Infinitely many homoclinic solutions for sublinear and nonperiodic Schrödinger lattice systems

Abstract

By using variational methods we obtain infinitely many nontrivial solutions for a class of nonperiodic Schrödinger lattice systems, where the nonlinearities are sublinear at both zero and infinity.

1 Introduction and main results

Schrödinger lattice systems are a class of very important discrete models, ranging from biology and condensed matter physics to solid state physics [8, 10, 11]. In fact, most results are about the periodic Schrödinger lattice systems, such as [2, 4, 12–14, 18, 19, 24]. However, there are only few results about the nonperiodic Schrödinger lattice systems [5, 9, 15, 16, 22, 23]. In particular, in [3, 6, 7] the authors recently obtained the existence and multiplicity of homoclinic solutions for a class of Schrödinger lattice systems with perturbed terms.

In this paper, we investigate the nonperiodic Schrödinger lattice system

$$ \textstyle\begin{cases} -(\Delta u)_{n}+v_{n} u_{n}= \mu \chi _{n} \vert u_{n} \vert ^{\mu -2}u_{n}, \quad n\in \mathbb {Z}, \\ \lim_{ \vert n \vert \rightarrow \infty }u_{n}=0, \end{cases} $$

(1.1)

where $\mu \in (1,2)$,

$$ (\Delta u)_{n}:=u_{n+1}+u_{n-1}-2u_{n}, $$

(1.2)

$\{u_{n}\}$, $\{v_{n}\}$, and $\{\chi _{n}\}$ are real-valued sequences, and the discrete potential $V=\{v_{n}\}_{n\in \mathbb {Z}}$ and $\{\chi _{n}\}$ are nonperiodic. A solution $u=\{u_{n}\}_{n\in \mathbb {Z}}$ is said to be nontrivial if $u_{n}\not \equiv 0$. Problem (1.1) appears when we look for standing wave (or breather) solutions of the Schrödinger lattice system

$$ i\dot{\psi }_{n}=-(\Delta \psi )_{n}+\widetilde{v}_{n} \psi _{n}-\mu \chi _{n} \vert \psi _{n} \vert ^{\mu -2}\psi _{n},\quad n\in \mathbb {Z}, $$

(1.3)

where $\{\psi _{n}\}$ is a real-valued sequence. Standing waves (or breathers) are the solutions for (1.3) of the form $\psi _{n}=u_{n}e^{-i\omega t}$, where $\omega \in \mathbb {R}$ is the temporal frequency, and $u_{n}$ satisfies $\lim_{|n|\rightarrow \infty }u_{n}=0$. By the standing wave ansatz $\psi _{n}=u_{n}e^{-i\omega t}$ we get that (1.3) reduces to (1.1) with $v_{n}\equiv \widetilde{v}_{n}-\omega$. Therefore we only need to study the existence of solutions of (1.1).

Let

$$ \Vert u \Vert _{l^{q}}:= \Biggl(\sum^{+\infty }_{n=-\infty } \vert u_{n} \vert ^{q} \Biggr)^{1/q}, \qquad \Vert u \Vert _{l^{\infty }}:=\sup_{n\in \mathbb {Z}} \vert u_{n} \vert , \qquad u=\{u_{n} \}_{n\in \mathbb {Z}}, $$

be the norms of the real sequence spaces $l^{q}:=l^{q}(\mathbb {Z})$ ($q \in [1,\infty )$). The following embedding between such spaces is well known:

$$ l^{q}\subset l^{p}, \qquad \Vert u \Vert _{l^{p}} \le \Vert u \Vert _{l^{q}}, \quad 1\le q \le p \le \infty . $$

We study solutions of (1.1) in $l^{2}$ since any $u=\{u_{n}\}_{n\in \mathbb {Z}}\in l^{2}$ satisfies $\lim_{|n|\rightarrow \infty }u_{n}=0$.

Note that the domain of (1.1) is $\mathbb {Z}$, and thus, to overcome the loss of compactness caused by the unboundedness of the domain $\mathbb {Z}$, we need the following condition:

${\mathbf{(V_{1})}}$:

$\lim_{|n|\rightarrow +\infty }v_{n}=+\infty$.

Then $(V_{1})$ implies that (see [21]) the spectrum $\sigma (-\triangle +V)$ is discrete and consists of simple eigenvalues accumulating to +∞, that is, we can assume that

$$ \lambda _{1} < \lambda _{2} < \cdots < \lambda _{k} < \cdots \to + \infty $$

are all eigenvalues of $-\triangle +V$, where $((-\triangle +V) u)_{n}:=-(\triangle u)_{n}+v_{n} u_{n}$ for $u=\{u_{n}\}\in l^{2}$.

Theorem 1.1

System (1.1) has infinitely many nontrivial solutions if $(V_{1})$, and the following conditions hold:

${\mathbf{(W_{1})}}$:

$0\notin \sigma (-\Delta +V)$.

