Geometrically distinct solutions of nonlinear elliptic systems with periodic potentials

Abstract

In this paper, we study the following nonlinear elliptic systems:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1=\partial _{u_1}F(x,u)&{}\quad x\in {\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2=\partial _{u_2}F(x,u)&{}\quad x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$

where $u=(u_1,u_2):{\mathbb {R}}^N\rightarrow {\mathbb {R}}^2$, F and $V_i$ are periodic in $x_1,\ldots ,x_N$ and $0\notin \sigma (-\,\Delta +V_i)$ for $i=1,2$, where $\sigma (-\,\Delta +V_i)$ stands for the spectrum of the Schrödinger operator $-\,\Delta +V_i$. Under some suitable assumptions on F and $V_i$, we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).

1 Introduction and main results

In this paper, we study the multiplicity of solutions for the nonlinear elliptic systems:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1=f_1(x,u),&{}\quad x\in {\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2=f_2(x,u),&{}\quad x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$

(1.1)

where $f:=(f_1,f_2)=\partial _uF$ and $F:{\mathbb {R}}^N\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}$. This type of systems arises when one considers standing wave solutions of time-dependent 2-coupled Schrödinger systems of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} i\frac{\partial \phi _1}{\partial t}=-\,\Delta \phi _1+a_1(x)\phi _1-g_1(x,|\phi |)\phi _1,\\ i\frac{\partial \phi _2}{\partial t}=-\,\Delta \phi _2+a_2(x)\phi _2-g_2(x,|\phi |)\phi _2, \end{array}\right. } \end{aligned}$$

(1.2)

where $\phi = (\phi _1, \phi _2)$, i is the imaginary unit, $a_i(x)$ is a potential function, $g_i$ is a coupled nonlinear function modeling various types of the interaction effect among many particles. System (1.2) has applications in many physical problems, especially in nonlinear optics and in Bose–Einstein condensates theory for multispecies Bose–Einstein condensates (see [1, 10, 14]). A standing wave solution of system (1.2) is a solution of the form

$$\begin{aligned} \phi _i(x,t)=e^{-i\lambda _it}u_i(x),~\lambda _i \in {\mathbb {R}},~t>0, \end{aligned}$$

and $(u_1,u_2)$ solves the system (1.1) with $V_i(x)=a_i(x)-\lambda _i, f_i(x,u)=g_i(x,|u|)u_i$ for $i=1,2$.

System (1.2) has been studied by some authors quite recently. In a bounded smooth domain $\Omega \subset {\mathbb {R}}^N$, the similar systems were extensively studied by some authors, see for instance [3, 5,6,7, 19] and the references therein. The problem settled on the whole space ${\mathbb {R}}^N$ was also considered recently in some works. One of the main difficulties of this problem is the lack of the compactness of the Sobolev embedding. And the second difficulty is that the negative definite space of the quadratic form which appears in the energy functional is infinitely dimensional, i.e., the energy functional is strongly indefinite. There are many different conditions and methods involved to avoid these difficulties, we refer to [4, 8, 9, 16, 24, 25] and the references therein.

Recall that the spectrum $\sigma (-\,\Delta +V)$ of $-\,\Delta +V$ is purely continuous and may contain gaps, i.e., open intervals free of spectrum (see [18]). In [21], Szulkin and Weth considered the following Schrödinger equation:

$$\begin{aligned} -\Delta u + V(x)u = f (x, u) ~~x\in {\mathbb {R}}^N, \end{aligned}$$

(1.3)

and proved that Eq. (1.3) possesses a ground state solution under the assumption $0\notin \sigma (-\,\Delta + V)$. Later, Mederski [12] considered the system of coupled Schrödinger equations as follows:

$$\begin{aligned} -\Delta u_i + V_i(x)u_i= \partial _{u_i}F(x,u),\quad x\in {\mathbb {R}}^N, i=1,2,\ldots ,K, \end{aligned}$$

where F and $V_i$ are periodic in x, $0\notin \sigma (-\,\Delta +V_i), i=1,2,\ldots ,K,$ and proved the existence of a ground state solution based on a new linking-type result involving the Nehari-Pankov manifold.

Inspired by the above facts, more precisely by [12, 21], the aim of this paper is to study the existence and multiplicity of nontrivial solutions to system (1.1) via variational methods. As far as we know, it seems that this problem was not considered in literature before.

We assume that V and f satisfy the following hypotheses:

(V):

$V_i\in C({\mathbb {R}}^N,{\mathbb {R}})$, 1-periodic in $x_1,\ldots ,x_N$, and $0\notin \sigma (-\,\Delta +V_i)$ for $i=1,2$;

$(f_1)$:

$f_i:{\mathbb {R}}^N\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}$ is continuous with respect to the second variable, 1-periodic in $x_1,\ldots ,x_N$;

$(f_2)$:

there are $2<p<2^*=\frac{2N}{(N-2)_+}$ and $c_0>0$ such that $|f(x,u)|\le c_0(1+|u|^{p-1})~\text {for~all}~(x,u)\in {\mathbb {R}}^N\times {\mathbb {R}}^2$;

$(f_3)$:

$\lim \limits _{|u|\rightarrow 0}\frac{|f(x,u)|}{|u|}=0$ and $\lim \limits _{|u|\rightarrow \infty }\frac{F(x,u)}{|u|^2}=\infty$ uniformly in $x\in {\mathbb {R}}^N$;

$(f_4)$:

$\frac{1}{2}f(x,u)u\ge F(x, u)>0$ for all $(x,u)\in {\mathbb {R}}^N \times {\mathbb {R}}^2{\setminus } \{0\}$;

$(f_5)$:

if $f(x,u)v=f(x,v)u>0$, then $F(x,u)-F(x,v)\le \frac{(f(x,u)u)^2-(f(x,u)v)^2}{2f(x,u)u}$. If in addition, $F(x,u)\ne F(x,v)$, then the strict inequality holds.

