Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains

1.   Introduction and main results

In this paper, we are concerned with the existence and multiplicity of weak solutions for the damped-like fractional differential system

where $_0D_t^{-\xi}$ and $_tD_T^{-\xi}$ are the left and right Riemann-Liouville fractional integrals of order $0\le \xi < 1$, respectively, $p, r, q\in C([0, T], \mathbb{R})$, $L(t): = \int_{0}^{t}(r(s)/p(s))ds$, $0 < m \leq e^{-L(t)} p(t)\leq M$ and $q(t)-p(t)\geq 0$ for a.e. $t \in [0, T]$, $u(t) = (u_1(t), u_2(t)\cdots, u_n(t))^{T}$, $(\cdot)^{T}$ denotes the transpose of a vector, $n \ge 1$ is a given positive integer, $\lambda > 0$ is a parameter, $\nabla F(t, x)$ is the gradient of $F$ with respect to $x = (x_1, \cdots, x_n)\in \mathbb{R}^n$, that is, $\nabla F(t, x) = (\frac{\partial F}{\partial x_1}, \cdots, \frac{\partial F}{\partial x_n})^{T}$, and there exists a constant $\delta\in(0, 1)$ such that $F:[0, T]\times \overline{B_0^\delta}\to \mathbb{R}$ (where $\overline{B_0^\delta}$ is a closed ball in $\mathbb{R}^N$ with center at $0$ and radius $\delta$) satisfies the following condition

$(F_0)$ $F(t, x)$ is continuously differentiable in $\overline{B_0^\delta}$ for a.e. $t \in [0, T]$, measurable in $t$ for every $x\in \overline{B_0^\delta}$, and there are $a\in C(\overline{B_0^\delta}, \mathbb{R}^+)$ and $b\in L^1([0, T];\mathbb{R}^+)$ such that

and

for all $x\in \overline{B_0^\delta}$ and a.e. $t \in [0, T]$.

In recent years, critical point theory has been extensively applied to investigate the existence and multiplicity of fractional differential equations. An successful application to ordinary fractional differential equations with Riemann-Liouville fractional integrals was first given by ^[1], in which they considered the system

They established the variational structure and then obtained some existence results for system (1.2). Subsequently, this topic attracted lots of attention and a series of existence and multiplicity results are established (for example, see ^{[2,3,4,5,6,7,8,9,10,11,12]} and reference therein). It is obvious that system (1.1) is more complicated than system (1.2) because of the appearance of damped-like term

In ^[13], the variational functional for system (1.1) with $\lambda = 1$ and $N = 1$ has been established, and in ^[14], they investigated system (1.1) with $\lambda = 1$, $N = 1$ and an additional perturbation term. By mountain pass theorem and symmetric mountain pass theorem in ^[15] and a local minimum theorem in ^[16], they obtained some existence and multiplicity results when $F$ satisfies superquadratic growth at infinity and some other reasonable conditions at origin.

In this paper, motivated by the idea in ^[17,18], being different from those in ^[13,14], we consider the case that $F$ has subquadratic growth only near the origin and no any growth condition at infinity. Our main tools are Ekeland's variational principle in ^[19], a variant of Clark's theorem in ^[17] and a cut-off technique in ^[18]. We obtain that system (1.1) has a ground state weak solution $u_\lambda$ if $\lambda$ is in some given interval and then some estimates of $u_\lambda$ are given, and when $F(t, x)$ is also even about $x$ near the origin for a.e. $t\in [0, T]$, for each given $\lambda > 0$, system (1.1) has infinitely many weak solutions $\{u_n^\lambda\}$ with $\|u_n^\lambda\|\to 0$ as $n\to \infty$. Next, we make some assumptions and state our main results.

$(f_{0})$ There exist constants $M_1 > 0$ and $0 < p_{1} < 2$ such that

for all $x\in \overline{B_0^\delta}$ and a.e. $t\in [0, T]$.

$(f_{1})$ There exist constants $M_2 > 0$ and $0 < p_{2} < p_{1} < 2$ such that

for all $x\in \overline{B_0^\delta}$ and a.e. $t\in [0, T]$.

$(f_{0})'$ There exist constants $M_1 > 0$ and $0 < p_{1} < 1$ such that (1.3) holds.

$(f_{1})'$ There exist constants $M_2 > 0$ and $0 < p_{2} < p_{1} < 1$ such that (1.4) holds.

