Multiple entire solutions of fractional Laplacian Schrödinger equations

1.   Introduction

We consider the following semi-linear fractional Schrödinger equation

where $(-\Delta)^s$, $s\in(0, 1)$, denotes the usual fractional Laplace operator, a Fourier multiplier of symbol $|\xi|^{2s}$. Here $H^{s}(\mathbb {R}^N)$ is the fractional Sobolev space

Suppose that $V:\mathbb {R}^N \to \mathbb R$ and $f : \mathbb {R}^N\times \mathbb{R} \to \mathbb R$ satisfy the following basic assumptions

$(V_s):$ $V\in C(\mathbb {R}^N, \mathbb R)$ is 1-periodic in each component $x_1, x_2, ..., x_N$ of $x$ and

where $\sigma[(-\Delta)^s+V]$ denotes the spectrum of $(-\Delta)^s+V$.

$(F_1)$: $f\in C(\mathbb R^N\times\mathbb R, \mathbb R)$ is 1-periodic in each of $x_1, x_2, ..., x_N$ and $|f(x, t)|\leq c_1(1+ |t|^{p-1})$ for some $c_1 > 0$ and $p\in (2, 2_s^\ast), \; \rm{where} \; 2_s^\ast = \frac{2N}{N-2s}$ if $N > 2s$, $2_s^\ast = +\infty$ if $N\leq 2s$.

$(F_2)$: $f(x, t) = \rm{o}(|t|)$ as $|t|\to 0$ uniformly in $x\in\mathbb{ R}^N$.

Denote ${ \Lambda_s}: = \inf \{\sigma[(-\Delta)^s+V]\cap(0, \infty) \}$. By $(V_s)$, one has $\Lambda_s > 0.$

Two simple examples of function satisfying the conditions $(F_1)$ and $(F_2)$ are the following:

where the function $P(x)$ is 1-periodic in each of $x_1, x_2, ..., x_N.$

The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was introduced by Laskin ^[16] and ^[17] as a result of expanding the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths, where the Feynman path integral leads to the classical Schrödinger equation, and the path integral over Lévy trajectories leads to the fractional Schrödinger equation.

The fractional Laplacian operator is defined as

Here P.V. stands for the Cauchy principal value and the positive constant $C(N, s)$ depends only on $N$ and $s$, which is not essential in our problem and we will omit it for simplicity of notation. For fractional Laplacian operators and fractional spaces, the reader can refer to ^[4] and ^[8]. The authors in ^[1] raise the following assumption (AR) of the nonlinear term to study a semi-linear elliptic boundary value problem

${\rm{(AR): There\; exists}}\; \mu > 2\; {\rm{such\; that} } \; 0 < \mu F(x, t)\leq tf(x, t), \; {\rm {for}}\; x\in \mathbb{R}^N, \; t\neq 0,$

where $F(x, t): = \int_0^t f(x, \tau)d \tau$. By a direct integration of (AR), one can deduce the existence of positive constants $A, B$ such that $F(x, t)\ge A|t|^\mu -B\; \rm {for\; any} \; t\in \mathbb{R}$. We first recall some main results of the particular case $s = 1$, namely the standard Laplacian case of (1.1). The existence of a nontrivial solution to (1.1) has been obtained in ^{[2,3,7,15,23,29,31]} under (AR) and some other standard assumptions of $f$. The authors of ^[21] introduce the following more natural super-quadratic condition to replace (AR)

${\rm{(SQ)}}: \lim_{|t| \to \infty} \frac{F(x, t)}{t^2} = \infty \; {\rm {uniformly \;in}}\; x\in \mathbb {R}^N,$

and obtain the existence of nontrivial solutions of $(1.1)$ under $\rm{(SQ)}$ and some other standard assumptions of $f$ by imposing some compact conditions on the potential function $V$.After that, condition (SQ) is also used in many papers, see ^{[5,9,19,20,25,30,32,33]}. In the definite cases where $\sigma (-\Delta +V) \subset (0, \infty)$, ^[20] obtains a ground state solution via a Nehari type argument for (1.1). The corresponding energy functional of (1.1) in the case $s = 1$ is

