Some hemivariational inequalities in the Euclidean space

Giovanni Molica Bisci; Dušan Repovš

doi:10.1515/anona-2020-0035

Article Open Access

Some hemivariational inequalities in the Euclidean space

Giovanni Molica Bisci and Dušan Repovš

Published/Copyright: August 6, 2019

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From the journal Advances in Nonlinear Analysis Volume 9 Issue 1

Abstract

The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝ^d (d ≥ 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 H¹(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 Schrödinger equations a concrete and meaningful example of an application is presented.

Keywords: Hemivariational inequalities; variational methods; principle of symmetric criticality; radial and non-radial solutions

MSC 2010: Primary: 35A15; 35J60; 35J65; 35J91; Secondary: 35A01; 45A15; 35P30

1 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 subdifferential of a locally Lipschitz function. The theory of hemivariational inequalities appears as a new field of Non-smooth Analysis; see [23, Part I - Chapter II] and the references therein.

More precisely, we study the following hemivariational inequality problem:

Find u ∈ H¹(ℝ^d) such that

∫ Rd∇u(x)⋅∇φ(x)dx+∫ R du(x)φ(x)dx+λ∫ RdW(x)F0(u(x);−φ(x))dx≥0,∀φ∈H1(Rd).

Here (ℝ^d, |⋅|) denotes the Euclidean space (with d ≥ 3), F : ℝ → ℝ is a locally Lipschitz continuous function, whereas

F0(s;z):=lim supy→st→0+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) ∩ L¹(ℝ^d) ∖ {0} is a non-negative radially symmetric map and λ is a positive real parameter.

We assume that there exist κ₁ > 0 and q ∈ (2, 2^*), where 2^* = 2d/(d – 2), such that

|ζ|≤κ1(1+|s|q−1),∀ζ∈∂F(s),for everys∈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.

Theorem 1

Let F : ℝ → ℝ be a locally Lipschitz continuous function with F(0) = 0 and satisfying the growth condition (1.1) for some q ∈ (2, 2^*), in addition to

lim sups→0+F(s)s2=+∞andlim infs→0+F(s)s2>−∞. (1.2)

Moreover, let W ∈ L^∞(ℝ^d) ∩ L¹(ℝ^d) ∖ {0} be a non-negative radially symmetric map. Then the following facts hold:

There exists a positive number λ^* such that, for every λ ∈ (0, λ^*), the problem (S_λ) admits at least one non-trivial radial weak solution u_λ ∈ H¹(ℝ^d) with |u_λ(x)| → 0 as |x| → ∞.
If d > 3 and F is even then there exists a positive number λ_* such that for every λ ∈ (0, λ_*), the problem (S_λ) admits at least

ζS(d):=1+(−1)d+d−32

pairs of non-trivial weak solutions {±uλ,i}i∈Jd′⊂H1(Rd) with |u_λ,i(x)| → 0, as |x| → ∞, for every i ∈ Jd′ := {1, …, ζS(d) }, and with different symmetries structure. More precisely, if d ≠ 5 problem (S_λ) admits at least

τd:=ζS(d)−1

pairs of sign-changing weak solutions.

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 H¹(ℝ^d) into Lebesgue spaces produces several difficulties for exploiting variational methods. In order to recover compactness, the first task is to construct certain subspaces of H¹(ℝ^d) containing invariant functions under special actions defined 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 (a₂) is obtained by using the group-theoretical approach developed by Kristály, Moroşanu, and O’Regan [22]; see Subsection 2.1. Thanks to this analysis, we are able to construct

ζS(d):=1+(−1)d+d−32

subspaces of H¹(ℝ^d) with different symmetries properties. In addition, when d ≠ 5, there are

τd:=(−1)d+d−32

of these subspaces which do not contain radial symmetric functions; see the quoted paper [8] due to Bartsch and Willem, as well as [22, Theorem 2.2].

We point out that some almost straightforward computations in [26] are adapted here to the non-smooth case. However, due to the non-smooth framework, our abstract procedure, as well as the setting of the main results, is different from the results contained in [26], where the continuous case was studied; see Section 4 additional comments and remarks.

The manuscript is organized as follows. In Section 2 we set some notations and recall some properties of the functional space we shall work in. In order to apply critical point methods to problem (S_λ), we need to work in a subspace of the functional space H¹(ℝ^d) in particular, we give some tools which will be useful in the paper (see Propositions 8 and Lemma 7). In Section 3 we study problem (S_λ) and we prove our existence result (see Theorem 1). Finally, we study the existence of multiple non-radial solutions to the problem (S_λ) for λ sufficiently small. In connection to classical Schrödinger equations in the continuous setting (see, among others, the papers [5, 6, 9, 10]) a meaningful example of an application is given in the last section.

We refer to the books [1, 23, 33] as general references on the subject treated in the paper.

2 Abstract framework

Let (X, ∥⋅∥_X) be a real Banach space. We denote by X^* the dual space of X, whereas 〈⋅, ⋅〉 denotes the duality pairing between X^* and X.

A function J : X → ℝ is called locally Lipschitz continuous if to every y ∈ X there correspond a neighborhood V_y of y and a constant L_y ≥ 0 such that

|J(z)−J(w)|≤Ly∥z−w∥X,(∀z,w∈Vy).

If y, z ∈ X, we write J⁰(y; z) for the generalized directional derivative of J at the point y along the direction z, i.e.,

J0(y;z):=lim supw→yt→0+J(w+tz)−J(w)t.

The generalized gradient of the function J at y ∈ X, denoted by ∂J(y), is the set

∂J(y):=y∗∈X∗:〈y∗,z〉≤J0(y;z),∀z∈X.

