EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński

DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 409 418 EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE Leszek Gasiński Jagiellonian University Institute of Computer Science Nawojki 11 30072 Kraków, Poland Abstract. In this paper we consider quasilinear hemivariational inequality at resonance. We obtain two existence theorems using a Landesman-Lazer type condition. The method of the proof is based on the nonsmooth critical point theory for locally Lipschitz functions. 1. Introduction. Let R N be a bounded domain with C 1 -boundary Γ and let 2 p < +. In this paper we study the following quasilinear hemivariational inequality at resonance: { p x λ 1 x p 2 x j(z, x(z)) for a.a. z, x Γ = 0. By λ 1 we denote the first eigenvalue of the negative p-laplacian p x = div( x p 2 x) with Dirichlet boundary conditions (i.e. of ( p, W 1,p 0 ())). Also j : R R is a functional measurable in the first variable and locally Lipschitz in the second variable. By j(z, ζ) we denote the subdifferential of j(z, ) in the sense of Clarke [2] (see Section 2). This paper continues in the direction of two papers by Gasiński-Papageorgiou (see [3], [4]). It is also related to the recent work of Goeleven-Motreanu-Panagiotopoulos [7], who examined semilinear (i.e. for p = 2) hemivariational inequalities of resonance. The first existence result extends Theorem 4 of Gasiński-Papageorgiou [3, p. 362], where the hypotheses on j(z, ζ) were more restrictive and assumed that the asymptotic values of the generalized potential j(z, ζ) existed as ζ ±. It also completes Theorem 3.3 of Gasiński-Papageorgiou [4, p. 1101] were we assumed a Landesman- Lazer condition of another type. The second result of this paper extends Theorem 4.8 of Gasiński-Papageorgiou [4, p. 1109] in the case when we look for at least one nontrivial solution of (HV I). In both cases the proof will be based on the generalized nonsmooth version of the well known mountain pass theorem. For other resonant quasilinear hemivariational inequalities we refer to Filippakis- Gasiński-Papageorgiou [8], Gasiński-Papageorgiou [5] and Kyritsi-Papageorgiou [9]. 2000 Mathematics Subject Classification. Primary: 49J40; Secondary: 34B15, 35J20. Key words and phrases. p-laplacian, strong resonance at infinity, first eigenvalue,clarke subdifferential, nonsmooth Palais-Smile condition, nonsmooth Cerami condition, Landesman-Lazer type conditions, mountain pass theorem. 409

410 LESZEK GASIŃSKI Our approach is variational and is based on the critical point theory for nonsmooth locally Lipschitz functionals of Chang [1]. For the convenience of the reader, in the next section, we recall some basic definitions and facts from that theory, which we will need in the sequel. 2. Preliminaries. Let X be a Banach space and X its topological dual. By we will denote the norm in X, by the norm in X, and by, the duality brackets for the pair (X, X ). In analogy with the directional derivative of a convex function, we define the generalized directional derivative of a locally Lipschitz function ϕ at x X in the direction h X, by ϕ 0 (x; h) = lim sup x x t 0 ϕ(x + th) ϕ(x ). t The function X h ϕ 0 (x; h) R is sublinear, continuous and by the Hahn- Banach theorem it is the support function of a nonempty, convex and w -compact set ϕ(x) = {x X : x, h ϕ 0 (x; h) for all h X}. The set ϕ(x) is called generalized or Clarke subdifferential of ϕ at x. If ϕ is strictly differentiable at x (in particular if ϕ is continuously Gâteaux differentiable at x), then ϕ(x) = {ϕ (x)}. Let ϕ: X R be a locally Lipschitz function on a Banach space X. A point x X is said to be a critical point of ϕ, if 0 ϕ(x). If x X is a critical point of ϕ, then the value c = ϕ(x) is called a critical value of ϕ. It is easy to see that, if x X is a local extremum of ϕ, then 0 ϕ(x). Moreover, the multifunction X x ϕ(x) 2 X is upper semicontinuous, where the space X is equipped with the w -topology, i.e. for any w -open set U X, the set {x X : ϕ(x) U} is open in X (see Gasiński-Papageorgiou [6, Proposition 1.3.9, p. 50]). For more details on the generalized subdifferential we refer to Clarke [2] and Gasiński-Papageorgiou [6, Section 1.3.4]. The critical point theory for smooth functions uses a compactness conditions known as the Palais-Smale condition and Cerami condition. In our present nonsmooth setting, the conditions take the following form. A locally Lipschitz function ϕ: X R satisfies the nonsmooth Palais-Smale condition at level c (nonsmooth PS c -condition for short), if any sequence {x n } n 1 X such that ϕ(x n ) c and m ϕ (x n ) 0, where m ϕ (x n ) = min{ x : x ϕ(x n )}, has a strongly convergent subsequence. Function ϕ satisfies the nonsmooth Palais-Smale condition (nonsmooth PS-condition for short), if it satisfies the nonsmooth PS c -condition for any c R. A locally Lipschitz function ϕ: X R satisfies the nonsmooth Cerami condition at level c (nonsmooth C c -condition for short), if any sequence {x n } n 1 X such that ϕ(x n ) c and (1 + x n )m ϕ (x n ) 0, has a strongly convergent subsequence. Function ϕ satisfies the nonsmooth Cerami condition (nonsmooth C-condition for short), if it satisfies the nonsmooth C c -condition for any c R. PS c -condition implies C c -condition and PS-condition implies C-condition. The proof of our main result will be based on the nonsmooth extension of the well-known mountain pass theorem (see Gasiński-Papageorgiou [6, Theorem 2.1.3, p. 140]). Theorem 1. If X is a reflexive Banach space, ϕ: X R is a locally Lipschitz functional satisfying the nonsmooth C c0 -condition, with c 0 = inf max ϕ( γ(t) ), γ Γ t [0,1] where Γ = { } γ C([0, 1]; X) : γ(0) = 0, γ(1) = x 1, there exist a real number r > 0 and a point x 1 X, such that x 1 > r and max { ϕ(0), ϕ(x 1 ) } < inf ϕ(x), x =r

QUASILINEAR HVI AT RESONANCE 411 then c 0 inf ϕ(x) and there exists x 0 X, such that 0 ϕ(x 0 ) and ϕ(x 0 ) = c 0 x =r (i.e. c 0 is a critical value of ϕ). In the formulation of (HV I), we encounter λ 1, which is the first eigenvalue of the negative p-laplacian with Dirichlet boundary conditions. More precisely, let us consider the following nonlinear eigenvalue problem: { p x = λ x (EP ) p 2 x for a.a. z, x Γ = 0. The least real number λ for which (EP ) has a nontrivial solution is called the first eigenvalue λ 1 of ( p, W 1,p 0 ()). This first eigenvalue λ 1 is positive, isolated and simple (i.e. the associated eigenspace is one-dimensional). Moreover, we have a variational characterization of λ 1 via the Rayleigh quotient, i.e. { x p } p λ 1 = min x p : x W 1,p 0 (), x 0. (1) p The above minimum is realized at the normalized eigenfunction u 1. Note that if u 1 minimizes the Rayleigh quotient, then so does u 1 and so we infer that the first eigenfunction u 1 does not change sign on. In fact we can show that u 1 (z) 0 for all z and so we can assume that u 1 > 0 on. In the sequel, we will assume that p 2, p is such that 1 p + 1 p = 1. By p, we will denote the Sobolev critical exponent and by p the number such that 1 p + 1 p = 1. Note that 1 p < p 2 p < p +. 