$\mathbf{(SG_{1})}$:

$\chi :=\{\chi _{n}>0\}_{n\in \mathbb {Z}}\in l^{ \frac{2}{2-\mu }}$, $\mu \in (1,2)$.

Clearly, condition $(W_{1})$ implies that we have the following two cases:

$\mathbf{(W'_{1})}$:

$0\in (\lambda _{k_{0}},\lambda _{k_{0}+1})$ for some $k_{0}\ge 1$ (the indefinite case);

$\mathbf{(W''_{1})}$:

$0<\lambda _{1}$ (the positive definite case).

Remark 1.1

To the best of our knowledge, there is no result published concerning the multiplicity of nontrivial solutions for (1.1) with sublinear nonlinearities at both zero and infinity. For the nonperiodic system (1.1), the main differences between our and known results [5, 9, 15, 16, 22, 23] are as follows:

(1) The nonlinearities $g_{n}(s)$ in [5, 15, 16, 22, 23] are superlinear as $|s|\to 0$ ($\lim_{|s|\to 0}\frac{g_{n}(s)}{s}=0$, $\forall n\in \mathbb {Z}$), and the nonlinearities $g_{n}(s)$ in [9] are superlinear or asymptotically linear ($\lim_{|s|\to 0}\frac{g_{n}(s)}{s}=l_{n}\in (0,+\infty )$, $\forall n\in \mathbb {Z}$) as $|s|\to 0$. However, our nonlinearities $g_{n}(s)=\mu \chi _{n}|u_{n}|^{\mu -2}u_{n}$ are sublinear as $|s|\to 0$ ($\lim_{|s|\to 0}\frac{g_{n}(s)}{s}=+\infty$, $\forall n\in \mathbb {Z}$).

(2) The nonlinearities $g_{n}(s)$ in [5, 9, 15, 22, 23] are superlinear as $|s|\to \infty$ ($\lim_{|s|\to \infty }\frac{g_{n}(s)}{s}=+ \infty$, $\forall n\in \mathbb {Z}$), and the nonlinearities $g_{n}(s)$ in [16] are asymptotically linear as $|s|\to \infty$ ($\lim_{|s|\to \infty }\frac{g_{n}(s)}{s}=c_{n}\in (0,+ \infty )$, $\forall n\in \mathbb {Z}$). However, our nonlinearities $g_{n}(s)=\mu \chi _{n}|u_{n}|^{\mu -2}u_{n}$ are sublinear as $|s|\to \infty$ ($\lim_{|s|\to \infty }\frac{g_{n}(s)}{s}=0$, $\forall n\in \mathbb {Z}$).

(3) Our method is based on the variant fountain theorem in [25], which is different from the methods used in the papers mentioned.

In Sect. 2, we give some lemmas and the proofs of our main result. In Appendix, we give the proofs of the conditions in the critical point theorem used in this paper.

2 Proof of the main result

The corresponding action functional Φ of (1.1) is defined as follows:

$$ \Phi (u)=\frac{1}{2}(Lu,u)_{l^{2}}-\sum ^{+\infty }_{n=-\infty }\chi _{n} \vert u_{n} \vert ^{\mu }, \quad u\in E, $$

where $(\cdot ,\cdot )_{l^{2}}$ is the inner product in $l^{2}$, $L:=-\triangle +V$, $E:=\mathcal {D}(|L|^{1/2})$ is the form domain of L (the domain of $|L|^{1/2}$). Since the operator −△ is bounded in $l^{2}$, we easily see that

$$ E=\bigl\{ u\in l^{2}: \vert V \vert ^{1/2}u\in l^{2} \bigr\} $$

with the inner product and norm

$$ (u,v):=\bigl( \vert L \vert ^{1/2}u, \vert L \vert ^{1/2}v\bigr)_{l^{2}}=(-\triangle u,v)_{l^{2}}+\bigl( \vert V \vert ^{1/2}u, \vert V \vert ^{1/2}v \bigr)_{l^{2}}, \qquad \Vert u \Vert :=(u,u)^{1/2}; $$

E is a Hilbert space, where $|V|^{1/2}u$ is defined by $(|V|^{1/2}u)_{n}:= |v_{n}|^{1/2} u_{n}$ ($n\in \mathbb {Z}$). By $(W_{1})$. We have the orthogonal decomposition

$$ E=E^{-}\oplus E^{+} $$

with respect to both inner products $(\cdot ,\cdot )$ and $(\cdot ,\cdot )_{l^{2}}$, where $E^{\pm }:=E\cap (l^{2})^{\pm }$, and $(l^{2})^{\pm }$ is the positive (negative) eigenspace of L.

