Some hemivariational inequalities in the Euclidean space

TL;DR

This paper investigates the existence of weak solutions for certain hemivariational inequalities in Euclidean space using variational methods and symmetry principles, also demonstrating multiple solutions with different symmetries.

Contribution

It introduces a variational approach combined with a non-smooth criticality principle to establish existence and multiplicity of solutions for hemivariational problems in high-dimensional spaces.

Findings

Existence of at least one weak solution established.

Multiple sign-changing solutions with distinct symmetries proved.

Application to classical Schrödinger equations demonstrated.

Abstract

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.

Equations

$$\left\{\begin{array}[]{l}\displaystyle\int_{\mathbb{R}^{d}}\nabla u(x)\cdot\nabla\varphi(x)dx+\int_{\mathbb{R}^{d}}u(x)\varphi(x))dx\\ \qquad\qquad\qquad\qquad\quad\quad+\displaystyle\lambda\int_{\mathbb{R}^{d}}W(x)F^{0}(u(x);-\varphi(x))dx\geq 0,\\ \forall\,\varphi\in H^{1}(\mathbb{R}^{d}).\end{array}\right.$$

$$F^0(s;z):=\underset{\begin{array}{c}y\to s\\ t\to 0^{+}\end{array}}{\mathrm{limsup}}\frac{F(y+tz)-F(y)}{t}$$

$$|\zeta |\le {\kappa }_1(1+|s{|}^{q-1}),\forall \zeta \in \partial F(s),\text{mbox}foreverys\in \mathbb{R},$$

$$\underset{s\to 0^{+}}{\mathrm{limsup}}\frac{F(s)}{s^2}=+\infty and\underset{s\to 0^{+}}{\mathrm{liminf}}\frac{F(s)}{s^2}>-\infty .$$

$${\zeta }_S^{(d)}:=1+(-1{)}^d+\left[\frac{d-3}{2}\right]$$

$${\tau }_d:={\zeta }_S^{(d)}-1$$

$${\zeta }_S^{(d)}:=1+(-1{)}^d+\left[\frac{d-3}{2}\right]$$

$${\tau }_d:=(-1{)}^d+\left[\frac{d-3}{2}\right]$$

$$|J(z)-J(w)|\le L_y\Vert z-w{\Vert }_X,(\forall z,w\in V_y).$$

$$J^0(y;z):=\underset{\begin{array}{c}w\to y\\ t\to 0^{+}\end{array}}{\mathrm{limsup}}\frac{J(w+tz)-J(w)}{t}.$$

$$\partial J(y):=\left\{y^{\ast }\in X^{\ast }:⟨y^{\ast },z⟩\le J^0(y;z),\forall z\in X\right\}.$$

$$J^0(y;z)\ge 0,$$

$$⟨u,v⟩:={\int }_{{\mathbb{R}}^d}\nabla u(x)\cdot \nabla v(x)dx+{\int }_{{\mathbb{R}}^d}u(x)v(x)dx,\forall u,v\in H^1({\mathbb{R}}^d)$$

$$\Vert u\Vert :={\left({\int }_{{\mathbb{R}}^d}|\nabla u(x){|}^{2}dx+{\int }_{{\mathbb{R}}^d}|u(x){|}^{2}dx\right)}^{1/2},$$

$$\phi (r):=\underset{u\in {\Phi }^{-1}((-\infty ,r))}{\inf }\frac{\left(\underset{v\in {\Phi }^{-1}((-\infty ,r))}{\sup }\Psi (v)\right)-\Psi (u)}{r-\Phi (u)}.$$

$$\ast :G\times X\to X:(g,y)\mapsto g\ast y,$$

$$1\ast y=y,(gh)\ast y=g\ast (h\ast y),y\mapsto g\ast y\mathrm{is}\mathrm{linear}.$$

$$Fi{x}_G(X):=\{y\in X:g\ast y=y,\forall g\in G\},$$

$$h(g\ast y)=h(y)$$

$$Fi{x}_G(H^1({\mathbb{R}}^d)):=\{u\in H^1({\mathbb{R}}^d):gu=u,\forall g\in G\}.$$

$$gu(x):=u(g^{-1}x),\forall x\in {\mathbb{R}}^d\mathrm{and}g\in O(d)$$

$$Fi{x}_{O(d)}(H^1({\mathbb{R}}^d))\hookrightarrow L^q({\mathbb{R}}^d)$$

$$H_{d,i}:=\left\{\begin{array}[]{ll}\displaystyle O(d/2)\times O(d/2)&\mbox{ if $i=\displaystyle\frac{d-2}{2}$}\\ \displaystyle O(i+1)\times O(d-2i-2)\times O(i+1)&\mbox{ if $i\neq\displaystyle\frac{d-2}{2}$},\end{array}\right.$$

$${\tau }_d:=(-1{)}^d+\left[\frac{d-3}{2}\right].$$

$$\eta_{{}_{H_{d,i}}}(x):=\left\{\begin{array}[]{ll}\displaystyle(x_{3},x_{1})&\mbox{ if $i=\displaystyle\frac{d-2}{2}$ and $x:=(x_{1},x_{3})\in\mathbb{R}^{d/2}\times\mathbb{R}^{d/2}$}\\ \displaystyle(x_{3},x_{2},x_{1})&\mbox{ if $i\neq\displaystyle\frac{d-2}{2}$ and $x:=(x_{1},x_{2},x_{3})\in\mathbb{R}^{i+1}\times\mathbb{R}^{d-2i-2}\times\mathbb{R}^{i+1}$},\end{array}\right.$$

