2. Variational solutions

For the convenience of the reader, we first briefly recall some notions and results in the frame of critical point theory for convex, lower semicontinuous perturbations of C ¹-functionals due to Szulkin [Reference Szulkin13].

Let $(Y, \| \cdot \|)$ be a real Banach space and $I:Y \to (-\infty, +\infty]$ be a functional of type

(2.1)\begin{eqnarray} I=\psi+\mathfrak{F}, \end{eqnarray}

where $\mathfrak{F} \in C^1(Y,\mathbb{R})$ and $\psi:Y \to (-\infty, +\infty]$ is proper, convex and lower semicontinuous. An element $u\in D(\psi)$ is said to be a critical point of I if it satisfies

\begin{equation*}\langle \mathfrak{F}'(u), v-u \rangle+\psi(v)-\psi(u) \geq 0, \quad \mbox{for all } \ v\in D(\psi).\end{equation*}

A sequence $\{u_n\}\subset D(\psi)$ is called a (PS)-sequence if $I(u_n)\to r\in\mathbb{R}$ and

\begin{equation*}\langle \mathfrak{F}'(u_n), v-u_n \rangle+\psi(v)-\psi(u_n) \geq -\varepsilon_n \|v-u_n\|, \quad \mbox{for all } \ v\in D(\psi),\end{equation*}

where $\varepsilon_n \to 0$. The functional I is said to satisfy the (PS) condition if any (PS)-sequence possesses a convergent subsequence in Y.

Let Σ be the collection of all symmetric and closed subsets of $Y\setminus\{0\}$. The Krasnoselskii genus of a nonempty set $A\in \Sigma$ – denoted $\gamma(A)$, is defined as being the smallest integer k with the property that there exists an odd continuous mapping $h:A\to\mathbb{R}^k\setminus\{0\}$. If such an integer does not exist, then $\gamma(A):=+\infty$. If $A\in \Sigma$ is homeomorphic to $S^{k-1}$ (k − 1 dimension unit sphere in the Euclidean space $\mathbb{R}^k$) by an odd homeomorphism, then $\gamma(A)=k$ (see e.g. [Reference Rabinowitz11, Corrolary 5.5]). The following theorem is an immediate consequence of [Reference Szulkin13, Theorem 4.3].

Theorem 2.1. Let $I$ be of type (2.1) with $\mathfrak{F}$ and ψ even. Also, suppose that $I$ is bounded from below, satisfies the (PS) condition and $I(0)=0$. If there exists a nonempty compact symmetric subset $A\subset Y\setminus \{0 \}$ with $\gamma(A)\geq k$, such that

\begin{equation*}\sup_{v\in A} I(v) \lt 0,\end{equation*}

then the functional $I$ has at least $k$ distinct pairs of nontrivial critical points.

Let $\mathrm{G}$ be a subgroup of Y and denote by $\pi:Y\to Y/\mathrm{G}$ the canonical projection. A function $\ell$ defined on Y is called $\mathrm{G}$-invariant if

\begin{equation*}\ell(u+g)=\ell(u),\quad \mbox{for all } u\in Y \mbox{and } g\in \mathrm{G}.\end{equation*}

If $\ell :Y \to \mathbb{R}$ is a differentiable and $\mathrm{G}$-invariant functional, then $\ell'$ is also $\mathrm{G}$-invariant. This implies that, if $\mathfrak{F}$ and ψ are $\mathrm{G}$-invariant and u is a critical point of I in (2.1), then $\pi^{-1}(\pi(u))$ is a set of critical points of I. In this case, the set $\pi^{-1}(\pi(u))$ is called a critical orbit of I.

The following result is proved in [Reference Bereanu and Jebelean1, Theorem 7.1].