We note that if $u_0$ is a solution of (1.1), then so are all elements of the orbit of $u_0$ under the action of ${\mathbb {Z}}^N$, $\mathcal {O}(u):=\{u(\cdot -k): k\in {\mathbb {Z}}^N\}$. Two solutions $u_1$ and $u_2$ are said to be geometrically distinct if $\mathcal {O}(u_1)$ and $\mathcal {O}(u_2)$ are disjoint.

We now describe our main result.

Theorem 1.1

Assume that (V) and $(f_1)$–$(f_5)$ hold, then the following conclusions are true:

• (i) system (1.1) has a ground state solution;

• (ii) if in addition F is even in u, that is, $F(x,u)=F(x,-u)$ for any $(x,u)\in {\mathbb {R}}^N\times {\mathbb {R}}^2$, then system (1.1) admits infinitely many pairs of geometrically distinct solutions.

Remark 1.1

The condition $(f_5)$ was first introduced by Bartsch and Mederski [2] to study the time-harmonic Maxwell equations and then as used in [12] for a Schrödinger system where our first conclusion was covered. In fact, this condition can be regarded as a vector version of the following monotonicity condition :

$$\begin{aligned} u\mapsto \frac{f(x,u)}{|u|}\quad \text {is strictly increasing on}\ (-\infty ,0)\ \text {and on}\ (0,\infty ), \end{aligned}$$

so that assumptions $(f_1)$–$(f_5)$ are more general than those in [21].

Remark 1.2

Here is an example which satisfies the assumptions $(f_1)$–$(f_5)$. For example, $F(x,u)=\Gamma (x)|Mu|^q$ for $q\in (2,2^*)$, where $\Gamma \in L^\infty ({\mathbb {R}}^N)$ is 1-periodic in $x_1,\ldots ,x_N$, positive, and bounded away from 0, $M\in GL(2)$ is an invertible $2\times 2$ matrix.

In the present paper, we are concerned with the existence of infinitely many solutions of (1.1). The main technical difficulty we need to overcome is that the energy functional is strongly indefinite. On the other hand, since f is only continuous, $\mathcal {M}$ (see Section 2) may not be of class $C^1$ in our case, so we cannot use standard arguments on the Nehari manifold in the standard way. To overcome the nondifferentiability of the Nehari manifold, we shall use the reduction method developed by Szulkin and Weth [21, 22].

2 Variational setting and preliminaries

First, $(f_2)$–$(f_3)$ imply that for every $\varepsilon > 0$, there is $C_\varepsilon > 0$ such that, for all $(x,u)\in {\mathbb {R}}^N\times {\mathbb {R}}^2$,

$$\begin{aligned} |f(x,u)|\le \varepsilon |u|+C_\varepsilon |u|^{p-1}~~\text {and}~~|F(x,u)|\le \varepsilon |u|^2+C_\varepsilon |u|^p. \end{aligned}$$

(2.1)

Since $0\notin \sigma (-\,\Delta +V_i)$, the spectral theory implies that there exist continuous projections $P_i^+$ and $P_i^-$ onto $E_i^+$ and $E_i^-$, respectively, such that $H^1({\mathbb {R}}^N)=E_i^+\oplus E_i^-$ for $i=1,2$ (see [17]). Denote $P_i^\pm u=u_i^\pm$ for $u\in H^1({\mathbb {R}}^N)$ in the sequel. Moreover, we introduce new inner products in $H^1({\mathbb {R}}^N)$ by

$$\begin{aligned} \langle u,v\rangle _i:=\int \limits _{{\mathbb {R}}^N}\nabla u_i^+ \nabla v_i^++V_i(x)u_i^+ v_i^+ dx-\int \limits _{{\mathbb {R}}^N}\nabla u_i^- \nabla v_i^-+V_i(x) u_i^-v_i^-dx \end{aligned}$$

and norms given by $\Vert u\Vert _i:=(\langle u,u\rangle _i)^{\frac{1}{2}}$ for $i=1,2$, which are equivalent to the $H^1$-norm. Let

$$\begin{aligned} E^+:=E_1^+\times E_2^+,~~E^-:=E_1^-\times E_2^-, \end{aligned}$$

and observe that any $u\in E:=H^1({\mathbb {R}}^N)\times H^1({\mathbb {R}}^N)$ admits a unique decomposition $u=u^++u^-$, where $u^+=(u_1^+,u_2^+)\in E^+$ and $u^-=(u_1^-,u_2^-)\in E^-$. We introduce a new norm in E given by

$$\begin{aligned} \Vert u\Vert ^2=\sum _{i=1}^2\big (\Vert u_i^+\Vert _i^2+\Vert u_i^-\Vert _i^2\big )=\sum _{i=1}^2\Vert u_i\Vert _i^2. \end{aligned}$$

Thus, the energy functional $\Phi$ corresponding to (1.1) is given by

$$\begin{aligned} \begin{aligned} \Phi (u)&=\frac{1}{2}\sum _{i=1}^2\int \limits _{{\mathbb {R}}^N}|\nabla u_i|^2+V_i(x)|u_i|^2dx-\int \limits _{{\mathbb {R}}^N}F(x,u)dx\\&=\frac{1}{2}\sum _{i=1}^2\big (\Vert u_i^+\Vert _i^2-\Vert u_i^-\Vert _i^2\big )-\int \limits _{{\mathbb {R}}^N}F(x,u)dx\\&=\frac{1}{2}\Vert u^+\Vert ^2-\frac{1}{2}\Vert u^-\Vert ^2-\int \limits _{{\mathbb {R}}^N}F(x,u)dx, \end{aligned} \end{aligned}$$

which is of $C^1$-class and its critical points correspond to solutions of (1.1).