$(f_{2})$ There exists a constant $\eta\in (0, 2)$ such that

for all $x\in \overline{B_0^\delta}$ and a.e. $t\in [0, T]$.

$(f_{3})$ $F(t, x) = F(t, -x)$ for all $x\in \overline{B_0^\delta}$ and a.e. $t\in [0, T]$.

Theorem 1.1. Suppose that $(F_0)$, $(f_{0})$, $(f_{1})$ and $(f_{2})$ hold. If

then system (1.1) has a ground state weak solution $u_\lambda$ satisfying

where

If $(f_{0})$ and $(f_{1})$ are replaced by the stronger conditions $(f_{0})'$ and $(f_{1})'$, then $(f_{2})$ is not necessary in Theorem 1.1. So we have the following result.

Theorem 1.2. Suppose that $(F_0)$, $(f_{0})'$ and $(f_{1})'$ hold. If

then system (1.1) has a ground state weak solution $u_\lambda$ satisfying

where $C^* = \max_{t\in [0, T]}e^{-L(t)} \max\left\{(1+\rho_0)a_0B\int_0^Tb(t)dt, M_1p_1TB^{p_1}, \rho_0M_1TB^{p_1+1}\right\}$, $a_0 = \max_{s\in [0, \delta]}a(s)$ and $\rho_0 = \max_{s\in [\frac{\delta}{2}, \delta]}|\rho'(s)|$ and $\rho(s)\in C^{1}(\mathbb{R}, [0, 1])$ is any given even cut-off function satisfying

Theorem 1.3. Suppose that $(F_0)$, $(f_{0})$, $(f_{1})$ and $(f_{3})$ hold. Then for each $\lambda > 0$, system (1.1) has a sequence of weak solutions $\{u_{n}^{\lambda}\}$ satisfying $\{u_{n}^{\lambda}\}\rightarrow0$, as $n\rightarrow\infty$.

Remark 1.1. Theorem 1.1-Theorem 1.3 still hold even if $r(t)\equiv 0$ for all $t\in [0, T]$, that is, the damped-like term disappears, which are different from those in ^{[2,3,4,5,6,7,8,9,10,11,12]} because all those assumptions with respect to $x$ in our theorems are made only near origin without any assumption near infinity.

The paper is organized as follows. In section 2, we give some preliminary facts. In section 3, we prove Theorem 1.1–Theorem 1.3.

2.   Preliminaries

In this section, we introduce some definitions and lemmas in fractional calculus theory. We refer the readers to ^{[1,9,20,21,22]}. We also recall Ekeland's variational principle in ^[19] and the variant of Clark's theorem in ^[17].

Definition 2.1. (Left and Right Riemann-Liouville Fractional Integrals ^[22]) Let $f$ be a function defined on $[a, b].$ The left and right Riemann-Liouville fractional integrals of order $\gamma$ for function $f$ denoted by $_aD_t^{-\gamma} f(t)$ and $_tD_b^{-\gamma} f(t),$ respectively, are defined by

provided the right-hand sides are pointwise defined on $[a, b],$ where $\Gamma > 0$ is the Gamma function.

Definition 2.2. (^[22]) For $n \in \mathbb {N},$ if $\gamma = n,$ Definition $2.1$ coincides with $n$th integrals of the form

Definition 2.3. (Left and Right Riemann-Liouville Fractional Derivatives ^[22]) Let $f$ be a function defined on $[a, b].$ The left and right Riemann-Liouville fractional derivatives of order $\gamma$ for function $f$ denoted by $_aD_t^{\gamma} f(t)$ and $_tD_b^{\gamma} f(t),$ respectively, are defined by

where $t \in [a, b], n-1 \leq \gamma < n$ and $n \in \mathbb {N}.$ In particular, if $0 \leq \gamma < 1,$ then

Remark 2.1. (^[9,13]) The left and right Caputo fractional derivatives are defined by the above-mentioned Riemann-Liuville fractional derivative. In particular, they are defined for function belonging to the space of absolutely continuous functions, which we denote by $AC([a, b], \mathbb {R^{N}})$. $AC^{k}([a, b], \mathbb {R^{N}})(k = 0, 1, ...)$ are the space of the function $f$ such that $f \in C^{k} ([a, b], \mathbb {R^{N}})$. In particular, $AC([a, b], \mathbb {R^{N}}) = AC^{1} ([a, b], \mathbb {R^{N}})$.