$${\eta }_{{}_{H_{d,i}}}{H}_{d,i}{\eta }_{{}_{H_{d,i}}}^{-1}=H_{d,i},\mathrm{and}{\eta }_{{}_{H_{d,i}}}^2={\text{id}}_{{\mathbb{R}}^d},$$

$$H_{d,{\eta }_i}:=⟨H_{d,i},{\eta }_{{}_{H_{d,i}}}⟩,$$

$$h\circledast_{i}u(x):=\left\{\begin{array}[]{ll}\displaystyle u(h^{-1}x)&\mbox{ if $h\in H_{d,i}$}\\ \displaystyle-u(g^{-1}\eta_{{}_{H_{d,i}}}^{-1}x)&\mbox{ if $h=\eta_{{}_{H_{d,i}}}g\in H_{d,\eta_{i}}\setminus H_{d,i}$},\,g\in H_{d,i}\end{array}\right.$$

$$Fi{x}_{H_{d,{\eta }_i}}(H^1({\mathbb{R}}^d)):=\{u\in H^1({\mathbb{R}}^d):h{⊛}_{i}u=u,\forall h\in H_{d,{\eta }_i}\},$$

$$Fi{x}_{H_{d,{\eta }_i}}(H^1({\mathbb{R}}^d))\hookrightarrow L^q({\mathbb{R}}^d)$$

Full text

aa 2010 AMS Subject Classification: Primary: 35A15, 35J60, 35J65, 35J91; Secondary: 35A01, 45A05, 35P30.

1. 1 Introduction

2. 2 Abstract framework

3. 2.1 Group-theoretical arguments

4. 2.2 Some test functions with symmetries

5. 3 Proof of the Main Result

6. 4 Some applications

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:

• $({S}_{\lambda})$

Find $u\in H^{1}(\mathbb{R}^{d})$ such that

[TABLE]

Here $(\mathbb{R}^{d},|\cdot|)$ denotes the Euclidean space (with $d\geq 3$), $F:\mathbb{R}\rightarrow\mathbb{R}$ is a locally Lipschitz continuous function, whereas

[TABLE]

is the generalized directional derivative of $F$ at the point $s\in\mathbb{R}$ in the direction $z\in\mathbb{R}$; see the classical monograph of Clarke [15] for details. Finally, $W\in L^{\infty}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})\setminus\{0\}$ is a non-negative radially symmetric map and $\lambda$ is a positive real parameter.

We assume that there exist $\kappa_{1}>0$ and $q\in(2,2^{*})$, where $2^{*}=2d/(d-2)$, such that

[TABLE]

where $\partial F(s)$ denotes the generalized gradient of the function $F$ at $s\in\mathbb{R}$ (see Section 2).

With the above notations the main result reads as follows.

Theorem 1**.**

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

[TABLE]

Moreover, let $W\in L^{\infty}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})\setminus\{0\}$ be a non-negative radially symmetric map. Then the following facts hold:

• $(a_{1})$

There exists a positive number $\lambda^{\star}$ such that, for every $\lambda\in(0,\lambda^{\star})$, the problem $(S_{\lambda})$ admits at least one non-trivial radial weak solution $u_{\lambda}\in H^{1}(\mathbb{R}^{d})$ with $|u_{\lambda}(x)|\rightarrow 0$ as $|x|\rightarrow\infty$.

• $(a_{2})$

If $d>3$ and $F$ is even then there exists a positive number $\lambda_{\star}$ such that for every $\lambda\in(0,\lambda_{\star})$, the problem $(S_{\lambda})$ admits at least

[TABLE]

pairs of non-trivial weak solutions $\{\pm u_{\lambda,i}\}_{i\in J^{\prime}_{d}}\subset H^{1}(\mathbb{R}^{d})$ with $|u_{\lambda,i}(x)|\rightarrow 0$, as $|x|\rightarrow\infty$, for every $i\in J^{\prime}_{d}:=\{1,...,\zeta^{(d)}_{S}\}$, and with different symmetries structure. More precisely, if $d\neq 5$ problem $(S_{\lambda})$ admits at least

[TABLE]

pairs of sign-changing weak solutions.

Here, the symbol $[\cdot]$ 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 $H^{1}(\mathbb{R}^{d})$ into Lebesgue spaces produces several difficulties for exploiting variational methods. In order to recover compactness, the first task is to construct certain subspaces of $H^{1}(\mathbb{R}^{d})$ 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 $(S_{\lambda})$. 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 $(a_{2})$ 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

[TABLE]

subspaces of $H^{1}(\mathbb{R}^{d})$ with different symmetries properties. In addition, when $d\neq 5$, there are

[TABLE]

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 for 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 $(S_{\lambda})$, we need to work in a subspace of the functional space $H^{1}(\mathbb{R}^{d})$ 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 $(S_{\lambda})$ and we prove our existence result (see Theorem 1). Finally, we study the existence of multiple non-radial solutions to the problem $(S_{\lambda})$ for $\lambda$ 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 $(X,\|\cdot\|_{X})$ be a real Banach space. We denote by $X^{*}$ the dual space of $X$, whereas $\langle\cdot,\cdot\rangle$ denotes the duality pairing between $X^{*}$ and $X$.

