Differential inclusions involving oscillatory terms

Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem \[ \begin{cases} -\Delta u(x)\in \partial F(u(x))+\lambda \partial G(u(x))\ \mbox{in}\ \Omega \newline u\geq 0\ \mbox{in}\ \Omega \newline u= 0\ \mbox{on}\ \partial\Omega, \end{cases} \ \ \ \ \ \ \ \ \ \ \ \ {({\mathcal D}_\lambda)} \] where $\Omega \subset {\mathbb R}^n$ is a bounded open domain, and $\partial F$ and $\partial G$ stand for the generalized gradients of the locally Lipschitz functions $F$ and $G$. In this paper we provide a quite complete picture on the number of solutions of $({\mathcal D}_\lambda)$ whenever $\partial F$ oscillates near the origin/infinity and $\partial G$ is a generic perturbation of order $p>0$ at the origin/infinity, respectively. Our results extend in several aspects those of Krist\'aly and Moro\c{s}anu [J. Math. Pures Appl., 2010].


Introduction
We consider the model Dirichlet problem in Ω; u = 0, on ∂Ω, where ∆ is the usual Laplace operator, Ω ⊂ R n is a bounded open domain (n ≥ 2), and f : R → R is a continuous function verifying certain growth conditions at the origin and infinity. Usually, such a problem is studied on the Sobolev space H 1 0 (Ω) and weak solutions of (P 0 ) become classical/strong solutions whenever f has further regularity. There are several approaches to treat problem (P 0 ), mainly depending on the behavior of the function f . When f is superlinear and subcritical at infinity (and superlinear at the origin), the seminal paper of Ambrosetti and Rabinowitz [2] guarantees the existence of at least a nontrivial solution of (P 0 ) by using variational methods. An important extension of (P 0 ) is its perturbation, i.e.,      −∆u(x) = f (u(x)) + λg(u(x)) in Ω; u ≥ 0, in Ω; u = 0, on ∂Ω, where g : R → R is another continuous function which is going to compete with the original function f . When both functions f and g are of polynomial type of sub-and super-unit degree, -the right hand side being called as a concave-convex nonlinearity -the existence of at least one or two nontrivial solutions of (P λ ) is guaranteed, depending on the range of λ > 0, see e.g. Ambrosetti, Brezis and Cerami [1], Autuori and Pucci [4], de Figueiredo, Gossez and Ubilla [8].
In these papers variational arguments, sub-and super-solution methods as well as fixed point arguments are employed.
Another important class of problems of the type (P λ ) is studied whenever f has a certain oscillation (near the origin or at infinity) and g is a perturbation. Although oscillatory functions seemingly call forth the existence of infinitely many solutions, it turns out that 'too classical' oscillatory functions do not have such a feature. Indeed, when f (s) = c sin s and g = 0, with c > 0 small enough, a simple use of the Poincaré inequality implies that problem (P λ ) has only the zero solution. However, when f strongly oscillates, problem (P 0 ) has indeed infinitely many different solutions; see e.g. Omari and Zanolin [19], Saint Raymond [21]. Furthermore, if g(s) = s p (s > 0), a novel competition phenomena has been described for (P λ ) by Kristály and Moroşanu [12]. We notice that several extensions of [12] can be found in the literature, see e.g. Ambrosio, D'Onofrio and Molica Bisci [3] and Molica Bisci and Pizzimenti [16] for nonlocal fractional Laplacians; Molica Bisci, Rȃdulescu and Servadei [17] for general operators in divergence form; Mȃlin and Rȃdulescu [15] for difference equations. We emphasize that in the aforementioned papers the perturbations are either zero or have a (smooth) polynomial form.