3. Existence Results. In this section we prove two existence results for the following quasilinear hemivariational inequality: { p x λ (HV I) 1 x p 2 x j(z, x(z)) for a.a. z, x Γ = 0. In our first theorem together with the function j : R R, we consider the following functions: v ± (z) = inf {v n} lim inf v n(z), V ± (z) = sup lim sup v n (z), {v n} for almost all z, where the infimum and supremum are taken over all sequences {v n } n 1 L p (), such that v n (z) j(z, ζ n ) for almost all z, with ζ n ± as n +. Our hypotheses on the generalized potential function j(z, ζ) are the following: H(j) 1 j : R R is a function such that: (i): for all ζ R the function z j(z, ζ) R is measurable; (ii): for almost all z the function R ζ j(z, ζ) R is locally Lipschitz with L r ()-Lipschitz constant, where r < p and 1 r + 1 r = 1; (iii): for almost all z, all ζ R and all η j(z, ζ) we have η a 1 (z), with some a 1 L q () +, with q > p p p ; j(, 0) L1 () and j(z, 0) dz 0; (iv): there exists µ > 0 such that lim sup ζ 0 z. j(z, ζ) ζ p µ, uniformly for almost all

412 LESZEK GASIŃSKI (v): we have v +, V +, v, V L 1 () and V + (z)u 1 (z) dz < p v + (z)u 1 (z) dz p V (z)u 1 (z) dz < v (z)u 1 (z) dz; Remark 1. (1) Hypothesis H(j) 1 (v) is a Landesman-Lazer type condition. For a comparison of the above Landesman-Lazer type condition with other existing in the literature we refer to Gasiński-Papageorgiou [6, pp. 470-474]. (2) Hypothesis H(j) 1 (v) is satisfied if e.g. V + (z) pv + (z) and pv (z) v (z), for almost all z and the inequalities are strict on sets of positive measure. (3) Hypothesis H(j) 1 (v) implies in particular that V + (z)u 1 (z) dz v + (z)u 1 (z) dz > 0 and v (z)u 1 (z) dz V (z)u 1 (z) dz < 0. (4) Hypothesis H(j) 1 (v) is satisfied if e.g. v + (z) = V + (z) 0, v (z) = V (z) 0 and the inequalities are strict on sets of positive measure. Let ϕ: W 1,p 0 () R be the energy functional defined by ϕ(x) = 1 p x p p λ 1 p x p p j(z, x(z)) dz. (2) Let ψ : W 1,p 0 () R be defined by ψ(x) = and j(z, x(z)) dz. (3) By virtue of hypothesis H(j) 1 (iii) we have that ψ is locally Lipschitz (see Gasiński-Papageorgiou [6, Theorem 1.3.10, p. 59]). Also functionals W 1,p 0 () x x p p R and W 1,p 0 () x x p p R are convex, continuous, hence locally Lipschitz on W 1,p 0 () (see Gasiński-Papageorgiou [6, Theorem 1.3.2, p. 35]). Therefore ϕ is locally Lipschitz. Proposition 1. If hypotheses H(j) 1 hold, then ϕ satisfies the nonsmooth PScondition. Proof. Let {x n } n 1 W 1,p 0 () be a sequence, such that ϕ(x n ) M 1 for n 1 and m ϕ (x n ) 0. We will show that {x n } n 1 W 1,p 0 () is bounded. Suppose that this is not true. Then by passing to a subsequence if necessary, we may assume that x n +. Let y n = xn x n for n 1. Passing to a next subsequence if necessary, we may assume that y n y in L p () and weakly in W 1,p 0 (). (4) From the choice of the sequence {x n } n 1 W 1,p 0 (), we have M 1 1 p x n p p λ 1 p x n p p j(z, ) dz M 1 n 1, (5) so pm 1 x n p p λ 1 x n p p p j(z, ) dz pm 1 n 1 (6)