The basic properties of generalized directional derivative and generalized gradient which we shall use here were studied in [13, 15].

The following lemma displays some useful properties of the notions introduced above.

Lemma 2

If I, J : X → ℝ are locally Lipschitz continuous functionals, then

J⁰(y; ⋅) is positively homogeneous, sub-additive, and continuous for every y ∈ X;
J⁰(y; z) = max{〈y^*, z〉 : y^* ∈ ∂J(z)} for every y, z ∈ X;
J⁰(y; –z) = (–J)⁰ (y; z) for every y, z ∈ X;
if J ∈ C¹(X), then J⁰(y; z) = 〈J′(y), z〉 for every y, z ∈ X;
(I + J)⁰(y; z) ≤ I⁰(y; z) + J⁰(y; z) for every y, z ∈ X. Moreover, if J is is continuously Gâteaux differentiable, then (I + J)⁰(y; z) = I⁰(y; z) + J′(y; z) for every y, z ∈ X.

See [17] for details.

Further, a point y ∈ X is called a (generalized) critical point of the locally Lipschitz continuous function J if 0_X^* ∈ ∂J(y), i.e.

J0(y;z)≥0,

for every z ∈ X.

Clearly, if J is a continuously Gâteaux differentiable at y ∈ X, then y becomes a (classical) critical point of J, that is J′(y) = 0_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 ≥ 3 and let H¹(ℝ^d) be the standard Sobolev space endowed by the inner product

〈u,v〉:=∫Rd∇u(x)⋅∇v(x)dx+∫Rdu(x)v(x)dx,∀u,v∈H1(Rd)

and the induced norm

∥u∥:=∫Rd|∇u(x)|2dx+∫Rd|u(x)|2dx1/2,

for every u ∈ H¹(ℝ^d).

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.

Theorem 3

Let X be a reflexive real Banach space and let Φ, Ψ : X → ℝ be locally Lipschitz continuous functionals such that Φ is sequentially weakly lower semicontinuous and coercive. Furthermore, assume that Ψ is sequentially weakly upper semicontinuous. For every r > inf_X Φ, put

φ(r):=infu∈Φ−1((−∞,r))supv∈Φ−1((−∞,r))Ψ(v)−Ψ(u)r−Φ(u).

Then for each r > inf_X Φ and each λ ∈ ]0, 1/φ(r)[, the restriction of 𝓙_λ := Φ – λΨ to Φ^–1((–∞, r)) admits a global minimum, which is a critical point (local minimum) of 𝓙_λ in X.

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 sufficiently 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

∗:G×X→X:(g,y)↦g∗y,

such that

1∗y=y,(gh)∗y=g∗(h∗y),y↦g∗yislinear.

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 defined by

FixG(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.

For details see, for instance, the book [23, Part I - Chapter 1] and Krawcewicz and Marzantowicz [19].

2.1 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 H¹(ℝ^d). Hence, the set of fixed points of H¹(ℝ^d), with respect to G, is clearly given by

FixG(H1(Rd)):={u∈H1(Rd):gu=u,∀g∈G}.

We note that, if G = O(d) and the action is the standard linear isometric map defined by

gu(x):=u(g−1x),∀x∈Rdandg∈O(d)

then Fix_O(d)(H¹(ℝ^d)) is exactly the subspace of radially symmetric functions of H¹(ℝ^d), also denoted by Hrad1 (ℝ^d). Moreover, the following embedding

FixO(d)(H1(Rd))↪Lq(Rd) (2.1)

is continuous (resp. compact), for every q ∈ [2, 2^*] (resp. q ∈ (2, 2^*)). See, for instance, the celebrated paper [24].

Let either d = 4 or d ≥ 6 and consider the subgroup H_d,i ⊂ O(d) given by

Hd,i:=O(d/2)×O(d/2) if i=d−22O(i+1)×O(d−2i−2)×O(i+1) if i≠d−22,

for every i ∈ J_d := {1, …, τ_d}, where

τd:=(−1)d+d−32.

Let us define the involution η_{H_d,i} : ℝ^d → ℝ^d as follows

ηHd,i(x):=(x3,x1) if i=d−22 and x:=(x1,x3)∈Rd/2×Rd/2(x3,x2,x1) if i≠d−22 and x:=(x1,x2,x3)∈Ri+1×Rd−2i−2×Ri+1,

for every i ∈ J_d.

By definition, one has η_{H_d,i} ∉ H_d,i, as well as

ηHd,iHd,iηHd,i−1=Hd,i,andηHd,i2=idRd,

for every i ∈ J_d.

Moreover, for every i ∈ J_d, let us consider the compact group

Hd,ηi:=〈Hd,i,ηHd,i〉,

that is H_{d,η_i} = H_d,i ∪ η_{H_d,i} H_d,i, and the action ⊛_i : H_{d,η_i} × H¹(ℝ^d) → H¹(ℝ^d) of H_{d,η_i} on H¹(ℝ^d) given by

h⊛iu(x):=u(h−1x) if h∈Hd,i−u(g−1ηHd,i−1x) if h=ηHd,ig∈Hd,ηi∖Hd,i,g∈Hd,i (2.2)

for every x ∈ ℝ^d.

We note that ⊛_i is defined for every element of H_{d,η_i}. Indeed, if h ∈ H_{d,η_i}, then either h ∈ H_d,i or h = τg ∈ H_{d,η_i} ∖ H_d,i, with g ∈ H_d,i. Moreover, set

FixHd,ηi(H1(Rd)):={u∈H1(Rd):h⊛iu=u,∀h∈Hd,ηi},

for every i ∈ J_d.