Let $E = H^1(\mathbb R^N)$. Recall that $E = E^-\oplus E^+$ corresponds to the spectral decomposition of $-\Delta +V$ with respect to the positive and negative part of the spectrum, and $u = u^-+u^+ \in E^-\oplus E^+$. (See Section 2 for more details.) The following set has been introduced in ^[22]

By definition, ${\mathcal{M}}$ contains all nontrivial critical points of $I.$ The authors of ^[25] develop an ingenious approach to find ground state solutions of (1.1). Their approach transforms, by a direct and simple reduction, the indefinite variational problem to a definite one, resulting in a new minimax characterization of the corresponding critical value. More precisely, they establish the following two propositions by introducing the strictly monotonicity assumption $\rm{(Mo)}$

$\rm{(Mo)}$: $t \to \frac{f(x, t)}{|t|}$ is strictly increasing on $(-\infty, 0)$ and on $(0, \infty)$.

Proposition 1.1. (^[25]) Assume $(V_1), (F_1)$, $(F_2), \mathit{{(Mo)}}, \mathit{{(SQ)}}$ are satisfied and let $m =$ $\inf\limits_{u\in \mathcal{M}}{\Phi}(u).$ Then $m$ is attained, $m > 0$ and if $u_0\in \mathcal{M}$ satisfies ${\Phi}(u_0) = m$, then $u_0$ is a solution of $(1.1).$

Proposition 1.2. (^[25]) Assume $(V_1), (F_1), (F_2), \mathit{{(Mo)}}, \mathit{{(SQ)}}$ are satisfied and $f(x, t)$ is odd in $t$. Then $(1.1)$ admits infinitely many pairs geometrically distinct solutions $\pm u$.

In ^[24], the author obtains nontrivial and ground solutions of Schrödinger equation (1.1) under weaker conditions than those of ^[25]. Via deformation arguments jointed with the notion of Cerami sequence(See Section 2 for concrete definition), ^[9] establishes the following proposition.

Proposition 1.3. (^[9]) Assume that $V$ and $f$ satisfy $(V_1), (F_1), (F_2), (SQ)$ and the following condition $(\mathit{{DL}}):\; F(x, t)\ge 0, G(x, t) = \frac{1}{2} f(x, t)t- F(x, t) > 0$ if $t\neq 0, G(x, t)\to +\infty \; \mathit{{as}} \; |t|\to \infty$ uniformly in $x\in \mathbb R^N$, and there exists $c_2, r_0 > 0$ and $\nu > \mathit{{max}} \{ 1, \frac{N}{2}\}$ such that

Then $(1.1)$ has a nontrivial solution. If, in addition, $f(x, t)$ is odd in $t$, then $(1.1)$ admits infinitely many pairs geometrically distinct solutions $\pm u$.

In ^[26], the author obtains the existence of ground state solutions by non-Nehari manifold method for (1.1) with periodic and asymptotically periodic potential function V, under $\rm{(SQ)}$ and some other standard assumptions of $f$. Recently, under the weaker super-quadratic condition ${\rm{(SQ)}}^\prime$ and some other standard assumptions of $f$, the authors in ^[28] obtain the existence of nontrivial solution for (1.1) with periodic and non-periodic potential function V. The authors of ^[27] further obtain the existence of ground state solutions and infinitely many geometrically distinct solutions under ${\rm{(SQ)}}^\prime$ and non-strictly monotonicity condition ${\rm{(Mo)}}^\prime$, and as a compensation, additional condition $(F_0)$ or $(F_0)^\prime$ is necessary. These conditions are defined as follows.

${\rm{(Mo)}}^\prime$: $u \to \frac{f(x, t)}{|t|}$ is nondecreasing on $(-\infty, 0)$ and on $(0, \infty)$.

