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Advances in Nonlinear Analysis

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Some hemivariational inequalities in the Euclidean space

Giovanni Molica Bisci
  • Corresponding author
  • Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica 13, 61029, Urbino, Italy
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/ Dušan Repovš
Published Online: 2019-08-06 | DOI: https://doi.org/10.1515/anona-2020-0035

Abstract

The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd (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(ℝd). 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:

  • (Sλ)

    Find uH1(ℝd) such that

    Rdu(x)φ(x)dx+Rdu(x)φ(x)dx+λRdW(x)F0(u(x);φ(x))dx0,φH1(Rd).

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

F0(s;z):=lim supyst0+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, WL(ℝd) ∩ L1(ℝd) ∖ {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|q1),ζF(s),for everysR,(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 sups0+F(s)s2=+andlim infs0+F(s)s2>.(1.2)

Moreover, let WL(ℝd) ∩ L1(ℝd) ∖ {0} be a non-negative radially symmetric map. Then the following facts hold:

  • (a1)

    There exists a positive number λ* such that, for every λ ∈ (0, λ*), the problem (Sλ) admits at least one non-trivial radial weak solution uλH1(ℝd) with |uλ(x)| → 0 as |x| → ∞.

  • (a2)

    If d > 3 and F is even then there exists a positive number λ* such that for every λ ∈ (0, λ*), the problem (Sλ) admits at least

    ζS(d):=1+(1)d+d32

    pairs of non-trivial weak solutions {±uλ,i}iJdH1(Rd) with |uλ,i(x)| → 0, as |x| → ∞, for every iJd := {1, …, ζS(d)}, and with different symmetries structure. More precisely, if d ≠ 5 problem (Sλ) 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(ℝd) 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(ℝd) 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 (Sλ). 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+d32

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

τd:=(1)d+d32

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 (Sλ), we need to work in a subspace of the functional space H1(ℝd) 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 (Sλ) and we prove our existence result (see Theorem 1). Finally, we study the existence of multiple non-radial solutions to the problem (Sλ) 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 yX there correspond a neighborhood Vy of y and a constant Ly ≥ 0 such that

|J(z)J(w)|LyzwX,(z,wVy).

If y, zX, 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 supwyt0+J(w+tz)J(w)t.

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

J(y):=yX:y,zJ0(y;z),zX.

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

  1. J0(y; ⋅) is positively homogeneous, sub-additive, and continuous for every yX;

  2. J0(y; z) = max{〈y*, z〉 : y*∂J(z)} for every y, zX;

  3. J0(y; –z) = (–J)0 (y; z) for every y, zX;

  4. if JC1(X), then J0(y; z) = 〈J′(y), zfor every y, zX;

  5. (I + J)0(y; z) ≤ I0(y; z) + J0(y; z) for every y, zX. Moreover, if J is is continuously Gâteaux differentiable, then (I + J)0(y; z) = I0(y; z) + J′(y; z) for every y, zX.

See [17] for details.

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

J0(y;z)0,

for every zX.

Clearly, if J is a continuously Gâteaux differentiable at yX, then y becomes a (classical) critical point of J, that is J′(y) = 0X*.

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(ℝd) be the standard Sobolev space endowed by the inner product

u,v:=Rdu(x)v(x)dx+Rdu(x)v(x)dx,u,vH1(Rd)

and the induced norm

u:=Rd|u(x)|2dx+Rd|u(x)|2dx1/2,

for every uH1(ℝd).

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 > infX Φ, put

φ(r):=infuΦ1((,r))supvΦ1((,r))Ψ(v)Ψ(u)rΦ(u).

Then for each r > infX Φ and each λ ∈ ]0, 1/φ(r)[, the restriction of 𝓙λ := ΦλΨ to Φ–1((–∞, r)) admits a global minimum, which is a critical point (local minimum) of 𝓙λ 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×XX:(g,y)gy,

such that

1y=y,(gh)y=g(hy),ygyislinear.

The action * is said to be isometric if ∥g*yX = ∥yX, for every gG and yX. Moreover, the space of G-invariant points is defined by

FixG(X):={yX:gy=y,gG},

and a map h : X → ℝ is said to be G-invariant on X if

h(gy)=h(y)

for every gG and yX.