In mechanical applications, however, the perturbation may occur in a discontinuous manner as a non-regular external force, see e.g. the gluing force in von Kármán laminated plates, cf. Bocea, Panagiotopoulos and Rȃdulescu [5], Motreanu and Panagiotopoulos [18] and Panagiotopoulos [20]. In order to give a reasonable reformulation of problem (P λ ) in such a non-regular setting, the idea is to 'fill the gaps' of the discontinuities, considering instead of the discontinuous nonlinearity a set-valued map appearing as the generalized gradient of a locally Lipschitz function. In this way, we deal with an elliptic differential inclusion problem rather than an elliptic differential equation, see e.g. Chang [6], Gazzolla and Rȃdulescu [9] and Kristály [10]; this problem can be formulated generically as in Ω; u = 0, on ∂Ω, where F and G are both nonsmooth, locally Lipschitz functions having various growths, while ∂F and ∂G stand for the generalized gradients of F and G, respectively. The main purpose of the present paper is to extend the main results of Kristály and Moroşanu [12] in two directions: (a) to allow the presence of nonsmooth nonlinear terms -reformulated into the inclusion (D λ ) -which are more suitable from mechanical point of view (mostly due to the perturbation term G, although we allow non-smoothness for the oscillatory term F as well); (b) to consider a generic p-order perturbation ∂G at the origin/infinity, not necessarily of polynomial growth as in [12], p > 0.
In the present paper we study the inclusion (D λ ) in two different settings, i.e., we analyze the number of distinct solutions of (D λ ) whenever ∂F oscillates near the origin/infinity and ∂G is of order p > 0 near the origin/infinity. Roughly speaking, when ∂F oscillates near the origin and ∂G is of order p > 0 at the origin, we prove that the number of distinct, nontrivial solutions of (D λ ) is • infinitely many whenever p > 1 (λ ≥ 0 is arbitrary) or p = 1 and λ is small enough (see Theorem 2.1); • at least (a prescribed number) k ∈ N whenever 0 < p < 1 and λ is small enough (see Theorem 2.2).
As we can observe, in the first case, the term ∂G(s) ∼ s p as s → 0 + with p > 1 has no effect on the number of solutions (i.e., the oscillatory term is the leading one), while in the second case, the situation changes dramatically, i.e., ∂G has a 'truth' competition with respect to the oscillatory term ∂F . We can state a very similar result as above whenever ∂F oscillates at infinity and ∂G is of order p > 0 at infinity by proving that the number of distinct, nontrivial solutions of the differential inclusion (D λ ) is • infinitely many whenever p < 1 (λ ≥ 0 is arbitrary) or p = 1 and λ is small enough (see Theorem 2.3); • at least (a prescribed number) k ∈ N whenever p > 1 and λ is small enough (see Theorem 2.4).
Contrary to the competition at the origin, in the first case the term ∂G(s) ∼ s p as s → ∞ with p < 1 has no effect on the number of solutions (i.e., the oscillatory term is the leading one), while in the second case, the perturbation term ∂G competes with the oscillator function ∂F . We admit that the line of the proofs is conceptually similar to that of Kristály and Moroşanu [12]; however, the presence of the nonsmooth terms ∂F and ∂G requires a deep argumentation by fully exploring the nonsmooth calculus of locally Lipschitz functions in the sense of Clarke [7]. In addition, the presence of the generic p-order perturbation ∂G needs a special attention with respect to [12]; in particular, the p-order growth of ∂G is new even in smooth settings.
The organization of the present paper is the following. In Section 2 we state our main assumptions and results, providing also some examples of functions fulfilling the assumptions. Section 3 contains a generic localization theorem for differential inclusions, while Sections 4 and 5 are devoted to the proof of our main results. In Section 6 we formulate some concluding remarks, while in the Appendix (Section 7) we collect those notions and results on locally Lipschitz functions that are used throughout our arguments.

Main theorems
Let F, G : R + → R be locally Lipschitz functions and as usual, let us denote by ∂F and ∂G their generalized gradients in the sense of Clarke (see the Appendix). Hereafter, R + = [0, ∞). Let p > 0, λ ≥ 0 and Ω ⊂ R n be a bounded open domain, and consider the elliptic differential inclusion problem We distinguish the cases when ∂F oscillates near the origin or at infinity.