In order to look for the ground state solutions of system (1.1), we consider the following set

$$\begin{aligned} \mathcal {M}:=\{u\in E{\setminus } E^-:\Phi ^\prime (u)u=0~\text {and}~\Phi ^\prime (u)v=0~\text {for~all}~v\in E^-\}, \end{aligned}$$

which has been introduced by Pankov [13]. Clearly, $\mathcal {M}$ contains all nontrivial critical points of $\Phi$.

We consider the following minimizing problem:

$$\begin{aligned} c_0:=\inf _{u\in \mathcal {M}}\Phi (u). \end{aligned}$$

Since $c_0$ is the lowest level for $\Phi$ at which there are nontrivial solutions of (1.1), $u_0$ will be called a least energy solution or ground state solution.

Before proving our result, we need some preliminary lemmas.

Lemma 2.1

Let $u\in E,v\in E^-$, and $s\ge 0$ with $u\ne su+v$, then

$$\begin{aligned} \Phi (u)>\Phi (su+v)-\Phi ^\prime (u)\big (\frac{s^2-1}{2}u+sv\big ). \end{aligned}$$

Proof

Let u, v, and s be as in the statement. Then we need to show that

$$\begin{aligned} \Phi (su+v)-\Phi ^\prime (u)\big (\frac{s^2-1}{2}u+sv\big )-\Phi (u)=-\frac{1}{2}\Vert v\Vert ^2+\int \limits _{{\mathbb {R}}^3}\varphi (s,x)dx<0, \end{aligned}$$

(2.2)

where

$$\begin{aligned} \varphi (s,x):=f(x,u)(\frac{s^2-1}{2}u+sv)+F(x,u)-F(x,su+v). \end{aligned}$$

We first claim that $\varphi (s,x)\le 0$ for $s\ge 0$ and $x\in {\mathbb {R}}^N$. Without loss of generality, we assume that $u\ne 0$. Then by $(f_4)$, we have $\varphi (0,x)\le 0$ and it follows from $(f_2)$ that

$$\begin{aligned} \lim \limits _{s\rightarrow \infty }\varphi (s,x)=-\,\infty . \end{aligned}$$

Let $s_0\ge 0$ be such that $\varphi (s_0,x)=\max \limits _{s\ge 0}\varphi (s,x)$. We may assume that $s_0>0$ and thus $\partial _s\varphi (s_0,x)=0$. Therefore,

$$\begin{aligned} f(x,u)(s_0u+v)=f(x,s_0u+v)u. \end{aligned}$$

If $f(x,u)(s_0u+v)\le 0$, then by $(f_4)$,

$$\begin{aligned} \begin{aligned} \varphi (s_0,x)&=\frac{-s_0^2-1}{2}f(x,u)u+s_0f(x,u)(s_0u+v)+F(x,u)-F(x,s_0u+v)\\&\le -\frac{s_0^2}{2}f(x,u)u-F(x,s_0u+v)\\&\le 0. \end{aligned} \end{aligned}$$

If $f(x,u)(s_0u+v)>0$, then by $(f_5)$,

$$\begin{aligned} \begin{aligned} \varphi (s_0,x)&\le f(x,u)\big (\frac{s_0^2-1}{2}u+s_0v\big )+\frac{(f(x,u)u)^2-\big (f(x,u)(s_0u+v)\big )^2}{2f(x,u)u}\\&=-\frac{(f(x,u)v)^2}{2f(x,u)u}\\&\le 0. \end{aligned} \end{aligned}$$

Then we infer that $\varphi (s,x)\le 0$ for any $s\ge 0$ and $x\in {\mathbb {R}}^N$. If $v\ne 0$, then (2.2) holds.

Now we consider the case $v=0$. If there exists $s_0>0$ and $s_0\ne 1$ such that $\varphi (s_0,x)=\max \limits _{s\ge 0}\varphi (s,x)$, then $\partial _s(s_0,x)=s_0f(x,u)u-f(x,s_0u)u=0$. Thus, it follows from $(f_5)$ that

$$\begin{aligned} \begin{aligned} \varphi (s_0,x)&=\frac{s_0^2-1}{2}f(x,u)u+F(x,u)-F(x,s_0u)\\&<\frac{s_0^2-1}{2}f(x,u)u+\frac{(f(x,u)u)^2-\big (f(x,u)(s_0u)\big )^2}{2f(x,u)u}\\&= 0, \end{aligned} \end{aligned}$$

which implies that $\varphi (s,x)$ has only one maximum point at $s=1$. Therefore, $\varphi (s,x)<\varphi (1,x)=0$. $\square$

From Lemma 2.1, we have the following lemma.

Lemma 2.2

Let $u\in \mathcal {M},v\in E^-$, and $s\ge 0$ with $u\ne su+v$, then

$$\begin{aligned} \Phi (u)>\Phi (su+v). \end{aligned}$$

We define for any $u\in E^+{\setminus }\{0\},$

$$\begin{aligned} {\hat{E}}(u):=E^-\oplus {\mathbb {R}}^+u. \end{aligned}$$

Thus, Lemma 2.2 implies that u is the unique global maximum of $\Phi \big |_{{\hat{E}}(u)}$. Applying Lemma 2.2, we can prove the following results, we omit the proof here.

Lemma 2.3

• (i) There is a constant $\rho >0$ such that $\inf \limits _{\mathcal {M}}\Phi \ge \inf \limits _{S_\rho }\Phi >0$, where $S_\rho =\{u\in E^+:\Vert u\Vert =\rho \}$.

