1/ x n )(x + n x n ), from the boundedness of {x n } it results that (1/ x n )x + w n y in W 1,p 0 (), and up to a subsequence (1/ x n )x + n (z) y(z) a.e.z, therefore y 0. We now show that y 0. To this end we write (4.2) A(y n ),y n y λ y n p 2 u n y n (y n y) dz x n p 1 (y n y) dz ε n y n y wherewehaveused x n 1 for sufficiently large n. From (4.1) we infer (4.3) y n p 2 y n (y n y) dz 0 as n. By virtue of hypothesis H(j) 4 (iii), {(1/ x n p 1 )u n } is bounded in L p () so u n (4.4) x n p 1 (y n y) dz 0 as n. Passing to the limit in (4.2) and using the convergences in (4.3) and (4.4), we obtain lim A(y n),y n y = 0. Then as in the proof of Theorem 3.1 (see Claim 1), we n obtain y n y in W 1,p 0 (). This entails y = 1, hence y 0,y 0. We notice that by hypothesis H(j) 4 (iii) we have u lim sup c uniformly for almost all z and all u j(z, x). x + xp 1 Given ε (0,ε 0 ), with ε 0 in H(j) 4 (iv), we introduce the set E ε,n = {z : x n (z) > 0, ε 0 ε< u n(z) c + ε}, n 1. x n (z) p 1 Note that for almost all z {y > 0}, we have x n (z) +. So hypothesis H(j) 4 (iv) implies that the characteristic function χ Eε,n of E ε,n has the property χ Eε,n (z) 1a.e.on{y >0}. Assumption H(j) 4 (iii) ensures u n (z) (4.5) x n p 1 a(z) x n p 1 + c y n(z) p 1 a.e. on. Then {(1/ x n p 1 )u n } is bounded in L p (), so up to a subsequence one has 1 x n p 1 u w n h in L p () asn, for some h L p (). Since using (4.5), ( u n 1 χeε,n) x n p 1 0 in L1 ({y >0}), it follows that u n w χ Eε,n h in L 1 x n p 1 ({y >0}) asn. The definition of the sets E ε,n shows χ Eε,n (z)(ε 0 ε)y n (z) p 1 χ Eε,n (z) u n (z) x n p 1 χ E ε,n (z)(c + ε)y n (z) p 1 for a.e. on {y >0}. Taking weak limits in L 1 ({y >0}), we obtain (ε 0 ε)y(z) p 1 h(z) (c + ε)y(z) p 1 a.e. on {y >0}.

NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2541 Because ε>0 was arbitrary, we conclude (4.6) ε 0 y(z) p 1 h(z) cy(z) p 1 a.e. on {y >0}. Also from (4.5) it is clear that (4.7) h(z) =0 a.e. on{y =0}. Taking into account that = {y =0} {y>0}, from (4.6) and (4.7) we infer (4.8) h(z) =g(z)y(z) p 1 with ε 0 g(z) c a.e. on. Thechoiceofthesequence{x n } W 1,p 0 () implies A(y n),v λ y n p 2 u n y n vdz vdz x n p 1 ε n v, v W 1,p 0 (), with ε n 0. Using y n y in W 1,p 0 () and (4.8), we obtain that in W 1,p () there holds (4.9) div ( Dy(z) p 2 Dy(z)) = (λ + g(z)) y(z) p 2 y(z). Exploiting (4.8), the assumption λ>λ 1 ε 0 and the decreasing monotonicity of the principal eigenvalue λ 1 (ĝ) > 0 of the weighted eigenvalue problem p v(z) = λĝ(z) v(z) p 2 v(z) a.e. on, v = 0, on the weight function ĝ L () +,we deduce (4.10) λ 1 (g + λ) <λ 1 (λ 1 )=1. By (4.9) in conjunction with y 0,wegetthaty is an eigenfunction whose corresponding eigenvalue is 1 for the problem with weight g + λ L () +. According to (4.10), y cannot be a principal eigenfunction that is associated to the first eigenvalue λ 1 (g + λ). Knowing that only the principal eigenfunctions have constant sign (see Anane [2]), we deduce that y must change sign, thus achieving a contradiction with y 0. This proves that {x n } W 1,p 0 () is bounded, from which as in Claim 1 of the proof of Theorem 3.1 we conclude that ϕ 1,λ satisfies the nonsmooth PS-condition. Claim 2. For every λ (λ 1 ε 0,λ 1 ), we can find v λ W 1,p 0 () of arbitrarily large norm such that ϕ 1,λ (v λ ) < 0=ϕ 1,λ (0). Fix λ (λ 1 ε 0,λ 1 ). For a.a. z and all s R \ D(z) with D(z) 1 =0there exists d ds j(z, s) and d ds j(z, s) j(z, s). So by H(j) 4(iv) for ε (0,ε 0 ), there is M = M(ε) > 0 such that d (4.11) ds j(z, s) (ε 0 ε)s p 1 for a.a. z, s M, s D(z). By H(j) 4 (iii) and (4.11), it is allowed to write j(z, x) = x 0 d j(z, s) ds = ds M 0 d j(z, s) ds + ds x M d j(z, s) ds ds ξ ε + 1 p (ε 0 ε)(x p M p ) for a.a. z, x M, for some constant ξ ε > 0. Thus there is c ε > 0 such that (4.12) j 1 (z, x) =j(z, x) 1 p (ε 0 ε) x p c ε for a.a. z, x 0.