In the sequel, we provide a quite complete picture about the competition concerning the terms s → ∂F (s) and s → ∂G(s), respectively. First, we are going to show that when p ≥ 1 then the 'leading' term is the oscillatory function ∂F ; roughly speaking, one can say that the effect of s → ∂G(s) is negligible in this competition. More precisely, we prove the following result.
, t ≥ 0, or some of its jumping variants; one has that F ∞ verifies the assumptions ( . For a fixed p > 0, let G ∞ (s) = s p max{0, sin s}, s ≥ 0; it is clear that G ∞ is a typically locally Lipschitz function on [0, ∞) (not being of class C 1 ) and verifies (G ∞ 1 ) with c = −1 and c = 1; see Figure 2 representing both f ∞ and G ∞ (for p = 2), respectively. Figure 2. Graphs of f ∞ and G ∞ at infinity, respectively.
In the sequel, we investigate the competition at infinity concerning the terms s → ∂F (s) and s → ∂G(s), respectively. First, we show that when p ≤ 1 then the 'leading' term is the oscillatory function F , i.e., the effect of s → ∂G(s) is negligible. More precisely, we prove the following result: Remark 2.3. Let 2 * be the usual critical Sobolev exponent. In addition to (2.3), we also have In the case when p > 1, it turns out that the perturbation term ∂G may compete with the oscillatory function ∂F ; more precisely, we have:

Localization: a generic result
We consider the following differential inclusion problem where k > 0 and For simplicity, we extend the function A by A(s) = 0 for s ≤ 0; the extended function is locally Lipschitz on the whole R. The natural energy functional T : H 1 0 (Ω) → R associated with the differential inclusion problem (D k A ) is defined by The energy functional T is well defined and locally Lipschitz on H 1 0 (Ω), while its critical points in the sense of Chang (see Definition 7.3 in the Appendix) are precisely the weak solutions of the differential inclusion problem note that at this stage we have no information on the sign of u.
where ξ x ∈ ∂A(u(x)) a.e. x ∈ Ω, see e.g. Motreanu and Panagiotopoulos [18]. By using the divergence theorem for the first term at the left hand side (and exploring the Dirichlet boundary condition), we obtain that Accordingly, we have that ) in the weak sense in Ω, as claimed before.
Let us consider the number η ∈ R from (H 2 A ) and the set Our localization result reads as follows (see [12, Theorem 2.1] for its smooth form): Theorem 3.1. Let k > 0 and assume that hypotheses (H 1 A ) and (H 2 A ) hold. Then (i) the energy functional T is bounded from below on W η and its infimum is attained at someũ ∈ W η ; (ii)ũ(x) ∈ [0, δ] for a.e. x ∈ Ω; (iii)ũ is a weak solution of the differential inclusion (D k A ).
Proof. The proof is similar to that of Kristály and Moroşanu [12]; for completeness, we provide its main steps.
(i) Due to (H 1 A ), it is clear that the energy functional T is bounded from below on H 1 0 (Ω). Moreover, due to the compactness of the embedding H 1 0 (Ω) ⊂ L q (Ω), q ∈ [2, 2 * ), it turns out that T is sequentially weak lower semi-continuous on H 1 0 (Ω). In addition, the set W η is weakly closed, being convex and closed in H 1 0 (Ω). Thus, there isũ ∈ W η which is a minimum point of T on the set W η , cf. Zeidler [24].

Let us introduce the sets
In particular, L = L 1 ∪ L 2 , and by definition, it follows that w(x) =ũ(x) for all x ∈ Ω \ L, w(x) = 0 for all x ∈ L 1 , and w(x) = δ for all x ∈ L 2 . In addition, one has On account of k > 0, we have By means of the Lebourg's mean value theorem, for a.e.
The latter relation implies in particular that m(L) = 0, which is a contradiction, completing the proof of (ii).
In the sequel, we need a truncation function of H 1 0 (Ω), see also [12]. To construct this function, let B(x 0 , r) ⊂ Ω be the n-dimensional ball with radius r > 0 and center x 0 ∈ Ω. For s > 0, define Note that that w s ∈ H 1 0 (Ω), w s L ∞ = s and hereafter ω n stands for the volume of B(0, 1) ⊂ R n .