Following Bartsch and Willem [8], for every i ∈ J_d, the embedding

FixHd,ηi(H1(Rd))↪Lq(Rd) (2.3)

is compact, for every q ∈ (2, 2^*).

Proposition 5

With the above notations, the following properties hold:

if d = 4 or d ≥ 6, then

FixHd,ηi(H1(Rd))∩FixO(d)(H1(Rd))={0}, (2.4)

for every i ∈ J_d;

if d = 6 or d ≥ 8, then

FixHd,ηi(H1(Rd))∩FixHd,ηj(H1(Rd))={0}, (2.5)

for every i, j ∈ J_d and i ≠ j.

See [22, Theorem 2.2] for details.

From now on, for every u ∈ L^ℓ(ℝ^d) and ℓ ∈ [2, 2^*), we shall denote

∥u∥ℓ:=∫Rd|u(x)|ℓdx1/ℓ,

and

∥W∥∞:=esssupx∈Rd|W(x)|,∥u∥p:=∫Rd|u(x)|pdx1/p,

for every p ∈ [2, 2^*).

Moreover, let Ψ : H¹(ℝ^d) → ℝ given by

Ψ(u):=∫RdW(x)F(u(x))dx,∀u∈H1(Rd).

The following locally Lipschitz property holds.

Lemma 6

Assume that condition (1.1) holds for some q ∈ (2, 2^*) and F(0) = 0. Furthermore, let W ∈ L^∞(ℝ^d) ∩ L¹(ℝ^d) ∖ {0}. Then the extended functional Ψ^e : L^q(ℝ^d) → ℝ defined by

Ψe(u):=∫RdW(x)F(u(x))dx,∀u∈Lq(Rd)

is well-defined and locally Lipschitz continuous on L^q(ℝ^d).

Proof

It is clear that Ψ^e is well-defined. Indeed, by using Lebourg’s mean value theorem, fixing t₁, t₂ ∈ ℝ, there exist θ ∈ (0, 1) and ζ_θ ∈ ∂F(θt₁ + (1 – θ)t₂) such that

F(t1)−F(t2)=ζθ(t1−t2). (2.6)

Since F(0) = 0, by using (2.6) and condition (1.1), our assumptions on W and the Hölder inequality gives that

∫RdW(x)F(u(x))dx≤κ1∫RdW(x)|u(x)|dx+∫RdW(x)|u(x)|qdx≤κ1∫Rd|W(x)|qq−1dxq−1q∫Rd|u(x)|qdx1/q+κ1∥W∥∞∫Rd|u(x)|qdx, (2.7)

for every u ∈ L^q(ℝ^d). Hence, inequality (2.7) yields

Ψe(u)≤κ1∥W∥qq−1∥u∥q+∥W∥∞∥u∥qq<+∞, (2.8)

for every u ∈ L^q(ℝ^d).

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 fixed number r > 0 and arbitrary elements u, v ∈ L^q(ℝ^d) with max{∥u∥_q, ∥v∥_q} ≤ r, the following estimate holds

|Ψe(u)−Ψe(v)|≤∫RdW(x)F(u(x))−F(v(x))dx≤κ1∫RdW(x)1+|u(x)|q−1+|v(x)|q−1|u(x)−v(x)|dx≤κ1(∥W∥qq−1∥u−v∥q+∥W∥∞(∥u∥qq−1+∥v∥qq−1)∥u−v∥q)≤κ2∥u−v∥q, (2.9)

where the Lipschitz constant κ₂ := 2^q–2 (∥W∥qq−1+2rq−1∥W∥∞)κ1 depends on r.

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 verified.□

A meaningful consequence of the above lemma is the following semicontinuity property.

Corollary 7

Assume that condition (1.1) holds for some q ∈ (2, 2^*) and let W ∈ L^∞(ℝ^d) ∩ L¹(ℝ^d) ∖ {0}. Then for every λ > 0, the functional

u↦12∥u∥2−λΨ|FixY(H1(Rd))(u),∀u∈FixY(H1(Rd))

is sequentially weakly lower semicontinuous on Fix_Y(H¹(ℝ^d)), where either Y = O(d) or Y = H_{d,η_i} for some i ∈ J_d.

Proof

First, on account of Brézis [12, Corollaire III.8], the functional u ↦ ∥u∥²/2 is sequentially weakly lower semicontinuous on Fix_Y(H¹(ℝ^d)). Now, we prove that Ψ|_{Fix_Y(H¹(ℝ^d))} is sequentially weakly continuous. Indeed, let {u_j}_j∈ℕ ⊂ Fix_Y(H¹(ℝ^d)) be a sequence which weakly converges to an element u₀ ∈ Fix_Y(H¹(ℝ^d)). Since Y is compactly embedded in L^q(ℝ^d), for every q ∈ (2, 2^*), passing to a subsequence if necessary, one has ∥u_j – u₀∥_q → 0 as j → ∞. According to Lemma 6, the extension of Ψ to L^q(ℝ^d) is locally Lipschitz continuous. Hence, there exists a constant L_u₀ ≥ 0 such that

|Ψ(uj)−Ψ(u0)|≤Lu0∥uj−u0∥q, (2.10)

for every j ∈ ℕ. Passing to the limit in (2.10), we conclude that Ψ is sequentially weakly continuous on Fix_Y(H¹(ℝ^d)). The proof is now complete.□

The next result will be crucial in the sequel; see [15, 20, 21, 27] for related results.

Proposition 8

Assume that condition (1.1) holds for some q ∈ (2, 2^*) and let W ∈ L^∞(ℝ^d) ∩ L¹(ℝ^d) ∖ {0}. Furthermore, let E be a closed subspace of H¹(ℝ^d) and denote by Ψ_E the restriction of Ψ to E. Then the following inequality holds

ΨE0(u;v)≤∫RdW(x)F0(u(x);v(x))dx, (2.11)

for every u, v ∈ E.