A function $J:X\to\mathbb{R}$ is called locally Lipschitz continuous if to every $y\in X$ there correspond a neighborhood $V_{y}$ of $y$ and a constant $L_{y}\geq 0$ such that

[TABLE]

If $y,z\in X$, we write $J^{0}(y;z)$ for the generalized directional derivative of $J$ at the point $y$ along the direction $z$, i.e.,

[TABLE]

The generalized gradient of the function $J$ at $y\in X$, denoted by $\partial J(y)$, is the set

[TABLE]

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 $I,J:X\to\mathbb{R}$ are locally Lipschitz continuous functionals, then

• $(i)$

$J^{0}(y;\cdot)$* is positively homogeneous, sub-additive, and continuous for every $y\in X;$*

• $(ii)$

$J^{0}(y;z)=\displaystyle\max\{\langle y^{*},z\rangle:{y^{*}\in\partial J(z)}\}$* for every $y,z\in X;$*

• $(iii)$

$J^{0}(y;-z)=(-J)^{0}(y;z)$* for every $y,z\in X;$*

• $(iv)$

if $J\in C^{1}(X)$, then $J^{0}(y;z)=\langle J^{\prime}(y),z\rangle$ for every $y,z\in X;$

• $(v)$

$(I+J)^{0}(y;z)\leq I^{0}(y;z)+J^{0}(y;z)$* for every $y,z\in X$. Moreover, if $J$ is is continuously Gâteaux differentiable, then $(I+J)^{0}(y;z)=I^{0}(y;z)+J^{\prime}(y;z)$ for every $y,z\in X$.*

See [17] for details.

Further, a point $y\in X$ is called a (generalized) critical point of the locally Lipschitz continuous function $J$ if $0_{X^{*}}\in\partial J(y)$, i.e.

[TABLE]

for every $z\in X$.

Clearly, if $J$ is a continuously Gâteaux differentiable at $y\in X$, then $y$ becomes a (classical) critical point of $J$, that is $J^{\prime}(y)=0_{X^{*}}$.

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 $d\geq 3$ and let $H^{1}(\mathbb{R}^{d})$ be the standard Sobolev space endowed by the inner product

[TABLE]

and the induced norm

[TABLE]

for every $u\in H^{1}(\mathbb{R}^{d})$.

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 $\Phi,\Psi:X\to\mathbb{R}$ be locally Lipschitz continuous functionals such that $\Phi$ is sequentially weakly lower semicontinuous and coercive. Furthermore, assume that $\Psi$ is sequentially weakly upper semicontinuous. For every $r>\inf_{X}\Phi$, put

[TABLE]

Then for each $r>\inf_{X}\Phi$ and each $\lambda\in\left]0,{1}/{\varphi(r)}\right[$, the restriction of $\mathcal{J}_{\lambda}:=\Phi-\lambda\Psi$ to $\Phi^{-1}((-\infty,r))$ admits a global minimum, which is a critical point $($local minimum$)$ of $\mathcal{J}_{\lambda}$ 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 $(X,\|\cdot\|_{X})$ is a continuous map

[TABLE]

such that

[TABLE]

The action $*$ is said to be isometric if $\|g*y\|_{X}=\|y\|_{X}$, for every $g\in G$ and $y\in X$. Moreover, the space of $G$-invariant points is defined by

[TABLE]

and a map $h:X\rightarrow\mathbb{R}$ is said to be $G$-invariant on $X$ if

[TABLE]

for every $g\in G$ and $y\in X$.

Theorem 4**.**

Let $X$ be a Banach space, let $G$ be a compact topological group acting linearly and isometrically on $X$, and $J:X\rightarrow\mathbb{R}$ a locally Lipschitz, $G$-invariant functional. Then every critical point of $\mathcal{J}:Fix_{G}(X)\rightarrow\mathbb{R}$ 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 $\mathbb{R}^{d}$ and let $G\subseteq O(d)$ be a subgroup. Assume that $G$ acts on the space $H^{1}(\mathbb{R}^{d})$. Hence, the set of fixed points of $H^{1}(\mathbb{R}^{d})$, with respect to $G$, is clearly given by

[TABLE]

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

[TABLE]

then $Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ is exactly the subspace of radially symmetric functions of $H^{1}(\mathbb{R}^{d})$, also denoted by $H^{1}_{\rm rad}(\mathbb{R}^{d})$. Moreover, the following embedding

[TABLE]

is continuous (resp. compact), for every $q\in[2,2^{*}]$ (resp. $q\in(2,2^{*})$). See, for instance, the celebrated paper [24].

Let either $d=4$ or $d\geq 6$ and consider the subgroup $H_{d,i}\subset O(d)$ given by

[TABLE]

for every $i\in J_{d}:=\{1,...,\tau_{d}\}$, where

[TABLE]

Let us define the involution $\eta_{{}_{H_{d,i}}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ as follows

[TABLE]

for every $i\in J_{d}$.

By definition, one has $\eta_{{}_{H_{d,i}}}\notin H_{d,i}$, as well as

[TABLE]

for every $i\in J_{d}$.