${\rm{(SQ)}}^\prime$: There exists a domain $\Omega\subseteq \mathbb{R}^N$, such that $\lim_{|t|\to\infty } \frac{F(x, t)}{t^2} = \infty,$ a.e$\; x\in \Omega.$

$(F_0)$: $G(x, t): = \frac{1}{2}f(x, t)t-F(x, t)\ge 0$, there exists $c_0 > 0, R_0 > 0$ and $\alpha \in (0, 1)$, such that

and for some $k\in (1, \frac{2}{1-\alpha}]$,

$(F_0)^\prime$: $\; G(x, t) = \frac{1}{2}f(x, t)t-F(x, t)\ge 0, \; F(x, t)\ge 0,$ and there exists $c_0 > 0, \delta_0\in(0, \Lambda_1)$ and $\alpha \in (0, 1)$, such that

and for some $k\in (1, \frac{2}{1-\alpha}]$,

The following propositions are established in ^[27].

Proposition 1.4. Assume that $V$ and $f$ satisfy $(V_1), (F_0)^\prime, (F_1), (F_2)$ and $(SQ)^\prime$. Then $(1.1)$ has a solution $u_0\in E\setminus \{0\}$ such that ${\Phi}(u_0) = \inf\limits_{u\in K}{\Phi}(u) > 0$, where $K: = \{u\in E\setminus\{0\}:{\Phi}^\prime(u) = 0\}$. If, in addition, $f(x, t)$ is odd in $t$, then $(1.1)$ admits infinitely many pairs geometrically distinct solutions $\pm u$.

Proposition 1.5. Assume that $V$ and $f$ satisfy $(V_1), (F_0), (F_1), (F_2), \mathit{{(Mo)}}^\prime$ and $(SQ)^\prime$. Then $(1.1)$ has a solution $u_0\in E$ such that ${\Phi}(u_0) = \inf\limits_{u\in \mathcal{M}}{\Phi}(u) > 0$. If, in addition, $f(x, t)$ is odd in $t$, then $(1.1)$ admits infinitely many pairs geometrically distinct solutions $\pm u$.

Existence of nontrivial solutions to a strongly indefinite Choquard equation with critical exponent is obtained in ^[13]. We also want to mention that the existence and some quantitative properties of periodic solutions of fractional equation with double well potential in one-dimensional case are established in ^[10,12,14].

In this paper we will generalize the existence of nontrivial solutions in ^[9] by replacing $\rm{(SQ)}, \rm{(DL)}$ by the weaker conditions $(F_3)$ and $(F_4)$, and generalize the existence of infinitely many geometrically different solutions in ^[27] by replacing $\rm{(Mo)}^\prime, {\rm{(SQ)}}^\prime$ by the weaker conditions $(F_3)$ and $(F_4)$.

The corresponding energy functional of (1.1) is

It is easy to verify that $\Phi_s$ is $C^1(H^s (\mathbb {R}^N), \mathbb R)$ and

From $(F_1)$ and $(F_2)$, for any given $\epsilon > 0$, there exists $C_\epsilon > 0$ such that

which yields

The following conditions are required to arrive at our results.

$(F_3)$: $F(x, t)\ge 0$ for any $(x, t)\in \mathbb{R}^N\times \mathbb{R}$, and there exists $r_1 > 0$ such that $F(x, t)\ge \Lambda_s t^2$ for any $|t|\ge r_1$ and $x\in\mathbb{R}^N$.

$(F_4)$: $G(x, t) = \frac{1}{2}f(x, t)t-F(x, t) > 0$ if $|t|\neq 0$, $G(x, t)\to +\infty$ as $|t|\to \infty \; {\rm{uniformly}}\; {\rm{in}}\; x$, and there exists $c_3, r_2 > 0$ and $\sigma > {\rm {max}}\{1, \frac{N}{2s}\}$ such that $\left|\frac{f(x, t)}{t}\right|^\sigma \leq c_3 G(x, t)$ for $|t|\ge r_2$ and $x\in\mathbb{R}^N$.

An example that satisfies the conditions $(F_1)$-$(F_4)$, but does not satisfy $\rm{(SQ)}$ is

where $h(x)$ is 1-periodic in each of $x_1, x_2, ..., x_N$ and $\inf h(x)\ge 4{\Lambda}_s$.

The followings are our main results.