• (ii) $\Vert u^+\Vert \ge \max \{\sqrt{2c_0},\Vert u^-\Vert \}>0$ for every $u\in \mathcal {M}$.

Lemma 2.4

If $\mathcal {W}$ is a compact subset of $E^+{\setminus }\{0\}$, then there exists $R>0$ such that $\Phi <0$ on $E(u){\setminus } B_R(0)$ for each $u\in \mathcal {W}$.

Proof

Assume that this is not true. Then there exist sequences $\{u_n\}\subset \mathcal {W}$ and $w_n\in E(u_n)$ such that $\Phi (w_n)\ge 0$ for all $n\in {\mathbb {N}}$ and $\Vert w_n\Vert \rightarrow \infty$ as $n\rightarrow \infty$. By the compactness of $\mathcal {W}$, we can assume that $u_n\rightarrow u\in \mathcal {W},\Vert u\Vert =1$. Set $v_n=\frac{w_n}{\Vert w_n\Vert }=s_nu_n+v_n^-$, then

$$\begin{aligned} 0\le \frac{\Phi (w_n)}{\Vert w_n\Vert ^2}=\frac{1}{2}(s_n^2-\Vert v_n^-\Vert ^2)-\int \limits _{{\mathbb {R}}^N}\frac{F(x,w_n)}{|w_n|^2}|v_n|^2dx, \end{aligned}$$

(2.3)

which implies that $\Vert v_n^-\Vert ^2\le s_n^2=1-\Vert v_n^-\Vert ^2$. So $\frac{1}{\sqrt{2}}\le s_n \le 1$ and then, up to a subsequence, $s_n\rightarrow s>0, v_n\rightharpoonup v$, and $v_n(x)\rightarrow v(x)$ for a.e. $x\in {\mathbb {R}}^N$. Thus $v=su+v^-\ne 0$. Set $\Omega :=\{x\in {\mathbb {R}}^N:v(x)\ne 0\}$. Then $\text {meas}(\Omega )>0$. Hence, for $x\in \Omega$, $|w_n(x)|\rightarrow +\infty$. Consequently, by Fatou’s lemma, one has

$$\begin{aligned} \int \limits _{{\mathbb {R}}^N}\frac{F(x,w_n)}{|w_n|^2}|v_n|^2dx\ge \int \limits _{\Omega }\frac{F(x,w_n)}{|w_n|^2}|v_n|^2dx\rightarrow \infty , \end{aligned}$$

which is a contradiction to (2.3). This completes the proof. $\square$

As a consequence of Lemmas 2.2–2.4, one has

Lemma 2.5

For any $u\in E{\setminus } E^-$, the set $\mathcal {M}\cap {\hat{E}}(u)$ consists of precisely one point ${\hat{m}}(u)$ which is the unique global maximum of $\Phi \big |_{{\hat{E}}(u)}$.

Lemma 2.6

$\Phi$ is coercive on $\mathcal {M}$, i.e., $\Phi (u)\rightarrow \infty$ as $\Vert u\Vert \rightarrow \infty ,u\in \mathcal {M}$.

Proof

If the conclusion is false, then there exist a sequence $\{u_n\}\subset \mathcal {M}$ such that $\Vert u_n\Vert \rightarrow \infty$ and $\Phi (u_n)\le d$ for some $d\in [c_0,\infty )$. Let $v_n = \frac{u_n}{\Vert u_n\Vert }$. Then, up to a subsequence, $v_n\rightharpoonup v$ in E and $v_n(x)\rightarrow v(x)$ for a.e. in $x \in {\mathbb {R}}^N$. By Lemma 2.3(ii), $\Vert v_n^+\Vert ^2\ge \frac{1}{2}$. By Lions’ concentration principle [11, Lemma 1.1], it is not difficult to check that $\{v_n^+\}$ is nonvanishing, that is, there exist $r,\delta >0$ and a sequence $\{y_n\}\subset {\mathbb {R}}^N$ such that

$$\begin{aligned} \int \limits _{B_r(y_n)}|v_n^+|^2dx\ge \delta . \end{aligned}$$

By the assumptions of periodicity, we may assume that $\{y_n\}$ is bounded in ${\mathbb {Z}}^N$. Thus, up to a subsequence, one has $v_n^+\rightarrow v^+$ in $L_{loc}^2({\mathbb {R}}^N)$ with $v^+\ne 0$. Set $\Omega =\{x\in {\mathbb {R}}^N:v(x)\ne 0\}$. Then $\text {meas}(\Omega )\ne 0$ and $|u_n(x)|\rightarrow \infty$ for each $x\in \Omega$. It follows from $(f_3)$ and Fatou’s lemma that

$$\begin{aligned} \int \limits _{{\mathbb {R}}^N}\frac{F(x,u_n)}{\Vert u_n\Vert ^2}dx\ge \int \limits _{\Omega }\frac{F(x,u_n)}{|u_n|^2}|v_n|^2dx\rightarrow \infty , \end{aligned}$$

and therefore,

$$\begin{aligned} 0\le \frac{\Phi (u_n)}{\Vert u_n\Vert ^2}=\frac{1}{2}(\Vert v_n^+\Vert ^2-\Vert v_n^-\Vert ^2)-\int \limits _{{\mathbb {R}}^N}\frac{F(x,u_n)}{\Vert u_n\Vert ^2}dx\rightarrow -\infty \end{aligned}$$

as $n\rightarrow \infty$, a contradiction. $\square$

Lemma 2.7

(see [21]). The map ${\hat{m}}:E^+{\setminus } \{0\}\rightarrow \mathcal {M}$ is continuous, and the restriction of the map ${\hat{m}}$ to $S^+$ is a homeomorphism with inverse given by

$$\begin{aligned} {\check{m}}:\mathcal {M}\rightarrow S^+,~~~{\check{m}}(u)=\frac{u^+}{\Vert u^+\Vert }, \end{aligned}$$

where $S^+:=\{u\in E^+:\Vert u\Vert =1\}$.