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: FixG(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 ℝd and let GO(d) be a subgroup. Assume that G acts on the space H1(ℝd). Hence, the set of fixed points of H1(ℝd), with respect to G, is clearly given by

FixG(H1(Rd)):={uH1(Rd):gu=u,gG}.

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

gu(x):=u(g1x),xRdandgO(d)

then FixO(d)(H1(ℝd)) is exactly the subspace of radially symmetric functions of H1(ℝd), also denoted by Hrad1(ℝd). 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 Hd,iO(d) given by

Hd,i:=O(d/2)×O(d/2) ifi=d22O(i+1)×O(d2i2)×O(i+1) ifid22,

for every iJd := {1, …, τd}, where

τd:=(1)d+d32.

Let us define the involution ηHd,i : ℝd → ℝd as follows

ηHd,i(x):=(x3,x1) ifi=d22andx:=(x1,x3)Rd/2×Rd/2(x3,x2,x1) ifid22andx:=(x1,x2,x3)Ri+1×Rd2i2×Ri+1,

for every iJd.

By definition, one has ηHd,iHd,i, as well as

ηHd,iHd,iηHd,i1=Hd,i,andηHd,i2=idRd,

for every iJd.

Moreover, for every iJd, let us consider the compact group

Hd,ηi:=Hd,i,ηHd,i,

that is Hd,ηi = Hd,iηHd,i Hd,i, and the action ⊛i : Hd,ηi × H1(ℝd) → H1(ℝd) of Hd,ηi on H1(ℝd) given by

hiu(x):=u(h1x) ifhHd,iu(g1ηHd,i1x) ifh=ηHd,igHd,ηiHd,i,gHd,i(2.2)

for every x ∈ ℝd.

We note that ⊛i is defined for every element of Hd,ηi. Indeed, if hHd,ηi, then either hHd,i or h = τgHd,ηiHd,i, with gHd,i. Moreover, set

FixHd,ηi(H1(Rd)):={uH1(Rd):hiu=u,hHd,ηi},

for every iJd.

Following Bartsch and Willem [8], for every iJd, 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 iJd;

if d = 6 or d ≥ 8, then

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

for every i, jJd and ij.

See [22, Theorem 2.2] for details.

From now on, for every uL(ℝd) and ∈ [2, 2*), we shall denote

u:=Rd|u(x)|dx1/,

and

W:=esssupxRd|W(x)|,up:=Rd|u(x)|pdx1/p,

for every p ∈ [2, 2*).

Moreover, let Ψ : H1(ℝd) → ℝ given by

Ψ(u):=RdW(x)F(u(x))dx,uH1(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 WL(ℝd) ∩ L1(ℝd) ∖ {0}. Then the extended functional Ψe : Lq(ℝd) → ℝ defined by

Ψe(u):=RdW(x)F(u(x))dx,uLq(Rd)

is well-defined and locally Lipschitz continuous on Lq(ℝd).

Proof

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

F(t1)F(t2)=ζθ(t1t2).(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))dxκ1RdW(x)|u(x)|dx+RdW(x)|u(x)|qdxκ1Rd|W(x)|qq1dxq1qRd|u(x)|qdx1/q+κ1WRd|u(x)|qdx,(2.7)

for every uLq(ℝd). Hence, inequality (2.7) yields

Ψe(u)κ1Wqq1uq+Wuqq<+,(2.8)

for every uLq(ℝd).

In order to prove that Ψe is locally Lipschitz continuous on Lq(ℝd) it is straightforward to establish that the functional Ψe is in fact Lipschitz continuous on Lq(ℝd). Now, for a fixed number r > 0 and arbitrary elements u, vLq(ℝd) with max{∥uq, ∥vq} ≤ r, the following estimate holds

|Ψe(u)Ψe(v)|RdW(x)F(u(x))F(v(x))dxκ1RdW(x)1+|u(x)|q1+|v(x)|q1|u(x)v(x)|dxκ1(Wqq1uvq+W(uqq1+vqq1)uvq)κ2uvq,(2.9)

where the Lipschitz constant κ2 := 2q–2(Wqq1+2rq1W)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 Ψe 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 WL(ℝd) ∩ L1(ℝd) ∖ {0}. Then for every λ > 0, the functional

u12u2λΨ|FixY(H1(Rd))(u),uFixY(H1(Rd))

is sequentially weakly lower semicontinuous on FixY(H1(ℝd)), where either Y = O(d) or Y = Hd,ηi for some iJd.

