The sub- and supersolution method for variational–hemivariational inequalities

Nonlinear Analysis 69 (2008) 816–822 www.elsevier.com/locate/na

The sub- and supersolution method for variational–hemivariational inequalities Siegfried Carl Institut f¨ur Mathematik, Martin-Luther-Universit¨at Halle-Wittenberg, D-06099 Halle, Germany

Dedicated to Professor V. Lakshmikantham on the occasion of his 84th birthday

Abstract The well-known method of sub- and supersolutions is a powerful tool for proving existence and comparison results for initial and/or boundary value problems of nonlinear ordinary differential equations as well as for nonlinear partial differential equations of elliptic and parabolic type. The main goal of this paper is to extend the idea of the sub- and supersolution method in a natural and systematic way to quasilinear elliptic variational–hemivariational inequalities. Owing to the intrinsic asymmetry of the latter (where the problems are stated as inequalities rather than as equalities) an appropriate generalization of the notion of sub- and supersolutions to variational–hemivariational inequalities is by no means straightforward. The obtained results of this paper complement the development of the sub- and supersolution method for nonsmooth variational problems presented in a recent monograph by S. Carl, Vy K. Le and D. Motreanu. c 2008 Elsevier Ltd. All rights reserved.

MSC: 35J85; 35R70; 47J20 Keywords: Sub- and supersolution; Existence; Comparison; Clarke’s gradient; Variational–hemivariational inequality; Multivalued pseudomonotone operator

1. Introduction 1, p

Let Ω ⊂ R N be a bounded domain with Lipschitz boundary Γ = ∂Ω , and let V = W 1, p (Ω ) and V0 = W0 (Ω ), 1 < p < ∞, denote the usual Sobolev spaces with their dual spaces V ∗ and V0∗ , respectively. The main purpose of this paper is to extend the method of sub- and supersolutions to the following class of quasilinear elliptic variational–hemivariational inequalities: Z Z u ∈ K : h−∆ p u + F(u), v − ui + j1o (u; v − u)dx + j2o (γ u; γ v − γ u)dΓ ≥ 0, ∀v ∈ K , (1.1) Ω

Γ

where K is a closed, convex subset of V , h·, ·i denotes the duality pairing, and ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian. The operator F stands for the Nemytskij operator associated with some Carath´eodory function

E-mail address: [email protected]. c 2008 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2008.02.046

S. Carl / Nonlinear Analysis 69 (2008) 816–822

817

f : Ω × R → R, and γ : V → L p (Γ ) denotes the trace operator which is known to be linear and compact from V into L p (Γ ). By jko (s; r ), k = 1, 2, we denote the generalized directional derivative of locally Lipschitz functions jk : R → R at s in the direction r given by jko (s; r ) = lim sup

y→s,t↓0

jk (y + tr ) − jk (y) , t

(1.2)