Proof

The map x ↦ W(x)F⁰(u(x);v(x)) is measurable on ℝ^d. Indeed, W ∈ L^∞(ℝ^d) and the function x ↦ F⁰(u(x);v(x)) is measurable as the countable limsup of measurable functions, see p. 16 of [27] for details. Moreover, condition (1.1) ensures that

∫RdW(x)F0(u(x);v(x))dx<∞.

Thus the map x ↦ W(x)F⁰(u(x);v(x)) belongs to L¹(ℝ^d).

The next task is to prove (2.11). To this goal, since E is separable, let us notice that there exist two sequences {t_j}_j∈ℕ ∈ ℝ and {w_j}_j∈ℕ ⊂ E such that t_j → 0⁺, ∥w_j – u∥ → 0 in E and

ΨE0(u;v)=limj→∞ΨE(wj+tjv)−ΨE(wj)tj.

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 : ℝ^d → ℝ ∪ {+∞} defined by

gj(x):=κ1|v(x)|(1+|wj(x)+tjv(x)|q−1+|wj(x)|q−1) −F(wj(x)+tjv(x))−F(wj(x))tj,

for a.e. x ∈ ℝ^d. Set

I:=lim supj→∞−∫RdW(x)gj(x)dx.

The inverse Fatou’s Lemma applied to the sequences {Wg_j}_j∈ℕ yields

I≤J:=∫RdW(x)lim supj→∞(αj(x)−βj(x))dx, (2.12)

where

αj(x)=F(wj(x)+tjv(x))−F(wj(x))tj,

and

βj(x):=κ1|v(x)|(1+|wj(x)+tjv(x)|q−1+|wj(x)|q−1)

for every j ∈ ℕ and a.e. x ∈ ℝ^d.

By setting

yj:=∫RdW(x)βj(x)dx,

one has

I=lim supj→∞∫RdW(x)αj(x)dx−yj. (2.13)

Now, it is easily seen that there exists a function k ∈ L¹(ℝ^d) such that

|βj(x)|≤k(x),

and

βj(x)→κ1|v(x)|(1+2|u(x)|q−1)

for a.e. x ∈ ℝ^d.

Consequently, the Lebesgue’s Dominated Convergence Theorem implies that

limj→∞yj=κ1∫RdW(x)|v(x)|(1+2|u(x)|q−1)dx. (2.14)

By (2.13) and (2.14) it follows that

I=lim supj→∞ΨE(wj+tjv)−ΨE(wj)tj−limj→∞yj=ΨE0(u;v)−κ1∫RdW(x)|v(x)|(1+2|u(x)|q−1)dx. (2.15)

Now

J≤Jα−κ1∫RdW(x)|v(x)|(1+2|u(x)|q−1)dx. (2.16)

where

Jα:=∫RdW(x)lim supj→∞αj(x)dx.

Inequality (2.12) in addition to (2.15) and (2.16) yield

ΨE0(u;v)≤Jα. (2.17)

Finally,

Jα=∫RdW(x)lim supj→∞F(wj(x)+tjv(x))−F(wj(x))tjdx≤∫RdW(x)limj→∞F(wj+tjv)−F(wj)tjdx≤∫RdW(x)F0(u(x);v(x))dx. (2.18)

By (2.17) and (2.18), inequality (2.11) now immediately follows.□

The next result is a direct and easy consequence of Proposition 8.

Proposition 9

Assume that condition (1.1) holds for some q ∈ (2, 2^*) and let W ∈ L^∞(ℝ^d) ∩ L¹(ℝ^d) ∖ {0}. Let J_λ : H¹(ℝ^d) → ℝ be the functional defined by

Jλ(u):=12∥u∥2−λΨ(u),∀u∈H1(Rd).

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 C¹(H¹(ℝ^d)) functional u ↦ ∥u∥²/2 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 u₀ ∈ H¹(ℝ^d) is a critical point of J_λ, a direct application of inequality (2.11) in Proposition 8 yields

0≤Jλ0(u0;φ)=〈u0,φ〉+λ(−Ψ)0(u0;φ)=〈u0,φ〉+λ(−Ψ)0(u0;φ)≤〈u0,φ〉+λ∫RdW(x)F0(u0(x);−φ(x))dx, (2.19)

for every φ ∈ H¹(ℝ^d). Since (2.19) holds, the function u₀ ∈ H¹(ℝ^d) solves (S_λ).□

2.2 Some test functions with symmetries

Following Kristály, Moroşanu, and O’Regan [22], we construct some special test functions belonging to Fix_O(d)(H¹(ℝ^d)) that will be useful for our purposes. If a < b, define

Aab:={x∈Rd:a≤|x|≤b}.

Since W ∈ L^∞(ℝ^d) ∖ {0} is a radially symmetric function with W ≥ 0, one can find real numbers R > r > 0 and α > 0 such that

essinfx∈ArRW(x)≥α>0. (2.20)

Hence, let 0 < r < R, such that (2.20) holds and take σ ∈ (0, (R – r)/2). Set v_σ ∈ Fix_O(d)(H¹(ℝ^d)) given by

vσ(x):=|x|−rσ+ if |x|≤r+σ1 if r+σ≤|x|≤R−σR−|x|σ+ if |x|≥R−σ

where z₊ := max{0, z}. With the above notation, we have:

supp(v_σ) ⊆ ArR ;
∥v_σ∥_∞ ≤ 1;
v_σ(x) = 1 for every x ∈ Ar+σR−σ .

