Skip to main content

Existence of ground state solutions for an asymptotically 2-linear fractional Schrödinger–Poisson system


In this paper, we investigate the following fractional Schrödinger–Poisson system:

$$\left \{ \textstyle\begin{array}{l@{\quad}l} (-\Delta)^{s} u + u + \phi u = f(u), & \text{in } \mathbb{R}^{3}, \\ (-\Delta)^{t} \phi= u^{2}, & \text{in } \mathbb{R}^{3}, \end{array}\displaystyle \right . $$

where \(\frac{3}{4} < s < 1\), \(\frac{1}{2} < t < 1\), and f is a continuous function, which is superlinear at zero, with \(f(\tau) \tau \ge3 F(\tau) \ge0\), \(F(\tau) = \int_{0}^{\tau} f(s) \,ds\), \(\tau \in\mathbb{R}\). We prove that the system admits a ground state solution under the asymptotically 2-linear condition. The result here extends the existing study.

1 Introduction

In this paper, we study the existence of ground state solutions for the following fractional Schrödinger–Poisson system:

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} (-\Delta)^{s} u + u + \phi u = f(u), & \text{in } \mathbb{R}^{3}, \\ (-\Delta)^{t} \phi= u^{2}, & \text{in } \mathbb{R}^{3}, \end{array}\displaystyle \right . $$

where \(\frac{3}{4} < s < 1\), \(\frac{1}{2} < t < 1\), \((-\Delta)^{s}\) and \((-\Delta)^{t}\) are the fractional Laplace operators, f satisfies the following conditions:


\(f \in C(\mathbb{R}, \mathbb{R})\), \(\lim_{\tau\to0} \frac {f(\tau)}{\tau} = 0\);

$$\lim_{ \vert \tau \vert \to\infty} \frac{f(\tau)}{ \vert \tau \vert ^{2}} = \mu \quad \text{with } \sqrt{ \frac{54}{\pi} C(3, s)^{-1} S^{2} \biggl(\frac{32 \pi}{3} \biggr)^{\frac{5}{3}}} < \mu< + \infty, $$

where the constant S and the function \(C(3, s)\) will be specified in Sect. 2;

$$f(\tau) \tau \ge3 F(\tau) \ge0, \quad\forall\tau \in\mathbb {R}, \text{where } F(\tau) = \int_{0}^{\tau} f(s) \,ds. $$

In recent years, the nonlinear fractional Schrödinger–Poisson systems have received a lot of attention. In [1], Gao, Tang and Chen studied the existence of ground state solutions of (1.1) in a mild assumption on f with super-quadratic nonlinearity. If u is replaced by \(V(x) u\) and \(f(u) = \mu|u|^{q-2} u + |u|^{2_{s}^{*} - 2} u\) (\(2_{s}^{*} = \frac {6}{3-2s}\)) in (1.1), the existence of a nontrivial ground state solution is given by Teng [2]. In [3], based on the symmetric mountain pass theorem, He and Jing investigated a class of fractional Schrödinger–Poisson system with superlinear terms, the existence and multiplicity of nontrivial solutions of such a system are obtained. Wang, Ma and Guan [4] studied the existence of a sign-changing solution of the following nonlinear fractional Schrödinger–Poisson system:

$$\left \{ \textstyle\begin{array}{l@{\quad}l} (-\Delta)^{s} u + V(x) u + \phi u = K(x) f(u), & \text{in } \mathbb{R}^{3}, \\ (-\Delta)^{t} \phi= u^{2}, & \text{in } \mathbb{R}^{3}, \end{array}\displaystyle \right . $$

by means of the constraint variational method and the quantitative deformation lemma.

When \(s=t=1\), system (1.1) reduces to the following Schrödinger–Poisson system:

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} -\Delta u + u + \phi u = g(u), & \text{in } \mathbb{R}^{3}, \\ -\Delta\phi= u^{2}, & \text{in } \mathbb{R}^{3}. \end{array}\displaystyle \right . $$

Yin, Wu and Tang [5] proved the existence of ground state solutions of (1.2) by using

$$\lim_{ \vert t \vert \to\infty} \frac{f(t)}{ \vert t \vert ^{2}} = \nu \quad \text{with } \sqrt{ \frac{189}{8 \pi S} \biggl(\frac{32 \pi}{3} \biggr)^{\frac{5}{3}}} < \nu< + \infty, $$

instead of the usual 2-superlinear condition \(\lim_{|t| \to\infty} \frac{G(t)}{|t|^{3}} = + \infty\) (\(G(t) = \int_{0}^{t} g(s) \,ds\)), which relaxed the conditions of nonlinearity in [68].