cf., e.g., [8, Chap. 2]. Only for the sake of simplifying our presentation and in order to emphasize the main concept and the fundamental ideas we consider problem (1.1) as a prototype of much more general problems that may include general Leray–Lions operators instead of −∆ p + F, and the functions j1 and j2 may depend, in addition, on x ∈ Ω and x ∈ Γ , respectively. The method of sub- and supersolutions has been proved to be a powerful tool for obtaining existence and comparison results for a wide range of nonlinear elliptic and parabolic boundary value problems. This method combined with variational and topological arguments (such as, critical point theory, Mountain–Pass Theorem, Second Deformation Lemma) has been successfully employed, e.g., in the qualitative analysis of nonlinear elliptic problems, in particular, in the study of multiple solutions, see [3,4]. Moreover, the constructive aspect of this method is reflected by the monotone iterative technique and its advanced version, the generalized quasilinearization method, see, e.g., [6,7,9,10,12]. The extension of the concept of the sub- and supersolution method to the variational–hemivariational inequality (1.1) is by no means straightforward, because of the asymmetry of the latter which is stated as an inequality rather than an equality. In the next section we introduce an appropriate notion for sub- and supersolution for (1.1), which in a natural way extends the well-known notions for sub- and supersolution for elliptic equations. 2. Notion of sub- and supersolution For functions w, z and sets W and Z of functions defined on Ω or Γ we use the notations: w ∧ z = min{w, z}, w ∨ z = max{w, z}, W ∧ Z = {w ∧ z : w ∈ W, z ∈ Z }, W ∨ Z = {w ∨ z : w ∈ W, z ∈ Z }, and w ∧ Z = {w} ∧ Z , w ∨ Z = {w} ∨ Z . Let q be the H¨older conjugate to p satisfying 1/ p + 1/q = 1. Next we introduce our basic notion of sub- and supersolution for the variational–hemivariational inequality (1.1). Definition 2.1. A function u ∈ V is called a subsolution of (1.1) if F(u) ∈ L q (Ω ) and the following hold: (i) u ∨ K ⊂ K , R R (ii) h−∆ p u + F(u), v − ui + Ω j1o (u; v − u)dx + Γ j2o (γ u; γ v − γ u)dΓ ≥ 0, for all v ∈ u ∧ K . Definition 2.2. A function u¯ ∈ V is called a supersolution of (1.1) if F(u) ¯ ∈ L q (Ω ) and the following hold: (i) u¯ ∧ K ⊂ K , R R (ii) h−∆ p u¯ + F(u), ¯ v − ui ¯ + Ω j1o (u; ¯ v − u)dx ¯ + Γ j2o (γ u; ¯ γ v − γ u)dΓ ¯ ≥ 0, for all v ∈ u¯ ∨ K . Remark. Note that the notions for sub- and supersolution defined in Definitions 2.1 and 2.2 have a symmetric structure, i.e., one obtains the definition for the supersolution u¯ from the definition of the subsolution by replacing u in Definition 2.1 by u, ¯ and interchanging ∨ by ∧. To justify that Definitions 2.1 and 2.2 are in fact natural extensions of the usual notions of sub- and supersolutions for elliptic boundary value problems let us discuss several special cases of (1.1) and their corresponding sub- and supersolutions according to the definitions given above. Example 1. K = V , jk : R → R smooth, i.e., jk ∈ C 1 (R), k = 1, 2. In this case we have jko (s; r ) = jk0 (s)r and (1.1) reduces to: Z Z h−∆ p u + F(u), vi + j10 (u)vdx + j20 (γ u)γ vdΓ = 0, Ω

Γ

∀v ∈ V,

(2.1)

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S. Carl / Nonlinear Analysis 69 (2008) 816–822

which corresponds to the weak formulation of the nonlinear elliptic boundary value problem ∂u + j20 (u) = 0 on Γ , (2.2) ∂ν where ∂/∂ν denotes the outward pointing conormal derivative associated with ∆ p . Let us consider the Definition 2.1 in this case. Condition (i) is trivially satisfied, and v ∈ u ∧ V has the form v = u ∧ ϕ = u − (u − ϕ)+ with ϕ ∈ V , where w + = max{w, 0}. Let χ = (u − ϕ)+ . Replacing v in Definition 2.1(ii) by its representation we obtain Z Z j10 (u)χdx + j20 (γ u)γ χ dΓ ≤ 0, ∀χ ∈ M, h−∆ p u + F(u), χi + (2.3) −∆ p u + F(u) + j10 (u) = 0

in Ω ,

Ω

Γ

p

ϕ)+ ,

ϕ ∈ V }. If L + (Ω ) denotes the positive cone of all nonnegative elements of where M := {χ ∈ V : χ = (u − p L p (Ω ), then we have apparently M = V ∩ L + (Ω ), and thus (2.3) is nothing but the usual definition of the (weak) subsolution of (2.2). Similar arguments apply to show that the supersolution defined in Definition 2.2 is the (weak) supersolution of (2.2) in the usual sense. Example 2. K = V0 , jk ∈ C 1 (R), k = 1, 2. One readily verifies that problem (1.1) in this case is equivalent to Z u ∈ V0 : h−∆ p u + F(u), vi + j10 (u)vdx = 0, ∀v ∈ V0 ,

(2.4)

Ω

which is nothing but the weak formulation of the homogeneous Dirichlet problem −∆ p u + F(u) + j10 (u) = 0

in Ω ,

u=0

on Γ .