Definition 2.4. (Left and Right Caputo Fractional Derivatives ^[22]) Let $\gamma \geq 0$ and $n \in \mathbb {N}.$

(ⅰ) If $\gamma \in (n-1, n)$ and $f \in AC^{n}([a, b], \mathbb {R^{N}}),$ then the left and right Caputo fractional derivatives of order $\gamma$ for function $f$ denoted by $_{a}^{c} D_{t}^{\gamma} f(t)$ and $_{t}^{c} D_{b}^{\gamma} f(t),$ respectively, exist almost everywhere on $[a, b].$ $_{a}^{c} D_{t}^{\gamma} f(t)$ and $_{t}^{c} D_{b}^{\gamma} f(t)$ are represented by

respectively, where $t\in[a, b].$ In particular, if $0 < \gamma < 1,$ then

(ⅱ) If $\gamma = n-1$ and $f \in AC^{n}([a, b], \mathbb {R}^{N}),$ then $_{a}^{c} D_{t}^{\gamma} f(t)$ and $_{t}^{c} D_{b}^{\gamma} f(t)$ are represented by

In particular, $_{a}^{c} D_{t}^{0} f(t) = \ _{t}^{c}D_{b}^{0} f(t) = f(t), \ t\in[a, b].$

Lemma 2.1. (^[22]) The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, i.e.

in any point $t\in[a, b]$ for continuous function $f$ and for almost every point in $[a, b]$ if the function $f \in L^{1}([a, b], \mathbb {R}^{N}).$

For $1 \leq r < \infty,$ define

and

Definition 2.5. (^[1]) Let $0 < \alpha \leq 1$ and $1 < p < \infty.$ The fractional derivative space $E_{0}^{\alpha, p}$ is defined by closure of $C_{0}^{\infty}\left([0, T], \mathbb{R^{N}}\right)$ with respect to the norm

Remark 2.2. (^[9]) $E_{0}^{\alpha, p}$ is the space of functions $u \in L^{p}([0, T], \mathbb {R^{N}})$ having an $\alpha$-order Caputo fractional derivative $_{0} ^{c} D_{t}^{\alpha} u(t) \in L^{p}([0, T], \mathbb {R^{N}})$ and $u(0) = u(T) = 0.$

Lemma 2.2. (^[1]) Let $0 < \alpha\leq 1$ and $1 < p < \infty$. $E_{0}^{\alpha, p}$ is a reflexive and separable Banach space.

Lemma 2.3. (^[1]) Assume that $1 < p < \infty$ and $\alpha > \frac{1}{p}.$ Then $E^{\alpha, p}_{0}$ compactly embedding in $C([0, T], \mathbb {R}^{N}).$

Lemma 2.4. (^[1]) Let $0 < \alpha\leq 1$ and $1 < p < \infty$. For all $u \in E_{0}^{\alpha, p},$ we have

Moreover, if $\alpha > \frac{1}{p}$ and $\frac{1}{p}+\frac{1}{q} = 1,$ then

Definition 2.6. (^[13]) Assume that $X$ is a Banach space. An operator $A : X \rightarrow X^{\ast}$ is of type $(S)_{+}$ if, for any sequence $\{u_{n}\}$ in $X$, $u_{n} \rightharpoonup u$ and $\limsup_{n\rightarrow+\infty}\langle A(u_{n}), u_{n}-u\rangle \leq 0$ imply $u_{n}\rightarrow u.$

Let $\varphi:X\to \mathbb{R}$. A sequence $\{u_{n}\}\subset X$ is called (PS) sequence if the sequence $\{u_{n}\}$ satisfies

Furthermore, if every (PS) sequence $\{u_{n}\}$ has a convergent subsequence in $X$, then one call that $\varphi$ satisfies (PS) condition.

Lemma 2.5. (^[19]) Assume that $X$ is a Banach space and $\varphi:X\to \mathbb{R}$ is G$\hat{\mbox{a}}$teaux differentiable, lower semi-continuous and bounded from below. Then there exists a sequence $\{x_n\}$ such that

Lemma 2.6. (^[17]) Let $X$ be a Banach space, $\varphi \in C^{1}(X, \mathbb {R}).$ Assume $\varphi$ satisfies the (PS) condition, is even and bounded below, and $\varphi(0) = 0.$ If for any $k\in \mathbb {N},$ there exist a $k$-dimensional subspace $X^{k}$ of $X$ and $\rho_{k} > 0$ such that $\sup _{X^{k} \cap S_{p_{k}}} \varphi < 0,$ where $S_{\rho} = \{ u\in X| \; \|u\| = \rho \},$ then at least one of the following conclusions holds.