Moreover, for every $i\in J_{d}$, let us consider the compact group

[TABLE]

that is $H_{d,\eta_{i}}=H_{d,i}\cup\eta_{{}_{H_{d,i}}}H_{d,i}$, and the action $\circledast_{i}:H_{d,\eta_{i}}\times H^{1}(\mathbb{R}^{d})\rightarrow H^{1}(\mathbb{R}^{d})$ of $H_{d,\eta_{i}}$ on $H^{1}(\mathbb{R}^{d})$ given by

[TABLE]

for every $x\in\mathbb{R}^{d}$.

We note that $\circledast_{i}$ is defined for every element of $H_{d,\eta_{i}}$. Indeed, if $h\in H_{d,\eta_{i}}$, then either $h\in H_{d,i}$ or $h=\tau g\in H_{d,\eta_{i}}\setminus H_{d,i}$, with $g\in H_{d,i}$. Moreover, set

[TABLE]

for every $i\in J_{d}$.

Following Bartsch and Willem [8], for every $i\in J_{d}$, the embedding

[TABLE]

is compact, for every $q\in(2,2^{*})$.

Proposition 5**.**

With the above notations, the following properties hold:

• if $d=4$ or $d\geq 6$, then

[TABLE]

for every $i\in J_{d}$;

• if $d=6$ or $d\geq 8$, then

[TABLE]

for every $i,j\in J_{d}$ and $i\neq j$.

See [22, Theorem 2.2] for details.

From now on, for every $u\in L^{\ell}(\mathbb{R}^{d})$ and $\ell\in[2,2^{*})$, we shall denote

[TABLE]

and

[TABLE]

for every $p\in[2,2^{*})$.

Moreover, let $\Psi:H^{1}(\mathbb{R}^{d})\rightarrow\mathbb{R}$ given by

[TABLE]

The following locally Lipschitz property holds.

Lemma 6**.**

Assume that condition (1.1) holds for some $q\in\left(2,2^{*}\right)$ and $F(0)=0$. Furthermore, let $W\in L^{\infty}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})\setminus\{0\}$. Then the extended functional $\Psi^{e}:L^{q}(\mathbb{R}^{d})\rightarrow\mathbb{R}$ defined by

[TABLE]

is well-defined and locally Lipschitz continuous on $L^{q}(\mathbb{R}^{d})$.

Proof. It is clear that $\Psi^{e}$ is well-defined. Indeed, by using Lebourg’s mean value theorem, fixing $t_{1},t_{2}\in\mathbb{R}$, there exist $\theta\in(0,1)$ and $\zeta_{\theta}\in\partial F(\theta t_{1}+(1-\theta)t_{2})$ such that

[TABLE]

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

[TABLE]

for every $u\in L^{q}(\mathbb{R}^{d})$. Hence, inequality (2.1) yields

[TABLE]

for every $u\in L^{q}(\mathbb{R}^{d})$.

In order to prove that $\Psi^{e}$ is locally Lipschitz continuous on $L^{q}(\mathbb{R}^{d})$ it is straightforward to establish that the functional $\Psi^{e}$ is in fact Lipschitz continuous on $L^{q}(\mathbb{R}^{d})$. Now, for a fixed number $r>0$ and arbitrary elements $u,v\in L^{q}(\mathbb{R}^{d})$ with $\max\{\|u\|_{q},\|v\|_{q}\}\leq r$, the following estimate holds

[TABLE]

where the Lipschitz constant $\kappa_{2}:=2^{q-2}(\|W\|_{\frac{q}{q-1}}+2r^{q-1}\|W\|_{\infty})\kappa_{1}$ 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 $\Psi^{e}$ is thus verified. $\Box$

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

Corollary 7**.**

Assume that condition (1.1) holds for some $q\in\left(2,2^{*}\right)$ and let $W\in L^{\infty}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})\setminus\{0\}$. Then for every $\lambda>0,$ the functional

[TABLE]

is sequentially weakly lower semicontinuous on $Fix_{Y}(H^{1}(\mathbb{R}^{d}))$, where either $Y=O(d)$ or $Y=H_{d,\eta_{i}}$ for some $i\in J_{d}$.

Proof. First, on account of Brézis [12, Corollaire III.8], the functional $u\mapsto\|u\|^{2}/2$ is sequentially weakly lower semicontinuous on $Fix_{Y}(H^{1}(\mathbb{R}^{d}))$. Now, we prove that $\Psi|_{Fix_{Y}(H^{1}(\mathbb{R}^{d}))}$ is sequentially weakly continuous. Indeed, let $\{u_{j}\}_{j\in\mathbb{N}}\subset Fix_{Y}(H^{1}(\mathbb{R}^{d}))$ be a sequence which weakly converges to an element $u_{0}\in Fix_{Y}(H^{1}(\mathbb{R}^{d}))$. Since $Y$ is compactly embedded in $L^{q}(\mathbb{R}^{d})$, for every $q\in(2,2^{*})$, passing to a subsequence if necessary, one has $\|u_{j}-u_{0}\|_{q}\rightarrow 0$ as $j\rightarrow\infty$. According to Lemma 6, the extension of $\Psi$ to $L^{q}(\mathbb{R}^{d})$ is locally Lipschitz continuous. Hence, there exists a constant $L_{u_{0}}\geq 0$ such that

[TABLE]

for every $j\in\mathbb{N}$. Passing to the limit in (2.10), we conclude that $\Psi$ is sequentially weakly continuous on $Fix_{Y}(H^{1}(\mathbb{R}^{d}))$. The proof is now complete. $\Box$

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 $q\in\left(2,2^{*}\right)$ and let $W\in L^{\infty}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})\setminus\{0\}$. Furthermore, let $E$ be a closed subspace of $H^{1}(\mathbb{R}^{d})$ and denote by $\Psi_{E}$ the restriction of $\Psi$ to $E$. Then the following inequality holds

[TABLE]

for every $u,v\in E$.