Then the functional Φ can be rewritten as

$$ \Phi (u)=\frac{1}{2} \bigl\Vert u^{+} \bigr\Vert ^{2}-\frac{1}{2} \bigl\Vert u^{-} \bigr\Vert ^{2} -\sum_{n=- \infty }^{+\infty }\chi _{n} \vert u_{n} \vert ^{\mu }, \quad u\in E, $$

where $u=u^{+}+u^{-} \in E=E^{+} \oplus E^{-}$. Let $I(u):=\sum_{n=-\infty }^{+\infty }\chi _{n}|u_{n}|^{\mu }$. Under our assumptions, $I,\Phi \in C^{1}(E,\mathbb {R})$ with derivatives

$$\begin{aligned}& \bigl\langle \Phi '(u),v\bigr\rangle =\bigl(u^{+},v^{+} \bigr)-\bigl(u^{-},v^{-}\bigr)-\bigl\langle I'(u),v \bigr\rangle ,\\& \bigl\langle I'(u),v\bigr\rangle =\sum_{n=-\infty }^{+\infty } \mu \chi _{n} \vert u_{n} \vert ^{\mu -2}u_{n} v_{n}, \quad u, v\in E, \end{aligned}$$

where $u=u^{+}+u^{-}$, $v=v^{+}+v^{-}\in E=E^{+} \oplus E^{-}$. The standard argument shows that nonzero critical points of Φ are nontrivial solutions of (1.1). We will use the following critical point theorem.

Lemma 2.1

([25])

Let E be a Banach space with norm $\|\cdot \|$ and suppose $E=\overline{\bigoplus_{j=1}^{\infty }X_{j}}$ with $\dim X_{j}<\infty$, $j\in \mathbb {N}$. Set $Y_{k}=\bigoplus_{j=1}^{k}X_{j}$ and $Z_{k}=\overline{\bigoplus_{j=k}^{\infty }X_{j}}$. Assume that the functional $\Phi _{\lambda }=A(u)-\lambda B(u)$ ($\Phi _{\lambda }\in C^{1}$, $\Phi _{\lambda }: E\rightarrow \mathbb {R}$, $\lambda \in [1,2]$) satisfies

$(F_{1})$:

$\Phi _{\lambda }$ maps bounded sets to bounded sets uniformly for $\lambda \in [1,2]$, and $\Phi _{\lambda }(-u)=\Phi _{\lambda }(u)$ for all $(\lambda ,u)\in [1,2]\times E$;

$(F_{2})$:

$B(u)\geq 0$, $\forall u\in E$; and $B(u)\to \infty$ as $\|u\|\to \infty$ on any finite-dimensional subspace of E.

$(F_{3})$:

There exist $\rho _{k}>r_{k}>0$ such that

$$ \alpha _{k}(\lambda ):=\inf_{u\in Z_{k},\|u\|=\rho _{k}}\Phi _{\lambda }(u)\geq 0>\beta _{k}(\lambda ):=\max _{u\in Y_{k},\|u\|=r_{k}} \Phi _{\lambda }(u),\quad \forall \lambda \in [1,2], $$

and

$$ \xi _{k}(\lambda ):=\inf_{u\in Z_{k},\|u\|\leq \rho _{k}}\Phi _{\lambda }(u)\to 0\quad \textit{as } k\to \infty \textit{ uniformly for } \lambda \in [1,2]. $$

Then there exist $\lambda _{j}\to 1$ and $u^{\lambda _{j}}\in Y_{j}$ such that

$$ \Phi '_{\lambda _{j}}|_{Y_{j}}\bigl(u^{\lambda _{j}} \bigr)=0,\qquad \Phi _{ \lambda _{j}}\bigl(u^{\lambda _{j}}\bigr)\to \eta _{k}\in \bigl[\xi _{k}(2),\beta _{k}(1)\bigr] \quad \textit{as } j\to \infty . $$

Particularly, if $\{u^{\lambda _{j}}\}$ has a convergent subsequence for every k, then $\Phi _{1}$ has infinitely many nontrivial critical points $\{u^{k}\}\subset E\setminus \{0\}$ satisfying $\Phi _{1}(u^{k})\to 0^{-}$ as $k\to \infty$.

From $(V_{1})$, $(W_{1})$, and [21] we have that the eigenvalues of L are as follows:

$$ \lambda _{1} < \lambda _{2}< \cdots < \lambda _{k_{0}}< 0< \lambda _{k_{0}+1} < \cdots \to +\infty . $$