Proof

First, on account of Brézis [12, Corollaire III.8], the functional u ↦ ∥u2/2 is sequentially weakly lower semicontinuous on FixY(H1(ℝd)). Now, we prove that Ψ|FixY(H1(ℝd)) is sequentially weakly continuous. Indeed, let {uj}j∈ℕFixY(H1(ℝd)) be a sequence which weakly converges to an element u0FixY(H1(ℝd)). Since Y is compactly embedded in Lq(ℝd), for every q ∈ (2, 2*), passing to a subsequence if necessary, one has ∥uju0q → 0 as j → ∞. According to Lemma 6, the extension of Ψ to Lq(ℝd) is locally Lipschitz continuous. Hence, there exists a constant Lu0 ≥ 0 such that

|Ψ(uj)Ψ(u0)|Lu0uju0q,(2.10)

for every j ∈ ℕ. Passing to the limit in (2.10), we conclude that Ψ is sequentially weakly continuous on FixY(H1(ℝd)). 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 WL(ℝd) ∩ L1(ℝd) ∖ {0}. Furthermore, let E be a closed subspace of H1(ℝd) and denote by ΨE 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, vE.

Proof

The map xW(x)F0(u(x);v(x)) is measurable on ℝd. Indeed, WL(ℝd) and the function xF0(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 xW(x)F0(u(x);v(x)) belongs to L1(ℝd).

The next task is to prove (2.11). To this goal, since E is separable, let us notice that there exist two sequences {tj}j∈ℕ ∈ ℝ and {wj}j∈ℕE such that tj → 0+, ∥wju∥ → 0 in E and

ΨE0(u;v)=limjΨE(wj+tjv)ΨE(wj)tj.

Without loss of generality we can also suppose that wj(x) → u(x) a.e. in ℝd as j → ∞.

Now, for every j ∈ ℕ, let us consider the measurable and non-negative function gj : ℝd → ℝ ∪ {+∞} defined by

gj(x):=κ1|v(x)|(1+|wj(x)+tjv(x)|q1+|wj(x)|q1)F(wj(x)+tjv(x))F(wj(x))tj,

for a.e. x ∈ ℝd. Set

I:=lim supjRdW(x)gj(x)dx.

The inverse Fatou’s Lemma applied to the sequences {Wgj}j∈ℕ yields

IJ:=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)|q1+|wj(x)|q1)

for every j ∈ ℕ and a.e. x ∈ ℝd.

By setting

yj:=RdW(x)βj(x)dx,

one has

I=lim supjRdW(x)αj(x)dxyj.(2.13)

Now, it is easily seen that there exists a function kL1(ℝd) such that

|βj(x)|k(x),

and

βj(x)κ1|v(x)|(1+2|u(x)|q1)

for a.e. x ∈ ℝd.

Consequently, the Lebesgue’s Dominated Convergence Theorem implies that

limjyj=κ1RdW(x)|v(x)|(1+2|u(x)|q1)dx.(2.14)

By (2.13) and (2.14) it follows that

I=lim supjΨE(wj+tjv)ΨE(wj)tjlimjyj=ΨE0(u;v)κ1RdW(x)|v(x)|(1+2|u(x)|q1)dx.(2.15)

Now

JJακ1RdW(x)|v(x)|(1+2|u(x)|q1)dx.(2.16)

where

Jα:=RdW(x)lim supjαj(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 supjF(wj(x)+tjv(x))F(wj(x))tjdxRdW(x)limjF(wj+tjv)F(wj)tjdxRdW(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 WL(ℝd) ∩ L1(ℝd) ∖ {0}. Let Jλ : H1(ℝd) → ℝ be the functional defined by

Jλ(u):=12u2λΨ(u),uH1(Rd).

Then the functional is locally Lipschitz continuous and its critical points solve (Sλ).