Define the mapping ${\hat{\Psi }}:E^+{\setminus }\{0\}\rightarrow {\mathbb {R}}$ and $\Psi :S^+\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} {\hat{\Psi }}(u)=\Phi ({\hat{m}}(u))~~~\text {and}~~~\Psi ={\hat{\Psi }}|_{S^+}, \end{aligned}$$

which are continuous by Lemma 2.7.

By [21, Proposition 2.9 and Corollary 2.10], we have the following Lemma 2.8.

Lemma 2.8

(see [21]).

• (i) ${\hat{\Psi }}\in C^1(E^+{\setminus }\{0\},{\mathbb {R}})$, and ${\hat{\Psi }}^\prime (w)z=\frac{\Vert {\hat{m}}(w)^+\Vert }{\Vert w\Vert } \Phi ^\prime ({\hat{m}}(w))z~~\text {for}~w,z\in E^+,w\ne 0$.

• (ii) $\Psi \in \mathcal {C}^1(S^+,{\mathbb {R}})$ and for each $w\in S^+$, one has

  $$\begin{aligned} \Psi ^\prime (w)z=\Vert {\hat{m}}(w)^+\Vert \Phi ^\prime ({\hat{m}}(w))z~~\text {for~all}~z\in T_w(S^+)=\{v\in E^+:\langle w,v\rangle =0\}. \end{aligned}$$

• (iii) If $\{w_n\}$ is a (PS)-sequence for $\Psi$, then $\{{\hat{m}}(w_n)\}$ is a (PS)-sequence for $\Phi$.

• (iv) $w\in S^+$ is a critical point of $\Psi$ if and only if ${\hat{m}}(w)\in \mathcal {M}$ is a critical point of $\Phi$. Moreover, the corresponding values of $\Psi$ and $\Phi$ coincide and $\inf \limits _{S^+}\Psi =\inf \limits _{\mathcal {M}}\Phi$.

3 Proof of the main result

Now we are in a position to give the proof of existence in Theorem 1.1.

Proof of Theorem 1.1

(i) If $u_0\in \mathcal {M}$ satisfies $\Phi (u_0)=c_0$, then ${\check{m}}(u_0)\in S^+$ is a minimizer of $\Psi$ and therefore a critical point of $\Psi$, thus $u_0$ is a critical point of $\Phi$ by Lemma 2.8. It remains to show that there exists a minimizer $u\in \mathcal {M}$ of $\Phi |_{\mathcal {M}}$. By Ekeland’s variational principle [23], there exists a sequence $\{w_n\}\subset S^+$ with $\Psi (w_n)\rightarrow c_0$ and $\Psi ^\prime (w_m)\rightarrow 0$. Set $u_n={\hat{m}}(w_n)\in \mathcal {M}$. Then $\Phi (u_n)\rightarrow c_0$ and $\Phi ^\prime (u_n)\rightarrow 0$. By Lemma 2.6, $\{u_n\}$ is bounded in E. Therefore, up to a subsequence, $u_n\rightharpoonup u$ in E. By Lions’ concentration principle [11, Lemma 1.1], it is not difficult to check that $\{u_n\}$ is nonvanishing, that is, there exist $r,\delta >0$ and a sequence $\{y_n\}\subset {\mathbb {R}}^N$ such that

$$\begin{aligned} \int \limits _{B_r(y_n)}|u_n|^2dx\ge \delta , \end{aligned}$$

here we may assume that $y_n\in {\mathbb {Z}}^N$ by taking a large r if necessary. By the assumptions of periodicity, we may assume that $\{y_n\}$ is bounded in ${\mathbb {Z}}^N$. Thus, one has $u\ne 0, \Phi ^\prime (u)=0$, and then $u\in \mathcal {M}$. Using a standard argument, one obtains that $\Phi (u)=c_0$. This completes the first part of Theorem 1.1.

In the following, we focus on the proof of the multiplicity of Theorem 1.1. Set

$$\begin{aligned} K:=\{u\in S^+:\Psi ^\prime (u)=0\},~~K_d:=\{u\in K:\Psi (u)=d\}, \end{aligned}$$

and

$$\begin{aligned} U_\delta (K_d):=\{u\in S^+:\text {dist}(u,K_d)<\delta \}. \end{aligned}$$

Let $\Sigma :=\{A\subset S^+:A~\text {is~closed~and}~A=-\,A\}$. For each $A\in \Sigma$, $\gamma (A)$ denotes the Krasnoselskii genus ($\text {see}$ [15, 20]) of A, which is defined as the least integer k such that there exists an odd continuous mapping $\sigma :A\rightarrow {\mathbb {R}}^k{\setminus }\{0\}$. If there is no such mapping for any k, then $\gamma (A)=\infty$. Moreover, $\gamma (\emptyset )=0$. Set

$$\begin{aligned} c_k:=\inf \{d\in {\mathbb {R}}:\gamma (\Psi ^d)\ge k\} \end{aligned}$$

for all $k\in {\mathbb {N}}$. It is easy to prove that $c_0\le c_k$ and $c_k\le c_{k+1}$. $\square$

Lemma 3.1

$c_k$ is a critical value of $\Psi$.