Inspired by [5], the main objective of this paper is to extend the main results of [1], by relaxing the condition of super-quadratic nonlinearity used in [1]. That is, the nonlinearity f is assumed to be asymptotically 2-linear. We deal with the nonlinear fractional Schrödinger–Poisson system (1.1) in view of variational method and some analysis technique. Our result also extends the main results of [5].

2 Preliminaries

The fractional Sobolev space \(H^{s}(\mathbb{R}^{3})\) can be described by means of the Fourier transform, i.e.

$$H^{s}\bigl(\mathbb{R}^{3}\bigr) = \biggl\{ u \in L^{2}\bigl(\mathbb{R}^{3}\bigr) : \int_{\mathbb {R}^{3}} \bigl( \vert \xi \vert ^{2s} \bigl\vert \widehat{u}(\xi) \bigr\vert ^{2} + \bigl\vert \widehat{u}( \xi) \bigr\vert ^{2}\bigr) \,d \xi < + \infty \biggr\} $$

endowed with the norm

$$\begin{aligned} \Vert u \Vert & := \Vert u \Vert _{H^{s}} = \biggl( \int_{\mathbb{R}^{3}} \bigl( \vert \xi \vert ^{2s} \bigl\vert \widehat {u}(\xi) \bigr\vert ^{2} + \bigl\vert \widehat{u}( \xi) \bigr\vert ^{2}\bigr) \,d \xi \biggr)^{\frac{1}{2}} \\ &= \biggl( \int_{\mathbb{R}^{3}} \bigl( \bigl\vert (-\Delta)^{\frac{s}{2}} u(x) \bigr\vert ^{2} + \bigl\vert u(x) \bigr\vert ^{2}\bigr) \,dx \biggr)^{\frac{1}{2}}, \quad\forall u \in H^{s}\bigl(\mathbb {R}^{3}\bigr).\end{aligned} $$

Since \(4s + 2t > 3\), we have \(2 \le\frac{12}{3+2t} \le\frac {6}{3-2t}\), thus \(H^{s}(\mathbb{R}^{3}) \hookrightarrow L^{\frac {12}{3+2t}}(\mathbb{R}^{3})\). From [1], we know that there exists a unique \(\phi_{u}^{t} \in\mathcal {D}^{t, 2}(\mathbb{R}^{3}) = \{u \in L^{2_{t}^{*}}(\mathbb{R}^{3}) : |\xi|^{t} \widehat{u}(\xi) \in L^{2}(\mathbb{R}^{3})\}\) which is a weak solution of \((-\Delta)^{t} \phi_{u}^{t} = u^{2}\), and it has the following representation:

$$\phi_{u}^{t}(x) = c_{t} \int_{\mathbb{R}^{3}} \frac{u^{2}(y)}{ \vert x-y \vert ^{3-2t}} \,dy, \quad x \in \mathbb{R}^{3}, $$

where \(c_{t} = \pi^{-\frac{3}{2}} 2^{-2t} \frac{\varGamma(3-2t)}{\varGamma (t)}\). Substituting \(\phi_{u}^{t}\) in (1.1), we obtain the following fractional Schrödinger equation:

$$ (-\Delta)^{s} u + u + \phi_{u}^{t} u = f(u), \quad x \in\mathbb{R}^{3}. $$

For the properties of \(\phi_{u}^{t}\), see [2]. By (2.1), we define the functional \(\mathcal{I} : H^{s}(\mathbb{R}^{3}) \to\mathbb{R}\) as follows:

$$ \mathcal{I}(u) = \frac{1}{2} \int_{\mathbb{R}^{3}} \bigl( \bigl\vert (-\Delta)^{\frac{s}{2}} u \bigr\vert ^{2} + u^{2} \bigr) \,dx + \frac{1}{4} \int _{\mathbb{R}^{3}} \phi_{u}^{t} u^{2} \,dx - \int_{\mathbb{R}^{3}} F(u) \,dx, $$