(2.5)

Consider Definition 2.1 in this case. For u ∈ V condition (i), i.e., u ∨ V0 ⊂ V0 is satisfied if and only if γu ≤ 0

i.e., u ≤ 0

on Γ ,

(2.6) − ϕ)+

and from (ii) we get by using the representation v = u − (u Z h−∆ p u + F(u), χi + j10 (u)χdx ≤ 0, ∀χ ∈ M0 ,

with ϕ ∈ V0 the following inequality: (2.7)

Ω

p

where M0 := {χ ∈ V : χ = (u − ϕ)+ , ϕ ∈ V0 } ⊂ V0 ∩ L + (Ω ). One can prove that the set M0 is a dense subset of p V0 ∩ L + (Ω ) (cf. [1]), which shows that (2.7) together with (2.6) is nothing but the weak formulation for the subsolution of the Dirichlet problem (2.5). Example 3. K = V0 or K = V , jk : R → R not necessarily smooth, k = 1, 2. In this case, (1.1) reduces to a hemivariational inequality, which for K = V0 contains as special case the following Dirichlet problem for the elliptic inclusion −∆ p u + F(u) + ∂ j1 (u) 3 0

in Ω ,

u=0

on Γ ,

(2.8)

and for K = V the inclusion −∆ p u + F(u) + ∂ j1 (u) 3 0

in Ω ,

∂u + ∂ j2 (u) 3 0 ∂ν

on Γ ,

(2.9)

where the multifunctions ∂ jk : R → 2R \ ∅ in (2.8) and (2.9) are given by Clarke’s generalized gradient of the locally Lipschitz functions jk defined by ∂ jk (s) := {ζ ∈ R : jko (s; r ) ≥ ζ r, ∀r ∈ R}. Problems of the form (2.9) have been treated in [5]. Under the assumptions on the functions jk specified in the next section one can show that the notion of sub- and supersolution for the inclusion problem (2.9) used in [5] can be deduced from the general notion given by Definitions 2.1 and 2.2.

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Example 4. K ⊂ V , jk = 0. Problem (1.1) reduces to the usual variational inequality of the form u ∈ K : h−∆ p u + F(u), v − ui ≥ 0,

∀v ∈ K .

(2.10)

As will be seen later, Definitions 2.1 and 2.2 provide an appropriate notion for sub- and supersolution, which allows to establish the method of sub- and supersolutions. Various other special cases can be deduced from the general setting of the variational–hemivariational inequality (1.1) by a suitable choice of the closed convex set K . 3. Hypotheses and preliminaries Let (u, u) ¯ be an ordered pair of sub- and supersolutions for problem (1.1). We impose the following hypotheses on the locally Lipschitz functions jk : R → R, and the Carath´eodory function f : Ω × R → R: (H1) There exist constants ck ≥ 0 such that ζ ∈ ∂ jk (s) :

|ζ | ≤ ck (1 + |s| p−1 ),

∀s ∈ R.

(H2) There exist constants dk ≥ 0 such that ζ1 ≤ ζ2 + dk (s2 − s1 ) p−1 for all ζi ∈ ∂ jk (si ), i = 1, 2, and for all s1 , s2 with s1 < s2 . q (H3) There is some k ∈ L + (Ω ) such that for a.a. x ∈ Ω and for all s ∈ [u(x), u(x)] ¯ the following local growth condition holds: | f (x, s)| ≤ k(x). By means of j1 and j2 we introduce integral functionals J1 and J2 defined on L p (Ω ) and L p (Γ ), respectively, and given by Z Z J1 (u) = j1 (u(x))dx, u ∈ L p (Ω ), J2 (v) = j2 (v(x))dΓ , v ∈ L p (Γ ). Ω

Γ

Due to hypothesis (H1) and Lebourg’s mean value theorem the functionals J1 : L p (Ω ) → R and J2 : L p (Γ ) → R are well-defined and Lipschitz continuous on bounded sets of L p (Ω ) and L p (Γ ) (cf. [8]), respectively, so that p ∗ p ∗ Clarke’s generalized gradients ∂ J1 : L p (Ω ) → 2(L (Ω )) and ∂ J2 : L p (Γ ) → 2(L (Γ )) are well-defined too. Moreover, Aubin–Clarke theorem (cf. [8, p. 83]) provides the following characterization of the generalized gradients. For u ∈ L p (Ω ) we have η ∈ ∂ J1 (u) H⇒ η ∈ L q (Ω )

with η(x) ∈ ∂ j1 (u(x)) for a.e. x ∈ Ω ,

(3.1)

with ξ(x) ∈ ∂ j2 (v(x)) for a.e. x ∈ Γ .