Proof

If $c_k$ is not a critical value of $\Psi$, then for any $w\in S^+$, one has $\Psi (w)\ne c_k$ or $\Psi ^\prime (w)\ne 0$. Hence, there exists $\delta > 0$ such that

$$\begin{aligned} N_{c_k,\delta }:=\{w\in S^+:|\Psi (w)-c_k|<\delta ,~\Vert \Psi ^\prime (w)\Vert <\delta \}=\emptyset . \end{aligned}$$

Otherwise, there exists a sequence $\{w_n\}\subset S^+$ such that $\Psi (w_k)\rightarrow c_k$ and $\Vert \Psi ^\prime (w_n)\Vert \rightarrow 0$. Set $v_n={\hat{m}}(w_n)$. Then, by Lemma 2.8(iii), $\{v_n\}\subset \mathcal {M}$ is a $(PS)_{c_k}$ sequence of $\Phi$. Lemma 2.6 implies that $\{v_n\}$ is bounded in E. Hence, up to a subsequence, one has $u_n\rightharpoonup u$ in E, $u_n\rightarrow u$ in $L_{loc}^t({\mathbb {R}}^N)$ for all $t\in [1,2^*)$, and $u_n(x)\rightarrow u(x)$ for a.e. $x\in {\mathbb {R}}^N$. As in the proof of Theorem 1.1, we can prove that $\Phi ^\prime (v)=0$ and $v_n\rightarrow v$ in E, and hence $\Phi (v)=c_k\ge c_0>0$. Consequently, $v\in \mathcal {M}$, and hence Lemma 2.8(iii) implies $w:=m^{-1}(v)\in K_{c_k}$, a contradiction. This show $N_{c_k,\delta }=\emptyset$. Therefore, by [20, Remark $\amalg \, 3.12$], there exists $\varepsilon _0>0$ such that for any $0<\varepsilon <{\bar{\varepsilon }}\le \varepsilon _0$, and there exists a continuous 1-parameter family of homeomorphisms $\eta (t,\cdot )$ of $S^+, 0\le t<\infty$, with the properties:

$(1^\circ )$:

$\eta (w,t)=w$ if $t=0$, or $\Psi ^\prime (w)=0$, or $|\Psi (w)-c_k|\ge {\bar{\varepsilon }}$;

$(2^\circ )$:

$\Psi (\eta (w,t))$ is nonincreasing in t for any $w\in S^+$;

$(3^\circ )$:

$\eta (\Psi ^{c_k+\varepsilon },1)\subset \Psi ^{c_k-\varepsilon }$;

$(4^\circ )$:

$\eta (\cdot ,s)\circ \eta (\cdot ,t)=\eta (\cdot ,s+t)$ for all $s,t\ge 0$;

$(5^\circ )$:

$\eta (w,t)$ is odd in w for $t\ge 0$.

Moreover, by $N_{c_k,\delta }=\emptyset$, we know that there exists $0<\varepsilon _1<\varepsilon _0$ such that

$$\begin{aligned} \Psi _{c_k-\varepsilon _1}^{c_k+\varepsilon _1}\bigcap K=\emptyset . \end{aligned}$$

For each $w\in \Psi ^{c_k+\varepsilon _1}$, by the property $(3^\circ )$ of $\eta$, we know that $\Psi (\eta (w,1))\le c_k-\varepsilon _1$. Let $e=e(w)$ be the infimum of the time for which $\Psi (\eta (w,t))\le c_k-\varepsilon _1$. It is easy to see that $e:\Psi ^{c_k+\varepsilon _1}\rightarrow [0,\infty )$ is a continuous mapping. Since $\Psi$ is even, so is e. Define a mapping $h:\Psi ^{c_k+\varepsilon _1}\rightarrow \Psi ^{c_k-\varepsilon _1}$ by $h(w):=\eta (w,e(w))$. Then h is odd and continuous. It follows from the mapping property of the genus and the definition of $c_k$ that

$$\begin{aligned} k\le \gamma (\Psi ^{c_k+\varepsilon _1})\le \gamma (\Psi ^{c_k-\varepsilon _1})\le k-1, \end{aligned}$$

a contradiction. $\square$

Now, set

$$\begin{aligned} {\tilde{K}}:=\bigcup _{k=1}^\infty K_{c_k}. \end{aligned}$$

Choose a subset $\mathcal {F}$ of ${\tilde{K}}$ such that $\mathcal {F}=-\,\mathcal {F}$ and each orbit has a unique representation. Arguing indirectly, from now on, we always assume that

$$\begin{aligned} \mathcal {F}~\text {is~a~finite~set}. \end{aligned}$$

The following lemma has been proved in [21, Lemma 2.13].

Lemma 3.2

$\kappa :=\inf \{\Vert v-w\Vert :v,w\in {\tilde{K}},v\ne w\}>0$.

The following key lemma gives the discreteness property of (PS)-sequences of $\Phi$.

Lemma 3.3

(Discreteness of (PS)-sequences). Let $d\ge c_0$. If $\{v_n^1\},\{v_n^2\}\subset \Phi ^d$ are two (PS)-sequences for $\Phi$, then either $\lim \limits _{n\rightarrow \infty }\Vert v_n^1-v_n^2\Vert =0$ or $\limsup \limits _{n\rightarrow \infty }\Vert v_n^1-v_n^2\Vert \ge \rho (d)>0$, where $\rho (d)$ depends only on d but not on the particular choice of the (PS)-sequences.

Proof

We put $u_n^1:={\hat{m}}(v_n^1),u_n^2:={\hat{m}}(v_n^2)$. Then by Lemmas 2.8(iii) and 2.6, both sequences $\{u_n^1\}, \{u_n^2\}\in \Phi ^d\cap \mathcal {M}$ are bounded (PS)-sequence for $\Phi$. We distinguish two cases.