Theorem 2.2. Let $\mathrm{G}$ be a discrete subgroup of $Y$ with $\mbox{span}\, \mathrm{G} = \mathbb{R}^d$ and $\widetilde{Y}$ be a closed subspace of $Y$ with $Y=\mathbb{R}^d\oplus \widetilde{Y}$. So, any $v\in Y$ can be uniquely decomposed as

\begin{equation*} v=\overline{v}+\widetilde{v}, \quad \mbox{with }\overline{v} \in \mathbb{R}^d\, \mbox{and }\,\widetilde{v} \in \widetilde{Y}. \end{equation*}

In addition, assume that:

• (A ₁) $\mathfrak{F}$ and ψ are $\mathrm{G}$-invariant, $\mathfrak{F}'$ takes bounded sets into bounded sets and $D(\psi)$ is a closed nonempty set;

• (A ₂) $\psi(0)=0$ and $\psi(v)=\psi(\tilde{v})$, for all $v\in Y$;

• (A ₃) there exist ρ > 0 such that

  \begin{equation*}\|\tilde{v}\|\leq\rho,\ \ |\psi(v)|\leq\rho,\quad \mbox{for all } v\in D(\psi);\end{equation*}

• (A ₄) any sequence $\{v_n\}\subset D(\psi)$ with $\{\overline{v}_n\}$ bounded, has a convergent subsequence.

Then functional $I$ in (2.1) is bounded from below and has at least d + 1 critical orbits.

Next, by $\| \cdot \| _{\infty}$, we denote the sup-norm on $C:=C([0,T]; \mathbb{R}^N)$ and the usual norm on $L^p:=L^p([0,T]; \mathbb{R}^N)$ will be denoted by $\| \cdot \|_{L^p}$ ( $1\leq p \leq \infty$). The Sobolev space $W^{1, \infty}:=W^{1, \infty}([0,T]; \mathbb{R}^N)$ will be considered as being endowed with the norm $\|u\|_{W^{1, \infty}}=\|u'\|_{L^\infty}+\|u\|_{L^\infty}.$ The following variational formulation of problem (1.1)-(1.2) is introduced in [Reference Jebelean3]. Let

\begin{equation*} \mathcal{K}:= \left \{v \in W^{1, \infty}\, : \, \|v^{\prime}\|_{_{L^{\infty}}}\leq 1 \right \} \end{equation*}

and $\Psi:C \to (-\infty , +\infty]$ be defined by

(2.2)\begin{equation} \Psi(v):=\displaystyle \left\{ \begin{array}{ll} \displaystyle \int_0^T \left [1-\sqrt{1-|v'|^2}\right ] , & \hbox{if}\ v\in {\mathcal K}; \\ \\ +\infty, & \hbox{otherwise}. \end{array} \right. \end{equation}

The convex set $\mathcal{K}$ is closed and Ψ is proper, convex and lower semicontinuous. Also, since j is proper, convex and lower semicontinuous, the same holds true for

(2.3)\begin{equation} J:C \to (-\infty , +\infty ], \qquad J(v):=j(v(0),v(T)), \quad v\in C. \end{equation}

On account of hypothesis $(H_{F})$, the mapping

\begin{equation*}{\mathcal F}:C \to \mathbb{R}, \qquad {\mathcal F}(v):=-\int_0^TF(t,v)dt, \quad v\in C\end{equation*}

is of class C ¹ on C and its derivative is given by

\begin{equation*}\langle {\mathcal F}^{\prime} (u),v \rangle= -\int_0^T \langle \nabla F(t,u) |v \rangle dt, \quad u,v\in C.\end{equation*}

Then, the energy functional ${\mathcal I}:C \to (-\infty , + \infty ]$ associated to problem (1.1)-(1.2) will be

(2.4)\begin{equation} {\mathcal I}:=\Psi + J + {\mathcal F} \end{equation}

and notice that it fits the structure required by Szulkin’s critical point theory with Y = C, $\psi=\Psi + J$ and $\mathfrak{F}=\mathcal F$. Thus, accordingly, an element $u\in D(\Psi + J)$ is called critical point of ${\mathcal I}$ in (2.4) if it satisfies

(2.5)\begin{equation} (\Psi + J)(v)-(\Psi + J)(u) + \langle {\mathcal F}^{\prime} (u),v-u \rangle \geq 0, \quad \mbox{for all }v \in D(\Psi + J). \end{equation}

From [Reference Jebelean3, Theorem 3.2] one has that any critical point of ${\mathcal I}$ is a solution of problem (1.1)-(1.2).