Proof of Theorems 2.1 and 2.2
Before giving the proof of Theorems 2.1 and 2.2, in the first part of this section we study the differential inclusion problem where k > 0 and the locally Lipschitz function A : R + → R verifies Proof. We may assume that where A(s) = 0 for s ≤ 0 and τ η : R → R denotes the truncation function τ η (s) = min(η, s), η > 0. For further use, we introduce the energy functional T i : H 1 0 (Ω) → R associated with problem (D k A i ). We notice that for s ≥ 0, the chain rule (see Proposition 7.4 in the Appendix) gives It turns out that on the compact set [0, η i ], the upper semicontinuous set-valued map s → ∂A i (s) attains its supremum (see Proposition 7.1 in the Appendix); therefore, there exists On account of relations (4.2), (4.4) and (4.5), u 0 i is a weak solution also for the differential inclusion problem (D k A ).
We are going to prove that there are infinitely many distinct elements in the sequence {u 0 i } i . To conclude it, we first prove that The left part of (H 0 1 ) implies the existence of some l 0 > 0 and ζ ∈ (0, η 1 ) such that A(s) ≥ −l 0 s 2 for all s ∈ (0, ζ). (4.8) One can choose L 0 > 0 such that where r > 0 and C(r, n) > 0 come from (3.2). Based on the right part of (H 0 1 ), one can find a sequence {s i } i ⊂ (0, ζ) such thats i ≤ δ i and (4.10) Let i ∈ N be a fixed number and let ws i ∈ H 1 0 (Ω) be the function from (3.1) corresponding to the values i > 0. Then ws i ∈ W η i , and due to (4.8), (4.10) and (3.2) one has Accordingly, with (4.3) and (4.9), we conclude that which completes the proof of (4.6). Now, we prove (4.7). For every i ∈ N, by using the Lebourg's mean value theorem, relations (4.2) and (4.4) and (H 0 0 ) , we have Since lim i→∞ δ i = 0, the latter estimate and (4.11) provides relation (4.7). Based on (4.2) and (4.4), we have that T i (u 0 i ) = T 1 (u 0 i ) for all i ∈ N. This relation with (4.6) and (4.7) means that the sequence {u 0 i } i contains infinitely many distinct elements. We now prove (4.1). One can prove the former limit by (4.4), i.e. u 0 i L ∞ ≤ δ i for all i ∈ N, combined with lim i→∞ δ i = 0. For the latter limit, we use k > 0, (4.11), (4.2) and (4.4) to get for all i ∈ N that which completes the proof.
Therefore, one has a sequence {s i } i ⊂ (0, 1) converging to 0 such that max{∂A(s i )} where we used the inclusion (4.14). In particular, u i solves problem (D λ ), i ∈ N, which completes the proof of (i).
In particular, since p > 1, then and for sufficiently small > 0 there exists γ = γ( ) > 0 such that which means that u i solves problem (D λ ), i ∈ N. This completes the proof of Theorem 2.1.  One can observe that ∂A λ (s) ⊆ ∂F (s) + λ 0 s + λ∂G(s) for every s ≥ 0. On account of (F 0 2 ), there is a sequence {s i } i ⊂ (0, 1) converging to 0 such that Thus, due to the upper semicontinuity of (s, λ) → ∂A λ (s), we can choose three sequences Without any loss of generality, we may choose For every i ∈ N and λ ∈ [0, λ i ], let A λ i : [0, ∞) → R be defined as (4.20) and the energy functional T i,λ : H 1 0 (Ω) → R associated with the differential inclusion problem(D k A λ i ) is given by One can easily check that for every i ∈ N and λ ∈ [0, λ i ], the function A λ i verifies the hypotheses of Theorem 3.1. Accordingly, for every i ∈ N and λ ∈ [0, λ i ]: By the choice of the function A λ and k > 0, u 0 i,λ is also a solution to the differential inclusion problem (D k A λ ), so (D λ ).

