VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS OF NONHOMOGENEOUS NEUMANN BOUNDARY CONDITIONS

VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS OF NONHOMOGENEOUS NEUMANN BOUNDARY CONDITIONS GABRIELE BONANNO, DUMITRU MOTREANU, AND PATRICK WINKERT Abstract. In this aer variational-hemivariational inequalities with nonhomogeneous Neumann boundary conditions are investigated. Under an aroriate oscillating behaviour of the nonlinear term, the existence of infinitely many solutions to this tye of roblems, even under small erturbations of nonhomogeneous Neumann boundary conditions, is established.. Introduction The aim of this aer is to investigate variational-hemivariational inequalities with a nonhomogeneous Neumann boundary condition. Precisely, let be a nonemty, bounded, oen subset of the Euclidian sace R N, N, with C -boundary, let ]N, + [, and let q L satisfy q, q. Our urose is to study the following roblem: Find u K such that, for all v K, ux 2 ux vx uxdx + qx ux 2 uxvx uxdx P + λαxf ux; vx uxdx + µβxg γux; γvx γuxdσ, where K is a closed convex subset of W, containing the constant functions, and α L, β L, with αx for a.a. x, α, βx for a.a. x, and λ, µ are real arameters, with λ > and µ. Here, F and G stand for Clarke s generalized directional derivatives of locally Lischitz functions F, G : R R given by F ξ = ξ ftdt, Gξ = ξ gtdt, ξ R, with f, g : R R locally essentially bounded functions, and γ : W, L denotes the trace oerator. If =]a, b[ R and h : {a, b} R, then hxdσ reads hb + ha, so roblem P makes sense even for N =. A rototye of P for K = W, is the following boundary value roblem with nonsmooth otential and nonhomogeneous, nonsmooth Neumann boundary 2 Mathematics Subject Classification. 35J87, 49J4, 49J52, 49J53. Key words and hrases. Ellitic variational-hemivariational inequality, Nonsmooth critical oint theory, -Lalacian.

2 GABRIELE BONANNO, DUMITRU MOTREANU, AND PATRICK WINKERT condition u qx u 2 u λαx F u in, 2 u u ν µβx Gγu on. N The main result of this aer is Theorem 3., which establishes, under an aroriate oscillating behavior of F and a suitable growth of G at infinity, the existence of a recise interval for the real arameter λ such that, rovided µ is small enough, roblem P admits infinitely many solutions. Some consequences and alications are also ointed out see Theorems 3.2, 3.3 and Section 4. Just as an examle, we illustrate the alicability of our aroach by stating the following consequence of our results in the secial case of ordinary differential equations. Theorem.. Let f : R R be a non-negative, locally essentially bounded function and set F ξ = ξ ft dt for all ξ R. Assume that F ξ F ξ lim inf ξ 2 = and lim su ξ 2 = +. Then, for each non-negative, continuous function g : R R such that gt g := lim < +, t + t ] and for every µ, 8g [, there is a sequence of airwise distinct functions {u n } W 2,2 ], [ such that for all n N one has u nx + u n x [f u n x, f + u n x] for a.a. x ], [ u n = µgu n ON u n = µgu n, where f t = lim ess inf fz and f + t = lim ess su fz for all t R. δ + t z <δ δ + t z <δ Clearly, if f is a continuous function, then Theorem. ensures the existence of infinitely many classical solutions to the Neumann boundary value roblem ON. It is worth noticing that our results are comletely novel, even for continuous nonlinearities f and g, because of the resence of nonhomogeneous boundary Neumann condition see N. We refer to [2] and [3], and the references therein, for smooth Neumann roblems in the homogeneous case. Moreover, we also observe that our results and those of [] are different since in [] the Neumann boundary condition is homogeneous and the tye of oscillating behavior at infinity required for f imlies that f cannot be of constant sign, which is not necessary the case here. Regarding the existence of infinitely many solutions for nonsmooth Neumann-tye roblems we also mention the aer of Candito [5] and the work of Kristály-Motreanu see [9] where in the second aer the authors don t require that W, is continuously embedded into C. Recently, Kristály-Moroşanu have been described a new cometition henomena between oscillatory and ure ower terms cf. [8] while existence results for variational-hemivariational inequalities of tye P were established in [5] alying an abstract nonsmooth critical oint result given in [].

VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS 3 2. Preliminaries In this Section we give a brief overview on some rerequisites on nonsmooth analysis which are needed in the sequel. Let X, be a real Banach sace. We denote by X the dual sace of X, while, stands for the duality airing between X and X. A function h : X R is called locally Lischitz continuous when to every x X there corresond a neighborhood V x of x and a constant L x such that hz hw L x z w, z, w V x. If x, z X, we write h x; z for the generalized directional derivative of h at the oint x along the direction z, i.e., h x; z := hw + tz hw lim su, w x, t t + see [7, Chater 2]. If h, h 2 : X R are locally Lischitz functions, we have h + h 2 x, z h x, z + h 2x, z, x, z X. 2. The generalized gradient of the function h at x, denoted by hx, is the set hx := {x X : x, z h x; z, z X}. We say that x X is a generalized critical oint of h when h x; z, z X, that clearly means hx see [6]. When a non-smooth function I : X ], + ] is exressed as a sum of a locally Lischitz function, h : X R, and a convex, roer and lower semicontinuous function, j : X ], + ], that is I := h + j, a generalized critical oint of I is every u X such that h u; v u + jv ju, for all v X see [3, Chater 3] and [4]. From now on, assume that X is a reflexive real Banach sace, Φ : X R is a sequentially weakly lower semicontinuous functional, Υ : X R is a sequentially weakly uer semicontinuous functional, λ is a ositive real arameter, j : X ], + ] is a convex, roer and lower semicontinuous functional and D j is the effective domain of j. Write Ψ := Υ j and J λ := Φ λψ = Φ λυ + λj. We also assume that Φ is coercive and Dj Φ ], r[ 2.2 for all r > inf X Φ. Moreover, by 2.2 and rovided r > inf X Φ, we can define and ϕr = inf u Φ ],r[ su Ψv v Φ ],r[ r Φu Ψu ϕ + := lim inf r + ϕr, ϕ := lim inf r inf X Φ + ϕr.,

4 GABRIELE BONANNO, DUMITRU MOTREANU, AND PATRICK WINKERT Assuming also that Φ and Υ are locally Lischitz continuous functionals, in [4] it is roved the following result, which is a version of [, Theorem.]. Theorem 2.. Under the above assumtions on X, Φ and Ψ, one gets: a If ϕ + < + then, for each λ ], ϕ + [, the following alternative holds: either a J λ ossesses a global minimum, or a 2 there is a sequence {u n } of critical oints local minima of J λ such that lim n Φu n = +. b If ϕ < + then, for each λ ], ϕ [, the following alternative holds: either b there is a global minimum of Φ which is also a local minimum of J λ, or b 2 there is a sequence {u n } of critical oints local minima of J λ, with lim Φu n = inf Φ, which weakly converges to a global minimum of n X Φ. We recall that the revious theorem is a non-smooth version of the Ricceri s variational rincile see [6]. On the sace W, we consider the norm u := ux + qx ux dx, which is equivalent to the usual one see for instance [2, Section..5]. Set c := u su u W, \{} u, 2.3 where u := max ux. From 2.3 we infer that c q. If is convex, an x exlicit uer bound for the constant c in 2.3 is c 2 max q, d N N q q, 2.4 where denotes the Lebesgue measure of the set and d := diam see, e.g., [, Remark ]. Finally, we set and A = lim inf Our main result is the following. max F t t ξ F ξ ξ, B = lim su ξ, λ = q α B, λ 2 = α c A. 2.5 3. Main Results

VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS 5 Theorem 3.. Let α L and β L be nonnegative and non-zero functions. Let f : R R be a locally essentially bounded function and set F ξ = ξ ft dt for all ξ R. Assume that A < c q B. 3. Then, for each λ ]λ, λ 2 [, where λ, λ 2 are given by 2.5, for each locally essentially bounded function g : R R, whose otential Gξ = satisfies G := lim su and for every µ [, δ[, where δ = δ g,λ := β c G ξ gt dt, ξ R max t ξ Gt ξ < +, 3.2 lim inf Gξ >, 3.3 λλ2, δ = + if G =, with β = βxdσ, roblem P admits a sequence of weak solutions that is unbounded in W,. Proof. Our aim is to aly Theorem 2.. To this end, fix λ ]λ, λ 2 [ and let g be a locally essentially bounded function satisfying our assumtions. Since λ < λ 2, one has δ := δ g, λ >, so we can consider µ < δ. It follows that λ α c A + µβ c G <, which imlies λ < α c A + µ λ. 3.4 β c G Let X be the Sobolev sace W, endowed with the norm. For any u X, set Φu := u, Υu := αx[ F ux]dx + µ λ βx[ Gγux]dσ, {, if u K, ju = Ψu := Υu ju, J +, otherwise, λ u := Φu λψu. Therefore, J λu = u + λ αxf uxdx + µ βxgγuxdσ + λju, 3.5 for all u X. Now, we claim that ϕ + < +. Let {ρ n } be a real sequence such that lim ρ n = + and n + lim n + max F t t ρ n ρ n = A. Denote r n = ρn and let v Φ ], r n [. Then, taking into account that c v < r n and v c v, one has vx ρ n for every x. Therefore, it