Proof. The map $x\mapsto W(x)F^{0}(u(x);v(x))$ is measurable on $\mathbb{R}^{d}$. Indeed, $W\in L^{\infty}(\mathbb{R}^{d})$ and the function $x\mapsto F^{0}(u(x);v(x))$ is measurable as the countable limsup of measurable functions, see p. 16 of [27] for details. Moreover, condition (1.1) ensures that

[TABLE]

Thus the map $x\mapsto W(x)F^{0}(u(x);v(x))$ belongs to $L^{1}(\mathbb{R}^{d})$.

The next task is to prove (2.11). To this goal, since $E$ is separable, let us notice that there exist two sequences $\{t_{j}\}_{j\in\mathbb{N}}\in\mathbb{R}$ and $\{w_{j}\}_{j\in\mathbb{N}}\subset E$ such that $t_{j}\rightarrow 0^{+}$, $\|w_{j}-u\|\rightarrow 0$ in $E$ and

[TABLE]

Without loss of generality we can also suppose that $w_{j}(x)\rightarrow u(x)$ a.e. in $\mathbb{R}^{d}$ as $j\rightarrow\infty$.

Now, for every $j\in\mathbb{N}$, let us consider the measurable and non-negative function $g_{j}:\mathbb{R}^{d}\rightarrow\mathbb{R}\cup\{+\infty\}$ defined by

[TABLE]

[TABLE]

for a.e. $x\in\mathbb{R}^{d}$. Set

[TABLE]

The inverse Fatou’s Lemma applied to the sequences $\{Wg_{j}\}_{j\in\mathbb{N}}$ yields

[TABLE]

where

[TABLE]

and

[TABLE]

for every $j\in\mathbb{N}$ and a.e. $x\in\mathbb{R}^{d}$.

By setting

[TABLE]

one has

[TABLE]

Now, it is easily seen that there exists a function $k\in L^{1}(\mathbb{R}^{d})$ such that

[TABLE]

and

[TABLE]

for a.e. $x\in\mathbb{R}^{d}$.

Consequently, the Lebesgue’s Dominated Convergence Theorem implies that

[TABLE]

By (2.13) and (2.14) it follows that

[TABLE]

Now

[TABLE]

where

[TABLE]

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

[TABLE]

Finally,

[TABLE]

By (2.17) and (2.1), inequality (2.11) now immediately follows. $\Box$

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

Proposition 9**.**

Assume that condition (1.1) holds for some $q\in\left(2,2^{*}\right)$ and let $W\in L^{\infty}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})\setminus\{0\}$. Let $J_{\lambda}:H^{1}(\mathbb{R}^{d})\rightarrow\mathbb{R}$ be the functional defined by

[TABLE]

Then the functional is locally Lipschitz continuous and its critical points solve $(S_{\lambda})$.

Proof. The functional $J_{\lambda}$ is locally Lipschitz continuous. Indeed, $J_{\lambda}$ is the sum of the $C^{1}(H^{1}(\mathbb{R}^{d}))$ functional $u\mapsto\|u\|^{2}/2$ and of the locally Lipschitz continuous functional $\Psi$, see Lemma 6. Now, every critical point of $J_{\lambda}$ is a weak solution of problem $(S_{\lambda})$. Indeed, if $u_{0}\in H^{1}(\mathbb{R}^{d})$ is a critical point of $J_{\lambda}$, a direct application of inequality (2.11) in Proposition 8 yields

[TABLE]

for every $\varphi\in H^{1}(\mathbb{R}^{d})$. Since (2.1) holds, the function $u_{0}\in H^{1}(\mathbb{R}^{d})$ solves $(S_{\lambda})$. $\Box$

2.2. Some test functions with symmetries

Following Kristály, Moroşanu, and O’Regan [22], we construct some special test functions belonging to $Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ that will be useful for our purposes. If $a<b$, define

[TABLE]

Since $W\in L^{\infty}(\mathbb{R}^{d})\setminus\{0\}$ is a radially symmetric function with $W\geq 0$, one can find real numbers $R>r>0$ and $\alpha>0$ such that

[TABLE]

Hence, let $0<r<R$, such that (2.20) holds and take $\sigma\in(0,(R-r)/2)$. Set $v_{\sigma}\in Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ given by

[TABLE]

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

• $(i_{1})$

$\textrm{supp}(v_{\sigma})\subseteq A_{r}^{R}$;

• $(i_{2})$

$\|v_{\sigma}\|_{\infty}\leq 1$;

• $(i_{3})$

$v_{\sigma}(x)=1$ for every $x\in A_{r+\sigma}^{R-\sigma}$.