Theorem 1.1. Assume $(V_s)$ and $(F_1)$-$(F_4)$ are satisfied. Then $(1.1)$ has a nontrivial solution.

Theorem 1.2. Assume $(V_s)$ and $(F_1)$-$(F_4)$ are satisfied and $f(x, t)$ is odd in $t$. Then $(1.1)$ admits infinitely many pairs geometrically distinct solutions $\pm u$.

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 ${\mathbb Z}^N$, $O(u) = \{k\ast u: k\in \mathbb Z^N \}$, where $k\ast u(x): = u(x+k)$. Two solutions $u_1$ and $u_2$ are said to be geometrically distinct if $O(u_1)\; \rm{and}\; O(u_2)$ are disjoint.

2.   Preliminaries

Denote $A_s = (-\Delta)^s+V$. Plainly $A_s$ is self-adjoint in $L^2(\mathbb{R}^N)$ with domain $\mathcal{D}(A_s) = H^{2s}(\mathbb {R}^N)$. Let $\{ \Upsilon_s(\lambda) : -\infty \leq \lambda \leq+\infty \}$ and $|A_s|$ be the spectral family and the absolute value of $A_s$ respectively, and $|A_s|^\frac{1}{2}$ be the square root of $|A_s|$. Let ${E_s} = \mathcal{D}(|A_s|^\frac{1}{2})$ and

For any $u\in {E_s}$, it is easy to see that $u = u^- +u^+$ and

where

Under assumption $(V_s)$, we can define an inner product

and the corresponding norm

By $(V_s), E_s = H^{s}(\mathbb R^N)$ with equivalent norms. Therefore $E_s$ embeds continuously in $L^p(\mathbb R^N)$ for all $2\leq p\leq 2_s^\ast$. Hence, there exists constant $\gamma_p > 0$ such that$\; \|u\|_{L^p}\leq \gamma_p\|u\|$. By the definitions of $\Lambda_s$ and ${ E}_s^+$ we also have

From $(2, 2)$–$(2.4)$, one has

Then

Let $X$ be a real Hilbert space. Recall that a functional $\psi\in C^1(X, \mathbb R)$ is said to be weakly sequentially lower semi-continuous if for any $u_n\rightharpoonup u$ in $X$ one has $\psi(u)\leq \liminf_{n\rightarrow \infty}\psi(u_n)$, and $\psi^\prime$ is said to be weakly sequentially continuous if $\lim_{n \to \infty}\langle\psi^\prime(u_n), v\rangle = \langle\psi^\prime(u), v\rangle$ for each $v\in X$. Let $\Psi(u) = \int_{\mathbb R^N} F(x, u)\, dx$. By $(F_1)$–$(F_3)$, one can easily get that $\Psi$ is weakly sequentially lower semi-continuous and $\Psi^\prime$ is weakly sequentially continuous.

We introduce the following generalized linking theorem.

Lemma 2.1. (^[15,18]) Let $X$ be a real Hilbert space, $\varphi\in C^1(X, \mathbb {R}), \varphi(0) = 0$ and

Suppose that the following assumptions are satisfied

$i):$ $\psi\in C^1(X, \mathbb {R})$ is bounded from below and weakly sequentially lower semi-continuous;

$ii):$ $\psi^\prime$ is weakly sequentially continuous;

$iii):$ there exists $\tau > 0$ such that

$iv):$ there exists $r > \tau > 0$, $e\in X^+ and\; \|e\| = 1,$ such that

where

Then there exists a constant $C_0\in [m_\tau, \mathit{{sup}}(Q_{e, r})]$ and a sequence $\{u_n\}\subseteq X$, such that $\varphi(u_n)\to C_0,$ $\|\varphi^\prime(u_n)\|(1+\|u_n\|) \to 0.$

A sequence $\{u_n\}$ is called Cerami sequence (denoted also as $(C_e)_c$-sequence) of the energy functional $\varphi$, if there exists constant $c$ such that $\varphi(u_n)\to c$ and $\|\varphi^\prime(u_n)\|(1+\|u_n\|) \to 0$.