Let $\{e_{j}\}_{j=1}^{k_{0}}$ and $\{e_{j}\}_{j=k_{0} +1}^{\infty }$ be the orthonormal bases of $E^{-}$ and $E^{+}$, respectively ($E^{-}=\{0 \}$ if $0<\lambda _{1}$). Then $\{e_{j}\}_{j\in \mathbb {N}}$ is an orthonormal basis of E. Let $X_{j}:=\operatorname{span}\{e_{j}\}$ for $j\in \mathbb {N}$. Then $Z_{k}$ and $Y_{k}$ can be defined as in Lemma 2.1. Let

$$ A(u):=\frac{1}{2} \bigl\Vert u^{+} \bigr\Vert ^{2},\qquad B(u):=\frac{1}{2} \bigl\Vert u^{-} \bigr\Vert ^{2}+ \sum_{n=-\infty }^{+\infty }\chi _{n} \vert u_{n} \vert ^{\mu }, $$

and

$$ \Phi _{\lambda }(u)=A(u)-\lambda B(u)=\frac{1}{2} \bigl\Vert u^{+} \bigr\Vert ^{2}-\lambda \Biggl(\frac{1}{2} \bigl\Vert u^{-} \bigr\Vert ^{2}+\sum _{n=-\infty }^{+\infty }\chi _{n} \vert u_{n} \vert ^{\mu } \Biggr) $$

for all $u=u^{+}+u^{-}$, $v=v^{+}+v^{-}\in E=E^{+} \oplus E^{-}$ and $\lambda \in [1,2]$. Obviously, $\Phi _{\lambda }\in C^{1}(E,\mathbb {R})$ for all $\lambda \in [1,2]$.

Proof of Theorem 1.1

Under our assumptions, the definition of $\Phi _{\lambda }$ implies that $\Phi _{\lambda }$ maps bounded sets to bounded sets uniformly for $\lambda \in [1, 2]$. Evidently, $\Phi _{\lambda }(-u)=\Phi _{\lambda }(u)$ for all $(\lambda , u)\in [1, 2]\times E$, and thus $(F_{1})$ of Lemma 2.1 holds. Besides, Ax 3.1 and Ax 3.2 in the Appendix show that $(F_{2})$ and $(F_{3})$ of Lemma 2.1 hold for all $k\ge k_{1}$. Therefore by Lemma 2.1, for each $k\ge k_{1}$, there exist $\lambda _{j} \to 1$ and $u^{\lambda _{j}}\in Y_{j}$ such that

$$ \Phi '_{\lambda _{j}}|_{Y_{j}}\bigl(u^{\lambda _{j}} \bigr)=0,\qquad \Phi _{ \lambda _{j}}\bigl(u^{\lambda _{j}}\bigr)\to \eta _{k}\in \bigl[\xi _{k}(2),\beta _{k}(1)\bigr] \quad \text{as } j\to \infty . $$

(2.1)

Let

$$ u^{j}:=u^{\lambda _{j}}, \quad \forall j\in \mathbb {N}. $$

By (2.1), $(SG_{1})$, and the definition of $\Phi _{\lambda _{j}}$,

$$ \begin{aligned} -\Phi _{\lambda _{j}}\bigl(u^{j}\bigr) &= \frac{1}{2}\bigl\langle \Phi '_{\lambda _{j}} |_{Y_{j}} \bigl(u^{j}\bigr),u^{j}\bigr\rangle -\Phi _{\lambda _{j}} \bigl(u^{j}\bigr) \\ &= \lambda _{j}\sum_{n=-\infty }^{+\infty } \biggl(1- \frac{\mu }{2}\biggr) \chi _{n} \bigl\vert u_{n}^{j} \bigr\vert ^{\mu } \\ &\ge \lambda _{j}\biggl(1- \frac{\mu }{2}\biggr)\theta \sum _{n=-\infty }^{+ \infty } \bigl\vert u_{n}^{j} \bigr\vert ^{\mu }, \quad \forall j\in \mathbb {N}. \end{aligned} $$

(2.2)

Relations (2.1), (2.2), and $\mu <2$ imply that $\|u^{j}\|_{l^{\mu }}= (\sum_{n=-\infty }^{+\infty } |u_{n}^{j}|^{\mu } )^{1/\mu }<\infty$. It follows from the equivalence of any two norms on finite-dimensional space $E^{-}$ and the Hölder inequality that

$$ \bigl\Vert \bigl(u^{j}\bigr)^{-} \bigr\Vert ^{2}_{l^{2}} = \bigl(\bigl(u^{j}\bigr)^{-},u_{j} \bigr)_{l^{2}}\le \bigl\Vert u^{j} \bigr\Vert _{l^{\mu }} \cdot \bigl\Vert \bigl(u^{j}\bigr)^{-} \bigr\Vert _{l^{\mu '}} \le C_{1} \bigl\Vert \bigl(u^{j}\bigr)^{-} \bigr\Vert _{l^{2}} $$

for some $C_{1}> 0$, where $\mu '$ satisfies $1/\mu +1/\mu ' =1$. Consequently, we have $\|(u^{j})^{-}\|_{l^{2}} \le C_{1}$, $\forall j \in \mathbb {N}$. In view of the equivalence of norms $\|\cdot \|_{l^{2}}$ and $\|\cdot \|$ on $E^{-}$ again, there exists $C_{2} > 0$ such that

$$ \bigl\Vert \bigl(u^{j}\bigr)^{-} \bigr\Vert \le C_{2} ,\quad \forall j \in \mathbb {N}. $$