(3.2)

and similarly for v ∈ L p (Γ ) ξ ∈ ∂ J2 (v) H⇒ ξ ∈ L q (Γ )

In the proof of our main result we make use of the following surjectivity result for multivalued pseudomonotone mappings perturbed by maximal monotone operators in reflexive Banach spaces (cf., e.g., [11, Theorem 2.12]). ∗

Theorem 3.1. Let X be a real reflexive Banach space with dual space X ∗ , Φ : X → 2 X a maximal monotone ∗ operator, and u 0 ∈ dom(Φ). Let A : X → 2 X be a pseudomonotone operator, and assume that either Au 0 is quasi∗ bounded or Φu 0 is strongly quasi-bounded. Assume further that A : X → 2 X is u 0 -coercive, i.e., there exists a real-valued function c : R+ → R with c(r ) → +∞ as r → +∞ such that for all (u, u ∗ ) ∈ graph(A) one has hu ∗ , u − u 0 i ≥ c(kuk X )kuk X . Then A + Φ is surjective, i.e., range (A + Φ) = X ∗ . The operators Au 0 and Φu 0 that appear in the theorem above are defined by Au 0 (v) := A(u 0 + v) and similarly for Φu 0 . As for the notion of quasi-bounded and strongly quasi-bounded we refer to [11, p.51]. In particular, any bounded operator is quasi-bounded and strongly quasi-bounded as well. The following proposition provides sufficient ∗ conditions for an operator A : X → 2 X to be pseudomonotone, which is suitable for our purpose.

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S. Carl / Nonlinear Analysis 69 (2008) 816–822 ∗

Proposition 3.1. Let X be a real reflexive Banach space, and assume that A : X → 2 X satisfies the following conditions: (i) For each u ∈ X we have that A(u) is a nonempty, closed and convex subset of X ∗ ; ∗ (ii) A : X → 2 X is bounded; (iii) If u n * u in X and u ∗n * u ∗ in X ∗ with u ∗n ∈ A(u n ) and if lim suphu ∗n , u n − ui ≤ 0, then u ∗ ∈ A(u) and hu ∗n , u n i → hu ∗ , ui. ∗

Then the operator A : X → 2 X is pseudomonotone. As for the proof of Proposition 3.1 we refer, e.g., to [11, Chap. 2]. By means of Clarke’s generalized gradient ∂ Jk we introduce the following multivalued operators: Φ1 (u) := (i ∗ ◦ ∂ J1 ◦ i)(u),

Φ2 (u) := (γ ∗ ◦ ∂ J2 ◦ γ )(u),

(3.3)

where i ∗ : L q (Ω ) → V ∗ and γ ∗ : L q (Γ ) → V ∗ denote the adjoint operators of the embedding i : V → L p (Ω ) and the trace operator γ : V → L p (Γ ), respectively, given by Z Z hi ∗ η, ϕi = ηϕdx, hγ ∗ ξ, ϕi = ξ γ ϕdΓ , ∀ϕ ∈ V. Ω

Γ

The operators Φk , k = 1, 2, possess the following properties, cf. [5]. ∗

Lemma 3.1. The operators Φk : V → 2V , k = 1, 2, are bounded and pseudomonotone. Let b1 : Ω × R → R and b2 : Γ × R → R be cut-off functions related to an ordered pair of sub- and supersolutions and defined by, respectively,  p−1 ¯ if s > u(x), ¯ (s − u(x)) if u(x) ≤ s ≤ u(x), ¯ b1 (x, s) = 0 (3.4)  −(u(x) − s) p−1 if s < u(x),  p−1 ¯ if s > γ u(x), ¯ (s − γ u(x)) if γ u(x) ≤ s ≤ γ u(x), ¯ (3.5) b2 (x, s) = 0  −(γ u(x) − s) p−1 if s < γ u(x). One readily verifies that b1 and b2 are Carath´eodory functions and their associated Nemytskij operators B1 : L p (Ω ) → L q (Ω ) and B2 : L p (Γ ) → L q (Γ ), respectively, are continuous and bounded. Taking into account the compactness of the embedding i : V → L p (Ω ) and the trace γ : V → L p (Γ ) one has the following result. Lemma 3.2. The operators Bˆ1 = i ∗ ◦ B1 ◦i : V → V ∗ , and Bˆ2 = γ ∗ ◦ B2 ◦ γ : V → V ∗ are bounded and completely continuous. Let F ◦ T be the composition of the Nemytskij operator F generated by f and the truncation operator T given by T u = max{u, u} + min{u, u} ¯ − u, then F ◦ T : L p (Ω ) → L q (Ω ) is continuous and bounded by (H3), and thus Fˆ := i ∗ ◦ (F ◦ T ) ◦ i : V → V ∗ is completely continuous. By a slight modification of [5, Lemma 3.3] one obtains the following result. ∗ Lemma 3.3. The operator A := −∆ p + Fˆ + λ1 Bˆ1 + λ2 Bˆ2 + Φ1 + Φ2 : V → 2V is bounded and pseudomonotone, and for λk > 0 sufficiently large, A is, in addition, u 0 -coercive for any u 0 ∈ K .