(ⅰ) There exist a sequence of critical points $\{u_k\}$ satisfying $\varphi(u_k) < 0$ for all $k$ and $\|u_k\|\to 0$ as $k\to \infty$.

(ⅱ) There exists a constant $r > 0$ such that for any $0 < a < r$ there exists a critical point $u$ such that $\|u\| = a$ and $\varphi(u) = 0$.

Remark 2.3. (^[17]) Lemma 2.6 implies that there exist a sequence of critical points $u_k\not = 0$ such that $\varphi(u_k)\le 0$, $\varphi(u_k)\to 0$ and $\|u_k\|\to 0$ as $k\to \infty$.

Now, we establish the variational functional defined on the space $E_{0}^{\alpha, 2}$ with $\frac{1}{2} < \alpha \leq 1.$ We follow the same argument as in ^[13] where the one-dimensional case $N = 1$ and $\lambda = 1$ for system (1.1) was investigated. For reader's convenience, we also present the details here. Note that $L(t): = \int_{0}^{t}(r(s) / p(s)) ds, 0 < m \leq e^{-L(t)}p(t) \leq M$ and $q(t)-p(t) \geq 0$ for a.e. $t \in [0, T].$ Then system (1.1) is equivalent to the system

By Lemma 2.1, for every $u \in AC([0, T], \mathbb{R}^{N}),$ it is easy to see that system (2.6) is equivalent to the system

where $\xi \in [0, 1).$

By Definition 2.4, we obtain that $u \in AC([0, T], \mathbb {R^{N}})$ is a solution of problem (2.7) if and only if $u$ is a solution of the following system

for a.e. $t \in [0, T],$ where $\alpha = 1-\frac{\xi}{2} \in (\frac{1}{2}, 1].$ Hence, the solutions of system (2.8) correspond to the solutions of system (1.1) if $u \in AC([0, T], \mathbb {R^{N}}).$

In this paper, we investigate system (2.8) in the Hilbert space $E^{\alpha, 2}_{0}$ with the corresponding norm

It is easy to see that $\|u\|$ is equivalent to $\|u\|_{\alpha, 2}$ and

So

and

where

(see ^[13]).

Lemma 2.7. (^[13]) If $\frac{1}{2} < \alpha \leq 1,$ then for every $u \in E^{\alpha, 2}_{0},$ we have

3.   Proofs

We follow the idea in ^[17] and ^[18]. We first modify and extend $F$ to an appropriate $\widetilde{F}$ defined by

where $\rho$ is defined by (1.5).

Lemma 3.1. Let $(F_0)$, $(f_{0})$, $(f_{1})$ (or $(f_{0})'$, $(f_{1})'$), $(f_{2})$ and $(f_{3})$ be satisfied. Then

$(\widetilde{F}_0)$ $\widetilde{F}(t, x)$ is continuously differentiable in $x\in\mathbb{R}^N$ for a.e. $t \in [0, T]$, measurable in $t$ for every $x\in \mathbb{R}^N$, and there exists $b\in L^1([0, T];\mathbb{R}^+)$ such that

for all $x\in \mathbb{R}^N$ and a.e. $t \in [0, T]$;

$(\widetilde{f}_{0})$ $\widetilde{F}(t, x)\ge M_1|x|^{p_{1}}$ for all $x\in \mathbb{R}^N$ and a.e. $t\in [0, T]$;

$(\widetilde{f}_{1})$ $\widetilde{F}(t, x) \leq \max\{M_1, M_2\}\left(|x|^{p_{1}}+|x|^{p_{2}}\right)$ for all $x\in \mathbb{R}^N$ and a.e. $t \in [0, T]$;

$(\widetilde{f}_{2})$ $(\nabla \widetilde{F}(t, x), x)\le\theta \widetilde{F}(t, x)$ for all $x\in \mathbb{R}^N$ and a.e. $t\in [0, T]$, where $\theta = \max\{p_1, \eta\}$;

$(\widetilde{f}_{3})$ $\widetilde{F}(t, x) = \widetilde{F}(t, -x)$ for all $x\in \mathbb{R}^N$ and a.e. $t\in [0, T]$.