QUASILINEAR HVI AT RESONANCE 413 and thus for all n 1, we have pm 1 x n p y n p p λ 1 y n p p x n p dz pm 1 x n p. (7) By virtue of the mean value theorem for locally Lipschitz functions (see Gasiński- Papageorgiou [6, Proposition 1.3.14, p. 53]), we know that for all n 1 and almost all z, we can find w n (z) j(z, t n ) with 0 < t n < 1, such that j(z, ) j(z, 0) = w n (z). From hypothesis H(j) 1 (iii), for almost all z, we have j(z, ) j(z, 0) + a 1 (z). (8) From hypothesis H(j) 1 (iii), we have that q < p p < p, so the space W 1,p 0 () is embedded continuously in L q () and we have j(z, ) x n p dz ( j(z, 0) x n p + a ) 1(z) x n p dz j(, 0) 1 x n p + a 1 q x n q x n p j(, 0) 1 x n p + c a 1 q x n p 1, with some constant c > 0 and thus j(z, ) x n p dz 0. (9) From the Rayleigh quotient (see (1)), convergence (4) and the weak lower semicontinuity of the norm functional, we have λ 1 y p p y p p lim inf y n p p. Thus, by passing to the limit in (7), using (9) and the fact that y n p p y p p (see (4)), we obtain y n p p λ 1 y p p. So y n p p y p p and y p p = λ 1 y p p (10) Since we already know that y n y weakly in L p (; R N ) (see (4)) and the space L p (; R N ) is uniformly convex, from the Kadec-Klee property (see Gasiński- Papageorgiou [6, Remark A.3.11, p. 722]), we have that y n y in L p (; R) and so y n y in W 1,p 0 (). Since y n = 1, we have y = 1, i.e. y 0. Therefore from (10), we infer that y = ±u 1 (see (1)). Without any loss of generality we can assume that y = u 1 (the case y = u 1 is treated similarly). Since u 1 (z) > 0 for all z, we have that + for all z. Let x n ϕ(x n ) be such that m ϕ (x n ) = x n for n 1. For every n 1, its existence is a consequence of the fact that ϕ(x n ) W 1,p () (where 1 p + 1 p = 1) is weakly compact and the norm functional is weakly lower semicontinuous. Let A: W 1,p 0 () W 1,p () be the nonlinear operator defined by Ax, v = x(z) p 2 R N ( x(z), v(z) ) R N dz x, v W 1,p 0 (). (11) (by, we denote the duality brackets for the pair ( W 1,p 0 (), W 1,p () ) ). It is straightforward to check that A is demicontinuous and strongly monotone, hence maximal monotone (see Gasiński-Papageorgiou [6, Corollary 1.4.2 and Remark 1.4.4, p. 75]). For every n 1, we have where u n ψ(x n ) and ψ is defined by (3). x n = Ax n λ 1 x n p 2 x n u n, (12)

414 LESZEK GASIŃSKI We know that u n L r () and u n(z) j(z, ) for almost all z (see Gasiński-Papageorgiou [6, Theorem 1.3.10, p. 59]). Because x n 0, using (12) and passing to a subsequence if necessary, for all n 1, we have 1 n x n x n p p + λ 1 x n p p + u n(z) dz 1 n x n. Adding (6) and the last inequalities and dividing by x n, we have pm 1 x n 1 [ n u n(z)y n (z) pj(z, x ] n(z)) dz pm 1 x n x n + 1 n. (13) Because + for all z and u n(z) j(z, ) for almost all z and all n 1, using hypothesis H(j) 1 (iii) and passing to a subsequence if necessary, we have that u n u weakly in L r (), (14) for some u L r () and so also u n(z)y n (z) dz From (13) and (15), we also have dz x n u (z)u 1 (z) dz. (15) u (z)u 1 (z) dz. (16) Let ε > 0. Again form the mean value theorem for locally Lipschitz functions, we know that for almost all z and all n 1, we can find w n (z) j(z, u n (z)) where u n (z) = (1 t n ) + t n ε with 0 < t n < 1, such that j(z, ) = j(z, ε) + w n (z)(1 ε). (17) Because u n (z) = t n (1 