Further, we introduce the first eigenvalue-like constant

\begin{equation*} \lambda_1=\lambda_1(j):= \inf \left \{\frac{\|v^{\prime} \|_{L^2}^2}{\|v \|_{L^2}^2} \, : \, v\in D(\Psi+J), \; v \not\equiv 0_{\mathbb{R}^N} \right \} \end{equation*}

and for $v\in C$, we denote

\begin{equation*}\overline{v}:=\frac{1}{T}\int_0^Tv(t)dt, \quad \tilde{v}:= v-\overline{v}.\end{equation*}

If $v\in \mathcal{K}$, then each component $\tilde{v}_i$ of $\tilde{v}$ vanishes at some $\xi_i \in [0,T]$. Thus, one has that

\begin{equation*}|\tilde{v}_i(t)|=|\tilde{v}_i(t)-\tilde{v}_i(\xi)|\leq \int_0^T|\tilde{v}'_i(\tau)|d \tau \leq T, \quad i=\overline{1,N}, \, t\in[0,T]\end{equation*}

and hence

(2.6)\begin{equation} \|\tilde{v}\|_{\infty} \leq T\sqrt{N}, \end{equation}

(2.7)\begin{equation} |\overline{v}|- T\sqrt{N} \leq |v(t)|\leq |\overline{v}|+ T\sqrt{N}, \quad t\in [0,T]. \end{equation}

Proof. If $\lambda_1 \gt 0$, the result follows from Theorem 4.1 in [Reference Jebelean3]. In case (ii) we apply [Reference Jebelean3, Theorem 4.4]. Accordingly, as $\mathcal{I}(0_{\mathbb{R}^N})=0$, it suffices to prove that for some δ > 0 one has

\begin{equation*} \mathcal{I}(u) \gt 0, \quad \mbox{for all }u\in D(\Psi + J) \mbox{ with }|\overline{u}| \gt \delta. \end{equation*}

From (2.8) there are constants $\sigma, \rho \gt 0$ such that

\begin{equation*} F(t,x)\leq-\sigma,\quad \mbox{for all}\ t\in[0,T] \mbox{ and }\ x\in\mathbb{R}^N \mbox{ with } |x| \gt \rho. \end{equation*}

Take $\delta:=\rho+T \sqrt{N}$ and let $u\in \mathcal{K}$ be with $|\overline{u}| \gt \delta.$ From (2.7) it follows $|u(t)| \gt \rho$ for all $t\in [0,T]$ and we get

\begin{equation*} \mathcal{F}(u)=-\int_0^TF(t,u(t))dt\geq\sigma T. \end{equation*}

Then, since $\Psi\geq0$ and $J\geq0$, we have that $\mathcal{I}(u)\geq\sigma T \gt 0$ for all $u\in \mathcal{K}\supset D(\Psi + J)$ with $|\overline{u}| \gt \delta.$

3. Multiple solutions for even potentials

To establish multiplicity of solutions for problem (1.1)-(1.2), we first consider the parameterized potential system

(3.1)\begin{equation} -\left[ \phi(u^{\prime}) \right] ^{\prime} = \lambda\nabla_u G(t,u),\quad \mbox{in } [0,T], \end{equation}

associated with the boundary condition (1.2). Here, λ is a positive real parameter and G is assumed to satisfy

• (H_G) $G:[0,T] \times \mathbb{R}^N \to \mathbb{R}$ is continuous, $\nabla_u G$ exists and is continuous on $[0,T] \times \mathbb{R}^N$ and $G(\, \cdot \, , 0_{\mathbb{R}^N})=0$.