Case 1 $\{u_n^1-u_n^2\}$ is vanishing. Then $\Vert u_n^1-u_n^2\Vert _p\rightarrow 0$ as $n\rightarrow \infty$. Thus, $\Vert (u_n^1-u_n^2)^+\Vert _p\rightarrow 0$ since the orthogonal projection of E on $E^+$ is continuous in the $L^p$-norm. By the Hölder inequality and (2.1), one has

$$\begin{aligned} \begin{aligned} \Vert (u_n^1-u_n^2)^+\Vert ^2&=\Phi ^\prime (u_n^1)(u_n^1-u_n^2)^+- \Phi ^\prime (u_n^2)(u_n^1-u_n^2)^+\\&\quad +\int \limits _{{\mathbb {R}}^3}\bigg (f(x,u_n^1)-f(x,u_n^2)\bigg )(u_n^1-u_n^2)^+dx\\&\le \varepsilon \Vert (u_n^1-u_n^2)^+\Vert \\&\quad +\int \limits _{{\mathbb {R}}^3}\bigg (\varepsilon (|u_n^1|+|u_n^2|)+C_\varepsilon (|u_n^1|^{p-1}+|u_n^2|^{p-1})\bigg )|(u_n^1-u_n^2)^+|dx\\&\le (1+C)\varepsilon \Vert (u_n^1-u_n^2)^+\Vert +D_\varepsilon \Vert (u_n^1-u_n^2)^+\Vert _p, \end{aligned} \end{aligned}$$

which implies that $\Vert (u_n^1-u_n^2)^+\Vert \rightarrow 0$ as $n\rightarrow \infty$. Similarly, $\Vert (u_n^1-u_n^2)^-\Vert \rightarrow 0$ as $n\rightarrow \infty$, so $\Vert u_n^1-u_n^2\Vert \rightarrow 0$ as $n\rightarrow \infty$. As a consequence, $\Vert v_n^1-v_n^2\Vert \rightarrow 0$ as $n\rightarrow \infty$ because ${\check{m}}$ is Lipschitz continuous on $\mathcal {M}$; indeed for $u,v\in \mathcal {M}$, by Lemma 2.3,

$$\begin{aligned} \begin{aligned} \Vert {\check{m}}(u)-{\check{m}}(v)\Vert&=\Vert \frac{u^+}{\Vert u^+\Vert }-\frac{v^+}{\Vert v^+\Vert }\Vert =\Vert \frac{u^+-v^+}{\Vert u^+\Vert }-\frac{(\Vert u^+\Vert -\Vert v^+\Vert )v^+}{\Vert u^+\Vert \Vert v^+\Vert }\Vert \\&\le \frac{2}{\Vert u^+\Vert }\Vert (u-v)^+\Vert \le \sqrt{\frac{2}{c_0}}\Vert u-v\Vert . \end{aligned} \end{aligned}$$

Case 2 $\{u_n^1-u_n^2\}$ is nonvanishing. That is, there exist $r,\delta >0$ and a sequence $\{y_n\}\subset {\mathbb {R}}^N$ such that

$$\begin{aligned} \int \limits _{B_r(y_n)}|u_n^1-u_n^2|^2dx\ge \delta . \end{aligned}$$

(3.1)

By the assumptions of periodicity, we may assume that $\{y_n\}$ is bounded in ${\mathbb {R}}^N$. Up to a subsequence, we may assume that $u_n^1\rightharpoonup u^1$ and $u_n^2\rightharpoonup u^2$ in E, where $u^1\ne u^2$ by (3.1) and $\Phi ^\prime (u^1)=\Phi ^\prime (u^2)=0$. Now we suppose

$$\begin{aligned} \Vert u_n^1\Vert \rightarrow \alpha ^1~~\text {and}~~\Vert u_n^2\Vert \rightarrow \alpha ^2, \end{aligned}$$

then it follows from Lemma 2.3(ii) that $\sqrt{2c_0}\le \alpha _i\le \nu (d):=\sup \{\Vert u\Vert :u\in \Phi ^d\cap \mathcal {M}\}$ (note that $\nu (d)<\infty$ by Lemma 2.6) for $i=1, 2$.

Suppose $u^1,u^2\ne 0$. Then $u^1,u^2\in \mathcal {M}$. We put $v^1:={\check{m}}(u^1)\in K,v^2:={\check{m}}(u^2)\in K,v^1\ne v^2$. Hence

$$\begin{aligned}&\liminf _{n\rightarrow \infty }\Vert v_n^1-v_n^2\Vert =\liminf _{n\rightarrow \infty }\bigg \Vert \frac{(v_n^1)^+}{\Vert (v_n^1)^+\Vert }-\frac{(v_n^2)^+}{\Vert (v_n^2)^+\Vert }\bigg \Vert \\&\quad \ge \bigg \Vert \frac{(u^1)^+}{\alpha ^1}-\frac{(u^2)^+}{\alpha ^2} \bigg \Vert \ge \Vert \beta _1v_1-\beta _2v_2\Vert , \end{aligned}$$

where

$$\begin{aligned} \beta _1:=\frac{\Vert (u^1)^+\Vert }{\alpha _1}\ge \frac{\sqrt{2c_*}}{\nu (d)}~~~\text {and}~~~\beta _2:=\frac{\Vert (u^2)^+\Vert }{\alpha _2}\ge \frac{\sqrt{2c_*}}{\nu (d)}. \end{aligned}$$

Since $\Vert v^1\Vert =\Vert v^2\Vert =1$, it is easy see from the above inequalities that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\Vert v_n^1-v_n^2\Vert \ge \Vert \beta _1v^1-\beta _2v^2\Vert \ge \min \{\beta ^1,\beta ^2\}\Vert v^1-v^2\Vert \ge \frac{\kappa \sqrt{2c_0}}{\nu (d)},\nonumber \\ \end{aligned}$$

(3.2)

where $\kappa$ is given by Lemma 3.2. Hence (3.2) implies $\liminf \limits _{n\rightarrow \infty }\Vert v_n^1-v_n^2\Vert \ge \rho (d)>0$, where $\rho (d)$ depends only on d.