where \(F(u) = \int_{0}^{u} f(x) \,dx\). It is easy to see that \((f_{1})\) and \((f_{2})\) imply that \(\mathcal{I}\) is a well-defined \(C^{1}\)-functional, and

$$\bigl\langle \mathcal{I}^{\prime}(u), v \bigr\rangle = \int_{\mathbb{R}^{3}} \bigl( (-\Delta)^{\frac{s}{2}} u (- \Delta)^{\frac{s}{2}} v + uv\bigr) \,dx + \int_{\mathbb{R}^{3}} \phi_{u}^{t} u v \,dx - \int_{\mathbb{R}^{3}} f(u) v \, dx, \quad \forall v \in H^{s} \bigl(\mathbb{R}^{3}\bigr). $$

Hence, if u is a critical point of \(\mathcal{I}\), then \((u, \phi _{u}^{t})\) is a solution of (1.1).


$$u_{R}(x) = \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{R}, & \vert x \vert \le R, \\ \frac{1}{R} (2 - \frac{ \vert x \vert }{R} ), & R < \vert x \vert \le2R, \\ 0, & \vert x \vert > 2R. \end{array}\displaystyle \right . $$

Hence \(u_{R} \in H^{s}(\mathbb{R}^{3})\). By Proposition 3.4 in [9], we have

$$ \Vert u_{R} \Vert _{H^{s}}^{2} = 2 C(3, s)^{-1} \int_{\mathbb{R}^{3}} \vert \xi \vert ^{2s} \bigl\vert \mathcal{F} u_{R}(\xi) \bigr\vert ^{2} \,d \xi, $$

where \(\mathcal{F}\) is the usual Fourier transform in \(\mathbb{R}^{3}\), and

$$C(3, s) = \biggl( \int_{\mathbb{R}^{3}} \frac{1-\cos(\zeta_{1})}{ \vert \zeta \vert ^{3+2s}} \,d\zeta \biggr)^{-1}, $$

here \(\zeta= (\zeta_{1}, \zeta_{2}, \zeta_{3})\). From the inequality \(|\xi |^{2s} \le1 + |\xi|^{2}\), \(s \in(0, 1]\), together with (2.3), we get

$$ \Vert u_{R} \Vert _{H^{\alpha}}^{2} \le2 C(3, s)^{-1} \int_{\mathbb {R}^{3}} \bigl(1 + \vert \xi \vert ^{2}\bigr) \bigl\vert \mathcal{F} u_{R}(\xi) \bigr\vert ^{2} \,d \xi = 2 C(3, s)^{-1} \Vert u_{R} \Vert _{H^{1}}^{2}. $$

If \(t > \frac{1}{2}\), then we have by Lemma 2.3 in [2]

$$\int_{\mathbb{R}^{3}} \phi_{u_{R}(x)}^{t} u_{R}(x)^{2} \,dx \le S_{t}^{2} \vert u_{R} \vert _{\frac{12}{3+2t}}^{4}, $$


$$S_{t} = \inf_{u \in\mathcal{D}^{t, 2}(\mathbb{R}^{3}) \setminus\{0\}} \frac{\int_{\mathbb{R}^{3}} \vert (-\Delta)^{\frac{t}{2}} u \vert ^{2} \,dx}{ (\int_{\mathbb{R}^{3}} \vert u(x) \vert ^{2_{t}^{*}} \,dx )^{\frac{2}{2_{t}^{*}}}}. $$

Remark 2.1

If \(t = 1\), then the above inequalities modifies to the following inequalities:

$$ \int_{\mathbb{R}^{3}} \phi_{u_{R}(x)} u_{R}(x)^{2} \,dx \le S^{2} \vert u_{R} \vert _{\frac{12}{5}}^{4}, $$


$$S = \inf_{u \in\mathcal{D}^{1, 2}(\mathbb{R}^{3}) \setminus\{0\}} \frac{\int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2} \,dx}{ (\int_{\mathbb{R}^{3}} \vert u(x) \vert ^{6} \,dx )^{\frac{1}{3}}}. $$