The energy functional ${\mathcal I}_\lambda:C \to (-\infty , + \infty ]$ associated to (3.1)-(1.2) will be now

\begin{equation*} {\mathcal I}_\lambda:=\Psi + J + {\mathcal G}_\lambda, \end{equation*}

where ${\mathcal G}_\lambda:C \to \mathbb{R}$ is given by

\begin{equation*} {\mathcal G}_\lambda(v):=-\lambda\int_0^TG(t,v)dt, \quad v\in C,\end{equation*}

and Ψ, J are introduced in (2.2), respectively, (2.3). We set

\begin{equation*}d_N^1:=\left \{(x,x) \, : \, x\in \mathbb{R}^N\right \}.\end{equation*}

Theorem 3.2. Assume that $G$ and $j$ are even and that one of the conditions (i) or (ii) in Proposition 3.1 are fulfilled. If, in addition, $G$ satisfies

(3.3)\begin{equation} \displaystyle\liminf_{|x|\to 0} \frac{2G(t,x)}{|x|^2}\geq 1,\quad \mbox{uniformly with } t\in[0,T] \end{equation}

and $\lambda \gt 2(m\pi/T)^2$ for some $m\in\mathbb{N}$, $m\geq 1$, then problem (3.1)-(1.2) has at least $mN$ distinct pairs of nontrivial solutions.

Proof. From Theorem 2.1 with Y = C, $I = \mathcal{I}_{\lambda}$ and Proposition 3.1, it suffices to show that there exists a nonempty compact symmetric set $A\subset C\setminus \{0_{\mathbb{R}^N}\}$ with $\gamma(A)\geq mN$, such that

(3.4)\begin{equation} \sup_{v\in A} I_{\lambda}(v) \lt 0. \end{equation}

With this aim, we introduce the functions $e_k(t):=\displaystyle\sin\frac{\pi k t}{T},$ ( $k=\overline{1,m}, \, t\in [0,T]$), and consider in C the subspace

\begin{equation*}X_m:=\left\{\alpha^1e_1+\alpha^2e_2\cdots+\alpha^m e_m:\ \alpha^1,\cdots,\alpha^m\in\mathbb{R}^N\right\}\end{equation*}

equipped with the norm

\begin{equation*}\left\|\alpha^1e_1+\alpha^2e_2\cdots+\alpha^m e_m\right\|_{X_m}=\left(|\alpha^1|^2+\cdots+|\alpha^m|^2\right)^{\frac{1}{2}}.\end{equation*}

Observe that $X_m=\mathbb{R}^N e_1 \oplus \dots \oplus \mathbb{R}^N e_m$ and hence, $\mbox{dim }X_m=mN$. Since $\lambda \gt 2(m\pi/T)^2$, we can find $\varepsilon\in(0,1)$ so that $\lambda \gt 2(m\pi/T)^2/(1-\varepsilon)$ and by virtue of (3.3), there exists δ > 0 such that

(3.5)\begin{equation} 2G(t,x)\geq (1-\varepsilon ) |x|^2,\quad \mbox{for all }t\in [0,T], \mbox{as } |x|\leq \delta. \end{equation}

Next, define A_m by

\begin{equation*} A_m=A_m(\rho):=\left\{v\in X_m:\ \|v\|_{X_m}=\rho\right\}, \end{equation*}

where ρ is a positive number, which is chosen $\displaystyle\leq\min\left\{\frac{T}{m^{3/2}\pi},\frac{\delta}{\sqrt{m}}\right\}$. Then, the odd mapping $H:A_m\to S^{mN-1}$ given by

\begin{equation*} H\left(\sum_{k=1}^{m}\alpha^k e_k\right)=\left(\frac{\alpha^1_1}{\rho},\ldots,\frac{\alpha^1_N}{\rho}, \cdots ,\frac{\alpha^m_1}{\rho}, \cdots , \frac{\alpha^m_N}{\rho} \right) \end{equation*}

is a homeomorphism between A_m and $S^{mN-1}$ and so, $\gamma(A_m)=\dim X_m=mN$.