ε) (1 ε) = ε and + for almost all z, we obtain u n (z) +. From definitions of v + and V +, we get that v + (z) ε w n (z) V + (z) + ε for almost all z and all n n 0, for some n 0 = n 0 (z) 1. Using (17), we have = pj(z, εx n(z)) + pw n (z)(1 ε). Similarly as before, we get that j(z, ε) j(z, 0) + a 1 (z)ε. So As ε > 0 was arbitrary, we get p j(z, 0) pa 1 (z)ε + p(v + (z) ε)(1 ε) pj(z, x n(z)) p j(z, 0) + pa 1(z)ε + p(v + (z) + ε)(1 ε). pv + (z) lim inf lim sup for almost all z. Using (18) and the Fatou lemma, we have p v + (z)u 1 (z) dz lim inf y n (z) dz pv + (z), (18)

QUASILINEAR HVI AT RESONANCE 415 lim dz x n lim sup y n (z) dz p So from (16), we get p v + (z)u 1 (z) dz From definitions of v + and V + we also know that v + (z)u 1 (z) dz u (z)u 1 (z) dz From (20) and (21), we get p v + (z)u 1 (z) dz V + (z)u 1 (z) dz. (19) u (z)u 1 (z) dz p V + (z)u 1 (z) dz. (20) V + (z)u 1 (z) dz. (21) V + (z)u 1 (z) dz0, (22) which contradict hypothesis H(j) 1 (v). Thus {x n } n 1 W 1,p 0 () is a bounded sequence. Hence, at least for a subsequence, we have that x n x in L r () (as r < p ) and weakly in W 1,p 0 (). As x n 0, at least for a subsequence, we have that x n, x n x 1 n. From (12), we obtain Ax n, x n x λ 1 ( x n p 2 x n, x n x) pp (u n, x n x) rr 1 n (by (, ) pp we denote the duality brackets for the pair (L p (), L p ()) and by (, ) rr the duality bracket for the pair (L r (), L r ()). From the continuity of the operator L p () x x p 2 x L p (), we have that x n p 2 x n x p 2 x in L p () and so ( x n p 2 x n, x n x) pp 0. From (14) we also have that (u n, x n x) rr 0. Thus we conclude that lim sup Ax n, x n x 0. As a maximal monotone operator, A is also generalized pseudomonotone (see Gasiński-Papageorgiou [6, Proposition 1.4.10, p. 84]) and so it follows that x n p p = Ax n, x n Ax, x = x p p. Employing the Kadec-Klee property of uniformly convex spaces and arguing as before, we have that x n x in W 1,p 0 (). So ϕ satisfies the nonsmooth PScondition. Proposition 2. If hypotheses H(j) 1 hold, then there exist two constants β 1, β 2 > 0 and p < ϑ < p, such that ϕ(x) β 1 x p β 2 x ϑ for all x W 1,p 0 (). Proof. Let κ > 1 be such that µ > λ1 κp. As q > p p p, so q < p p (where 1 q + 1 q = 1). Let ϑ (p, p ) be such that ϑq < p. From hypothesis H(j) 1 (iv) we can find δ > 0 such that for almost all z and all ζ such that ζ δ we have j(z, ζ) λ1 κp ζ p. On the other hand from the proof of Proposition 1 (see (8)), we know that for almost all z and all ζ such that ζ > δ we have j(z, ζ) j(z, 0) + a 1 (z) ζ. Thus for almost all z and all ζ R, we have j(z, ζ) λ 1 κp ζ p + γ(z) ζ ϑ, (23) with γ L q (), γ(z) = ( j(z, 0) + a 1 (z)δ)δ ϑ + λ1 Rayleigh quotient, we obtain that ϕ(x) = 1 p x p p λ 1 p x p p j(z, x(z)) dz κp δp ϑ. Using (23) and the

416 LESZEK GASIŃSKI 1 p x p p λ 1 p x p p + λ 1 κp x p p γ q x ϑ q = 1 p x p p λ 1(κ 1) x p p γ q x ϑ q 1 κp κp x p p γ q x ϑ ϑq. Because ϑq < p, from the Sobolev embedding theorem we have that W 1,p 0 () is embedded continuously in L ϑq (). So using Poincaré inequality we finish the proof. Theorem 2. If hypotheses H(j) 1 hold, then problem (HV I) has a nontrivial solution x 0 W 1,p 0 (). Proof. By virtue of hypothesis H(j) 1 (iv) and Proposition 2, we know that there exist β 1, β 2 > 0, such that ϕ(x) β 1 x p β 2 x ϑ for all x W 1,p 0 (), with some p < ϑ < p. Evidently, if we