(2.3)

Obviously, the definition of $\Phi _{\lambda _{j}}$ implies

$$ \bigl\Vert \bigl(u^{j}\bigr)^{+} \bigr\Vert ^{2}=2\Phi _{\lambda _{j}}\bigl(u^{j}\bigr)+\lambda _{j} \bigl\Vert \bigl(u^{j}\bigr)^{-} \bigr\Vert ^{2}+2\lambda _{j} \sum_{n=-\infty }^{+\infty } \chi _{n} \bigl\vert u_{n}^{j} \bigr\vert ^{\mu }. $$

It follows from $\|u^{j}\|^{2}=\|(u^{j})^{+}\|^{2}+\|(u^{j})^{-}\|^{2}$ that

$$ \bigl\Vert u^{j} \bigr\Vert ^{2}=2\Phi _{\lambda _{j}} \bigl(u^{j}\bigr)+(\lambda _{j}+1) \bigl\Vert \bigl(u^{j}\bigr)^{-} \bigr\Vert ^{2}+2\lambda _{j} \sum_{n=-\infty }^{+\infty }\chi _{n} \bigl\vert u_{n}^{j} \bigr\vert ^{\mu }, $$

which, together with (2.1), (2.3), $(SG_{1})$, and the fact E is compactly embedded into $l^{2}$ (see [21]), implies that

$$ \begin{aligned} \bigl\Vert u^{j} \bigr\Vert ^{2}&\le C_{3}+4 \Vert \chi \Vert _{l^{\frac{2}{2-\mu }}} \bigl\Vert u^{j} \bigr\Vert _{l^{2}}^{\mu } \\ &\le C_{3}+C_{4} \bigl\Vert u^{j} \bigr\Vert ^{\mu }\end{aligned} $$

for some $C_{3},C_{4}> 0$. This implies that $\{u^{j}\}$ is bounded in E since $\mu <2$.

Thus, without loss of generality, we can assume that

$$ u^{j} \rightharpoonup u \quad \text{as } j\to \infty $$

(2.4)

for some $u\in E$. By the Riesz representation theorem, $\Phi '_{\lambda _{j}} |_{Y_{j}}: Y_{j} \to Y_{j}^{\ast }$ and $I': E \to E^{\ast }$ can be viewed as $\Phi '_{\lambda _{j}} |_{Y_{j}}: Y_{j} \to Y_{j}$ and $I': E \to E$, respectively, where $Y_{j}^{\ast }$ and $E^{\ast }$ are the dual spaces of $Y_{j}$ and E, respectively. Note that (2.1) implies that

$$ 0=\Phi '_{\lambda _{j}}\bigl(u^{j}\bigr)|_{Y_{j}}=u^{j} -\lambda _{j}P_{j}I'\bigl(u^{j} \bigr), \quad \forall j \in \mathbb {N}, $$

where $P_{j} : E\to Y_{j}$ is the orthogonal projection for all $j \in \mathbb {N}$, that is,

$$ u^{j}=\lambda _{j}P_{j}I' \bigl(u^{j}\bigr) ,\quad \forall j \in \mathbb {N}. $$

(2.5)

By the standard argument (see [1, 17]) we know that $I': E \to E^{\ast }$ is compact. Therefore $I': E \to E$ is also compact. It follows from (2.4) that the right-hand side of (2.5) converges strongly in E, and hence $u^{j} \to u$ in E.

Therefore $\{u^{\lambda _{j}}\}$ has a convergent subsequence in E for every $k\ge k_{1}$, and then Lemma 2.1 implies that Φ has infinitely many nontrivial solutions. □

3 Conclusion

We obtain infinitely many nontrivial solutions for a class of non-periodic Schrödinger lattice systems with nonlinearities sublinear at both zero and infinity.

Appendix

Ax 3.1

$B(u)\geq 0$, $\forall u\in E$, $B(u)\rightarrow \infty$ as $\|u\|\rightarrow \infty$ on any finite-dimensional subspace of E.

Proof

Obviously, $B(u)\geq 0$ for all $u\in E$ by $(SG_{1})$ and the definition of $B(u)$.