3.   The existence of nontrivial solutions

In this section, we will prove Theorem 1.1 by applying Lemma 2.1.

Lemma 3.1. Under the assumptions $(V_s), (F_1)\rm \; {and}\; (F_2)$, there exists $\rho > 0$ such that

Proof. By (2.6), for $u\in E_s^+$, we have $\Phi_s(u) = \frac{1}{2}\|u\|^2- \int_{\mathbb R^N} F(x, u)\, dx$. Inequality (1.3) shows that for any given $\epsilon > 0$ the inequality $|F(x, u)|\leq \epsilon |u|^2$ holds for small $|u|$. So $|\int_{\mathbb R^N} F(x, u)\, dx|\leq \epsilon\|u\|^2$, and the conclusion follows if $\rho$ is sufficiently small.

Lemma 3.2. (^[11], Theorem 1.1) The fractional Schrödinger operator $A_s = (-\Delta)^s +V$ has purely continuous spectrum, which is bounded below and consists of closed disjoint intervals.

Since $\Upsilon_s (A_s)$ is purely continuous, for any given $\mu > {\Lambda}_s$, the space $Y_\mu: = \big((\Upsilon_s)_{{\mu}}-(\Upsilon_s)_0\big)_{L^2}$ is infinitely dimensional, where $((\Upsilon_s)_{\lambda})_{\lambda\in \mathbb R}$ denotes spectrum family of $A_s$. By $(2.5)$, for ${ {\Lambda}_s} < \mu < 2{ {\Lambda}_s}$ we have

Lemma 3.3. Suppose that $(V_s)\; \mathit{{and}}\; (F_3)$ are satisfied. Then for any $\; e\in Y_\mu$, $sup\; \Phi_s$ $(E_s^-\oplus \mathbb R^+ e) < \infty$ and there is $r_e > 0$ such that

Proof. Arguing indirectly, assume that for some sequence $\{u_n \}\subseteq E_s^- \oplus \mathbb R^+ e, \; e\in Y_\mu$ with $\|u_n\|$ $\to \infty\; \rm{and}\; \Phi_s(u_n) > 0$. Setting $v_n = \frac{u_n}{\|u_n\|}$, then $\|v_n\| = 1.$ Hence there exists $v = v^++v^-$ such that $v_n\rightharpoonup v, \; v_n^-\rightharpoonup v^-, \; v_n^+ \to v^+ \in\ \mathbb R^+e$. Here the strong convergence of $\{v_n^+\}$ is due to the reason that $\mathbb R^+e$ is finite dimensional. We have

We claim that $v \neq 0$. Suppose not, then

where the first inequality use the fact that $F\geq0$. The above relation gives $\|v_n^-\|\to 0$, hence $1 = \|v_n\|^2 = \|v_n^+\|^2+\|v_n^-\|^2\to 0$, which is a contradiction, and the claim is true. By $(3.1)$

Hence, there exists a bounded set $\Omega \subseteq \mathbb{R}^N$ such that

Note that

where the last inequality use the assumption $(F_3)$. Here $|\Omega|$ denotes Lebesgue's measure of $\Omega$. By the weak lower-semi continuity of the norm, we have $\|v^-\|^2\leq \liminf_{n\rightarrow \infty}\|v_n^-\|^2.$ Thus

a contradiction follows.

Lemma 3.4. Under the assumptions of $(V_s)$, $(F_2)$ and $(F_4)$, any $(C_e)_c$-sequence is bounded.

Proof. Let $\{ u_n\}\subseteq E_s$ be a $(C_e)_c$-sequence. Suppose that $u_n$ is unbounded, define $v_n$ $= \frac{u_n}{\|u_n\|}, \; \rm {then}\; \|v_n\| = 1$. Passing to subsequence, we may assume that $v_n\rightharpoonup v$ in $E_s$, $v_n\rightarrow v$ in $L_{\rm{loc}}^p\; (\mathbb{R}^N)$, $2\leq p < 2_s^\ast,$ and $v_n\to v$ a.e. in $\mathbb{R}^N$. Moreover $\|v_n\|_{L^p}\leq \gamma_p\|v_n\| = \gamma_p$. Note that

hence

since $\{ u_n\}$ $\subseteq$ $E_s$ is a $(C_e)_c$-sequence. For $R > 0, 0 < a < b$, we define

and

By $(F_4)$, $G_R > 0$ for $R > 0$ and $G_R\to +\infty$ as $R \to +\infty$. Since $G(x, u) > 0$ for $u\neq 0$ and depends periodically on $x$, we have