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4. The sub- and supersolution method Our main result is given in the next theorem which can be regarded as an appropriate extension of the method of sub- and supersolutions to the variational–hemivariational inequality (1.1). Theorem 4.1. Let hypotheses (H1)–(H3) be satisfied, and assume the existence of sub- and supersolutions u and u, ¯ respectively, satisfying u ≤ u. ¯ Then there exist solutions of (1.1) within the ordered interval [u, u]. ¯ Proof. Let I K : V → R ∪ {+∞} denote the indicator function related to the given closed convex set K 6= ∅, i.e.,  0 if u ∈ K , I K (u) = +∞ if u 6∈ K , which is known to be proper, convex, and lower semicontinuous. By means of the indicator function the variational–hemivariational inequality (1.1) can be rewritten in the following form: Find u ∈ K such that Z Z o h−∆ p u + F(u), v − ui + I K (v) − I K (u) + j1 (u; v − u)dx + j2o (γ u; γ v − γ u)dΓ ≥ 0, ∀v ∈ V. Ω

Γ

(4.1) We next associate to (4.1) an auxiliary problem: Find u ∈ K such that ˆ h−∆ p u + F(u) + λ1 Bˆ1 (u) + λ2 Bˆ2 (u), v − ui + I K (v) − I K (u) Z Z + j1o (u; v − u)dx + j2o (γ u; γ v − γ u)dΓ ≥ 0, ∀v ∈ V, Ω

(4.2)

Γ

ˆ Bˆk are defined in Section 3, and λk ∈ R+ are positive parameters to be specified later. We note that the proof where F, of the theorem is accomplished once we have shown that, first, the auxiliary problem (4.2) has solutions, and, second, that any solution of (4.2) belongs to the interval [u, u]. ¯ As for the existence we consider the multivalued operator A + ∂ I K : V → 2V ∗ ,

(4.3)

where A is as in Lemma 3.3, and ∂ I K is the usual subdifferential of I K which is known to be a maximal monotone operator, cf., e.g., [13]. In view of Lemma 3.3, for λk > 0 sufficiently large, we may apply the surjectivity result of Theorem 3.1, which implies the existence of u ∈ K such that 0 ∈ A(u) + ∂ I K (u). By definition of A and ∂ I K the latter inclusion implies the existence of η∗ ∈ Φ1 (u), ξ ∗ ∈ Φ2 (u), and θ ∗ ∈ ∂ I K (u) such that equation ˆ −∆ p u + F(u) + λ1 Bˆ1 (u) + λ2 Bˆ2 (u) + η∗ + ξ ∗ + θ ∗ = 0, η∗

i ∗η

ξ∗

in V ∗ γ ∗ξ

(4.4)

holds, where in view of (3.1)–(3.3) we have = and = with η ∈ and η(x) ∈ ∂ j1 (u(x)), and ξ ∈ L q (Γ ) and ξ(x) ∈ ∂ j2 (γ u(x)). By using the definition of Clarke’s generalized gradient and of ∂ I K (u) Eq. (4.4) readily implies (4.2), and thus the existence result for the auxiliary problem. To complete the proof we need to show that any solution of (4.2) is contained in [u, u]. ¯ Therefore, let u be any solution of (4.2) which is equivalent to ˆ u ∈ K : h−∆ p u + F(u) + λ1 Bˆ1 (u) + λ2 Bˆ2 (u), v − ui Z Z o + j1 (u; v − u)dx + j2o (γ u; γ v − γ u)dΓ ≥ 0, ∀v ∈ K . Ω