Now, assume r ≥ R5+42 and set σ ∈ (0, 1). Define vσi∈H1(Rd) as follows

vσi(x):=vσd−22(x) if i=d−22 and x:=(x1,x3)∈Rd/2×Rd/2viσ(x) if i≠d−22 and x:=(x1,x2,x3)∈Ri+1×Rd−2i−2×Ri+1,

for every x ∈ ℝ^d, where:

vσd−22(x1,x3):=[(R−r4−max|x1|2−R+3r42+|x3|2,σR−r4)+−(R−r4−max|x1|2−R+3r42+|x3|2,σR−r4)+] ×4(R−r)(1−σ),∀(x1,x3)∈Rd/2×Rd/2,

and

viσ(x1,x2,x3):=[(R−r4−max|x1|2−R+3r42+|x3|2,σR−r4)+−(R−r4−max|x3|2−R+3r42+|x1|2,σR−r4)+]×R−r4−max|x2|,σR−r4+4(R−r)2(1−σ)2,

for every (x₁, x₂, x₃) ∈ ℝ^d/2 × ℝ^d–2i–2 × ℝ^d/2, and i≠d−22 .

Now, it is possible to prove that vσi∈FixHd,ηi(H1(Rd)). Moreover, for every σ ∈ (0, 1], let

Qσ(1):=(x1,x3)∈Ri+1×Ri+1:|x1|2−R+3r42+|x3|2≤σR−r4

and

Qσ(2):=(x1,x3)∈Ri+1×Ri+1:|x3|2−R+3r42+|x1|2≤σR−r4.

Define

Dσi:=Dσd−22 if i=d−22Diσ if i≠d−22,

where

Dσd−22:=(x1,x3)∈Rd/2×Rd/2:(x1,x3)∈Qσ(1)∩Qσ(2),

and

Diσ:=(x1,x2,x3)∈Rd/2×Rd−2i−2×Rd/2:(x1,x3)∈Qσ(1)∩Qσ(2),and|x2|≤σR−r4,

for every i≠d−22.

The sets Dσi have positive Lebesgue measure and they are H_{d,η_i}-invariant. Moreover, for every σ ∈ (0, 1), one has vσi∈FixHd,ηi(H1(Rd)) and the following facts hold:

supp (vσi)=D1i⊆A[r,R];
∥vσi∥∞≤1;
|vσi(x)|=1 for every x∈Dσi.

3 Proof of the Main Result

Part (a₁) - The main idea of the proof consists of applying Theorem 3 to the functional

Jλ(u)=Φ(u)−λΨ|FixO(d)(H1(Rd))(u),∀u∈FixO(d)(H1(Rd)),

with

Φ(u):=12∫Rd|∇u(x)|2dx+∫Rd|u(x)|2dx,

as well as

Ψ(u):=∫RdW(x)F(u(x))dx.

Successively, the existence of one non-trivial radial solution of problem (S_λ) follows by the symmetric criticality principle due to Krawcewicz and Marzantowicz and recalled above, in Theorem 4.

To this aim, first notice that the functionals Φ and Ψ|_{Fix_O(d)(H¹(ℝ^d))} have the regularity required by Theorem 3, according to Corollary 7. On the other hand, the functional Φ is clearly coercive in Fix_O(d)(H¹(ℝ^d)) and

infu∈FixO(d)(H1(Rd))Φ(u)=0.

Now, let us define

λ⋆:=1κ1cqmaxy>0y2∥W∥qq−1+2q/2cqq−1∥W∥∞yq−1, (3.1)

where κ₁ = and

cℓ:=sup∥u∥ℓ∥u∥:u∈FixO(d)(H1(Rd))∖{0},

for every q ∈ (2, 2^*) and take 0 < λ < λ^*.

Thanks to (3.1), there exists ȳ > 0 such that

λ<λ⋆(y¯):=y¯κ1cq12∥W∥qq−1+2q/2cqq−1∥W∥∞y¯q−1. (3.2)

Arguing as in [26], let us define the function χ : (0, +∞) → [0, +∞) as

χ(r):=supu∈Φ−1((−∞,r))Ψ|FixO(d)(H1(Rd))(u)r,

for every r > 0.

It follows by (2.8) that

Ψ|FixO(d)(H1(Rd))(u)≤κ1∥W∥qq−1∥u∥q+∥W∥∞∥u∥qq, (3.3)

for every u ∈ Fix_O(d)(H¹(ℝ^d)).

Moreover, one has

∥u∥<2r, (3.4)

for every u ∈ Φ^–1((–∞, r)).

Now, by using (3.4), the Sobolev embedding (2.1) and (3.3) yield

Ψ|FixO(d)(H1(Rd))(u)<κ1cq∥W∥qq−12r+cqq−1∥W∥∞(2r)q/2,

for every u ∈ Φ^–1((–∞, r)).

Consequently,

supu∈Φ−1((−∞,r))Ψ|FixO(d)(H1(Rd))(u)≤κ1cq∥W∥qq−12r+cqq−1∥W∥∞(2r)q/2.

The above inequality yields

χ(r)≤κ1cq∥W∥qq−12r+2q/2cqq−1∥W∥∞rq/2−1, (3.5)

for every r > 0.

Evaluating inequality (3.5) in r = ȳ², it follows that

χ(y¯2)≤κ1cq2∥W∥qq−1y¯+2q/2cqq−1∥W∥∞y¯q−2. (3.6)

Now, we notice that

φ(y¯2):=infu∈Φ−1((−∞,y¯2))supv∈Φ−1((−∞,y¯2))Ψ|FixO(d)(H1(Rd))(v)−Ψ|FixO(d)(H1(Rd))(u)r−Φ(u)≤χ(y¯2),

owing to z₀ ∈ Φ^–1((–∞, ȳ²)) and Φ(z₀) = Ψ|_{Fix_O(d)(H¹(ℝ^d))}(z₀) = 0, where z₀ ∈ Fix_O(d)(H¹(ℝ^d)) is the zero function.