Proof

The functional Jλ is locally Lipschitz continuous. Indeed, Jλ is the sum of the C1(H1(ℝd)) functional u ↦ ∥u2/2 and of the locally Lipschitz continuous functional Ψ, see Lemma 6. Now, every critical point of Jλ is a weak solution of problem (Sλ). Indeed, if u0H1(ℝd) is a critical point of Jλ, a direct application of inequality (2.11) in Proposition 8 yields

0Jλ0(u0;φ)=u0,φ+λ(Ψ)0(u0;φ)=u0,φ+λ(Ψ)0(u0;φ)u0,φ+λRdW(x)F0(u0(x);φ(x))dx,(2.19)

for every φH1(ℝd). Since (2.19) holds, the function u0H1(ℝd) solves (Sλ).□

2.2 Some test functions with symmetries

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

Aab:={xRd:a|x|b}.

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

essinfxArRW(x)α>0.(2.20)

Hence, let 0 < r < R, such that (2.20) holds and take σ ∈ (0, (Rr)/2). Set vσFixO(d)(H1(ℝd)) given by

vσ(x):=|x|rσ+ if|x|r+σ1 ifr+σ|x|RσR|x|σ+ if|x|Rσ

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

  • (i1)

    supp(vσ) ⊆ ArR;

  • (i2)

    vσ ≤ 1;

  • (i3)

    vσ(x) = 1 for every xAr+σRσ.

Now, assume rR5+42 and set σ ∈ (0, 1). Define vσiH1(Rd) as follows

vσi(x):=vσd22(x) ifi=d22andx:=(x1,x3)Rd/2×Rd/2viσ(x) ifid22andx:=(x1,x2,x3)Ri+1×Rd2i2×Ri+1,

for every x ∈ ℝd, where:

vσd22(x1,x3):=[(Rr4max|x1|2R+3r42+|x3|2,σRr4)+(Rr4max|x1|2R+3r42+|x3|2,σRr4)+]×4(Rr)(1σ),(x1,x3)Rd/2×Rd/2,

and

viσ(x1,x2,x3):=[(Rr4max|x1|2R+3r42+|x3|2,σRr4)+(Rr4max|x3|2R+3r42+|x1|2,σRr4)+]×Rr4max|x2|,σRr4+4(Rr)2(1σ)2,

for every (x1, x2, x3) ∈ ℝd/2 × ℝd–2i–2 × ℝd/2, and id22.

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

Qσ(1):=(x1,x3)Ri+1×Ri+1:|x1|2R+3r42+|x3|2σRr4

and

Qσ(2):=(x1,x3)Ri+1×Ri+1:|x3|2R+3r42+|x1|2σRr4.

Define

Dσi:=Dσd22 ifi=d22Diσ ifid22,

where

Dσd22:=(x1,x3)Rd/2×Rd/2:(x1,x3)Qσ(1)Qσ(2),

and

Diσ:=(x1,x2,x3)Rd/2×Rd2i2×Rd/2:(x1,x3)Qσ(1)Qσ(2),and|x2|σRr4,

for every id22.

The sets Dσi have positive Lebesgue measure and they are Hd,ηi-invariant. Moreover, for every σ ∈ (0, 1), one has vσiFixHd,ηi(H1(Rd)) and the following facts hold:

  • (j1)

    supp(vσi)=D1iA[r,R];

  • (j2)

    vσi1;

  • (j3)

    |vσi(x)|=1 for every xDσi.

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),uFixO(d)(H1(Rd)),

with

Φ(u):=12Rd|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 (Sλ) 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|FixO(d)(H1(ℝd)) have the regularity required by Theorem 3, according to Corollary 7. On the other hand, the functional Φ is clearly coercive in FixO(d)(H1(ℝd)) and

infuFixO(d)(H1(Rd))Φ(u)=0.

Now, let us define

λ:=1κ1cqmaxy>0y2Wqq1+2q/2cqq1Wyq1,(3.1)

where κ1 = and

c:=supuu:uFixO(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¯κ1cq12Wqq1+2q/2cqq1Wy¯q1.(3.2)

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

χ(r):=supuΦ1((,r))Ψ|FixO(d)(H1(Rd))(u)r,

for every r > 0.