choose r > 0 small enough, we will have that ϕ(x) β 3 > 0 for all x W 1,p 0 (), with x = r and some β 3 > 0. Next let t > 0 and consider the quantity ϕ(tu 1 ). Using the fact that u 1 p p = λ 1 u 1 p p, we have ϕ(tu 1 ) = tp p u 1 p p λ 1t p p u 1 p p j(z, tu 1 (z)) dz = j(z, tu 1 (z)) dz. By a simple modification of the argumentation for (18) in the proof of Proposition j(z, tu 1 (z)) 1, we can verify that lim inf v + (z) for almost all z. If t +, t + tu 1 (z) j(z,tu using Fatou lemma, we have that lim inf 1(z)) t + tu 1(z) u 1 (z) dz v +(z)u 1 (z) dz. Hypothesis H(j) 1 (v) implies that v + (z)u 1 (z) dz > 0 and thus we infer that j(z, tu 1 (z)) dz = t j(z, tu 1 (z)) u 1 (z) dz +, tu 1 (z) as t +. Therefore, if t > 0 is large enough, we have ϕ(tu 1 ) 0. Also ϕ(0) 0 (see hypothesis H(j) 1 (iii)). By Proposition 1, ϕ satisfies the nonsmooth PS-condition, thus also C c -condition for any c R. So we can apply Theorem 1 and obtain x 0 W 1,p 0 (), x 0 0, such that 0 φ(x 0 ) and so Ax 0 λ 1 x 0 p 2 x 0 = u in W 1,p (), with u ψ(x 0 ), hence u L r () and u (z) j(z, x 0 (z)) for almost all z. We have that Ax 0, φ = λ 1 ( x 0 p 2 x 0, φ) pp + (u, φ) rr for all φ C0 () and by the Green theorem div( x 0 p 2 x R N 0 ), φ = λ 1 ( x 0 p 2 x 0, φ) pp + (u, φ) rr, for all φ C0 (). Note that from the representation theorem for the elements in the dual space W 1,p () = ( W 1,p 0 () ), we have that div( x0 p 2 x 0 ) W 1,p (). Since C0 () is dense in W 1,p 0 (), we deduce that div( x 0 (z) p 2 x R N 0 (z)) λ 1 x 0 (z) p 2 x 0 (z) = u (z) j(z, x 0 (z)) for a.a. z x 0 Γ = 0, and so x 0 is a nontrivial solution of (HV I). We can have another existence result for (HV I). Our hypotheses on the generalized potential function j(z, ζ) are now the following:

QUASILINEAR HVI AT RESONANCE 417 H(j) 2 : j : R R is a function such that: (i): for all ζ R the function z j(z, ζ) R is measurable; (ii): for almost all z the function R ζ j(z, ζ) R is locally Lipschitz with L r ()-Lipschitz constant, where r < p and 1 r + 1 r = 1; (iii): for almost all z, all ζ R and all η j(z, ζ) we have η a 1 (z), with some a 1 L q () +, with q > p p p ; j(, 0) L1 () and j(z, 0) dz 0; (iv): there exists function ĵ L 1 (), such that lim inf j(z, ζ) = ĵ(z) and ζ + ĵ(z) dz 0; (v): there exists µ > 0 such that lim sup ζ 0 all z ; pj(z, ζ) ζ p µ, uniformly for almost (vi): for any sequences {x n } n 1 W 1,p 0 () and {u n} n 1 L p () such that u n(z) j(z, ) and + for almost all z, we have lim sup u n(z) dz 0; (vii): there exists t 0 0, such that j(z, t 0 u 1 (z)) dz 0. Remark 2. (1) Hypothesis H(j) 2 (iv) classifies the problem as strongly resonant. It also incorporates the problem when j(z, ζ) ĵ(z) as ζ +, with ĵ L 1 () +. (2) In hypothesis H(j) 2 (iv) we do not assume anything about lim sup j(z, ζ). ζ + (3) Hypotheses H(j) 2 (iv) (vii) are more general then those of Gasiński-Papageorgiou [4, Hypotheses H(j) 4, p. 1104]. As before, we define a locally Lipschitz functional ϕ: W 1,p 0 () R by (2). Proposition 3. If hypotheses H(j) 2 hold, then ϕ satisfies the nonsmooth C c - condition at any level c > 0. Proof. Let c > 0. Let {x n } n 1 W 1,p 0 () be a sequence, such that ϕ(x n ) c and (1 + x n )m ϕ (x n ) 0 as n +. We will show that the sequence {x n } n 1 W 1,p 0 () is bounded. Suppose that this is not true. Then by passing to a subsequence if necessary, we may