Now, let $v=\displaystyle\sum_{k=1}^{m}\alpha^ke_k \in A_m$. It is clear that $v(0)=v(T)=0_{\mathbb{R}^N}$ and hence $J(v)=j(0_{\mathbb{R}^N \times \mathbb{R}^N})=0$. Also, by the choice of ρ, one gets

\begin{equation*} |v'(t)|=\left|\sum_{k=1}^{m}\alpha^k\frac{k\pi}{T}\cos\frac{k\pi t}{T}\right| \leq\frac{m\pi}{T} \sum_{k=1}^{m} |\alpha^k| \leq\frac{m\pi}{T}\left(\sum_{k=1}^{m}|\alpha^k|^2\right)^{\frac{1}{2}}m^{1/2}=\rho\frac{m^{3/2}\pi}{T}\leq1, \end{equation*}

meaning that $v\in \mathcal{K}$. Also, we obtain

(3.6)\begin{align} \int_0^T|v'(t)|^2dt&= \int_0^T\left|\sum_{k=1}^{m}\alpha^k\frac{k\pi}{T}\cos\frac{k\pi t}{T}\right|^2dt=\int_0^T\sum_{i=1}^N\left(\sum_{k=1}^{m}\alpha^k_i\frac{k\pi}{T}\cos\frac{k\pi t}{T}\right)^2dt\nonumber\\ &=\int_0^T\sum_{i=1}^N\left(\sum_{k=1}^{m}\left(\alpha^k_i \frac{k\pi}{T}\cos\frac{k\pi t}{T}\right)^2+\sum_{\substack{k,j=1\\k \neq j}}^{m}\alpha^k_i\alpha^j_i \frac{kj\pi^2}{T^2}\cos\frac{k\pi t}{T}\cos\frac{j\pi t}{T}\right)dt\nonumber\\ &=\sum_{k=1}^{m}\sum_{i=1}^N\left(\alpha^k_i\right)^2\int_0^T \left(\frac{k\pi}{T}\cos\frac{k\pi t}{T}\right)^2dt\nonumber\\ &\quad +\sum_{i=1}^N\sum_{\substack{k,j=1\\k \neq j}}^{m}\alpha^k_i\alpha^j_i \int_0^T\frac{kj\pi^2}{T^2}\cos\frac{k\pi t}{T}\cos\frac{j\pi t}{T}dt\nonumber\\ &=\sum_{k=1}^{m}\sum_{i=1}^N\left(\alpha^k_i\right)^2\left(\frac{k\pi}{T}\right)^2\frac{T}{2}\leq\frac{m^2\pi^2}{2T}\sum_{k=1}^{m}|\alpha^k|^2= \frac{m^2\pi^2}{2T}\rho^2 \end{align}

and

(3.7)\begin{align} \int_0^T|v(t)|^2dt&=\int_0^T\left|\sum_{k=1}^{m}\alpha^k\sin\frac{k\pi t}{T}\right|^2dt=\int_0^T\sum_{i=1}^N\left(\sum_{k=1}^{m}\alpha^k_i\sin\frac{k\pi t}{T}\right)^2dt\nonumber\\ &=\int_0^T\sum_{i=1}^N\left(\sum_{k=1}^{m}\left(\alpha^k_i\sin\frac{k\pi t}{T}\right)^2+\sum_{\substack{k,j=1\\k\neq j}}^{m}\alpha^k_i\alpha^j_i \sin\frac{k\pi t}{T}\sin\frac{j\pi t}{T}\right)dt \nonumber\\ &=\frac{T}{2}\sum_{k=1}^{m}|\alpha^k|^2=\frac{T\rho^2}{2}. \end{align}