We claim that for any finite-dimensional subspace $H\subset E$, there exists a constant $\epsilon > 0$ such that

$$ \sharp \bigl(\bigl\{ n\in \mathbb {Z}: \chi _{n} \vert u_{n} \vert ^{\mu }\geq \epsilon \Vert u \Vert ^{\mu }\bigr\} \bigr)\geq 1,\quad \forall u\in H\backslash \{0\}, $$

(A.1)

where $\sharp (\{n\in \mathbb {Z}: \chi _{n}|u_{n}|^{\mu }\geq \epsilon \|u\|^{\mu }\})$ denotes the number of integers n such that $\chi _{n}|u_{n}|^{\mu }\geq \epsilon \|u\|^{\mu }$. If not, then for any $j\in \mathbb {N}$, there exists $u^{j}\in H\backslash \{0\}$ such that

$$ \sharp \bigl(\bigl\{ n\in \mathbb {Z}: \chi _{n} \bigl\vert u_{n}^{j} \bigr\vert ^{\mu }\geq \bigl\Vert u^{j} \bigr\Vert ^{\mu }/j \bigr\} \bigr)=0. $$

Let $v^{j}:=\frac{u^{j}}{\|u^{j}\|}\in H$. Then $\|v^{j}\|=1$, and

$$ \sharp \bigl(\bigl\{ n\in \mathbb {Z}: \chi _{n} \bigl\vert v_{n}^{j} \bigr\vert ^{\mu }\geq 1/j\bigr\} \bigr)=0, \quad \forall j\in \mathbb {N}. $$

(A.2)

Since $\{v^{j}\}$ is bounded, passing to a subsequence if necessary, we may assume that $v^{j}\to v$ in E for some $v \in H$ (H is finite dimensional). Evidently, $\|v\|=1$. Since any two norms on H are equivalent, we have

$$ \bigl\Vert v^{j}-v \bigr\Vert _{l^{2}}= \Biggl(\sum _{n=-\infty }^{+\infty } \bigl\vert v^{j}_{n}-v_{n} \bigr\vert ^{2} \Biggr)^{\frac{1}{2}}\to 0 \quad \text{as } j\to \infty . $$

It follows by the Hölder inequality and $\chi \in l^{\frac{2}{2-\mu }}$ (see $(SG_{1})$) that

$$ \sum_{n=-\infty }^{+\infty }\chi _{n} \bigl\vert v^{j}_{n}-v_{n} \bigr\vert ^{\mu }\le \Vert \chi \Vert _{l^{\frac{2}{2-\mu }}} \bigl\Vert v^{j}-v \bigr\Vert _{l^{2}}^{\mu }\to 0 \quad \text{as } j\to \infty . $$

(A.3)

In fact, since $\|v\|=1$, there is a constant $\delta _{0}> 0$ such that

$$ \sharp \bigl(\bigl\{ n\in \mathbb {Z}: \chi _{n} \vert v_{n} \vert ^{\mu }\geq \delta _{0}\bigr\} \bigr) \geq 1. $$

(A.4)

If not, then

$$ \sharp \bigl(\bigl\{ n\in \mathbb {Z}: \chi _{n} \vert v_{n} \vert ^{\mu }\geq 1/j\bigr\} \bigr)=0, \quad \forall j \in \mathbb {N}. $$

It implies that

$$ 0\le \sum_{n=-\infty }^{+\infty }\chi _{n} \vert v_{n} \vert ^{\mu +2}=\sum _{n \in \{n\in \mathbb {Z}: \chi _{n} \vert v_{n} \vert ^{\mu }< 1/j\} }\chi _{n} \vert v_{n} \vert ^{ \mu +2}\leq \frac{ \Vert v \Vert ^{2}_{l^{2}}}{j}\to 0 \quad \text{as }j\to \infty , $$

which, together with $(SG_{1})$, implies that $v=0$. It is a contradiction to $\|v\|=1$. Thus (A.4) holds. For any $j\in \mathbb {N}$, let

$$ \Lambda _{j}:=\bigl\{ n\in \mathbb {Z}: \chi _{n} \bigl\vert v^{j}_{n} \bigr\vert ^{\mu }< 1/j\bigr\} \quad \text{and}\quad \Lambda _{j}^{c}:=\mathbb {Z} \backslash \Lambda _{j}=\bigl\{ n \in \mathbb {Z}: \chi _{n} \bigl\vert v^{j}_{n} \bigr\vert ^{\mu }\geq 1/j \bigr\} . $$

Set $\Lambda _{0}:=\{n\in \mathbb {Z}: \chi _{n}|v_{n}|^{\mu }\geq \delta _{0} \}$. Then for j large enough, by (A.2), (A.4), and the definitions of $\Lambda _{0}$ and $\Lambda _{j}^{c}$ we have

$$ \sharp (\Lambda _{j}\cap \Lambda _{0})\geq \sharp (\Lambda _{0})- \sharp \bigl(\Lambda _{j}^{c}\bigr)\geq 1-0=1. $$