Proof. We only prove $(\widetilde{f}_{0})$, $(\widetilde{f}_{1})$ and $(\widetilde{f}_{2})$. $(\widetilde{F}_0)$ can be proved by a similar argument by $({F}_0)$. By the definition of $\widetilde{F}(t, x)$, $(f_{0})$ and $(f_{1})$ (or $(f_{0})'$ and $(f_{1})'$), we have

and

Hence, $(\widetilde{f}_{1})$ holds. Note that

It is obvious that the conclusion holds for $0\leq|x|\leq\delta/2$ and $|x| > \delta$. If $\delta/2 < |x|\le\delta$, by using $\theta \ge p_{1}$, $({f}_{2})$, $(\widetilde{f}_{1})$ and the fact $s\rho'(s)\leq 0$ for all $s\in \mathbb{R}$, we can get the conclusion $(\widetilde{f}_{2})$. Finally, since $\rho(|x|)$ is even for all $x\in \mathbb{R}^N$, by $(f_{3})$ and the definition of $\widetilde{F}(t, x)$, it is easy to get $(\widetilde{f}_{3})$.

Remark 3.1. From the proof of Lemma 3.1, it is easy to see that $({F}_0)$, $({f}_{0})$ (or $({f}_{0})'$) and $({f}_{1})$ (or $({f}_{1})'$) independently imply $(\widetilde{F}_{0})$, $(\widetilde{f}_{0})$ and $(\widetilde{f}_{1})$, respectively.

Consider the modified system

for a.e. $t \in [0, T],$ where $\alpha = 1-\frac{\xi}{2} \in (\frac{1}{2}, 1].$

If the equality

holds for every $v\in E^{\alpha, 2}_{0}$, then we call $u\in E^{\alpha, 2}_{0}$ is a weak solution of system (3.1).

Define the functional $\widetilde{J}: E^{\alpha, 2}_{0}\rightarrow \mathbb {R}$ by

Then $(\widetilde{F}_0)$ and Theorem 6.1 in ^[9] imply that $\widetilde{J}\in C^1(E^{\alpha, 2}_{0}, \mathbb{R})$, and for every $u, v \in E^{\alpha, 2}_{0},$ we have

Hence, a critical point of $\tilde{J}(u)$ corresponds to a weak solution of problem (3.1).

Let

Lemma 3.2. (^[13])

where $\gamma_1 = |\cos(\pi\alpha)|$ and $\gamma_2 = \left(\max\left\{\frac{M}{m|\cos\pi\alpha|}, 1\right\}+\max_{t\in [0, T]}(q(t)-p(t))\right)$.

Lemma 3.3. Assume that $(F_{0})$, $(f_{0})$ and $(f_{1})$ (or $(f_{0})'$ and $(f_{1})'$) hold. Then for each $\lambda > 0$, $\widetilde{J}$ is bounded from below on $E_0^{\alpha, 2}$ and satisfies (PS) condition.

Proof. By $(\widetilde{f}_{1})$, (2.9) and (3.2), we have

It follows from $0 < p_{2} < p_{1} < 2$ that

Hence, $\widetilde{J}$ is coercive and then is bounded from below. Now we prove that $\widetilde{J}$ satisfies the (PS) condition. Assume that $\{u_n\}$ is a (PS) sequence of $\widetilde{J}$, that is,

Then by the coercivity of $\widetilde{J}$ and (3.3), there exists $C_0 > 0$ such that $\|u_n\|\le C_0$ and then by Lemma 2.3, there exists a subsequence (denoted again by $\{u_{n}\}$) such that

Therefore, the boundness of $\{u_{n}\}$ and (3.3) imply that

where $(E_0^{\alpha, 2})^*$ is the dual space of $E_0^{\alpha, 2}$, and $(\widetilde{F}_0)$, (2.9) together with (3.5) imply that

Note that

Then (3.6) and (3.7) imply that $\lim_{n \rightarrow \infty}\left\langle A u_{n}, u_{n}-u\right\rangle = 0$. Moreover, by (3.4), we have

Therefore

Since $A$ is of type $(S)_{+}$ (see ^[13]), by Definition 2.6, we obtain $u_{n}\rightarrow u$ in $E^{\alpha, 2}_{0}$.