Then, from

\begin{eqnarray*} |v|\leq \sum_{k=1}^m|\alpha^k|\leq\left(\sum_{k=1}^{m}|\alpha^k|^2\right)^{\frac{1}{2}}\sqrt{m}=\rho\sqrt{m}\leq\delta, \end{eqnarray*}

together with (3.5), (3.6) and (3.7), we estimate $\mathcal{I}_{\lambda}$ on A_m as follows

\begin{eqnarray*} \mathcal{I}_{\lambda}(v)&=&\Psi(v)+J(v)+\mathcal{G}_{\lambda}(v)=\Psi(v)+\mathcal{G}_{\lambda}(v)\leq\int_0^T|v'|^2- \frac{\lambda}{2}(1-\varepsilon)\int_0^T |v|^2\\ &\leq& \frac{m^2\pi^2}{2T}\rho^2-\lambda(1-\varepsilon)\frac{T\rho^2}{4}=T \rho^2\frac{2(m \pi /T)^2-\lambda(1-\varepsilon)}{4} \lt 0, \end{eqnarray*}

which shows that (3.4) holds true with $A=A_m$.

Theorem 3.2 can be employed to derive multiplicity of solutions for the non-parametric problem (1.1)-(1.2).

Proof. (a) By (3.8) there exists $\lambda^* \gt 0$ such that

\begin{equation*}\liminf_{|x|\to 0}\frac{2F(t,x)}{|x|^2}\geq \lambda^* \gt 2\left(\frac{m\pi}{T}\right)^2\end{equation*}

and from Theorem 3.2 with $G=F/ \lambda^*$, we have that system

\begin{equation*}-\left[\phi(u')\right]'= \lambda^* \ \nabla_u \left(\frac{F(t,u)}{\lambda^*}\right),\quad \mbox{in } [0,T],\end{equation*}

associated with boundary condition (1.2), has at least mN distinct pairs of nontrivial solutions.

(b) This is immediate from (a) and (3.9).

Example 3.4. Let $a,b \in \mathbb{R}$ be with $|a|+|b| \gt 0$, q > 0, $\alpha \in (-1,0]$ and $w:[0,T] \to \mathbb{R}$ be continuous. We consider a system of type

(3.10)\begin{equation} -\left[ \phi(u^{\prime}) \right] ^{\prime} = u \left ( |u|^{\alpha}-w(t) |u|^q \right ), \quad \mbox{in } [0,T], \end{equation}

in the presence of the boundary condition

(3.11)\begin{equation} a u(0)= b u(T), \qquad b \phi(u')(0)=a \phi (u')(T). \end{equation}

Notice that (3.10) is of type (1.1), with

\begin{equation*}F(t,x)=\frac{|x|^{\alpha +2}}{\alpha +2}-w(t)\frac{|x|^{q +2}}{q+2}, \qquad (t,x)\in [0,T] \times \mathbb{R}^N,\end{equation*}

while (3.11) is obtained from (1.2) by taking $j=I_Y$ - the indicator function of the subspace

\begin{equation*}Y=\{(x,y) \in \mathbb{R}^N \times \mathbb{R}^N \, : \, ax=by \}.\end{equation*}

Clearly, if a = b on has that $Y=d_N^1$, $\lambda_1=0$ (because any constant function belongs to $D(\Psi+J)$) and (3.10) becomes the periodic boundary condition

(3.12)\begin{equation} u(0)- u(T)= 0_{\mathbb{R}^N}=u'(0)- u'(T). \end{equation}

In the opposite case, when a ≠ b, one has $\lambda _1 \gt 0$. Indeed, by [Reference Jebelean and Precup5, Theorem 3.1] we have that

\begin{equation*} \underline{\lambda}_1:=\inf\left \{\frac{\|v^{\prime} \|_{L^2}^2}{\|v \|_{L^2}^2} \, : \, v\in W^{1,2}([0,T]; \mathbb{R}^N), \, (v(0),v(T))\in Y , \; v \not\equiv 0_{\mathbb{R}^N} \right \} \gt 0\end{equation*}

and the conclusion follows from $\lambda_1 \geq \underline{\lambda}_1$.