Thanks to (3.2), the above inequality in addition to (3.6) give

φ(y¯2)≤χ(y¯2)≤κ1cq2∥W∥qq−1y¯+2q/2cqq−1∥W∥∞y¯q−2<1λ. (3.7)

In conclusion,

λ∈0,y¯κ1cq12∥W∥qq−1+2q/2cqq−1∥W∥∞y¯q−1⊆(0,1/φ(y¯2)).

Invoking Theorem 3, there exists a function u_λ ∈ Φ^–1((–∞, ȳ²)) such that

J0(uλ;φ)≥0,∀φ∈FixO(d)(H1(Rd)).

More precisely, the function u_λ is a global minimum of the restriction of the functional 𝓙_λ to the sublevel Φ^–1((–∞, ȳ²)).

Hence, let u_λ be such that

Jλ(uλ)≤Jλ(u),for anyu∈FixO(d)(H1(Rd))such thatΦ(u)<y¯2 (3.8)

and

Φ(uλ)<y¯2, (3.9)

and also u_λ is a critical point of 𝓙_λ in Fix_O(d)(H¹(ℝ^d)). Now, the orthogonal group O(d) acts isometrically on H¹(ℝ^d) and, thanks to the symmetry of the potential W, one has

∫RdW(x)F((gu)(x))dx=∫RdW(x)F(u(g−1x))dx=∫RdW(z)F(u(z))dz,

for every g ∈ O(d). Then the functional J_λ is O(d)-invariant on H¹(ℝ^d).

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 ≢ 0 in Fix_O(d)(H¹(ℝ^d)), first we claim that there exists a sequence of functions {w_j}_j∈ℕ in Fix_O(d)(H¹(ℝ^d)) such that

lim supj→+∞Ψ|FixO(d)(H1(Rd))(wj)Φ(wj)=+∞. (3.10)

By the assumption on the limsup in (1.2), there exists a sequence {s_j}_j∈ℕ ⊂ (0, +∞) such that s_j → 0⁺ as j → +∞ and

limj→+∞F(sj)sj2=+∞, (3.11)

namely, we have that for any M > 0 and j sufficiently large

F(sj)>Msj2. (3.12)

Now, define w_j := s_jv_σ for any j ∈ ℕ, where the function v_σ is given in Subsection 2.2. Since v_σ ∈ Fix_O(d)(H¹(ℝ^d)) of course, one has w_j ∈ Fix_O(d)(H¹(ℝ^d)) for any j ∈ ℕ. Bearing in mind that the functions v_σ satisfy (i₁)–(i₃), thanks to F(0) = 0 and (3.12) we have

Ψ|FixO(d)(H1(Rd))(wj)Φ(wj)=∫Ar+σR−σW(x)F(wj(x))dx+∫ArR∖Ar+σR−σW(x)F(wj(x))dxΦ(wj)=∫Ar+σR−σW(x)F(sj)dx+∫ArR∖Ar+σR−σW(x)F(sjvσ(x))dxΦ(wj)≥2M|Ar+σR−σ|αsj2+∫ArR∖Ar+σR−σW(x)F(sjvσ(x))dxsj2∥vσ∥2, (3.13)

for j sufficiently large.

Now, we have to consider two different cases.

lims→0+F(s)s2=+∞.

Then there exists ρ_M > 0 such that for any s with 0 < s < ρ_M

F(s)≥Ms2. (3.14)

Since s_j → 0⁺ and 0 ≤ v_σ(x) ≤ 1 in ℝ^d, it follows that w_j(x) = s_jv_σ(x) → 0⁺ as j → +∞ uniformly in x ∈ ℝ^d. Hence, 0 ≤ w_j(x) < ρ_M for j sufficiently large and for any x ∈ ℝ^d. Hence, as a consequence of (3.13) and (3.14), we have that

Ψ|FixO(d)(H1(Rd))(wj)Φ(wj)≥2M|Ar+σR−σ|αsj2+∫ArR∖Ar+σR−σW(x)F(sjvσ(x))dxsj2∥vσ∥2≥2Mα|Ar+σR−σ|+∫ArR∖Ar+σR−σ|vσ(x)|2dx∥vσ∥2,

for j sufficiently large. The arbitrariness of M gives (3.10) and so the claim is proved.
lim infs→0+F(s)s2=ℓ∈R.

Then for any ε > 0 there exists ρ_ε > 0 such that for any s with 0 < s < ρ_ε

F(s)≥(ℓ−ε)s2. (3.15)

Arguing as above, we can suppose that 0 ≤ w_j(x) = s_jv_σ(x) < ρ_ε for j large enough and any x ∈ ℝ^d. Thus, by (3.13) and (3.15) we get

provided that j is sufficiently large.

Let

M>max0,−2ℓ|Ar+σR−σ|∫ArR∖Ar+σR−σ|vσ(x)|2dx,

and

0<ε<M2|Ar+σR−σ|+ℓ∫ArR∖Ar+σR−σ|vσ(x)|2dx∫ArR∖Ar+σR−σ|vσ(x)|2dx.

By (3.16) we have

for j sufficiently large. Hence, assertion (3.10) is clearly verified.

Now, we notice that

∥wj∥=sj∥vσ∥→0,

as j → +∞, so that for j large enough

∥wj∥<2y¯.

Hence

wj∈Φ−1((−∞,y¯2)), (3.17)

and on account of (3.10), also

Jλ(wj)=Φ(wj)−λΨ|FixO(d)(H1(Rd))(wj)<0, (3.18)

for j sufficiently large.