From [5], we have

$$ \int_{\mathbb{R}^{3}} \bigl\vert \nabla u_{R}(x) \bigr\vert ^{2} \,dx = \frac{28\pi }{3R}, \qquad \int_{\mathbb{R}^{3}} \bigl\vert u_{R}(x) \bigr\vert ^{3} \,dx \ge\frac{4 \pi}{3}, $$


$$ \vert u_{R} \vert _{\frac{12}{5}}^{4} = \biggl( \frac{32 \pi}{3} \biggr)^{\frac{5}{3}} R. $$

Lemma 2.1

If\((f_{1})\)and\((f_{2})\)hold, then

  1. (i)

    there exists a\(v \in H^{s}(\mathbb{R}^{3}) \setminus\{0\}\)such that\(\mathcal{I}(v) \le0\);

  2. (ii)

    \(c := \inf_{\gamma\in\varGamma} \max_{t \in[0, 1]} \mathcal {I}(\gamma(t)) > 0\), where

    $$\varGamma= \bigl\{ \gamma\in C\bigl([0, 1], H^{s}\bigl( \mathbb{R}^{3}\bigr)\bigr) : \gamma(0) = 0, \gamma(1) = v\bigr\} . $$


Set \(R = \frac{8 \pi\mu_{0}}{3 S^{2} (\frac{32 \pi}{3} )^{\frac {5}{3}}}\), where

$$\mu_{0} = \sqrt{ \frac{45}{8 \pi} C(3, s)^{-1} S^{2} \biggl(\frac{32 \pi }{3} \biggr)^{\frac{5}{3}}} < \sqrt{ \frac{6}{\pi} C(3, s)^{-1} S^{2} \biggl(\frac{32 \pi}{3} \biggr)^{\frac{5}{3}}}. $$

Denote \(u_{R, \theta} = \theta^{2} u_{R}(\theta x)\), from (2.2), (2.4), Fatou’s lemma, \((f_{2})\) and (2.5)–(2.7), we obtain

$$\begin{aligned} \lim_{\theta\to+ \infty} \frac{\mathcal{I}(u_{R, \theta})}{\theta ^{3}} &= \lim_{\theta\to+ \infty} \frac{1}{\theta^{3}} \biggl(\frac{1}{2} \int_{\mathbb{R}^{3}} \bigl( \bigl\vert (-\Delta)^{\frac{s}{2}} u_{R, \theta} \bigr\vert ^{2} + u_{R, \theta}^{2} \bigr) \,dx \\ &\quad+ \frac{1}{4} \int_{\mathbb{R}^{3}} \phi_{u_{R, \theta}}^{t} u_{R, \theta}^{2} \,dx - \int_{\mathbb{R}^{3}} F(u_{R, \theta}) \,dx \biggr) \\ &\le\lim_{\theta\to+ \infty} \frac{1}{\theta^{3}} \biggl( C(3, s)^{-1} \int_{\mathbb{R}^{3}} \bigl( \vert \nabla u_{R, \theta} \vert ^{2} + u_{R, \theta}^{2} \bigr) \,dx \\ &\quad+ \frac{1}{4} \int_{\mathbb{R}^{3}} \phi_{u_{R, \theta}}^{t} u_{R, \theta}^{2} \,dx - \int_{\mathbb{R}^{3}} F(u_{R, \theta}) \,dx \biggr) \\ &= \lim_{\theta\to+ \infty} \frac{1}{\theta^{3}} \biggl( C(3, s)^{-1} \biggl[\theta^{3} \int_{\mathbb{R}^{3}} \bigl\vert \nabla u_{R}(x) \bigr\vert ^{2} \,dx + \theta \int_{\mathbb{R}^{3}} u_{R}(x)^{2} \,dx \biggr] \\ &\quad + \frac{\theta^{1+2t}}{4} \int_{\mathbb{R}^{3}} \phi_{u_{R}(x)}^{t} u_{R}(x)^{2} \,dx - \int_{\mathbb{R}^{3}} F\bigl(\theta^{2} u_{R}(\theta x)\bigr) \,dx \biggr) \\ &= C(3, s)^{-1} \int_{\mathbb{R}^{3}} \bigl\vert \nabla u_{R}(x) \bigr\vert ^{2} \,dx + \frac{1}{4} \int_{\mathbb{R}^{3}} \phi_{u_{R}(x)}^{t} u_{R}(x)^{2} \,dx \cdot\lim_{\theta\to+ \infty} \frac{1}{\theta ^{2(1-t)}} \\ &\quad- \lim_{\theta\to+ \infty} \int_{\mathbb{R}^{3}} \frac{F(\theta^{2} u_{R})}{ \vert \theta^{2} u_{R} \vert ^{3}} \vert u_{R} \vert ^{3} \,dx \\ &\le \left\{ \textstyle\begin{array}{l@{\quad}l} \frac{28 \pi}{3R} C(3, s)^{-1} + \frac{S^{2}}{4} \vert u_{R} \vert _{\frac{12}{5}}^{4} - \frac{\mu}{3} \int_{\mathbb{R}^{3}} \vert u_{R} \vert ^{3} \,dx, & t = 1, \\ \frac{28 \pi}{3R} C(3, s)^{-1} - \frac{\mu}{3} \int_{\mathbb{R}^{3}} \vert u_{R} \vert ^{3} \,dx, & t < 1 \end{array}\displaystyle \right . \\ &< \frac{28 \pi}{3R} C(3, s)^{-1} + \frac{S^{2}}{4} \vert u_{R} \vert _{\frac{12}{5}}^{4} - \sqrt{ \frac{6}{\pi} C(3, s)^{-1} S^{2} \biggl(\frac{32 \pi}{3} \biggr)^{\frac{5}{3}}} \int_{\mathbb{R}^{3}} \vert u_{R} \vert ^{3} \,dx \\ &< \frac{28 \pi}{3R} C(3, s)^{-1} + \frac{S^{2}}{4} \biggl( \frac{32 \pi }{3} \biggr)^{\frac{5}{3}} R - \frac{4 \pi}{3} \mu_{0} \\ &= - \frac{2 \pi}{3R} C(3, s)^{-1} < 0.\end{aligned} $$