We first refer to the situation α = 0, when system (3.10) involves a Fisher-Kolmogorov type nonlinearity:

(3.13)\begin{equation} -\left[ \phi(u^{\prime}) \right] ^{\prime} = u \left ( 1-w(t) |u|^q \right ), \quad \mbox{in } [0,T], \end{equation}

From Theorem 3.3 (a) we obtain the following. Assume that $m\in \mathbb{N}$ and T > 0 are such that ${m \pi \sqrt{2}}/{T} \lt 1.$ If a ≠ b then problem (3.13)-(3.11) has at least $\mathit{mN}$ distinct pairs of nontrivial solutions. The same conclusion holds true for the periodic problem (3.13)-(3.12) provided that in addition w > 0.

In the remaining case $\alpha \in (-1,0)$, from Theorem 3.3 (b) we get that if a ≠ b then (3.10)-(3.11) has infinitely many distinct pairs of solutions and the same holds true for the periodic problem (3.10)-(3.12) provided that in addition w > 0.

4. Geometrically distinct solutions

In this section, we provide a result concerning the existence of geometrically distinct solutions for systems of type

(4.1)\begin{equation} -\left[ \phi(u^{\prime}) \right] ^{\prime} = \nabla_u F(t,u) + h(t),\quad \mbox{in } [0,T], \end{equation}

associated with boundary condition (1.2). Here, besides (H_F) and (H_j), we assume the following hypotheses:

• ( $H_{F}^1$) $F(t, u)$ is ω_i-periodic $(\omega_i \gt 0)$ with respect to each u_i $(i =\overline{1,N})$, for all $t\in[0,T]$;

• ( $H_j^1$) $j|_{D(j)}$ is bounded and if $z\in D(j)$ then $j(z+\zeta)=j(z)$ for all $\zeta \in d_N^1$;

• (H_h) $h:[0,T]\to\mathbb{R}^N$ is continuous with $\overline{h}=0$.

From ( $H_{F}^1$) we have that $\nabla_uF(t, u)$ is ω_i-periodic with respect to each u_i ( $i =\overline{1,N},\, t\in[0,T]$) and assumption $j(z+\zeta)=j(z)$ in ( $H_j^1$) ensures that $\partial j(z+\zeta)= \partial j(z)$ for all $z \in D(j),$ $\zeta \in d_N^1$. Let $(\mathbf{e}^1, \dots, \mathbf{e}^N)$ be the canonical basis in $\mathbb{R}^N$ and consider the discrete group

\begin{equation*}\mathbf{G}:=\left\{\sum_{i=1}^{N}k_i \, \omega_i \, \mathbf{e}^i:\ k_i\in\mathbb{Z},\ 1\leq i\leq N\right\}.\end{equation*}

Observe that if u is a solution of (4.1)-(1.2) then the same is true for $u + \mathbf{g}$, when $\mathbf{g} \in \mathbf{G}.$ Two solutions u and v of (4.1)-(1.2) are said to be geometrically distinct if $u-v \notin \mathbf{G}.$

The corresponding action functional to problem (4.1)-(1.2) will be ${\mathcal I}_h:C \to (-\infty , + \infty ]$ given by

\begin{equation*} {\mathcal I}_h:=\Psi + J + {\mathcal F}_h, \end{equation*}

where Ψ, J are introduced in (2.2), respectively, (2.3) and ${\mathcal F}_h:C \to \mathbb{R}$ is defined by

\begin{equation*} {\mathcal F}_h(v):=-\int_0^T\left[F(t,v)+\langle h(t)|v\rangle\right] dt, \quad v\in C. \end{equation*}

Proof. We apply Theorem 2.2 with Y = C, $\psi=\Psi+J$, $\mathfrak{F}={\mathcal F}_h$, $\mathrm{G}=\mathbf{G}$ and $\widetilde{Y}=\widetilde{C}:=\{u\in C:\ \overline{u}=0_{\mathbb{R}^N}\}$. Notice that $\mbox{span}\, \mathbf{G} = \mathbb{R}^N$, $C=\mathbb{R}^N\oplus \widetilde{C}$ and thus the conclusion will follow provided that we show that assumptions $(A_1)$- $(A_4)$ hold true with these choices.