assume that x n +. Let y n = xn x for n n 1. Similarly as in the proof of Proposition 1 we can show that y n ±u 1 and thus ± for almost all z. Let x n ϕ(x n ) be such that m ϕ (x n ) = x n for all n 1. Since (1 + x n )m ϕ (x n ) 0, we also have x n x n 0 and at least for a subsequence, we have 1 n x n, x n 1 n for all n 1. We have x n = Ax n λ 1 x n p 2 x n u n (see (12)), where A: W 1,p 0 () W 1,p () is defined as in the proof of Proposition 1 (see (11)) and u n L r () is such that u n (z) j(z, ) for almost all z. Thus we have 1 n x n p p λ 1 x n p p u n(z) dz 1 n. By virtue of hypothesis H(j) 2 (vi), we have that lim sup u n(z) dz 0, [ ] so lim sup xn p p λ 1 x n p p 0 and thus by the Rayleigh quotient we have

418 LESZEK GASIŃSKI [ lim xn p p λ 1 x n p ] p = 0. So, by the Fatou lemma (note that we can use it as our assumptions guarantee that there exists b L 1 (), such that b(z) j(z, ζ) for almost all z and all ζ R), we have c = lim ϕ(x n) = lim inf j(z, ) dz ĵ(z) dz 0, which contradicts our choice of c. So we have proved that the sequence {x n } W 1,p 0 () is bounded. Then as in the proof of Proposition 1, we prove the existence of a strongly convergent subsequence. Theorem 3. If hypotheses H(j) 2 hold, then problem (HV I) has a nontrivial solution x 0 W 1,p 0 (). Proof. As in the proof of Theorem 2 (note that Proposition 2 still holds), we conclude that for r > 0 small enough, we have that ϕ(x) β 4 > 0 for all x W 1,p 0 (), with x = r and some β 4 > 0. Using hypothesis H(j) 2 (vii) we have that ϕ(t 0 u 1 ) = tp 0 p u 1 p p λ 1t p 0 p u 1 p p j(z, t 0 u 1 (z)) dz = j(z, t 0 u 1 (z)) dz 0. Also ϕ(0) 0 (see hypothesis H(j) 1 (iii)). By Proposition 3, ϕ satisfies the nonsmooth C c -condition for any c > 0, so also for c 0 = inf ϕ(x) β 4 > 0. So we can x =r apply Theorem 1 and obtain x 0 W 1,p 0 (), x 0 0, such that 0 φ(x 0 ). As in the proof of Theorem 2 we show that x 0 is a solution of (HV I). REFERENCES [1] K. C. Chang, Variational Methods for Nondifferentiable Functionals and their Applications to Partial Differential Equations, J. Math. Anal. Appl., 80 (1981), 102 129. [2] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [3] L. Gasiński and N. S. Papageorgiou, Nonlinear Hemivariational Inequalities at Resonance, Bull. Australian Math. Soc., 60 (1999), 353 364. [4] L. Gasiński and N. S. Papageorgiou, Solutions and Multiple Solutions for Quasilinear Hemivariational Inequalities at Resonance, Proc. Roy. Soc. Edinburgh, 131A (2001), 1091 1111. [5] L. Gasiński and N. S. Papageorgiou, Strongly Resonant Semilinear and Quasilinear Hemivariational Inequalities, Acta Sci. Math. (Szeged), 68 (2002), 727 750. [6] L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman & Hall/CRC, Boca Raton, 2005. [7] D. Goeleven, D. Motreanu and P. D. Panagiotopoulos, Eigenvalue Problems for Variational- Hemivariational Inequalities at Resonance, Nonlin. Anal., 33 (1998), 161 180. [8] M. E. Filippakis, L. Gasiński and N. S. Papageorgiou, Quasilinear Hemivariational Inequalities with strong Resonance at Infinity, Nonlin. Anal., 56 (2004), 331 345. [9] S. Kyritsi and N. S. Papageorgiou, Nonlinear Hemivariational Inequalities with the Generalized Potential Going Beyond the Principal Eigenvalue, Bull. Austr. Math. Soc., 64 (2001), 381 393. Received July 2006; revised May 2007. E-mail address: gasinski@softlab.ii.uj.edu.pl