It follows by (2.8) that

Ψ|FixO(d)(H1(Rd))(u)κ1Wqq1uq+Wuqq,(3.3)

for every uFixO(d)(H1(ℝd)).

Moreover, one has

u<2r,(3.4)

for every uΦ–1((–∞, r)).

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

Ψ|FixO(d)(H1(Rd))(u)<κ1cqWqq12r+cqq1W(2r)q/2,

for every uΦ–1((–∞, r)).

Consequently,

supuΦ1((,r))Ψ|FixO(d)(H1(Rd))(u)κ1cqWqq12r+cqq1W(2r)q/2.

The above inequality yields

χ(r)κ1cqWqq12r+2q/2cqq1Wrq/21,(3.5)

for every r > 0.

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

χ(y¯2)κ1cq2Wqq1y¯+2q/2cqq1Wy¯q2.(3.6)

Now, we notice that

φ(y¯2):=infuΦ1((,y¯2))supvΦ1((,y¯2))Ψ|FixO(d)(H1(Rd))(v)Ψ|FixO(d)(H1(Rd))(u)rΦ(u)χ(y¯2),

owing to z0Φ–1((–∞, ȳ2)) and Φ(z0) = Ψ|FixO(d)(H1(ℝd))(z0) = 0, where z0FixO(d)(H1(ℝd)) is the zero function.

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

φ(y¯2)χ(y¯2)κ1cq2Wqq1y¯+2q/2cqq1Wy¯q2<1λ.(3.7)

In conclusion,

λ0,y¯κ1cq12Wqq1+2q/2cqq1Wy¯q1(0,1/φ(y¯2)).

Invoking Theorem 3, there exists a function uλΦ–1((–∞, ȳ2)) such that

J0(uλ;φ)0,φFixO(d)(H1(Rd)).

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

Hence, let uλ be such that

Jλ(uλ)Jλ(u),for anyuFixO(d)(H1(Rd))such thatΦ(u)<y¯2(3.8)

and

Φ(uλ)<y¯2,(3.9)

and also uλ is a critical point of 𝓙λ in FixO(d)(H1(ℝd)). Now, the orthogonal group O(d) acts isometrically on H1(ℝd) and, thanks to the symmetry of the potential W, one has

RdW(x)F((gu)(x))dx=RdW(x)F(u(g1x))dx=RdW(z)F(u(z))dz,

for every gO(d). Then the functional Jλ is O(d)-invariant on H1(ℝd).

So, owing to Theorem 4, uλ is a weak solution of problem (Sλ). In this setting, in order to prove that uλ ≢ 0 in FixO(d)(H1(ℝd)), first we claim that there exists a sequence of functions {wj}j∈ℕ in FixO(d)(H1(ℝd)) 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 {sj}j∈ℕ ⊂ (0, +∞) such that sj → 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 wj := sjvσ for any j ∈ ℕ, where the function vσ is given in Subsection 2.2. Since vσFixO(d)(H1(ℝd)) of course, one has wjFixO(d)(H1(ℝd)) for any j ∈ ℕ. Bearing in mind that the functions vσ satisfy (i1)–(i3), thanks to F(0) = 0 and (3.12) we have

Ψ|FixO(d)(H1(Rd))(wj)Φ(wj)=Ar+σRσW(x)F(wj(x))dx+ArRAr+σRσW(x)F(wj(x))dxΦ(wj)=Ar+σRσW(x)F(sj)dx+ArRAr+σRσW(x)F(sjvσ(x))dxΦ(wj)2M|Ar+σRσ|αsj2+ArRAr+σRσW(x)F(sjvσ(x))dxsj2vσ2,(3.13)

for j sufficiently large.

Now, we have to consider two different cases.

  • Case 1

    lims0+F(s)s2=+.