Observe that for $n$ large

Denote $\sigma^\prime$ as the conjugate number of $\sigma$, namely $\frac{1}{\sigma}+\frac{1}{\sigma^\prime} = 1$. Set $\tau_\sigma = \frac{2\sigma}{\sigma-1}$, then $\tau_\sigma = 2\sigma^\prime \in(2, 2_s^\ast)$, since $\sigma > \rm{max}\{ 1, \frac {N}{2s}\}$. Fix a $\tau \in (\tau_\sigma, 2_s^\ast)$. By (3.4), we can see

uniformly in $n$ as$\; b\to +\infty$. By using Hölder inequality we have

uniformly in $n$ as$\; b\to \infty$. For any given $0 < \delta < \frac{1}{3}$, let $b_{\delta}\ge r_2$. From $(3.4), (3.5)$ and $(F_4)$, we have

where we use the relation $\tau_\sigma = 2\sigma^\prime$. By $(F_2)$, there exist $a_{\delta} > 0$ such that $|f(x, t)| < \frac{\delta}{(\gamma_2)^2}|t|$ ${\rm{for \;any}}\; |t|\leq a_\delta$ and $x\in\mathbb{R}^N$, hence

From (3.4) we have

Note that there exists $\gamma = \gamma(\delta) > 0$ such as $|f(x, u_n)|\leq \gamma |u_n|\; {\rm {for}}\; x\in \Omega_n(a_\delta, b_\delta)$. By (3.8), there exists $n_0 > 0$, for $n\ge n_0$ we have

Combining (3.6), (3.7) and (3.9) we have

which contradicts with (3.2).

Proof of Theorem 1.1 From Lemmas 3.1 and 3.3, we verify that all the conditions of Lemma 2.1 hold true. Hence there exist a Cerami sequence such that $\Phi_s(u_n)\to C_0, \quad \|\Phi_s^\prime(u_n)\|(1+\|u_n\|)\to 0$. By Lemma 3.4 and Sobolev imbedding theorem, there exists $C \ge 0$ such that $\|u_n\|_{L^2}\leq C.$ If

then by Lions' concentration compactness principle, $u_n \to 0$ in $L^p(\mathbb R^N)$ for $2 < p < 2_s^\ast\;$. For $\epsilon = \frac{C_0}{4C^2}$, from (1.2) and (1.3) it follows that

We obtain

a contradiction follows, and so $\delta > 0.$

Passing to a subsequence, we may assume the existence of $k_n\in \mathbb {Z}^N$ such that

Let us define $v_n(x) = u_n(x + k_n)$, then

Since $V (x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$, then $\|u_n\| = \|v_n\|$ and

Passing to a subsequence, we have $v_n\rightharpoonup \bar{v}$ in $E_s.$ Obviously, $(3.10)$ implies that $\bar {v}\neq 0$. By a standard argument, one has $\Phi_s^\prime(\bar v) = 0$. We complete the proof of Theorem 1.1.

4.   The existence of infinite many solutions

In this section, we give the proof of Theorem 1.2.

We need to introduce some notations. For $d_2\ge d_1 > -\infty$, we set

Lemma 4.1. Assume that $(V_s), \; (F_1), \; (F_2), \; (F_4)$ hold, then

Proof. ⅰ) Assume $b_1 = 0$, then there is a sequence $\{ u_n\} \subset K$ with $\|u_n\|\to 0$, and

This and (1.2) yield that

By this and Sobolev imbedding theorem we deduce $\|u_n\|^{2-p}\leq \bar {C_\epsilon}$, which contradicts with the assumption $\|u_n\|\to 0$.