Define a Nehari manifold by

Lemma 3.4. Assume that $(F_{0})$ and $(f_{0})$ (or $(f_{0})'$) hold. For each $\lambda > 0$, $\widetilde{J}_\lambda$ has a nontrivial least energy (ground state) weak solution $u_\lambda$, that is, $u_\lambda\in \mathcal{N}_\lambda$ and $\widetilde{J}_\lambda(u_\lambda) = \inf\limits_{\mathcal{N}_\lambda}\widetilde{J}_\lambda$. Moreover, the least energy can be estimated as follows

where $w_0 = \frac{w}{\|w\|}$, and $w = \left(\frac{T}{\pi}\sin\frac{\pi t}{T}, 0, \cdots, 0\right)\in E_0^{\alpha, 2}.$

Proof. By Lemma 3.3 and $\widetilde{J}\in C^1(E^{\alpha, 2}_{0}, \mathbb{R})$, for each $\lambda > 0,$ Lemma 2.5 implies that there exists some $u_{\lambda}\in E^{\alpha, 2}_{0}$ such that

By (3.2) and $(\widetilde{f}_{0})$, we have

for all $s\in [0, \infty)$. Define $g:[0, +\infty)\to \mathbb{R}$ by

Then $g(s)$ achieves its minimum at

and

Note that $p_1 < 2$. So $g(s_{0, \lambda}) < 0$. Hence, (3.9) implies that

and then $u_{\lambda}\not = 0$ which together with (3.8) implies that $u_\lambda\in \mathcal{N}_\lambda$ and $\widetilde{J}_\lambda(u_\lambda) = \inf\limits_{\mathcal{N}_\lambda}\widetilde{J}_\lambda$.

Lemma 3.5. Assume that $({F}_{0})$, $({f}_{1})$ and $({f}_{2})$ hold. If $0 < \lambda\le\frac{|\cos(\pi\alpha)|}{2 C}$, then the following estimates hold

Proof. It follows from Lemma 3.1, (2.9) and $\langle\widetilde{J}'(u_{\lambda}), u_{\lambda}\rangle = 0$ that

We claim that $\|u_{\lambda}\|\leq 1$ uniformly for all $0 < \lambda\le\frac{|\cos(\pi\alpha)|}{2 C}$. Otherwise, we have a sequence of $\{\lambda_{n}\le \frac{|\cos(\pi\alpha)|}{2 C}\}$ such that $\|u_{\lambda_{n}}\| > 1$. Thus $\|u_{\lambda_{n}}\|^{p_{2}} < \|u_{\lambda_{n}}\|^{p_{1}}$ since $p_{2} < p_{1} < 2.$ By (2.10) and (3.10), we obtain

By (3.10) and (3.11), we obtain

Since $q(t)-p(t) > 0,$

Then

which contradicts with the assumption $\|u_{\lambda_{n}}\| > 1.$ Now, from (3.10) we can get

So

By (2.9), we can obtain

Observe that, in the proof of Lemma 3.5, $(\widetilde{f}_2)$ is used only in (3.10). If we directly use $(\widetilde{F}_0)$ to rescale $(\nabla \widetilde{F}(t, u_{\lambda}(t)), u_{\lambda}(t))$ in (3.10). Then the assumption $({f}_2)$ is not necessary but we have to pay the price that $p\in (0, 1)$. To be precise, we have the following lemma.

Lemma 3.6. Assume that $({F}_{0})$ and $(f_{0})'$ hold. If $0 < \lambda\le\frac{|\cos(\pi\alpha)|}{3C^*}$, then the following estimates hold

Proof. It follows from $({F}_{0})$, Lemma 3.1, Remark 3.1, (2.9) and $\langle\widetilde{J}'(u_{\lambda}), u_{\lambda}\rangle = 0$ that

We claim that $\|u_{\lambda}\|\leq 1$ uniformly for all $0 < \lambda\le\frac{|\cos(\pi\alpha)|}{3 C^*}$. Otherwise, we have a sequence of $\{\lambda_{n}\le \frac{|\cos(\pi\alpha)|}{3 C^*}\}$ such that $\|u_{\lambda_{n}}\| > 1$. Thus $\|u_{\lambda_{n}}\|^{p_{1}} < \|u_{\lambda_{n}}\| < \|u_{\lambda_{n}}\|^{p_{1}+1}$ since $p_{1} < 1.$ By (2.10) and (3.12), we obtain