Proof of Theorems 2.3 and 2.4
We consider again the differential inclusion problem where k > 0 and the locally Lipschitz function A : The counterpart of Theorem 4.1 reads as follows. Proof. The proof is similar to the one performed in Theorem 4.1; we shall show the differences only. We associate the energy functional T i : By (5.2), u ∞ i turns to be a weak solution also for differential inclusion problem (D k A ). We shall prove that there are infinitely many distinct elements in the sequence {u ∞ i } i by showing that lim By the left part of (H ∞ 1 ) we can find l A ∞ > 0 and ζ > 0 such that Let us choose L A ∞ > 0 large enough such that On account of the right part of (H ∞ 1 ), one can fix a sequence {s i } i ⊂ (0, ∞) such that lim i→∞si = ∞ and (5.9) We know from (H ∞ 2 ) that lim i→∞ δ i = ∞, therefore one has a subsequence {δ m i } i of {δ i } i such thats i ≤ δ m i for all i ∈ N. Let i ∈ N, and recall w si ∈ H 1 0 (Ω) from (3.1) with s i :=s i > 0. Then ws i ∈ W ηm i and according to (3.2), (5.7) and (5.9) we have It follows by (5.10) that lim i→∞ T m i (u ∞ m i ) = −∞. We notice that the sequence {T i (u ∞ i )} i is non-increasing. Indeed, let i < k; due to (5.2) one has that which completes the proof of (5.6). The proof of (5.1) goes in a similar way as in [12].
Proof of Theorem 2.3. We split the proof into two parts. (i) Case p = 1. Let λ ≥ 0 with λc < −l ∞ and fixλ ∞ ∈ R such that λc <λ ∞ < −l ∞ . With these choices, we define It is clear that A(0) = 0, i.e., (H ∞ 0 ) is verified. A similar argument for the p-order perturbation ∂G as before shows that it turns out that By using the upper semicontinuity of s → ∂A(s), one may fix two sequences i.e., u i solves problem (D λ ), i ∈ N.
which means that u i solves problem (D λ ), i ∈ N, which completes the proof.

Appendix: Locally Lipschitz functions
In this part we collect those notions and properties of locally Lipschitz functions which are used in the proofs; for details, see Clarke [7] and Chang [6]. Let (X, · ) be a real Banach space and U ⊂ X be an open set; we denote by ·, · the duality mapping between X and X.
Definition 7.1. (see [7]) A function f : X → R is locally Lipschitz if, for every x ∈ X, there exist a neighborhood U of x and a constant L > 0 such that Definition 7.2. (see [7]) Let f be a locally Lipschitz function near the point x and let v be any arbitrary vector in X. The generalized directional derivative in the sense of Clarke of f at the point x ∈ X in the direction v ∈ X is The generalized gradient of f at x ∈ X is the set For all x ∈ X, the functional f • (x, ·) is finite and positively homogeneous. Moreover, we have the following properties.
Remark 7.1. (see [7]) (a) u ∈ X is a critical point of f if f • (u; v) ≥ 0 for all v ∈ X.
(b) If x ∈ U is a local minimum or maximum of the locally Lipschitz function f : X → R on an open set of a Banach space, then x is a critical point of f. Proposition 7.3. (see [7]) (Lebourg's mean value theorem) Let X be a Banach space, x, y ∈ X and f : X → R be Lipschitz on an open set containing the line segment [x, y]. Then there is a point a ∈ (x, y) such that f (y) − f (x) ∈ ∂f (a), y − x .
Proposition 7.4. (see [7]) (Chain Rule) Let X be Banach space, let us consider the composite function f = g • h where h : X → R n and g : R n → R are given functions. Let denote h i , i ∈ {1, ..., n} be the component functions of h. We assume h i is locally Lipschitz near x and g is too near h(x). Then f is locally Lipschitz near x as well. Let us denote by α i the elements of ∂g, and let α = (α 1 , ..., α n ); then where co denotes the weak-closed convex hull.