Is easy to see that assumptions $(H_F)$, $(H_F^1)$ and $(H_h)$ imply that ${\mathcal F}_h'$ takes bounded sets into bounded sets and

\begin{equation*}{\mathcal F}_h(u+\mathbf{g})={\mathcal F}_h(u),\quad \mbox{for all } u\in C,\ \mathbf{g}\in\mathbf{G}.\end{equation*}

Also, by $(H_j^1)$ one has

(4.2)\begin{equation} J(u+\mathbf{c})=J(u),\quad \mbox{for all } u\in C,\ \mathbf{c}\in \mathbb{R}^N \supset\mathbf{G} \end{equation}

and we get that $\Psi+J$ is G-invariant. Since $\mathcal{K}$ and D(j) are closed, the same is true for $D(\Psi+J)=\mathcal{K} \cap D(J)$. Assumption $(A_1)$ is thus satisfied. Checking $(A_2)$ is straightforward from (4.2). Next, for $u\in D(\Psi+J) \subset {\mathcal K}$ we have $\|\widetilde{u}\|_\infty\leq T\sqrt{N}$ (see (2.6)) and, since $j|_{D(j)}$ is bounded, there exists $\rho_0 \gt 0$ with $|(\Psi+J)(u)|\leq\rho_0$ for all $u\in D(\Psi+J)$, which yields that $(A_3)$ holds with $\rho=\max\{\rho_0,T\sqrt{N}\}$. Finally, if $\{u_n\}\subset D(\Psi+J)$ is a sequence with $\{\overline{u}_n\}$ bounded, as $\|\widetilde{u}_n\|_\infty\leq T\sqrt{N}$, we get that $\{u_n\}$ is bounded in $W^{1,\infty}$. Then, $(A_4)$ follows by the compactness of the embedding $W^{1, \infty }\hookrightarrow C$ and the proof is complete.

Corollary 4.3. Assume $(H_{F})$, $(H_{F}^1)$, $(H_h)$ and let $f:\mathbb{R}^N \to \mathbb{R}$ be of class $C$¹, convex, with $f(0_{\mathbb{R}^N})=0$ and $\nabla f (0_{\mathbb{R}}^N)=0_{\mathbb{R}^N}.$ Then system (4.1) has at least $N$ + 1 geometrically distinct solutions satisfying the boundary condition

(4.3)\begin{equation} \phi \left( u^{\prime }\right)(0)=\nabla f ( u(0)- u(T))=\phi \left( u^{\prime }\right)(T). \end{equation}

Example 4.4. For any $a\in C$ and $h\in C$ with $\overline{h}=0$, system

\begin{equation*} -\left[ \frac{u_i'}{\sqrt{1-|u'|^2}} \right] ^{\prime} = a_i(t) \sin u_i +h_i(t),\, \, \mbox{in } [0,T], \quad i=\overline{1,N} \end{equation*}

has at least N + 1 geometrically distinct solutions satisfying the boundary condition

\begin{equation*} \phi \left( u^{\prime }\right)(0)=\displaystyle e^{|u(0)-u(T)|^2} ( u(0)- u(T))=\phi \left( u^{\prime }\right)(T). \end{equation*}

To see this, Corollary 4.3 applies with $f(x)=(e^{|x|^2}-1)/2 $ and $F(t,x) = -\sum _{i=1}^{N}a_i(t) \cos x_i$, $t\in [0,T], \, x\in \mathbb{R}^N$; here, $\omega_i=2 \pi$, $i=\overline{1,N}$.