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

    F(s)Ms2.(3.14)

    Since sj → 0+ and 0 ≤ vσ(x) ≤ 1 in ℝd, it follows that wj(x) = sjvσ(x) → 0+ as j → +∞ uniformly in x ∈ ℝd. Hence, 0 ≤ wj(x) < ρM for j sufficiently large and for any x ∈ ℝd. Hence, as a consequence of (3.13) and (3.14), we have that

    Ψ|FixO(d)(H1(Rd))(wj)Φ(wj)2M|Ar+σRσ|αsj2+ArRAr+σRσW(x)F(sjvσ(x))dxsj2vσ22Mα|Ar+σRσ|+ArRAr+σRσ|vσ(x)|2dxvσ2,

    for j sufficiently large. The arbitrariness of M gives (3.10) and so the claim is proved.

  • Case 2

    lim infs0+F(s)s2=R.

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

    F(s)(ε)s2.(3.15)

Arguing as above, we can suppose that 0 ≤ wj(x) = sjvσ(x) < ρε for j large enough and any x ∈ ℝd. Thus, by (3.13) and (3.15) we get

Ψ|FixO(d)(H1(Rd))(wj)Φ(wj)2M|Ar+σRσ|αsj2+ArRAr+σRσW(x)F(sjvσ(x))dxsj2vσ22αM|Ar+σRσ|+(ε)ArRAr+σRσ|vσ(x)|2dxvσ2,(3.16)

provided that j is sufficiently large.

Let

M>max0,2|Ar+σRσ|ArRAr+σRσ|vσ(x)|2dx,

and

0<ε<M2|Ar+σRσ|+ArRAr+σRσ|vσ(x)|2dxArRAr+σRσ|vσ(x)|2dx.

By (3.16) we have

Ψ|FixO(d)(H1(Rd))(wj)Φ(wj)2αM|Ar+σRσ|+(ε)ArRAr+σRσ|vσ(x)|2dxvσ22αvσ2M|Ar+σRσ|+ArRAr+σRσ|vσ(x)|2dxεArRAr+σRσ|vσ(x)|2dxαM|Ar+σRσ|vσ2,

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

Now, we notice that

wj=sjvσ0,

as j → +∞, so that for j large enough

wj<2y¯.

Hence

wjΦ1((,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 uλ is a global minimum of the restriction 𝓙λ|Φ–1((–∞,ȳ2)), by (3.17) and (3.18) we have that

Jλ(uλ)Jλ(wj)<0=Jλ(0),(3.19)

so that uλ ≢ 0 in FixO(d)(H1(ℝd)).

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

Part (a2) - Let

ci,:=supuu:uFixHd,ηi(H1(Rd)){0},

for every ∈ (2, 2*), with iJd and set

λi,q:=1κ1ci,qmaxy>0y2Wqq1+2q/2ci,qq1Wyq1.(3.20)

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

λ:=λifd=5min{λ,λi,q:iJd}ifd5.

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

ζS(d):=1+(1)d+d32

pairs of non-trivial weak solutions {±uλ,i}iJdH1(Rd), where Jd:={1,...,ζS(d)}, such that |uλ,i(x)| → 0, as |x| → ∞, for every iJd.

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

τd:=(1)d+d32

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),uFixO(d)(H1(Rd)),

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

Part 2: dimension d > 3 and d ≠ 5. For every λ > 0 and iJd, consider the restriction 𝓗λ,i := Jλ|FixHd,ηi(H1(ℝd)) : FixHd,ηi(H1(ℝd)) → ℝ defined by

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

where

ΦHd,ηi(u):=12u2andΨ|FixHd,ηi(H1(Rd))(u):=RdW(x)F(u(x))dx,

for every uFixHd,ηi(H1(ℝd)).

In order to obtain the existence of

τd:=(1)d+d32

pairs of sign-changing weak solutions {±zλ,i}iJdH1(ℝd), where Jd := {1, …, τd}, the main idea of the proof consists in applying Theorem 3 to the functionals 𝓗λ,i, for every iJd. We notice that, since d > 3 and d ≠ 5, τd ≥ 1. Consequently, the cardinality |Jd| ≥ 1.

Since 0<λ<λi,q, with iJd, there exists ȳi > 0 such that

λ<λ(i)(y¯i):=y¯iκ1ci,q12Wqq1+2q/2ci,qq1Wy¯iq1.(3.21)

Similar arguments used for proving (3.7) yield

φ(y¯i2)χ(y¯i2)κ1cq2Wqq1y¯i+2q/2cqq1Wy¯iq2<1λ.(3.22)

Thus,

λ0,y¯iκ1cq12Wqq1+2q/2cqq1Wy¯iq1(0,1/φ(y¯i2)).