ⅱ) By

we have $b_2\geq0$. Assume $b_2 = 0$, then there is a sequence $\{ u_n\} \subset K$ such that $\Phi_s(u_n)\to 0$. Since $\{ u_n\}$ is a $(C_e)_{c = 0}$ sequence, by lemma 3.4, $u_n$ is bounded. By Sobolev imbedding theorem, there exists $C \ge 0$ such that $\|u_n\|_{L^2}^2\leq C$. Note that

By (4.3), for any $0 < a < b$, we have

which gives

as $n\to \infty$. Similar as the derivation of (3.5), for any $p\in (2, 2_s^\ast)$ we have

Next, we prove $\|u_n\| \to 0$. For any given $\epsilon > 0$, by $(F_2)$, there exist $a_{\epsilon} > 0$ such that $|f(x, t)| < \frac{\epsilon}{3C}|t|$ ${\rm{for \;any}}\; |t|\leq a_\epsilon$ and $x\in\mathbb{R}^N$. Hence,

There exists $b_{\epsilon} > a_{\epsilon} > 0$ such as $|f(x, t)|\leq \bar C_{\epsilon} |t|^{p-1}\; {\rm{for}}\; |t|\ge b_\epsilon$ and $x\in\mathbb{R}^N$, and for large $n$ we have

where the last inequality follows from (4.6). Note that there exists $\bar \gamma = \bar \gamma(\epsilon) > 0$ such as $|f(x, t)| \leq\bar \gamma |t|\; {\rm {for}}\; |t|\in (a_\epsilon, b_\epsilon)$ and $x\in\mathbb{R}^N$. So for large $n$

Therefore, it follows from $(4.7)$–$(4.9)$ and $(4.4)$, we have that

which contradicts with the result $\|u_n\|\ge b_1 > 0$ of i).

Lemma 4.2. Assume that $(F_1)\; \mathit{{and}}\; (F_2)$ hold. If $u_n\rightharpoonup \bar u \; in\; H^s(\mathbb R^N)$, then along a subsequence of $\{u_n\}$,

Proof. We can refer to Lemma 4.1 of ^[27]. The only difference is that the space $H ^1(\mathbb R^N)$ there is replaced by $H^s(\mathbb R^N)$ in this lemma, the rest argument is almost similar as the proof of Lemma 4.1 in ^[27] and we omit it.

Applying Lemma 4.2, we can obtain the next lemma.

Lemma 4.3. Assume that $(V_s), (F_1)\; and \; (F_2)$ hold. If $u_n \rightharpoonup \bar u$ in $E_s$, then

Proof. The proof is rather similar as that of lemma 4.2 of ^[9]. The main differences are that the space $E$ and the energy functional $\Phi$ there are replaced by $E_s$ and $\Phi_s$ respectively. We omit it here.

Remark Theorem 1.1 shows that equation (1.1) has a nontrivial solution $\overline u \in E_s$, and so $K\neq \emptyset$. We choose a subset $\mathcal{Q}$ of $K$ such that $\mathcal{Q} = -\mathcal{Q}$ (here $-\mathcal{Q}: = \{ w: -w\in \mathcal{Q}\})$ and each orbit $O(u)\subseteq K$ has a unique representative in $\mathcal{Q}$. It suffices to show that the set $\mathcal{Q}$ is infinite, so from now on we assume by contradiction that

Let $[a]$ stands for the largest integer not exceeding $a$. As a consequence of Lemmas 3.4, 4.1, 4.3, we have the following lemma (see ^[9] lemma 4.4, ^[15] proposition 4.2 and ^[27] lemma 4.4. The only difference is that the space $E$ is replaced by $E_s$, and the energy functional $\Phi$ is replaced by $\Phi_s$.)