L q (Ω )

(4.5)

Γ

To prove the inequality u ≤ u¯ we apply the special test function v = u¯ ∨ u = u¯ + (u − u) ¯ + in Definition 2.2(ii), and + v = u¯ ∧ u = u − (u − u) ¯ ∈ K in (4.5), and get by adding the resulting inequalities the following: Z Z

j1o (u; ¯ (u − u) ¯ + ) + j1o (u; −(u − u) ¯ + ) dx + j2o (γ u; ¯ (γ u − γ u) ¯ + ) + j2o (γ u; −(γ u − γ u) ¯ + ) dΓ Ω Z Γ Z + ≥ ¯ (u − u) ¯ dx + λ1 B1 (u)(u − u) ¯ + dx ((F ◦ T )(u) − F(u)) Ω Ω Z + λ2 B2 (γ u)(γ u − γ u) ¯ + dΓ + h−∆ p u − (−∆ p u), ¯ (u − u) ¯ + i. (4.6) Γ

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By (H2) the left-hand side (LHS) of (4.6) can be estimated above as follows: Z Z p (γ u − γ u) ¯ p dΓ , (u − u) ¯ dx + d2 L H S ≤ d1 {u>u} ¯

(4.7)

{γ u>γ u} ¯

where {u > u} ¯ = {x ∈ Ω : u(x) > u(x)} ¯ and {γ u > γ u} ¯ = {x ∈ Γ : γ u(x) > γ u(x)}. ¯ As the first term on the right-hand side (RHS) of (4.6) is zero and the last term on the RHS is nonnegative, by applying the definition of bk we get an estimate of the RHS in the form below: Z Z (γ u − γ u) ¯ p dΓ . (4.8) (u − u) ¯ p dx + λ2 R H S ≥ λ1 {u>u} ¯

{γ u>γ u} ¯

From (4.6)–(4.8) we finally obtain Z Z ¯ + ) p dΓ ≤ 0. ¯ + ) p dx + (λ2 − d2 ) ((γ u − γ u) (λ1 − d1 ) ((u − u) Ω

(4.9)

Γ

Selecting λk > dk , and λk large enough such that also Lemma 3.3 holds, then from (4.9) we get (u − u) ¯ + = 0, i.e., u ≤ u. ¯ In a similar way one shows u ≤ u which completes the proof.  Remarks. (i) Based on the results obtained in [2] conditions (H1) and (H2) can be relaxed by assuming (H1) and (H2) to hold only locally. Moreover, applying the technique developed in [2] of appropriately modifying jk outside the interval of sub-supersolution eventually allows us to omit hypothesis (H2). (ii) Using techniques developed in [1] one can show that the set S ⊂ [u, u] ¯ of all solutions of (1.1) enclosed by the sub- and supersolution is compact in V , and, moreover, S possesses extremal (i.e., greatest and smallest) elements provided that K fulfills the following conditions: K ∧ K ⊂ K and K ∨ K ⊂ K . This latter condition holds for a number of relevant K , see, e.g., [1, p. 216]. References [1] S. Carl, V.K. Le, D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer, New York, 2007. [2] S. Carl, D. Motreanu, Quasilinear elliptic inclusions of Clarke’s gradient type under local growth conditions, Appl. Anal. 85 (2006) 1527–1540. [3] S. Carl, D. Motreanu, Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the p-Laplacian, Differential Integral Equations 20 (2007) 309–324. [4] S. Carl, D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal. 68 (2008) 2668–2676. [5] S. Carl, Existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems, Nonlinear Anal. 65 (2006) 1532–1546. [6] S. Carl, V. Lakshmikantham, Generalized quasilinearization for quasilinear parabolic equations with nonlinearities of dc-type, J. Optim. Theory Appl. 109 (2001) 27–50. [7] S. Carl, V. Lakshmikantham, Generalized quasilinearization method for reaction-diffusion equations under nonlocal flux conditions, J. Math. Anal. Appl. 271 (2002) 182–205. [8] F.H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990. [9] G.S. Ladde, V. Lakshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985. [10] V. Lakshmikantham, S. K¨oksal, Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations, Taylor & Francis, London, 2003. [11] Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995. [12] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [13] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II B, Springer-Verlag, Berlin, 1990.