6 GABRIELE BONANNO, DUMITRU MOTREANU, AND PATRICK WINKERT follows ϕr n su αx[ F wx]dx + µ λ w <r n r n α max t ρ n F t + µ λ β max t ρ n Gt r n = c α max t ρ n F t ρ n + c µ λ max Gt β t ρ n. ρ n βx[ Gγwx]dσ jw Hence, ϕ + lim su ϕr n c α A + c µ λ β G. From 3. and 3.2 we n + obtain ϕ + < +, and our claim is roved. Moreover, taking into account 3.4, we get λ < ϕ +. Next, we show that the function J λ in 3.5 is unbounded from below. Let {d n } be a real sequence such that lim d n = + and n + F d n lim n + d = B. 3.6 n Set w n x = d n for all x and n N. Clearly, w n K W, for each n N. We see that and = d n q w n = d n q Φw n λψw n = w n λ αx[ F w n x]dx + µ λ λ α F d n + µ λ β Gd n, thus J λw n = d n q βx[ Gγw n x]dσ + λjw n λ α F d n + µ λ β Gd n. 3.7 If B < +, by 2.5 and since λ > λ, we can take ɛ 3.6 there exists ν ɛ such that Combining with 3.7, one has J λw n < d q n F d n > B ɛd n, n > ν ɛ. λ α B ɛ µβ Gd n. ], B q [. From α λ

VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS 7 Since q λ α B ɛ < and, as known from 3.3, { Gd n } is bounded from below, it follows that lim J λw n =. If B = +, fix M > q n + α λ. Then from 3.6 there exists ν M such that Arguing as before, one obtains J λw n < d q n F d n > Md n, n > ν M. λ α M µβ Gd n. By the choice of M, we have lim J λw n =, which comletes the roof that n + J λ is unbounded from below. Then, from art a of Theorem 2., we know that the function J λ admits a sequence of critical oints {u n } X such that u n as n. The fact that u n X is a critical oint of J λ reads as Φ λυ u n ; v u n + λjv λju n for all v X. 3.8 It remains to rove that u n solves roblem P. From 3.8 it follows that u n K and Φ λυ u n ; v u n for all v K. 3.9 By 3.9 and 2. we infer that or, equivalently, Φ u n ; v u n + λ Υ u n ; v u n for all v K u n x 2 u n x vx u n xdx + qx u n x 2 u n xvx u n xdx + λ Υ u n, v u n, v K. By using 2. and formula 2 on. 77 in [7], we have λ Υ u n ; v u n λ αxf u n x; vx u n xdx + µ 3. βxg γu n x; γvx γu n xdσ. Inserting this into 3. leads to u n x 2 u n x vx u n xdx + qx u n x 2 u n xvx u n xdx [ + λ αxf u n x; vx u n xdx + µ λ ] βxg γu n x; γvx γu n xdσ for every v K, which comletes the roof.