Since u_λ is a global minimum of the restriction 𝓙_λ|_{Φ^–1((–∞,ȳ²))}, by (3.17) and (3.18) we have that

Jλ(uλ)≤Jλ(wj)<0=Jλ(0), (3.19)

so that u_λ ≢ 0 in Fix_O(d)(H¹(ℝ^d)).

Thus, u_λ is a non-trivial weak solution of problem (S_λ). The arbitrariness of λ gives that u_λ ≢ 0 for any λ ∈ (0, λ^*). By a Strauss-type estimate (see Lions [24]) we have that |u_λ(x)| → 0 as |x| → ∞. This concludes the proof of part (a₁) of Theorem 1.

Part (a₂) - Let

ci,ℓ:=sup∥u∥ℓ∥u∥:u∈FixHd,ηi(H1(Rd))∖{0},

for every ℓ ∈ (2, 2^*), with i ∈ J_d and set

λi,q⋆:=1κ1ci,qmaxy>0y2∥W∥qq−1+2q/2ci,qq−1∥W∥∞yq−1. (3.20)

Assume d > 3 and suppose that the potential F is even. Let

λ⋆:=λ⋆ifd=5min{λ⋆,λi,q⋆:i∈Jd}ifd≠5.

We claim that for every λ ∈ (0, λ_*) problem (S_λ) admits at least

ζS(d):=1+(−1)d+d−32

pairs of non-trivial weak solutions {±uλ,i}i∈Jd′⊂H1(Rd), where Jd′:={1,...,ζS(d)}, such that |u_λ,i(x)| → 0, as |x| → ∞, for every i∈Jd′.

Moreover, if d ≠ 5 problem (S_λ) admits at least

τd:=(−1)d+d−32

pairs of sign-changing weak solutions.

We divide the proof into two parts.

Part 1: dimension d = 5. Since F is symmetric, the energy functional

Jλ(u):=Φ(u)−λΨ|FixO(d)(H1(Rd))(u),∀u∈FixO(d)(H1(Rd)),

is even. Owing to Theorem 1, for every λ ∈ (0, λ^*), problem (S_λ) admits at least one (that is ζS(5) = 1) non-trivial pair of radial weak solutions {±u_λ} ⊂ H¹(ℝ^d). Furthermore, the functions ±u_λ are homoclinic.

Part 2: dimension d > 3 and d ≠ 5. For every λ > 0 and i ∈ J_d, consider the restriction 𝓗_λ,i := J_λ|_{Fix_{H_{d,η_i}}(H¹(ℝ^d))} : Fix_{H_{d,η_i}}(H¹(ℝ^d)) → ℝ defined by

Hλ,i:=ΦHd,ηi(u)−λΨ|FixHd,ηi(H1(Rd))(u),

where

ΦHd,ηi(u):=12∥u∥2andΨ|FixHd,ηi(H1(Rd))(u):=∫RdW(x)F(u(x))dx,

for every u ∈ Fix_{H_{d,η_i}}(H¹(ℝ^d)).

In order to obtain the existence of

τd:=(−1)d+d−32

pairs of sign-changing weak solutions {±z_λ,i}_{i∈J_d} ⊂ H¹(ℝ^d), where J_d := {1, …, τ_d}, the main idea of the proof consists in applying Theorem 3 to the functionals 𝓗_λ,i, for every i ∈ J_d. We notice that, since d > 3 and d ≠ 5, τ_d ≥ 1. Consequently, the cardinality |J_d| ≥ 1.

Since 0<λ<λi,q⋆, with i ∈ J_d, there exists ȳ_i > 0 such that

λ<λ⋆(i)(y¯i):=y¯iκ1ci,q12∥W∥qq−1+2q/2ci,qq−1∥W∥∞y¯iq−1. (3.21)

Similar arguments used for proving (3.7) yield

φ(y¯i2)≤χ(y¯i2)≤κ1cq2∥W∥qq−1y¯i+2q/2cqq−1∥W∥∞y¯iq−2<1λ. (3.22)

Thus,

λ∈0,y¯iκ1cq12∥W∥qq−1+2q/2cqq−1∥W∥∞y¯iq−1⊆(0,1/φ(y¯i2)).

Thanks to Theorem 3, there exists a function zλ,i∈ΦHd,ηi−1((−∞,y¯i2)) such that

J0(zλ,i;φ)≥0,∀φ∈FixHd,ηi(H1(Rd))

and, in particular, z_λ,i is a global minimum of the restriction of 𝓗_λ,i to ΦHd,ηi−1((−∞,y¯i2)).

Due to the evenness of J_λ, bearing in mind (2.2), and thanks to the symmetry assumptions on the potential W, we have that the functional J_λ is H_{d,η_i}-invariant on H¹(ℝ^d), i.e.

Jλ(h⊛iu)=Jλ(u),

for every h ∈ H_{d,η_i} and u ∈ H¹(ℝ^d). Indeed, the group H_{d,η_i} acts isometrically on H¹(ℝ^d) and, thanks to the symmetry assumption on W, it follows that

∫RdW(x)F((hu)(x))dx=∫RdW(x)F(u(h−1x))dx=∫RdW(z)F(u(z))dz,

if h ∈ H_d,i, and

∫RdW(x)F((hu)(x))dx=∫RdW(x)F(u(g−1ηHd,i−1x))dx=∫RdW(z)F(u(z))dz,

if h = η_{H_d,i} g ∈ H_{d,η_i} ∖ H_d,i.

On account of Theorem 4, the critical point pairs {±z_λ,i} of 𝓗_λ,i are also (generalized) critical points of J_λ.