Now, assume $r\geq\displaystyle\frac{R}{5+4\sqrt{2}}$ and set $\sigma\in(0,1)$. Define $v_{\sigma}^{i}\in H^{1}(\mathbb{R}^{d})$ as follows

[TABLE]

for every $x\in\mathbb{R}^{d}$, where:

\displaystyle v_{\sigma}^{\frac{d-2}{2}}(x_{1},x_{3}):=\Bigg{[}\Bigg{(}\frac{R-r}{4}-\max\left\{\sqrt{\left(|x_{1}|^{2}-\frac{R+3r}{4}\right)^{2}+|x_{3}|^{2}},\sigma\frac{R-r}{4}\right\}\Bigg{)}_{+}

[TABLE]

[TABLE]

and

\displaystyle v_{i}^{\sigma}(x_{1},x_{2},x_{3}):=\Bigg{[}\Bigg{(}\frac{R-r}{4}-\max\left\{\sqrt{\left(|x_{1}|^{2}-\frac{R+3r}{4}\right)^{2}+|x_{3}|^{2}},\sigma\frac{R-r}{4}\right\}\Bigg{)}_{+}

[TABLE]

[TABLE]

for every $(x_{1},x_{2},x_{3})\in\mathbb{R}^{d/2}\times\mathbb{R}^{d-2i-2}\times\mathbb{R}^{d/2}$, and $i\neq\displaystyle\frac{d-2}{2}$.

Now, it is possible to prove that $v_{\sigma}^{i}\in Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$. Moreover, for every $\sigma\in(0,1]$, let

[TABLE]

and

[TABLE]

Define

[TABLE]

where

[TABLE]

and

[TABLE]

for every $i\neq\displaystyle\frac{d-2}{2}$.

The sets $D^{i}_{\sigma}$ have positive Lebesgue measure and they are $H_{d,\eta_{i}}$-invariant. Moreover, for every $\sigma\in(0,1)$, one has $v_{\sigma}^{i}\in Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$ and the following facts hold:

• $(j_{1})$

$\textrm{supp}(v_{\sigma}^{i})=D_{1}^{i}\subseteq A[r,R]$;

• $(j_{2})$

$\|v_{\sigma}^{i}\|_{\infty}\leq 1$;

• $(j_{3})$

$|v_{\sigma}^{i}(x)|=1$ for every $x\in D_{\sigma}^{i}$.

3. Proof of the Main Result

Part $(a_{1})$ - The main idea of the proof consists of applying Theorem 3 to the functional

[TABLE]

with

[TABLE]

as well as

[TABLE]

Successively, the existence of one non-trivial radial solution of problem $(S_{\lambda})$ 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 $\Phi$ and $\Psi|_{Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))}$ have the regularity required by Theorem 3, according to Corollary 7. On the other hand, the functional $\Phi$ is clearly coercive in $Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ and

[TABLE]

Now, let us define

[TABLE]

where $\kappa_{1}=$ and

[TABLE]

for every $q\in(2,2^{*})$ and take $0<\lambda<\lambda^{\star}$.

Thanks to (3.1), there exists $\bar{\gamma}>0$ such that

[TABLE]

Arguing as in [26], let us define the function $\chi:(0,+\infty)\rightarrow[0,+\infty)$ as

[TABLE]

for every $r>0$.

It follows by (2.8) that

[TABLE]

for every $u\in Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$.

Moreover, one has

[TABLE]

for every $u\in\Phi^{-1}((-\infty,r))$.

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

[TABLE]

for every $u\in\Phi^{-1}((-\infty,r))$.

Consequently,

[TABLE]

The above inequality yields

[TABLE]

for every $r>0$.

Evaluating inequality (3.5) in $r=\bar{\gamma}^{2}$, it follows that

[TABLE]

Now, we notice that

[TABLE]

owing to $z_{0}\in\Phi^{-1}((-\infty,\bar{\gamma}^{2}))$ and $\Phi(z_{0})=\Psi|_{Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))}(z_{0})=0$, where $z_{0}\in Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ is the zero function.

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

[TABLE]

In conclusion,

[TABLE]

Invoking Theorem 3, there exists a function $u_{\lambda}\in\Phi^{-1}((-\infty,\bar{\gamma}^{2}))$ such that

[TABLE]

More precisely, the function $u_{\lambda}$ is a global minimum of the restriction of the functional $\mathcal{J}_{\lambda}$ to the sublevel $\Phi^{-1}((-\infty,\bar{\gamma}^{2}))$.

Hence, let $u_{\lambda}$ be such that

[TABLE]

and

[TABLE]

and also $u_{\lambda}$ is a critical point of $\mathcal{J}_{\lambda}$ in $Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$. Now, the orthogonal group $O(d)$ acts isometrically on $H^{1}(\mathbb{R}^{d})$ and, thanks to the symmetry of the potential $W$, one has

[TABLE]

for every $g\in O(d)$. Then the functional $J_{\lambda}$ is $O(d)$-invariant on $H^{1}(\mathbb{R}^{d})$.

So, owing to Theorem 4, $u_{\lambda}$ is a weak solution of problem $(S_{\lambda})$ . In this setting, in order to prove that $u_{\lambda}\not\equiv 0$ in $Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ , first we claim that there exists a sequence of functions \big{\{}w_{j}\big{\}}_{j\in\mathbb{N}} in $Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ such that

[TABLE]

By the assumption on the limsup in (1.2), there exists a sequence $\{s_{j}\}_{j\in\mathbb{N}}\subset(0,+\infty)$ such that $s_{j}\to 0^{+}$ as $j\to+\infty$ and

[TABLE]

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

[TABLE]

Now, define $w_{j}:=s_{j}v_{\sigma}$ for any $j\in\mathbb{N}$, where the function $v_{\sigma}$ is given in Subsection 2.2. Since $v_{\sigma}\in Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ of course, one has $w_{j}\in Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$ for any $j\in\mathbb{N}$. Bearing in mind that the functions $v_{\sigma}$ satisfy $(i_{1})$–$(i_{3})$, thanks to $F(0)=0$ and (3.12) we have

[TABLE]

for $j$ sufficiently large.