Thanks to Theorem 3, there exists a function zλ,iΦHd,ηi1((,y¯i2)) such that

J0(zλ,i;φ)0,φFixHd,ηi(H1(Rd))

and, in particular, zλ,i is a global minimum of the restriction of 𝓗λ,i to ΦHd,ηi1((,y¯i2)).

Due to the evenness of Jλ, bearing in mind (2.2), and thanks to the symmetry assumptions on the potential W, we have that the functional Jλ is Hd,ηi-invariant on H1(ℝd), i.e.

Jλ(hiu)=Jλ(u),

for every hHd,ηi and uH1(ℝd). Indeed, the group Hd,ηi acts isometrically on H1(ℝd) and, thanks to the symmetry assumption on W, it follows that

RdW(x)F((hu)(x))dx=RdW(x)F(u(h1x))dx=RdW(z)F(u(z))dz,

if hHd,i, and

RdW(x)F((hu)(x))dx=RdW(x)F(u(g1ηHd,i1x))dx=RdW(z)F(u(z))dz,

if h = ηHd,i gHd,ηiHd,i.

On account of Theorem 4, the critical point pairs {±zλ,i} of 𝓗λ,i are also (generalized) critical points of Jλ.

Let zλ,iFixHd,ηi(H1(ℝd)) be a critical point of 𝓗λ,i in FixHd,ηi(H1(ℝd)) such that

Hλ,i(zλ,i)Hλ,i(u),for anyuFixHd,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 zλ,i ≢ 0 in FixHd,ηi(H1(ℝd)), we claim that there exists a sequence {wji}jN in FixHd,ηi(H1(ℝd)) such that

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

The sequence {wji}jNFixHd,ηi(H1(ℝd)), 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, wjiFixHd,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+ArRDσiW(x)F(wji(x))dxΦ(wji)=DσiW(x)F(sj)dx+ArRDσiW(x)F(sjvσi(x))dxΦ(wji)2M|Dσi|αsj2+ArRDσiW(x)F(sjvσi(x))dxsj2vσi2,(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 zλ,i ≢ 0 in FixHd,ηi(H1(ℝd)). In addition, |zλ,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 (Sλ) admits at least one non-trivial pair of radial weak solutions {±uλ} ⊂ H1(ℝd). Moreover, |uλ(x)| → 0 as |x| → ∞.

In conclusion, since λ < λ*, there exist τd + 1 positive numbers ȳ, ȳ1, …, ȳτd such that

±uλΦ1((,y¯2)){0}FixO(d)(H1(Rd)),

and

±zλ,iΦHd,ηi1((,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

Φ1((,y¯2))ΦHd,ηi1((,y¯i2)){0}=,

for every iJd and

ΦHd,ηi1((,y¯i2))ΦHd,ηj1((,y¯j2)){0}=,

for every i, jJd and ij. Consequently problem (Sλ) admits at least

ζS(d):=τd+1,

pairs of non-trivial weak solutions {±uλ,i}iJdH1(Rd), where Jd:={1,...,ζS(d)}, such that |uλ,i(x)| → 0, as |x| → ∞, for every iJd. Moreover, by construction, it follows that

τd:=(1)d+d32

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

supsR|f(s)|1+|s|q1<+,(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 fLloc(Rd). Thus, invoking [27, Proposition 1.7] it follows that

F(s)=[f_(s),f¯(s)](4.2)

where

f_(s):=limδ0+essinf|ts|<δf(t),

and

f¯(s):=limδ0+esssup|ts|<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 (Sλ) assumes the simple and significative form:

(Sλ) Find uH1(ℝd) such that

Rdu(x)φ(x)dx+Rdu(x)φ(x)dxλRdW(x)f(u(x))φ(x)dx=0,φH1(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)inRduH1(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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About the article

Received: 2019-04-28

Accepted: 2019-06-11

Published Online: 2019-08-06

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 958–977, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0035.

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© 2020 Giovanni Molica Bisci and Dušan Repovš, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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