Let z_λ,i ∈ Fix_{H_{d,η_i}}(H¹(ℝ^d)) be a critical point of 𝓗_λ,i in Fix_{H_{d,η_i}}(H¹(ℝ^d)) such that

Hλ,i(zλ,i)≤Hλ,i(u),for anyu∈FixHd,ηi(H1(Rd))such thatΦHd,ηi(u)<y¯i2 (3.23)

and

ΦHd,ηi(zλ,i)<y¯i2. (3.24)

In order to prove that z_λ,i ≢ 0 in Fix_{H_{d,η_i}}(H¹(ℝ^d)), we claim that there exists a sequence {wji}j∈N in Fix_{H_{d,η_i}}(H¹(ℝ^d)) such that

lim supj→+∞Ψ|FixHd,ηi(H1(Rd))(wji)Φ(wji)=+∞. (3.25)

The sequence {wji}j∈N ⊂ Fix_{H_{d,η_i}}(H¹(ℝ^d)), for which (3.25) holds, can be constructed by using the test functions introduced in [22] and recalled in Subsection 2.2. Thus, let us define wji:=sjvσi for any j ∈ ℕ. Clearly, wji∈FixHd,ηi(H1(Rd)) for any j ∈ ℕ. Moreover, taking into account the properties of vσi displayed in (j₁)–(j₃), simple computations show that

Ψ|FixHd,ηi(H1(Rd))(wji)Φ(wji)=∫DσiW(x)F(wji(x))dx+∫ArR∖DσiW(x)F(wji(x))dxΦ(wji)=∫DσiW(x)F(sj)dx+∫ArR∖DσiW(x)F(sjvσi(x))dxΦ(wji)≥2M|Dσi|αsj2+∫ArR∖DσiW(x)F(sjvσi(x))dxsj2∥vσi∥2, (3.26)

for j sufficiently large.

Arguing as in the proof of Theorem 1, inequality (3.26) yields (3.25) and consequently, we conclude that

Hλ,i(zλ,i)≤Hλ,i(wji)<0=Hλ,i(0),

so that z_λ,i ≢ 0 in Fix_{H_{d,η_i}}(H¹(ℝ^d)). In addition, |z_λ,i(x)| → 0 as |x| → ∞.

On the other hand, since λ < λ^* and F is even, Theorem 1 and the principle of symmetric criticality (recalled in Theorem 4) ensure that problem (S_λ) admits at least one non-trivial pair of radial weak solutions {±u_λ} ⊂ H¹(ℝ^d). Moreover, |u_λ(x)| → 0 as |x| → ∞.

In conclusion, since λ < λ_*, there exist τ_d + 1 positive numbers ȳ, ȳ₁, …, ȳ_{τ_d} such that

±uλ∈Φ−1((−∞,y¯2))∖{0}⊂FixO(d)(H1(Rd)),

and

±zλ,i∈ΦHd,ηi−1((−∞,y¯i2))∖{0}⊂FixHd,ηi(H1(Rd)).

Bearing in mind relations (2.4) and (2.5) of Proposition 5 (see also [22, Theorem 2.2] for details) we have that

Φ−1((−∞,y¯2))∩ΦHd,ηi−1((−∞,y¯i2))∖{0}=∅,

for every i ∈ J_d and

ΦHd,ηi−1((−∞,y¯i2))∩ΦHd,ηj−1((−∞,y¯j2))∖{0}=∅,

for every i, j ∈ J_d and i ≠ j. Consequently problem (S_λ) admits at least

ζS(d):=τd+1,

pairs of non-trivial weak solutions {±uλ,i}i∈Jd′⊂H1(Rd), where Jd′:={1,...,ζS(d)}, such that |u_λ,i(x)| → 0, as |x| → ∞, for every i∈Jd′. Moreover, by construction, it follows that

τd:=(−1)d+d−32

pairs of the attained solutions are sign-changing.

The proof is now complete.□

4 Some applications

A simple prototype of a function F fulfilling the structural assumption (1.1) can be easily constructed as follows. Let f : ℝ → ℝ be a measurable function such that

sups∈R|f(s)|1+|s|q−1<+∞, (4.1)

for some q ∈ (2, 2^*). Furthermore, let F be the potential defined by

F(s):=∫0sf(t)dt,

for every s ∈ ℝ. Of course F is a Carathéodory function that is locally Lipschitz with F(0) = 0. Since the growth condition (4.1) is satisfied, f is locally essentially bounded, that is f∈Lloc∞(Rd). Thus, invoking [27, Proposition 1.7] it follows that

∂F(s)=[f_(s),f¯(s)] (4.2)

where

f_(s):=limδ→0+essinf|t−s|<δf(t),

and

f¯(s):=limδ→0+esssup|t−s|<δf(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 significative form:

(Sλ′) Find u ∈ H¹(ℝ^d) such that

∫Rd∇u(x)⋅∇φ(x)dx+∫Rdu(x)φ(x)dx−λ∫RdW(x)f(u(x))φ(x)dx=0,∀φ∈H1(Rd).

See [18] for related topics.

Of course, the solutions of (Sλ′) are exactly the weak solutions of the following Schrödinger equation

−Δu+u=λW(x)f(u)inRdu∈H1(Rd),

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 differs to the above, we will treat it in a forthcoming paper.

Acknowledgements

The paper was realized with the auspices of the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT 009) and the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025, N1-0083, and N1-0064.

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Received: 2019-04-28

Accepted: 2019-06-11

Published Online: 2019-08-06

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/anona-2020-0035

Keywords for this article

Hemivariational inequalities; variational methods; principle of symmetric criticality; radial and non-radial solutions

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