Some hemivariational inequalities in the Euclidean space

The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space $\mathbb{R}^d$ ($d\geq 3$). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group $O(d)$ and their actions on the Sobolev space $H^1(\mathbb{R}^d)$. Moreover, under an additional hypotheses on the dimension $d$ and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schr\"{o}dinger equations a concrete and meaningful example of an application is presented.


Introduction
The aim of this paper is to study some nonlinear eigenvalue problems for certain classes of hemivariational inequalities that depend on a real parameter. For instance, the motivation for such a study comes from the investigation of perturbations, usually determined in terms of parameters. The hemivariational inequalities appears as a generalization of the variational inequalities and their study is based on the notion of Clarke subdi erential of a locally Lipschitz function. The theory of hemivariational inequalities appears as a new eld of Non-smooth Analysis; see [23, Part I -Chapter II] and the references therein. More precisely, we study the following hemivariational inequality problem: Here (R d , | · |) denotes the Euclidean space (with d ≥ ), F : R → R is a locally Lipschitz continuous function, whereas F (s; z) := lim sup y→s t→ +

F(y + tz) − F(y) t
is the generalized directional derivative of F at the point s ∈ R in the direction z ∈ R; see the classical monograph of Clarke [15] for details. Finally, W ∈ L ∞ (R d ) ∩ L (R d ) \ { } is a non-negative radially symmetric map and λ is a positive real parameter. We assume that there exist κ > and q ∈ ( , * ), where * = d/(d − ), such that |ζ | ≤ κ ( + |s| q− ), ∀ζ ∈ ∂F(s), for every s ∈ R, (1.1) where ∂F(s) denotes the generalized gradient of the function F at s ∈ R (see Section 2). With the above notations the main result reads as follows. Here, the symbol [·] denotes the integer function.
The proof of the above result is based on variational method in the nonsmooth setting. As it is well known, the lack of a compact embeddings of the Sobolev space H (R d ) into Lebesgue spaces produces several difculties for exploiting variational methods. In order to recover compactness, the rst task is to construct certain subspaces of H (R d ) containing invariant functions under special actions de ned by means of carefully chosen subgroups of the orthogonal group O(d). Subsequently, a locally Lipschitz continuous function is constructed which is invariant under the action of suitable subgroups of O(d), whose restriction to the appropriate subspace of invariant functions admits critical points.
Thanks to a nonsmooth version of the principle of symmetric criticality obtained by Krawcewicz and Marzantowicz [19], these points will also be critical points of the original functional, and they are exactly weak solutions of problem (S λ ). The abstract critical point result that we employ here is a nonsmooth version of the variational principle established by Ricceri [31]; see Bonanno and Molica Bisci [11] for details.
Moreover, we also emphasize that the multiplicity property stated in Theorem 1 -part (a ) is obtained by using the group-theoretical approach developed by Kristály Further, a point y ∈ X is called a (generalized) critical point of the locally Lipschitz continuous function J if X * ∈ ∂J(y), i.e. J (y; z) ≥ , for every z ∈ X. Clearly, if J is a continuously Gâteaux di erentiable at y ∈ X, then y becomes a (classical) critical point of J, that is J (y) = X * .
For an exhaustive overview of the non-smooth calculus we refer to the monographs [13,15,27,28]. Further, we cite the book [23] as a general reference on this subject.
To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. Assume d ≥ and let H (R d ) be the standard Sobolev space endowed by the inner product and the induced norm In order to prove Theorem 1 we apply the principle of symmetric criticality together with the following critical point theorem proved in [11] by Bonanno and Molica Bisci.
The above result represents a nonsmooth version of a variational principle established by Ricceri in [31].
For completeness, we also recall here the principle of symmetric criticality of Krawcewicz and Marzantowicz which represents a non-smooth version of the celebrated result proved by Palais in [29]. We point out that the result proved in [19] was established for su ciently smooth Banach G-manifolds. We will use here a particular form of this result that is valid for Banach spaces.
An action of a compact Lie group G on the Banach space (X, · X ) is a continuous map such that * y = y, (gh) * y = g * (h * y), y → g * y is linear.
The action * is said to be isometric if g * y X = y X , for every g ∈ G and y ∈ X. Moreover, the space of G-invariant points is de ned by Fix G (X) := {y ∈ X : g * y = y, ∀g ∈ G}, and a map h : X → R is said to be G-invariant on X if h(g * y) = h(y) for every g ∈ G and y ∈ X. Theorem 4. Let X be a Banach space, let G be a compact topological group acting linearly and isometrically on X, and J : X → R a locally Lipschitz, G-invariant functional. Then every critical point of J : Fix G (X) → R is also a critical point of J.