It follows from $(SG_{1})$ and the definitions of $\Lambda _{j}$ and $\Lambda _{0}$ that for j large enough,

$$ \begin{aligned} \sum_{n=-\infty }^{+\infty } \chi _{n} \bigl\vert v^{j}_{n}-v_{n} \bigr\vert ^{\mu }& \geq \sum_{n\in \Lambda _{j}\cap \Lambda _{0}}\chi _{n} \bigl\vert v^{j}_{n}-v_{n} \bigr\vert ^{\mu } \\ &\geq \sum_{n\in \Lambda _{j}\cap \Lambda _{0}} \biggl( \frac{1}{2^{\mu }}\chi _{n} \vert v_{n} \vert ^{\mu }-\chi _{n} \bigl\vert v^{j}_{n} \bigr\vert ^{\mu } \biggr) \\ &\geq \sharp (\Lambda _{j}\cap \Lambda _{0}) \biggl( \frac{\delta _{0}}{2^{\mu }}-1/j \biggr) \\ &\geq \frac{\delta _{0}}{2^{\mu +1}}>0. \end{aligned} $$

This is a contradiction to (A.3). Therefore (A.1) holds.

For ϵ given in (A.1), let

$$ \Lambda _{u}:=\bigl\{ n\in \mathbb {Z}: \chi _{n} \vert u_{n} \vert ^{\mu }\geq \epsilon \Vert u \Vert ^{\mu }\bigr\} , \quad \forall u\in H\backslash \{0\}. $$

It follows from $(SG_{1})$, (A.1), and the definition of $\Lambda _{u}$ that

$$ B(u)= \sum_{n=-\infty }^{+\infty }\chi _{n} \vert u_{n} \vert ^{\mu }\ge \sum _{n \in \Lambda _{u}}\chi _{n} \vert u_{n} \vert ^{\mu }\ge \epsilon \Vert u \Vert ^{\mu }\cdot \sharp ( \Lambda _{u}) \ge \epsilon \Vert u \Vert ^{\mu }, \quad \forall u\in H \backslash \{0\}. $$

This implies that $B(u)\to \infty$ as $\|u\|\to \infty$ on any finite-dimensional subspace $H\subset E$. The proof is finished. □

Ax 3.2

There exist a positive integer $k_{1}$ and two sequences $0< r_{k}<\rho _{k}\to 0$ as $k\to \infty$ such that

$$\begin{aligned}& \alpha _{k}(\lambda ):=\inf_{u\in Z_{k},\|u\|=\rho _{k}}\Phi _{\lambda }(u)>0,\quad \forall k\ge k_{1}, \end{aligned}$$

(A.5)

$$\begin{aligned}& \xi _{k}(\lambda ):=\inf_{u\in Z_{k},\|u\|\le \rho _{k}}\Phi _{\lambda }(u)\to 0 \quad \textit{as } k\to \infty \textit{ uniformly for } \lambda \in [1,2], \end{aligned}$$

(A.6)

and

$$ \beta _{k}(\lambda ):=\max_{u\in Y_{k},\|u\|=r_{k}}\Phi _{\lambda }(u)< 0, \quad \forall k\in \mathbb {N}, $$

(A.7)

where $Y_{k}=\bigoplus_{m=1}^{k} X_{m}=\operatorname{span}\{e_{1},\ldots ,e_{k}\}$ and $Z_{k}=\overline{\bigoplus_{m=k}^{\infty }X_{m}}=\overline{ \operatorname{span}\{e_{k},\ldots \}}$ for $k\in \mathbb {N}$.

Proof

$(a)$ First, we show that (A.5) holds.

Note first that $Z_{k}\subset E^{+}$ for all $k\ge k_{1}:=k_{0}+1$, where $k_{0}$ is the integer defined in the paragraph just before the proof of Theorem 1.1. Thus by the definition of $\Phi _{\lambda }$ and and the Hölder inequality we have

$$ \begin{aligned} \Phi _{\lambda }(u)&\geq \frac{1}{2} \Vert u \Vert ^{2}-2\sum_{n=- \infty }^{+\infty } \chi _{n} \vert u_{n} \vert ^{\mu } \\ &\geq \frac{1}{2} \Vert u \Vert ^{2}-2 \Vert \chi \Vert _{l^{\frac{2}{2-\mu }}} \Vert u \Vert _{l^{2}}^{ \mu },\quad \forall ( \lambda ,u)\in [1,2]\times Z_{k}, \end{aligned} $$

(A.8)

for any $k\ge k_{1}$. Let

$$ l(k):=\sup_{u\in Z_{k}\backslash \{0\}}\frac{ \Vert u \Vert _{l^{2}}}{ \Vert u \Vert }, \quad \forall k\in \mathbb {N}. $$

(A.9)

From [20] and the fact that E is compactly embedded into $l^{2}$ (see [21]) we get

$$ l(k)\to 0 \quad \text{as } k\to \infty . $$

(A.10)

By (A.8) and (A.9) we have

$$ \Phi _{\lambda }(u)\geq \frac{1}{2} \Vert u \Vert ^{2}-2 \Vert \chi \Vert _{l^{ \frac{2}{2-\mu }}} l^{\mu }(k) \Vert u \Vert ^{\mu }, \quad \forall (\lambda ,u) \in [1,2]\times Z_{k}, $$