8 GABRIELE BONANNO, DUMITRU MOTREANU, AND PATRICK WINKERT The solutions obtained in Theorem 3. for roblem P corresonding to the arameters λ and µ are local minima of the functional J λ in 3.5 which is associated to P. The following corollary demonstrates that, if the functional J λ satisfies the Palais-Smale condition, there are solutions to P which are not local minima of J λ. Corollary 3.. Under the hyotheses of Theorem 3., assume in addition that min{f t s F t; t, Gt s G t; t} c t θ d for all t R, 3. with constants θ < < s and c, d. Let u W, be a solution rovided by Theorem 3. for P corresonding to λ ]λ, λ 2 [ and µ < δ, so u is a local minimum of the functional J λ in 3.5. If u is isolated, there exists another solution w W, for P corresonding to λ and µ which is not a local minimum of J λ. Proof. Let us check that the functional J λ : W, ], + ] given in 3.5 satisfies the Palais-Smale condition in the sense of [3,. 64]. This amounts to saying that whenever a sequence {u n } K is such that Ju n is bounded and J λ u n; v u n ε n v u n for all v K, 3.2 with ε n +, contains a convergent subsequence. Setting v = in 3.2 and combining with the inequality J λ u n M, for a constant M >, yield u n + s λ αx[f u n x s F u n ; u n ]dx + µ βx[gγu n x s G γu n ; γu n ]dσ M + ε n s u n. Using hyothesis 3., it is straightforward to rove that the sequence {u n } is bounded in W,. Furthermore, since on W, fulfills the S + roerty, by handling 3.2 we find that {u n } contains a convergent subsequence, so the Palais-Smale condition for the functional J λ is satisfied. We are thus in a osition to aly [, Theorem 4.2] to the functional J λ from which we achieve the desired conclusion. Remark 3.. Relation 2.4 is useful to verify inequality 3. and to estimate the numbers λ 2 and δ in Theorem 3. rovided the bounded domain is convex. Remark 3.2. Actually, Theorem 3. ensures that the sequence {u n } of solutions of roblem P satisfies the following sharer inequality: u n x 2 u n x vx u n xdx + qx u n x 2 u n xvx u n xdx + λu u n ; v u n + µv u n ; v u n, v K, where Uu = αxf uxdx and V u = βxgγuxdσ for all u W,.

VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS 9 Remark 3.3. In Theorem 3. the function g may not have an oscillating behaviour at infinity see, for instance, Examle 4. where gu = u. On the other hand, g must satisfy 3.2, namely it must have a suitable growth at infinity. It is worth noticing that when 3.2 fails, that is, lim su max t ξ Gt ξ = +, 3.3 the existence of infinitely many solutions to P can be again guaranteed, rovided that G max t ξ Gt := lim inf ξ < +, 3.4 for which g is then oscillating at infinity and assuming that f, ossibly even not oscillating at infinity, satisfies the following conditions: B + := lim su max F t t ξ ξ < +, lim inf F t >. 3.5 To be recise, the following result holds: Let α, β, f be as in the statement ] of Theorem 3. and assume that f satisfies 3.5. Then, for each λ, [ α c, B + for each locally essentially bounded function g : R R satisfying 3.3 and 3.4, and for each µ ], δ[, where δ := δ g,λ := β c G λ α c B +, δ = + if G =, the roblem P admits a sequence of weak solutions which is unbounded in W,. Now we oint out two significant secial cases of Theorem 3.. Theorem 3.2. Let α L and β L be nonnegative and non-zero. Let f : R R be a non-ositive, locally essentially bounded function, and set F ξ = ξ ft dt for every ξ R. Assume that F ξ F ξ lim inf ξ < c lim su q ξ. 3.6 Then, for each λ ]λ, λ 2 [, where λ, λ 2 are given by 2.5, for each non-ositive, locally essentially bounded function g : R R, whose otential Gξ = ξ R, satisfies and for every µ [, δ[, where G := lim su δ = δ g,λ := βxdσc G Gξ ξ < +, λ α c lim inf ξ F ξ ξ, gt dt, roblem P admits a sequence of weak solutions that is unbounded in W,.

GABRIELE BONANNO, DUMITRU MOTREANU, AND PATRICK WINKERT Remark 3.4. In Theorem 3.2 the assumtion 3.6 can be written as well as and δ = ]λ, λ 2 [ = F ξ lim inf ξ < c F ξ q lim su ξ, q α lim inf βxdσc lim inf F ξ ξ Gξ ξ, α c lim su F ξ ξ + λ α c lim su F ξ ξ. Theorem 3.3. Let α L and β L be nonnegative and non-zero. Let f : R R be a non-ositive, locally essentially bounded function and set F ξ = ξ ft dt for every ξ R. Assume that F ξ F ξ lim inf ξ = and lim su ξ = +. Then, for each non-ositive, locally essentially bounded function g : R R such that gξ g := lim ξ < +, and for every µ [, δ[, where δ = δ g := βxdσc g, there is an unbounded sequence {u n } W, such that u n x 2 u n x vx u n xdx + qx u n x 2 u n xvx u n xdx + αxf u n x; vx u n xdx + µ βxg γu n x; γvx γu n xdσ for all v K. Remark 3.5. We exlicitly observe that in Theorem 3. no symmetry assumtions on the nonlinear term is done. On the other hand, the case N cannot be studied by this method since the embedding of W, in C fails and ϕ + cannot be uer estimated, without further assumtions on the nonlinear term. Remark 3.6. An examle of alication of revious results is given in the next section see Examle 4..

VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS If F oscillates at zero, we can give an analogous result as in Theorem 3.. To this end, let max F t t ξ F ξ A = lim inf ξ + ξ, B = lim su ξ ξ +, and Then, our result reads as follows. λ = q α B, λ 2 = α c A. 3.7 Theorem 3.4. Let α L and β L be nonnegative and non-zero functions. Let f : R R be a locally essentially bounded function and set F ξ = ξ ft dt for all ξ R. Assume that A < c q B. Then, for each λ ]λ, λ 2 [, where λ, λ 2 are given by 3.7, for each locally essentially bounded function g : R R, whose otential Gξ = satisfies G := lim su ξ + and for every µ [, δ[, where δ = δ g,λ := β c G max t ξ Gt ξ lim inf Gξ, ξ + < +, ξ λλ2, δ = + if G =, gt dt, ξ R with β = βxdσ, roblem P admits a sequence of distinct weak solutions converging uniformly to zero. Proof. The roof can be done similarly as the roof of Theorem 3. by alying art b of Theorem 2. instead of art a, thus obtaining the assertion. 4. Alications and Examles Here we resent an alication of Theorem 3. to an ordinary differential roblem with discontinuous nonlinearities. Theorem 4.. Let α L ], [ be a non-negative and non-zero function and let β, β be non-negative constants such that at least one of them is ositive. Let f : R R be a non-ositive, locally essentially bounded function and set F ξ = ξ ft dt for every ξ R. Assume that F ξ lim inf ξ 2 < F ξ lim su 2 ξ 2. 4.

2 GABRIELE BONANNO, DUMITRU MOTREANU, AND PATRICK WINKERT Then, for each λ α 2 lim su, F ξ ξ 2 α 4 lim inf F ξ, for each nonositive, continuous function g : R R, whose otential Gξ = satisfies and for every µ [, δ[, where G := lim su δ = δ g,λ := β + β 4G Gξ ξ 2 < +, λ α 4 lim inf ξ 2 ξ F ξ ξ, gt dt, ξ R, there is a sequence of airwise distinct functions {u n } W 2,2 ], [ such that for all n N one has u nx u n x [λαxf u n x, λαxf + u n x] for a.a. x ], [ u n = µβ gu n N u n = µβ gu n. Proof. The result is a consequence of Theorem 3.2. For the sake of clarity, we first oint out three facts secific for the ordinary differential case that enable us to adat the roof of Theorem 3. to this situation. The first one is the inequality βx[ Gγux]dσ = β[ Gu] + β[ Gu] β max ξ ρ n [ Gξ] + β max ξ ρ n [ Gξ] = β + β max ξ ρ n [ Gξ] for all u < r n, from which we derive ϕ + lim su ϕr n c α A + c µ λ β + βg. n + The second one is the estimate βx[ Gγw n x]dσ = β + β[ Gd n ] β + β lim inf Gξ, from which we deduce lim J λw n =. The last one is n + [ βxgγu n x; γvx γu n xdσ = [βgu n ; v u n + βgu n ; v u n ] βg u n ; v u n + βg u n ; v u n, ]

VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS 3 from which it turns out + + λ u nx 2 u nx v x u nxdx qx u n x 2 u n xvx u n xdx αxf u n x; vx u n xdx + µ [βg u n ; v u n + βg u n ; v u n ], v K. The roof of Theorem 4. is carried out as follows. Fix λ and µ as in the conclusion of Theorem 4.. We may aly Theorem 3.2 see also Remark 3.2 by choosing =], [, = 2, q, K = W.2 ], [, and noticing that hyothesis 4. in conjunction with 2.4 imlies that 3.6 holds true. Then there exists an unbounded sequence {u n } W.2 ], [ such that u nxv xdx + u n xvxdx + λu u n ; v + µ [β G u n ; v + β G u n ; v], v W.2 ], [, where β = β and β = β, while the function U was introduced in Remark 3.2. Setting [ ] T n v = u nxv xdx + u n xvxdx µ [β gu n v + β gu n v] for all v W.2 ], [, we see that T n is linear and continuous on W.2 ], [, and T n λ Uu n. Taking into account that W.2 ], [ is continuously and densely embedded in L 2 ], [, from [6, Theorem 2.2] we know that there is h n L 2 ], [ such that [ ] u nxv xdx + u n xvxdx µ [β gu n v + β gu n v] = h n xvxdx for all v W.2 ], [. This ensures that u n is the unique solution of the roblem u u = h n x in ], [ u = µβ gu u = µβ gu and, in addition, u n W 2.2 ], [. through [6, Corollary,. ] that Moreover, since T n λ Uu n, we deduce h n x [ λαxf u n x, λαxf + u n x] for a.a. x ], [. Hence the conclusion regarding roblem N is obtained with u n = u n. Remark 4.. Theorem. in the Introduction is a direct consequence of Theorem 4..