Lemma 4.4. Suppose that $(V_s)$ and $(F_1)$–$(F_4)$ are satisfied. Let $\{ u_n\}$ be a $(C_e)_c$ sequence of $I_s$ in $E_s$. Then either

$(i)$ $u_n \to 0$ in $E_s$ $(\mathit{{and\; hence}}\; c = 0)$; or

$(ii)$ $c\ge b_2$ and there exists a positive integer $\ell \leq [\frac{c}{b_2}]$, points ${\bar{u}_1}, {\bar{u}_2}, ..., {\bar{u}_\ell} \in K$, a subsequence denoted again by $\{ u_n\}$ and sequences $\{ a_n^{i}\}\subset \mathbb {Z}^N$, such that

and

For any $c\ge b_2$, as in ^{[6,7,9,13,15]}, we let

Plainly $\mathcal{Q}_{c^\prime}\subseteq \mathcal{Q}_c$ for any $c\ge c^\prime \ge b_2$.

Following the argument of Proposition 1.55 in ^[6], we have the next lemma.

Lemma 4.5. Let $c\ge b_2$. Then $\kappa_c: = \mathit{{inf}}\; \{ \|u_1-u_2\|: u_1, u_2 \in \mathcal{Q}_c, u_1\neq u_2 \} > 0$.

To prove Theorem 1.2, we need to establish the following lemmas 4.6–4.10. The proofs of lemmas 4.6–4.10 are rather similar to the the proofs of lemmas 4.6–4.10 in ^[27]. The main differences are that the space $E$ and the energy functional $\Phi$ there are replaced by $E_s$ and $\Phi_s$ respectively. We omit them here.

Lemma 4.6. Let $c\ge b_2$. If $\{ u_n^{1}\}, \{u_n^2\}\subseteq \Phi_{s, b_2}^{c}$ are two $(C_e)$-sequence for $\Phi_s$, then either $\lim_{n \to \infty}$ $\|u_n^1-u_n^2\| = 0$ or $\limsup_{n \to \infty} \|u_n^1-u_n^2\|\ge \kappa_c$.

Lemma 4.7. Let $c > b_2$, $\alpha\in(0, \alpha_0]$ $\big(\alpha_0 \in (0, (c-b_2)/2 ]\; \big)$and $u\in E\setminus(K\cup \{0\})$ be such that $c-\alpha \leq \Phi_s(\eta(t, u))\leq c+\alpha$ for all $t\in [0, \infty)$. Then $u_\infty : = \lim_{t \to \infty} \eta(t, u)$ exists and $u_\infty\in \Phi_{s, c-\alpha}^{c+\alpha}\cap K$.

Lemma 4.8. Let $c > b_2$. If $K_c = \emptyset$, then there exists $\epsilon > 0$ such that $\; \lim_{t\to \infty} \Phi_s(\eta(t, u)) < c-\epsilon$ for $u\in \Phi_s^{c+\epsilon}$.

Lemma 4.9. Let $c > b_2$. Then for every $\delta \in (0, \kappa_c/4)$, there exists $\epsilon = \epsilon(c, \delta) > 0$ and an odd and continuous map $\varphi: \Phi_s^{c+\epsilon}\setminus U_\delta(\mathcal{Q}_c)\to \Phi_s^{c-\epsilon}$, where $U_\delta(\mathcal{Q}_c): = \{v\in E_s: \mathit{{dist}}(v, \mathcal{Q}_c) < \delta\}$.

Lemma 4.10. Let $c\ge b_2$. Then for every $\delta \in (0, \kappa_c/4)$, $\gamma\overline{(U_\delta(\mathcal{Q}_c))} = 1$, where $\gamma\overline{(U_\delta(\mathcal{Q}_c))}$ denotes the usual Krasnoselskki genus of $\overline{U_\delta(\mathcal{Q}_c)}$.

Proof of Theorem 1.2 We can prove Theorem 1.2 by applying Lemmas 4.6–4.10. Since the proof is rather similar that of the second part of theorem 1.4 and 1.5 in ^[27], we omit it here.

5.   Conclusions

In this paper we obtained the existence of nontrivial solutions to a semi-linear fractional Schrödinger equation by using a generalized linking theorem. Based on this existence results, infinitely many geometrically distinct solutions are further established under weaken conditions of the nonlinearity of the equation.

Acknowledgments

The authors received no specific funding for this work.

Conflict of interest

The authors declare no conflicts of interest in this paper.