Now, we have to consider two different cases.

Case 1: ${\displaystyle\lim_{s\to 0^{+}}\frac{\displaystyle F(s)}{s^{2}}=+\infty}$.

Then there exists $\rho_{M}>0$ such that for any $s$ with $0<s<\rho_{M}$

[TABLE]

Since $s_{j}\to 0^{+}$ and $0\leq v_{\sigma}(x)\leq 1$ in $\mathbb{R}^{d}$, it follows that $w_{j}(x)=s_{j}v_{\sigma}(x)\to 0^{+}$ as $j\to+\infty$ uniformly in $x\in\mathbb{R}^{d}$. Hence, $0\leq w_{j}(x)<\rho_{M}$ for $j$ sufficiently large and for any $x\in\mathbb{R}^{d}$. Hence, as a consequence of (3) and (3.14), we have that

[TABLE]

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

Case 2: ${\displaystyle\liminf_{s\to 0^{+}}\frac{\displaystyle F(s)}{s^{2}}=\ell\in\mathbb{R}}$ .

Then for any $\varepsilon>0$ there exists $\rho_{\varepsilon}>0$ such that for any $s$ with $0<s<\rho_{\varepsilon}$

[TABLE]

Arguing as above, we can suppose that $0\leq w_{j}(x)=s_{j}v_{\sigma}(x)<\rho_{\varepsilon}$ for $j$ large enough and any $x\in\mathbb{R}^{d}$. Thus, by (3) and (3.15) we get

[TABLE]

provided that $j$ is sufficiently large.

Let

[TABLE]

and

[TABLE]

By (3) we have

[TABLE]

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

Now, we notice that

[TABLE]

as $j\to+\infty$ , so that for $j$ large enough

[TABLE]

Hence

[TABLE]

and on account of (3.10), also

[TABLE]

for $j$ sufficiently large.

Since $u_{\lambda}$ is a global minimum of the restriction $\mathcal{J}_{\lambda}|_{\Phi^{-1}((-\infty,\bar{\gamma}^{2}))}$, by (3.17) and (3.18) we have that

[TABLE]

so that $u_{\lambda}\not\equiv 0$ in $Fix_{O(d)}(H^{1}(\mathbb{R}^{d}))$.

Thus, $u_{\lambda}$ is a non-trivial weak solution of problem $(S_{\lambda})$. The arbitrariness of $\lambda$ gives that $u_{\lambda}\not\equiv 0$ for any $\lambda\in(0,\lambda^{\star})$. By a Strauss-type estimate (see Lions [24]) we have that $|u_{\lambda}(x)|\rightarrow 0$ as $|x|\rightarrow\infty$. This concludes the proof of part $(a_{1})$ of Theorem 1.

Part $(a_{2})$ - Let

[TABLE]

for every $\ell\in(2,2^{*})$, with $i\in J_{d}$ and set

[TABLE]

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

[TABLE]

We claim that for every $\lambda\in(0,\lambda_{\star})$ problem $(S_{\lambda})$ admits at least

[TABLE]

pairs of non-trivial weak solutions $\{\pm u_{\lambda,i}\}_{i\in J^{\prime}_{d}}\subset H^{1}(\mathbb{R}^{d})$, where $J^{\prime}_{d}:=\{1,...,\zeta^{(d)}_{S}\}$, such that $|u_{\lambda,i}(x)|\rightarrow 0$, as $|x|\rightarrow\infty$, for every $i\in J^{\prime}_{d}$.

Moreover, if $d\neq 5$ problem $(S_{\lambda})$ admits at least

[TABLE]

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

[TABLE]

is even. Owing to Theorem 1, for every $\lambda\in(0,\lambda^{\star})$, problem $(S_{\lambda})$ admits at least one (that is $\zeta^{(5)}_{S}=1$) non-trivial pair of radial weak solutions $\{\pm u_{\lambda}\}\subset H^{1}(\mathbb{R}^{d})$. Furthermore, the functions $\pm u_{\lambda}$ are homoclinic.

Part 2: dimension $d>3$ and $d\neq 5$. For every $\lambda>0$ and $i\in J_{d}$, consider the restriction $\mathcal{H}_{\lambda,i}:=J_{\lambda}|_{Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))}:Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))\to\mathbb{R}$ defined by

[TABLE]

where

[TABLE]

for every $u\in Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$.

In order to obtain the existence of

[TABLE]

pairs of sign-changing weak solutions $\{\pm z_{\lambda,i}\}_{i\in J_{d}}\subset H^{1}(\mathbb{R}^{d})$, where $J_{d}:=\{1,...,\tau_{d}\}$, the main idea of the proof consists in applying Theorem 3 to the functionals $\mathcal{H}_{\lambda,i}$, for every $i\in J_{d}$. We notice that, since $d>3$ and $d\neq 5$, $\tau_{d}\geq 1$. Consequently, the cardinality $|J_{d}|\geq 1$.