Thus, \(\mathcal{I}(u_{R, \theta}) \le0\) if θ is sufficiently large.

(ii) By \((f_{1})\) and \((f_{2})\), for \(\varepsilon= \frac{1}{4} > 0\), there exists \(C > 0\) such that

$$ f(\theta) \le\frac{1}{4} \theta+ C \theta^{2}. $$

From (2.8) and by using the Sobolev inequality, we obtain

$$\mathcal{I}(u) \ge\frac{1}{2} \Vert u \Vert ^{2} - \frac{1}{4} \vert u \vert _{2}^{2} - C \vert u \vert _{3}^{3} \ge\frac{1}{4} \Vert u \Vert ^{2} - C S_{s,3}^{-\frac{3}{2}} \Vert u \Vert ^{3}, $$


$$S_{s, 3} = \inf_{u \in\mathcal{D}^{s, 2}} \frac{\int_{\mathbb{R}^{3}} ( \vert (-\Delta)^{\frac{s}{2}} u \vert ^{2} + u^{2}) \,dx}{ (\int_{\mathbb{R}^{3}} \vert u(x) \vert ^{3} \,dx )^{\frac{2}{3}} }. $$

For sufficiently small \(\rho> 0\), we have \(I(u) > 0\) with \(\|u\| = \rho\). □

Similar to the proof of Lemma 2.2 in [5], we have the following lemma.

Lemma 2.2

Suppose that\((f_{1})\), \((f_{2})\)hold. If\(\{u_{n}\} \subset H^{s}(\mathbb{R}^{3})\)is a bounded\((PS)_{c\neq0}\)sequence of\(\mathcal{I}\), then there exists\(u_{0} \neq 0\)such that\(\mathcal{I}^{\prime}(u_{0}) = 0\).

3 Main result

Theorem 3.1

Assume the conditions\((f_{1})\)\((f_{3})\)are satisfied, then system (1.1) has at least a ground state solution.