(A.11)

for any $k\ge k_{1}$. Let

$$ \rho _{k}:= \bigl(8 \Vert \chi \Vert _{l^{\frac{2}{2-\mu }}}l^{\mu }(k) \bigr)^{ \frac{1}{2-\mu }}, \quad \forall k\in \mathbb {N}. $$

(A.12)

By (A.10) and the fact that $1<\mu <2$ we have

$$ \rho _{k}\to 0 \quad \text{as } k\to \infty . $$

(A.13)

Therefore by (A.11) and (A.12) we have

$$ \alpha _{k}(\lambda ):=\inf_{u\in Z_{k},\|u\|=\rho _{k}}\Phi _{\lambda }(u)\ge \rho _{k}^{2}/4>0,\quad \forall k \ge k_{1}. $$

$(b)$ Second, we show that (A.6) holds.

By (A.11) we have

$$ \Phi _{\lambda }(u)\geq -2 \Vert \chi \Vert _{l^{\frac{2}{2-\mu }}} l^{\mu }(k) \Vert u \Vert ^{\mu }\ge -2 \Vert \chi \Vert _{l^{\frac{2}{2-\mu }}} l^{\mu }(k)\rho _{k}^{\mu }, \quad \forall \lambda \in [1,2], $$

for all $k\ge k_{1}$ and $u\in Z_{k}$ with $\|u\|\le \rho _{k}$. Therefore we get

$$ -2 \Vert \chi \Vert _{l^{\frac{2}{2-\mu }}} l^{\mu }(k)\rho _{k}^{\mu }\le \inf_{u \in Z_{k}, \Vert u \Vert \le \rho _{k}}\Phi _{\lambda }(u) \le 0, \quad \forall \lambda \in [1,2], \forall k\ge k_{1}. $$

It follows from (A.10) and (A.13) that

$$ \xi _{k}(\lambda ):=\inf_{u\in Z_{k},\|u\|\le \rho _{k}}\Phi _{\lambda }(u) \to 0 \quad \text{as }k\to \infty \text{ uniformly for } \lambda \in [1,2]. $$

$(c)$ Finally, we show that (A.7) holds.

Note that $Y_{k}$ is finite dimensional, and thus (A.1) implies that for any $k\in \mathbb {N}$, there exists a constant $\epsilon _{k}> 0$ such that

$$ \sharp \bigl(\bigl\{ n\in \mathbb {Z}: \chi _{n} \vert u_{n} \vert ^{\mu }\geq \epsilon _{k} \Vert u \Vert ^{\mu }\bigr\} \bigr)\geq 1,\quad \forall u\in Y_{k} \backslash \{0\}. $$

(A.14)

For any $k\in \mathbb {N}$ and $u\in Y_{k}$ with $\|u\|\le \epsilon _{k}^{\frac{1}{2-\mu }}$, by the definition of $\Phi _{\lambda }$ and (A.14) we have

$$ \begin{aligned}[b] \Phi _{\lambda }(u)&\leq \frac{1}{2} \bigl\Vert u^{+} \bigr\Vert ^{2}-\sum _{n=- \infty }^{+\infty } \chi _{n} \vert u_{n} \vert ^{\mu } \\ &\leq \frac{1}{2} \Vert u \Vert ^{2}-\sum _{n\in \{n\in \mathbb {Z}: \chi _{n} \vert u_{n} \vert ^{\mu }\geq \epsilon _{k} \Vert u \Vert ^{\mu }\}}\epsilon _{k} \Vert u \Vert ^{\mu } \\ &\leq \frac{1}{2} \Vert u \Vert ^{2}-\epsilon _{k} \Vert u \Vert ^{\mu }\cdot \sharp \bigl(\bigl\{ n \in \mathbb {Z}: \chi _{n} \vert u_{n} \vert ^{\mu } \geq \epsilon _{k} \Vert u \Vert ^{\mu }\bigr\} \bigr) \\ &\leq \frac{1}{2} \Vert u \Vert ^{2}-\epsilon _{k} \Vert u \Vert ^{\mu }\le -\frac{1}{2} \Vert u \Vert ^{2}, \quad \forall \lambda \in [1,2]. \end{aligned} $$

(A.15)

Now for any $k\in \mathbb {N}$, if we choose

$$ 0< r_{k}< \min \bigl\{ \rho _{k},\epsilon _{k}^{\frac{1}{2-\mu }}\bigr\} , $$

then (A.15) implies that

$$ \beta _{k}(\lambda ):=\max_{u\in Y_{k},\|u\|=r_{k}}\Phi _{\lambda }(u) \le -r_{k}^{2}/2< 0,\quad \forall k\in \mathbb {N}. $$

Therefore the proof is finished by $(a)$, $(b)$, and $(c)$. □