Since $0<\lambda<\lambda_{i,q}^{\star}$, with $i\in J_{d}$, there exists $\bar{\gamma}_{i}>0$ such that

[TABLE]

Similar arguments used for proving (3.7) yield

[TABLE]

Thus,

[TABLE]

Thanks to Theorem 3, there exists a function $z_{\lambda,i}\in\Phi_{H_{d,\eta_{i}}}^{-1}((-\infty,\bar{\gamma}^{2}_{i}))$ such that

[TABLE]

and, in particular, $z_{\lambda,i}$ is a global minimum of the restriction of $\mathcal{H}_{\lambda,i}$ to $\Phi_{H_{d,\eta_{i}}}^{-1}((-\infty,\bar{\gamma}^{2}_{i}))$.

Due to the evenness of $J_{\lambda}$, bearing in mind (2.2), and thanks to the symmetry assumptions on the potential $W$, we have that the functional $J_{\lambda}$ is $H_{d,\eta_{i}}$-invariant on $H^{1}(\mathbb{R}^{d})$, i.e.

[TABLE]

for every $h\in H_{d,\eta_{i}}$ and $u\in H^{1}(\mathbb{R}^{d})$. Indeed, the group $H_{d,\eta_{i}}$ acts isometrically on $H^{1}(\mathbb{R}^{d})$ and, thanks to the symmetry assumption on $W$, it follows that

[TABLE]

if $h\in H_{d,i}$, and

[TABLE]

if $h=\eta_{{}_{H_{d,i}}}g\in H_{d,\eta_{i}}\setminus H_{d,i}$.

On account of Theorem 4, the critical point pairs $\{\pm z_{\lambda,i}\}$ of $\mathcal{H}_{\lambda,i}$ are also (generalized) critical points of $J_{\lambda}$.

Let $z_{\lambda,i}\in Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$ be a critical point of $\mathcal{H}_{\lambda,i}$ in $Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$ such that

[TABLE]

and

[TABLE]

In order to prove that $z_{\lambda,i}\not\equiv 0$ in $Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$ , we claim that there exists a sequence \big{\{}w_{j}^{i}\big{\}}_{j\in\mathbb{N}} in $Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$ such that

[TABLE]

The sequence \big{\{}w_{j}^{i}\big{\}}_{j\in\mathbb{N}}\subset Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{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 $w_{j}^{i}:=s_{j}v_{\sigma}^{i}$ for any $j\in\mathbb{N}$. Clearly, $w_{j}^{i}\in Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$ for any $j\in\mathbb{N}$. Moreover, taking into account the properties of $v_{\sigma}^{i}$ displayed in $(j_{1})$–$(j_{3})$, simple computations show that

[TABLE]

for $j$ sufficiently large.

Arguing as in the proof of Theorem 1, inequality (3) yields (3.25) and consequently, we conclude that

[TABLE]

so that $z_{\lambda,i}\not\equiv 0$ in $Fix_{H_{d,\eta_{i}}}(H^{1}(\mathbb{R}^{d}))$. In addition, $|z_{\lambda,i}(x)|\rightarrow 0$ as $|x|\rightarrow\infty$.

On the other hand, since $\lambda<\lambda^{\star}$ and $F$ is even, Theorem 1 and the principle of symmetric criticality (recalled in Theorem 4) ensure that problem $(S_{\lambda})$ admits at least one non-trivial pair of radial weak solutions $\{\pm u_{\lambda}\}\subset H^{1}(\mathbb{R}^{d})$. Moreover, $|u_{\lambda}(x)|\rightarrow 0$ as $|x|\rightarrow\infty$.

In conclusion, since $\lambda<\lambda_{\star}$, there exist $\tau_{d}+1$ positive numbers $\bar{\gamma}$, $\bar{\gamma}_{1}$,…,$\bar{\gamma}_{\tau_{d}}$ such that

[TABLE]

and

[TABLE]

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

[TABLE]

for every $i\in J_{d}$ and

[TABLE]

for every $i,j\in J_{d}$ and $i\neq j$. Consequently problem $(S_{\lambda})$ admits at least

[TABLE]

pairs of non-trivial weak solutions $\{\pm u_{\lambda,i}\}_{i\in J^{\prime}_{d}}\subset H^{1}(\mathbb{R}^{d})$, where $J^{\prime}_{d}:=\{1,...,\zeta^{(d)}_{S}\}$, such that $|u_{\lambda,i}(x)|\rightarrow 0$, as $|x|\rightarrow\infty$, for every $i\in J^{\prime}_{d}$. Moreover, by construction, it follows that

[TABLE]

pairs of the attained solutions are sign-changing.

The proof is now complete. $\Box$

4. Some applications

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

[TABLE]

for some $q\in\left(2,2^{*}\right)$. Furthermore, let $F$ be the potential defined by

[TABLE]

for every $s\in\mathbb{R}$. 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 $f\in L^{\infty}_{\rm loc}(\mathbb{R}^{d})$. Thus, invoking [27, Proposition 1.7] it follows that

[TABLE]

where

[TABLE]

and

[TABLE]

for every $s\in\mathbb{R}$.

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 $(S_{\lambda})$ assumes the simple and significative form:

• $({S}^{\prime}_{\lambda})$

Find $u\in H^{1}(\mathbb{R}^{d})$ such that

[TABLE]

See [18] for related topics.

Of course, the solutions of $({S}^{\prime}_{\lambda})$ are exactly the weak solutions of the following Schrödinger equation

[TABLE]

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. This research was realized under 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.