. Group-theoretical arguments
Let O(d) be the orthogonal group in R d and let G ⊆ O(d) be a subgroup. Assume that G acts on the space H (R d ). Hence, the set of xed points of H (R d ), with respect to G, is clearly given by We note that, if G = O(d) and the action is the standard linear isometric map de ned by is exactly the subspace of radially symmetric functions of H (R d ), also denoted by H rad (R d ). Moreover, the following embedding is continuous (resp. compact), for every q ∈ [ , * ] (resp. q ∈ ( , * )). See, for instance, the celebrated paper [24]. Let either d = or d ≥ and consider the subgroup Let us de ne the involution η H d,i : R d → R d as follows Moreover, for every i ∈ J d , let us consider the compact group for every x ∈ R d .
We note that i is de ned for every element of for every i ∈ J d . Following Bartsch and Willem [8], for every i ∈ J d , the embedding is compact, for every q ∈ ( , * ).

Proposition 5.
With the above notations, the following properties hold: for every i, j ∈ J d and i ≠ j.

See [22, Theorem 2.2] for details.
From now on, for every u ∈ L (R d ) and ∈ [ , * ), we shall denote The following locally Lipschitz property holds.
is well-de ned and locally Lipschitz continuous on L q (R d ).
Proof. It is clear that Ψ e is well-de ned. Indeed, by using Lebourg's mean value theorem, xing t , t ∈ R, there exist θ ∈ ( , ) and ζ θ ∈ ∂F(θt + ( − θ)t ) such that Since F( ) = , by using (2.6) and condition (1.1), our assumptions on W and the Hölder inequality gives that In order to prove that Ψ e is locally Lipschitz continuous on L q (R d ) it is straightforward to establish that the functional Ψ e is in fact Lipschitz continuous on L q (R d ). Now, for a xed number r > and arbitrary elements The above inequalities have been derived by using (2.6), assumption (1.1) and Hölder's inequality. The Lipschitz property on bounded sets for Ψ e is thus veri ed.
A meaningful consequence of the above lemma is the following semicontinuity property.

Corollary 7.
Assume that condition (1.1) holds for some q ∈ , * and let is sequentially weakly lower semicontinuous on Proof. First, on account of Brézis [12, Corollaire III.8], the functional u → u / is sequentially weakly lower semicontinuous on Fix Y (H (R d )). Now, we prove that Ψ| Fix Y (H (R d )) is sequentially weakly continuous. Indeed, let {u j } j∈N ⊂ Fix Y (H (R d )) be a sequence which weakly converges to an element u ∈ Fix Y (H (R d )).
Since Y is compactly embedded in L q (R d ), for every q ∈ ( , * ), passing to a subsequence if necessary, one has u j − u q → as j → ∞. According to Lemma 6, the extension of Ψ to L q (R d ) is locally Lipschitz continuous. Hence, there exists a constant Lu ≥ such that for every j ∈ N. Passing to the limit in (2.10), we conclude that Ψ is sequentially weakly continuous on Fix Y (H (R d )). The proof is now complete.
The next result will be crucial in the sequel; see [15,20,21,27] for related results.
The next task is to prove (2.11). To this goal, since E is separable, let us notice that there exist two sequences Without loss of generality we can also suppose that w j (x) → u(x) a.e. in R d as j → ∞. Now, for every j ∈ N, let us consider the measurable and non-negative function g j : for a.e. x ∈ R d . Set The inverse Fatou's Lemma applied to the sequences {Wg j } j∈N yields for every j ∈ N and a.e. x ∈ R d . By setting Now, it is easily seen that there exists a function k ∈ L (R d ) such that for a.e. x ∈ R d . Consequently, the Lebesgue's Dominated Convergence Theorem implies that (2.14) By (2.13) and (2.14) it follows that
The next result is a direct and easy consequence of Proposition 8.