2542 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU Then, from (3.3), (2.2) and (4.12), for all t>0 we get with some constant ĉ ε > 0 that ( ϕ 1,λ (tu 1 ) tp 1 λ + ε ) 0 ε Du 1 p p +ĉ ε. p λ 1 Choosing ε<λ λ 1 + ε 0, we obtain that ϕ 1,λ (tu 1 ) as t +, which establishes Claim 2. Using H(j) 1 (v), as in Claim 2 of the proof of Theorem 3.1 we find ρ>0with (4.13) inf{ϕ 1,λ (x) : x = ρ} = β>0. Claims 1, 2 and assertion (4.13) enable us to make use of Theorem 2.1, which gives for all λ (λ 1 ε 0,λ 1 ) the existence of an x = x(λ) W 1,p 0 () such that ϕ 1,λ (x) β>0=ϕ 1,λ (0) (thus x 0)and0 ϕ 1,λ (x). The reasoning in the final part of the proof of Theorem 3.1, based on hypothesis H(j) 1 (vi), allows us to conclude that x is a positive solution of problem (1.1). Example 4.2. Consider problem (1.1) with the locally Lipschitz potential j(z, x) = j(x) givenby { } 1 1 j(x) =min p x p, r x r, with 1 <p<r<p. A direct verification shows that the assumptions of Theorem 4.1 are fulfilled. We obtain the existence of a positive solution to problem (1.1) with j(z, x) =j(x) as above and λ (λ 1 1,λ 1 ). 5. Multiple solutions with near resonance from the right We now examine the eigenvalue problem (1.1) near resonance from the right of λ 1. The hypotheses on the nonsmooth potential j(z, x) are the following: H(j) 5 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i) (iii), and (iv) there exists ˆη L () such that ˆη(z) 0a.e.on with strict inequality on a set of positive measure and pj(z, x) lim sup x + x p ˆη(z) uniformly for almost all z ; (v) there exists δ>0such that for almost all z and all 0 x δ or δ x 0, we have j(z, x) 0. Theorem 5.1. If hypotheses H(j) 5 hold, then there exists ˆε >0 such that for all λ (λ 1,λ 1 +ˆε) problem (1.1) has a nontrivial solution x C 1 0(). Proof. Suppose that λ = λ 1 + ε with ε>0. From hypotheses H(j) 1 (iii), H(j) 5 (iv) and the mean value theorem for locally Lipschitz functions, we can find a constant c ε > 0 such that (5.1) j(z, x) 1 p (ˆη(z)+ε) x p + c ε for a.a. z, x R. Consider the locally Lipschitz functional ϕ λ : W 1,p 0 () R defined by (5.2) ϕ λ (x) = 1 p Dx p p λ p x p p j(z, x(z)) dz, x W 1,p 0 ().

NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2543 By (5.2), (5.1) and Lemma 3.4 we derive the estimate ϕ λ (x) 1 p Dx p p 1 p 1 ( ξ 0 2ε p λ 1 (λ 1 +ˆη(z)) x(z) p dz 2ε p x p p ĉ ε ) Dx p p ĉ ε, x W 1,p 0 (), with a constant ĉ ε > 0andwhereξ 0 > 0 is independent of ε>0. We see that if ε (0, ˆε) withˆε = 1 2 ξ 0λ 1,thenϕ λ is coercive, and being weakly lower semicontinuous, we obtain x = x(λ) W 1,p 0 () such that ϕ λ (x) = inf W 1,p 0 () ϕ λ. Therefore x = x(λ) solves problem (1.1). We claim ϕ λ (x) < 0=ϕ λ (0). Suppose that the first option in hypothesis H(j) 5 (v) is valid (the reasoning is similar in the other case). Since u 1 C0(), 1 we can find t>0 such that tu 1 δ. Then, by (5.2) and H(j) 5 (v),wehave ϕ λ (tu 1 ) tp p (1 λλ1 ) Du 1 p p < 0. The claim is verified and so x 0. From nonlinear regularity theory we obtain that x C0(), 1 thus completing the proof. Example 5.2. Consider problem (1.1) with { x if x 1, j(z, x) = x +ˆη(z)( x 1) p if x 1 where ˆη L () is such that ˆη(z) 0a.e.on with strict inequality on a set of positive measure. One can easily check that hypotheses H(j) 5 are satisfied and so Theorem 5.1 applies to problem (1.1) with j(z, x) as defined above and λ near λ 1 from the right. Finally, strengthening hypothesis H(j) 5 (v) we prove a theorem on the existence of multiple nontrivial solutions for problem (1.1) under near resonance at λ 1 > 0 from the right. Consider the direct sum decomposition (5.3) W 1,p 0 () =Ru 1 V with V = {v W 1,p 0 () : up 1 1 vdz =0}. Since λ 1 > 0 is isolated, there is ˆλ 2 >λ 1 such that (5.4) ˆλ2 v p p Dv p p, v V. We formulate the assumptions H(j) 6 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i) (iii), H(j) 5 (iv) and there exists δ>0such that for almost all z and all x [ δ, δ], one has (v) 0 j(z, x) β p x p with β<ˆλ 2 λ 1. Theorem 5.3. If hypotheses H(j) 6 hold, then there exists ˆε 0 > 0 such that for all λ (λ 1,λ 1 +ˆε 0 ) problem (1.1) has at least two nontrivial solutions x 1,x 2 C 1 0().

2544 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU Proof. In the proof of Theorem 5.1 we obtained an ˆε >0 such that for all ε (0, ˆε), the functional ϕ λ in (5.2) is coercive with λ = λ 1 +ε. Therefore it is bounded below and satisfies the nonsmooth PS-condition. Moreover, again in the proof of Theorem 5.1, it was established that inf W 1,p 0 () ϕ λ <ϕ λ (0) = 0. Since u 1 C0(), 1 we can find t 0 > 0 such that if t t 0,then tu 1 (z) δ for all z, withδ>0determined in hypothesis H(j) 6 (v). By (5.2), H(j) 6 (v) and since λ>λ 1,weget ) (5.5) ϕ λ (tu 1 ) (1 tp λλ1 Du 1 p p 0 for t t 0. p On the other hand, from hypotheses H(j) 1 (iii) and H(j) 6 (v) we have (5.6) j(z, x) β p x p +ĉ x r for a.a. z, x R, with a constant ĉ>0. Then from (5.2) (5.4), (5.6) we obtain (5.7) ϕ λ (v) 1 ( 1 λ ) 1 + ε + β Dv p ˆλ p p ĉ 1 Dv r p, v V, λ = λ 1 +ε, ε>0, 2 for a new constant ĉ 1 > 0. Since ˆλ 2 λ 1 β>0(cf. hypothesish(j) 6 (v) ), we may take ε (0, ˆλ 2 λ 1 β). Hence by (5.7), recalling r>p, we can find ˆδ >0 such that (5.8) ϕ λ (v) 0 for all v V with v ˆδ. So if ˆε 0 =min{ˆε, ˆλ 2 λ 1 β}, then due to (5.5), (5.8), we can apply Theorem 2.2 that provides two nontrivial critical points of ϕ λ. This amounts to saying that there exist two nontrivial solutions of problem (1.1). Through the nonlinear regularity theory, they belong to C0 1 (). Example 5.4. Consider problem (1.1) with the locally Lipschitz potential j(z, x) = j(x) introduced as follows: β p ln( x p +1) if x 1, j(x) = (2 x ) β p ln( x p +1)+( x 1) ˆη(z) p x p if 1 < x < 2, ˆη(z) p x p if x 2. Here ˆη L () is such that ˆη(z) 0a.e. on with strict inequality on a set of positive measure and 0 β<ˆλ 2 λ 1. It is easily seen that hypotheses H(j) 6 are verified. Hence Theorem 5.3 can be applied to problem (1.1) with the above potential j(z, x) =j(x) in the case of near resonance from the right at λ 1. References 1. A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349 381. MR0370183 (51:6412) 2. A. Anane, Simplicité et isolation de la première valeur propre du p-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 725 728. MR920052 (89e:35124) 3. G. Barletta and S.A. Marano, Some remarks on critical point theory for locally Lipschitz functions, Glasg. Math. J. 45 (2003), 131 141. MR1972703 (2004e:58016)

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