For convenience, we introduce a functional on \(H^{s}(\mathbb{R}^{3})\) as follows:

$$\begin{aligned}[b] \mathcal{J}(u) ={}& \frac{1+2s}{2} \int_{\mathbb{R}^{3}} \bigl\vert (-\Delta)^{\frac {s}{2}} u \bigr\vert ^{2} \,dx + \frac{1}{2} \int_{\mathbb{R}^{3}} u^{2} \,dx + \frac{5-2t}{4} \int_{\mathbb {R}^{3}} \phi_{u}^{t} u^{2} \,dx \\ &+ \int_{\mathbb{R}^{3}} \bigl[(1+2t) F(u) - 2 f(u) u\bigr] \,dx.\end{aligned} $$

Inspired by the idea of Jeanjean [10], we define the map \(\varPsi: \mathbb{R} \times H^{s} (\mathbb{R}^{3}) \to H^{s} (\mathbb{R}^{3})\) by \(\varOmega(\lambda, w)(x) = e^{2 \lambda} w(e^{\lambda} x)\). For each λ and \(w \in H^{s} (\mathbb{R}^{3})\), we can compute the functional \(\mathcal{I} \circ\varOmega\) as follows:

$$\begin{aligned}[b] \mathcal{I}\bigl(\varOmega(\lambda, w)\bigr) = {}&\frac{e^{(1+2s)}\lambda}{2} \int _{\mathbb{R}^{3}} \bigl\vert (-\Delta)^{\frac{s}{2}} w \bigr\vert ^{2} + \frac{e^{\lambda}}{2} \int_{\mathbb{R}^{3}} w^{2} \\ &+ \frac{e^{(5-2t)\lambda}}{4} \int_{\mathbb{R}^{3}} \phi _{w}^{t} w^{2} - \frac{1}{e^{3\lambda}} \int_{\mathbb{R}^{3}} F\bigl(e^{2\lambda} w\bigr).\end{aligned} $$

By (3.2), \((f_{1})\) and \((f_{2})\), we see that \(\mathcal{I} \circ\varOmega\) is continuously Fréchet-differentiable on \(\mathbb{R} \times H^{s}(\mathbb{R}^{3})\). By virtue of Lemma 2.1, there exists \(\lambda^{*} \in\mathbb{R}\) such that \((\mathcal{I} \circ\varOmega)(\lambda^{*}, u_{R}) < 0\). The mountain pass level of \(\mathcal{I} \circ\varOmega\) is given as follows:

$$ \bar{c} = \inf_{\bar{\gamma} \in\bar{\varGamma}} \sup_{t \in[0, 1]} (\mathcal{I} \circ\varOmega) \bigl(\bar{\gamma}(t)\bigr), $$

where the family of paths is denoted by

$$\bar{\varGamma} = \bigl\{ \bar{\gamma} \in C\bigl([0, 1]; \mathbb{R} \times H^{s}(\mathbb{R})\bigr) : \bar{\gamma}(0) = (0, 0), \ (\mathcal{I} \circ\varOmega) \bigl(\bar{\gamma}(1)\bigr) < 0\bigr\} . $$

For \(\varGamma= \{\varOmega\circ\bar{\gamma}: \bar{\gamma} \in\bar{\varGamma }\}\), we have \(c \le\bar{c}\). Obviously, \(\{0\} \times\varGamma\subset\bar{\varGamma}\) and then \(\bar{c} \le c\). Thus, \(\bar{c} = c\). It follows for each \((\eta, u) \in\mathbb {R} \times H^{s}(\mathbb{R}^{3})\) that

$$\begin{gathered}\begin{aligned}[b] \mathcal{I}^{\prime}\bigl(\varOmega(\lambda_{n}, w_{n})\bigr) \bigl[\varOmega(\lambda_{n}, u)\bigr] ={}& e^{(1+2s) \lambda_{n}} \int_{\mathbb{R}^{3}} (-\Delta)^{\frac{s}{2}} w_{n} (- \Delta)^{\frac{s}{2}}u \\ &+ e^{\lambda_{n}} \int_{\mathbb{R}^{3}} w_{n} u + e^{(5-2t)\lambda_{n}} \int _{\mathbb{R}^{3}} \phi_{w_{n}}^{t} w_{n} u - \frac{1}{e^{\lambda_{n}}} \int_{\mathbb{R}^{3}} f\bigl(e^{2 \lambda_{n}} w_{n}\bigr) u,\end{aligned} \\ (\mathcal{I} \circ\varOmega)^{\prime}(\lambda_{n}, w_{n})[\eta, u] = \mathcal {I}^{\prime}\bigl(\varOmega( \lambda_{n}, w_{n})\bigr)\bigl[\varOmega( \lambda_{n}, u)\bigr] + \mathcal{J}\bigl(\varOmega( \lambda_{n}, w_{n})\bigr)\eta.\end{gathered} $$