If $u^2=0$, then $u^1\ne 0$ and

$$\begin{aligned} \liminf _{n\rightarrow \infty }\Vert v_n^1-v_n^2\Vert =\liminf _{n\rightarrow \infty }\bigg \Vert \frac{(v_n^1)^+}{\Vert (v_n^1)^+\Vert }-\frac{(v_n^2)^+}{\Vert (v_n^2)^+\Vert }\bigg \Vert \ge \frac{\Vert (u^1)^+\Vert }{\alpha ^1}\ge \frac{\sqrt{2c_0}}{\nu (d)}. \end{aligned}$$

The case $u^1=0$ can be treated similarly. The proof is completed. $\square$

It is known that $\Psi$ admits a pseudo-gradient vector field $H:S^+{\setminus } K\rightarrow TS^+$. Let $\eta _1: \mathcal {G}\rightarrow S^+{\setminus } K$ be the corresponding flow defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d}{dt}\eta _1(t,w)=-\,H(\eta _1(t,w)),&{}\\ \eta _1(0,w)=w, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \mathcal {G}:=\{(t,w):w\in S^+{\setminus } K,T^-(w)<t<T^+(w)\}\subset {\mathbb {R}}\times (S^+{\setminus } K) \end{aligned}$$

and $(T^-(w), T^+(w))$ are the maximal existence times of the trajectory $t\rightarrow \eta _1(t,w)$ in negative and positive direction. Here $\eta _1(t,w)$ is odd in w and $\Psi (\eta _1(t,w))$ is strictly decreasing in t.

We then have the following important deformation type results which play a crucial role in our proof.

Lemma 3.4

Let $d\ge c_0$. Then for every $\delta >0$, there exists $\varepsilon =\varepsilon (\delta )>0$ such that

• (i) $\Psi _{d-\varepsilon }^{d+\varepsilon }\cap {\tilde{K}}=K_d$;

• (ii) $\lim \limits _{t\rightarrow T^+(w)}\Psi (\eta _1(t,w))<d-\varepsilon$ for $w\in \Psi ^{d+\varepsilon }{\setminus } U_\delta (K_d)$.

Proof of Theorem 1.1 (completed)

(ii) If $\mathcal {F}$ is an infinite set, then $\Psi$ admits infinitely many pairs $\pm \,v$ of geometrically distinct critical points. So, Theorem 1.1(ii) follows from Lemmas 2.8 and 3.1.

By Lemma 3.2, $K_{c_k}$ is either empty or a discrete set, hence $\gamma (K_{c_k})=0$ or 1. By the continuity property of the genus, there exists $\delta >0$ such that $\gamma ({\bar{U}})=\gamma (K_{c_k})$, where $U:=U_\delta (K_{c_k})$ and $\delta <\frac{\kappa }{2}$. For such $\delta$, choose $\varepsilon >0$ so that the conclusions of Lemma 3.4 hold with $d=c_k$. Then for each $w\in \Psi ^{c_k+\varepsilon }{\setminus } U$, there exists $t\in [0,T^+(w))$ such that $\Psi (\eta _1(t,w))<c_k-\varepsilon$. Let $e=e(w)$ be the infimum of the time for which $\Psi (\eta _1(w,t))\le c_k-\varepsilon$. Since $c_k-\varepsilon$ is not a critical value of $\Psi$, it is easy to see by the implicit function theorem that e is a continuous mapping and since $\Psi$ is even, $e(-w)=e(w)$. Define a mapping $h:\Psi ^{c_k+\varepsilon }{\setminus } U\rightarrow \Psi ^{c_k-\varepsilon }$ by setting $h(w):=\eta _1(e(w),w)$. Then h is odd and continuous, so it follows from the properties of the genus and the definition of $c_k$ that

$$\begin{aligned} \gamma (\Psi ^{c_k+\varepsilon })\le \gamma ({\bar{U}})+\gamma (\Psi ^{c_k-\varepsilon })\le \gamma ({\bar{U}})+k-1=\gamma (K_{c_k})+k-1. \end{aligned}$$

(3.3)

If $\gamma (K_{c_k})=0$, then $\gamma (\Psi ^{c_k+\varepsilon })\le k-1$, in contrast to the definition of $c_k$. Therefore, $\gamma (K_{c_k})=1$ and $K_{c_k}\ne \emptyset$.

If $c_k=c_{k+1}$, then by the definition of $c_{k+1}$, there exists $r<c_{k+1}+\varepsilon$ such that $\gamma (\Psi ^r)\ge k+1$. Therefore, $\gamma (\Psi ^{c_{k+1}+\varepsilon })\ge \gamma (\Psi ^r)\ge k+1$, and hence, by (3.3), one has

$$\begin{aligned} \gamma (K_{c_k})\ge \gamma (\Psi ^{c_k+\varepsilon })-k+1\ge 2. \end{aligned}$$

But, by Lemma 3.2, $\gamma (K_{c_k})=1$, a contradiction. Therefore, $c_k<c_{k+1}$. This contradicts the fact that $\mathcal {F}$ is a finite set. Therefore, $\Psi$ admits infinitely many pairs $\pm v$ of geometrically distinct critical points. Consequently, Theorem 1.1(ii) follows from Lemmas 2.8 and 3.1. $\square$