By (3.12) and (3.13), we obtain

Since $q(t)-p(t) > 0,$

Then

which contradicts with the assumption $\|u_{\lambda_{n}}\| > 1.$ Now, we can get from (3.12) that

So

By (2.9), we can obtain

Proof of Theorem 1.1. Since $0 < \lambda\le\min\left\{\frac{|\cos(\pi\alpha)|}{2 C}, \left(\frac{1}{B}\right)^{2-p_{2}}\left(\frac{\delta}{2}\right)^{2-p_2}\frac{|\cos(\pi\alpha)|}{2C}\right\}$, Lemma 3.5 implies that

Therefore, for all $0 < \lambda\le\min\left\{\frac{|\cos(\pi\alpha)|}{2 C}, \left(\frac{1}{B}\right)^{2-p_{2}}\left(\frac{\delta}{2}\right)^{2-p_2}\frac{|\cos(\pi\alpha)|}{2C}\right\},$ we have $\widetilde{F}(t, u(t)) = F(t, u(t))$ and then $u_{\lambda}$ is a nontrivial weak solution of the original problem (1.1). Moreover, Lemma 3.5 implies that $\lim_{\lambda\to 0}\|u_\lambda\| = 0$ as $\lambda\to 0$ and

Proof of Theorem 1.2. Note that $0 < \lambda\le\min\left\{\frac{|\cos(\pi\alpha)|}{3 C^*}, \left(\frac{1}{B}\right)^{2-p_{1}}\left(\frac{\delta}{2}\right)^{2-p_1}\frac{|\cos(\pi\alpha)|}{3C^*}\right\}$. Similar to the proof of Theorem 1.1, by Lemma 3.6, it is easy to complete the proof.

Proof of Theorem 1.3. By Lemma 3.1 and Lemma 3.3, we obtain that $\widetilde{J}$ satisfies (PS) condition and is even and bounded from below, and $\widetilde{J}(0) = 0.$ Next, we prove that for any $k \in \mathbb {N},$ there exists a subspace $k$-dimensional subspace $X_{k}\subset E^{\alpha, 2}_{0}$ and $\rho_{k} > 0$ such that

In fact, for any $k \in \mathbb {N},$ assume that $X^{k}$ is any subspace with dimension $k$ in $E^{\alpha, 2}_{0}$. Then by (2.10) and Lemma 3.1, there exist constants $C_1, C_2 > 0$ such that

Since all norms on $X^{k}$ are equivalent and $p_{1} < 2,$ for each fixed $\lambda > 0$, we can choose $\rho_{k} > 0$ small enough such that

Thus, by Lemma 2.6 and Remark 2.3. $\widetilde{J}_\lambda$ has a sequence of nonzero critical points $\{u^{\lambda}_{n}\}\subset E^{\alpha, 2}_{0}$ converging to $0$ and $\widetilde{J}_\lambda(u^{\lambda}_{n})\le 0$. Hence, for each fixed $\lambda > 0$, (3.1) has a sequence of weak solutions $\{u^{\lambda}_{n}\}\subset E^{\alpha, 2}_{0}$ with $\|u^{\lambda}_{n}\|\rightarrow0$, as $n\to \infty$. Furthermore, there exists $n_0$ large enough such that $\|u^{\lambda}_{n}\|\le \frac{\delta}{2B}$ for all $n\ge n_0$ and then (2.9) implies that $\|u^{\lambda}_{n}\|_\infty\le \frac{\delta}{2}$ for all $n\ge n_0$. Thus, $\widetilde{F}(t, u(t)) = F(t, u(t))$ and then $\{u_n^{\lambda}\}_{n_0}^\infty$ is a sequence of weak solutions of the original problem (1.1) for each fixed $\lambda > 0$.

4.   Conclusions

When the nonlinear term $F(t, x)$ is local subquadratic only near the origin with respect to $x$, system (1.1) with $\lambda$ in some given interval has a ground state weak solution $u_\lambda$. If the nonlinear term $F(t, x)$ is also locally even near the origin with respect to $x$, system (1.1) with $\lambda > 0$ has infinitely many weak solutions $\{u_n^\lambda\}$.

Acknowledgments

This project is supported by Yunnan Ten Thousand Talents Plan Young & Elite Talents Project and Candidate Talents Training Fund of Yunnan Province (No: 2017HB016).

Conflict of interest

The authors declare that they have no conflicts of interest.