4 GABRIELE BONANNO, DUMITRU MOTREANU, AND PATRICK WINKERT Examle 4.. Set a n := 2n!n + 2! 2n!n + 2! +, b n := 4n +! 4n +! for every n N and define the nonnegative discontinuous function f : R R by { 2n +![n n +! n n! ] if t n N ]a n, b n [, f t = otherwise. Denoting F ξ = F ξ lim inf ξ f t dt for every ξ R, a simle comutation shows that F ξ ξ = +. Owing to Theorem 3.3, there is a ξ = and lim su sequence of airwise distinct functions {u n } W, such that u n x 2 u n x vx u n xdx + u n x 2 u n xvx u n xdx + F u n x; vx u n xdx + [ γu n x 2 ]γvx γun xdσ, v K. In articular, Theorem 4. ensures that there is a sequence of airwise distinct functions {u n } W 2,2 ], [ such that for all n N there holds u nx + u n x [f2 ux, f 2 + ux] for a.a. x ], [ u n = u n u n = u n. Acknowledgement: The authors are grateful to the referees for the careful reading and helful comments. References [] G. Bonanno and P. Candito, Three solutions to a Neumann roblem for ellitic equations involving the Lalacian, Arch. Math. Basel 8 23, 424 429. [2] G. Bonanno and G. D Aguì, A Neumann boundary value roblem for the Sturm-Liouville equation, Al. Math. Comut. 28 29, 38-327. [3] G. Bonanno and G. D Aguì, On the Neumann Problem for ellitic equations involving the Lalacian, J. Math. Anal. Al. 358 29, 223-228,. [4] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value roblem with discontinuous nonlinearities, Bound. Value Probl. 29 29, 2. [5] P. Candito, Infinitely many solutions to the Neumann roblem for ellitic equations involving the -Lalacian and with discontinuous nonlinearities, Proc. Edinb. Math. Soc. 2 45 22, 397 49. [6] K. C. Chang, Variational methods for nondifferentiable functionals and their alications to artial differential equations, J. Math. Anal. Al. 8 98, 2 29. [7] F. H. Clarke, Otimization and Nonsmooth Analysis, Classics in Alied Mathematics, vol. 5, SIAM, Philadelhia, 99. [8] A. Kristály and G. Moroşanu, New cometition henomena in Dirichlet roblems, J. Math. Pures Al. 94 2, 555 57. [9] A. Kristály and D. Motreanu, Nonsmooth Neumann-tye roblems involving the -Lalacian, Numer. Funct. Anal. Otim. 28 27, 39 326.

VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH SMALL PERTURBATIONS 5 [] R. Livrea and S. A. Marano, Existence and classification of critical oints for nondifferentiable functions, Adv. Differential Equations 9 24, 96 978. [] S. A. Marano and D. Motreanu, Infinitely many critical oints of non-differentiable functions and alications to a Neumann tye roblem involving the Lalacian, J. Differential Equations 82 22, 8 2. [2] V. G. Maz ja, Sobolev saces, Sringer Series in Soviet Mathematics, Sringer-Verlag, Berlin, 985. [3] D. Motreanu and P. D. Panagiotooulos, Minimax Theorems and Qualitative Proerties of the Solutions of Hemivariational Inequalities, Nonconvex Otim. Al. 29 Kluwer, Dordrecht, 998. [4] D. Motreanu and V. Rădulescu, Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Otimization and Alications, Kluwer Academic Publishers 23. [5] D. Motreanu and P. Winkert, Variational-hemivariational inequalities with nonhomogeneous Neumann boundary condition, Matematiche Catania, in ress. [6] B. Ricceri, A general variational rincile and some of its alications, J. Comut. Al. Math. 33 2, 4-4. Deartment of Science for Engineering and Architecture Mathematics Section, Engineering Faculty, University of Messina, 9866 - Messina, Italy E-mail address: bonanno@unime.it Déartement de Mathématiques, Université de Perignan, Avenue Paul Alduy 52, 6686 Perignan Cedex, France E-mail address: motreanu@univ-er.fr Technische Universität Berlin, Institut für Mathematik, Straße des 7. Juni 36, 623 Berlin, Germany E-mail address: winkert@math.tu-berlin.de