Proposition 9.
Assume that condition (1.1) holds for some q ∈ , * and let

Then the functional is locally Lipschitz continuous and its critical points solve (S λ ).
Proof. The functional J λ is locally Lipschitz continuous. Indeed, J λ is the sum of the C (H (R d )) functional u → u / and of the locally Lipschitz continuous functional Ψ, see Lemma 6. Now, every critical point of J λ is a weak solution of problem (S λ ). Indeed, if u ∈ H (R d ) is a critical point of J λ , a direct application of inequality (2.11) in Proposition 8 yields where z+ := max{ , z}. With the above notation, we have: Now, it is possible to prove that v i σ ∈ Fix H d,η i (H (R d )). Moreover, for every σ ∈ ( , ], let De ne The sets D i σ have positive Lebesgue measure and they are H d,η i -invariant. Moreover, for every σ ∈ ( , ), one has v i σ ∈ Fix H d,η i (H (R d )) and the following facts hold:

Proof of the Main Result
Part (a ) -The main idea of the proof consists of applying Theorem 3 to the functional Successively, the existence of one non-trivial radial solution of problem (S λ ) follows by the symmetric criticality principle due to Krawcewicz Now, let us de ne where κ = and for every q ∈ ( , * ) and take < λ < λ * .
Thanks to (3.1), there existsγ > such that Arguing as in [26], let us de ne the function χ : ( , +∞) → [ , +∞) as for every r > . It follows by (2.8) that Moreover, one has u < √ r, (3.4) for every u ∈ Φ − ((−∞, r)). Now, by using (3.4), the Sobolev embedding (2.1) and (3.3) yield The above inequality yields for every r > . Evaluating inequality (3.5) in r =γ , it follows that Now, we notice that is the zero function. Thanks to (3.2), the above inequality in addition to (3.6) give In conclusion, Invoking Theorem 3, there exists a function u λ ∈ Φ − ((−∞,γ )) such that More precisely, the function u λ is a global minimum of the restriction of the functional J λ to the sublevel Φ − ((−∞,γ )). Hence, let u λ be such that and Φ(u λ ) <γ , (3.9) and also u λ is a critical point of J λ in Fix O(d) (H (R d )). Now, the orthogonal group O(d) acts isometrically on H (R d ) and, thanks to the symmetry of the potential W, one has So, owing to Theorem 4, u λ is a weak solution of problem (S λ ) . In this setting, in order to prove that u λ ≢ in Fix O(d) (H (R d )) , rst we claim that there exists a sequence of functions w j j∈N in Fix O(d) for j su ciently large. Now, we have to consider two di erent cases. Since s j → + and ≤ vσ(x) ≤ in R d , it follows that w j (x) = s j vσ(x) → + as j → +∞ uniformly in x ∈ R d . Hence, ≤ w j (x) < ρ M for j su ciently large and for any x ∈ R d . Hence, as a consequence of (3.13) and (3.14), we have that for j su ciently large. The arbitrariness of M gives (3.10) and so the claim is proved.
Then for any ε > there exists ρε > such that for any s with < s < ρε Arguing as above, we can suppose that ≤ w j (x) = s j vσ(x) < ρε for j large enough and any x ∈ R d . Thus, by (3.13) and (3.15) we get provided that j is su ciently large. Let for j su ciently large. Hence, assertion (3.10) is clearly veri ed. Now, we notice that w j = s j vσ → , as j → +∞ , so that for j large enough w j < √ γ. Hence and on account of (3.10), also for j su ciently large.

Part (a ) -Let
for every ∈ ( , * ), with i ∈ J d and set Assume d > and suppose that the potential F is even. Let We claim that for every λ ∈ ( , λ * ) problem (S λ ) admits at least Moreover, if d ≠ problem (S λ ) admits at least pairs of sign-changing weak solutions. We divide the proof into two parts. Part 1: dimension d = .
Since F is symmetric, the energy functional

Some applications
A simple prototype of a function F ful lling the structural assumption (1.1) can be easily constructed as follows. Let f : R → R be a measurable function such that sup s∈R |f (s)| + |s| q− < +∞, (4.1) for some q ∈ , * . Furthermore, let F be the potential de ned by where f (s) := lim δ→ + essinf |t−s|<δ f (t), and f (s) := lim δ→ + esssup |t−s|<δ f (t), for every s ∈ R.
On account of (4.1) and (4.2), inequality (1.1) immediately follows. Furthermore, if f is a continuous function and (4.1) holds, then problem (S λ ) assumes the simple and signi cative form: See [18] for related topics.
Of course, the solutions of (S λ ) are exactly the weak solutions of the following Schrödinger equation which has been widely studied in the literature. In particular, Theorem 1 can be viewed as a non-smooth version of the results contained in [26]. See, among others, the papers [1][2][3][4]7] as well as [14,16,25,30].
We point out that the approach adopted here can be used in order to study the existence of multiple solutions for hemivariational inequalities on a strip-like domain of the Euclidean space (see [21] for related topics). Since this approach di ers to the above, we will treat it in a forthcoming paper.