From Theorem 2.9 of [11], (3.3) and setting \(u_{n} = \varOmega(\lambda_{n}, w_{n})\), one has

$$ \mathcal{I}(u_{n}) \to c > 0, \qquad \mathcal{I}^{\prime }(u_{n}) \to0,\qquad \mathcal{J}(u_{n}) \to0. $$

By (3.4) and \((f_{3})\), we derive that

$$\begin{aligned}[b] &c \ge\mathcal{I}(u_{n}) - \frac{1}{5-2t} \mathcal{J}(u_{n}) + o(1) \\ &\quad= \frac{2-(s+t)}{5-2t} \int_{\mathbb{R}^{3}} \bigl\vert (-\Delta)^{\frac{s}{2}} u_{n} \bigr\vert ^{2} \,dx + \frac{2-t}{5-2t} \int_{\mathbb{R}^{3}} u_{n}^{2} \,dx \\ &\qquad+ \frac{2}{5-2t} \int_{\mathbb{R}^{3}} \bigl[f(u_{n}) u_{n} - 3 F(u_{n})\bigr] \,dx + o(1) \\ &\quad\ge\frac{2-t}{5-2t} \int_{\mathbb{R}^{3}} u_{n}^{2} \,dx + o(1),\end{aligned} $$

which implies that \(\{u_{n}\}\) is bounded in \(L^{2}(\mathbb{R}^{3})\). According to (3.4), we obtain

$$\begin{aligned}[b] \int_{\mathbb{R}^{3}} f(u_{n}) u_{n} \,dx + o\bigl( \Vert u_{n} \Vert \bigr) &= \int_{\mathbb{R}^{3}} \bigl( \bigl\vert (-\Delta)^{\frac{s}{2}} u_{n} \bigr\vert ^{2} + u_{n}^{2} \bigr) \,dx + \int_{\mathbb{R}^{3}} \phi_{u_{n}}^{t} u_{n}^{2} \,dx \\ &\ge \int_{\mathbb{R}^{3}} \bigl\vert (-\Delta)^{\frac{s}{2}} u_{n} \bigr\vert ^{2} \,dx.\end{aligned} $$

Combining \((f_{1})\) and \((f_{2})\), we have

$$ f(\tau) \tau\le C \vert \tau \vert ^{3} + \varepsilon \tau^{2}. $$

By means of (3.7), the interpolation and Sobolev inequalities, we get

$$\begin{aligned}[b] \int_{\mathbb{R}^{3}} f(u_{n}) u_{n} \,dx &\le C \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{3} \,dx + \varepsilon \int_{\mathbb{R}^{3}} u_{n}^{2} \,dx \\ &\le C \biggl( \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{2} \,dx \biggr)^{\frac{6s-3}{4s}} \cdot \biggl( \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{2_{s}^{*}} \,dx \biggr)^{\frac{3-2s}{4s}} + C \varepsilon \\ &\le C \biggl( \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{2} \,dx \biggr)^{\frac{6s-3}{4s}} \cdot \biggl( \int_{\mathbb{R}^{3}} \bigl\vert (-\Delta)^{\frac {s}{2}} u_{n} \bigr\vert ^{2} \,dx \biggr)^{\frac{3}{4s}} + C \varepsilon.\end{aligned} $$

Since \(s > \frac{3}{4}\), by (3.6) and (3.8), we know that \(\{(-\Delta )^{\frac{s}{2}} u_{n}\}\) is bounded in \(L^{2}(\mathbb{R}^{3})\). Hence, \(\{ u_{n}\}\) is bounded in \(H^{s}(\mathbb{R}^{3})\).


$$m = \inf_{M} \mathcal{I}(u), \quad M = \bigl\{ u \in H^{s}\bigl(\mathbb{R}^{3}\bigr) \setminus\{ 0\} | \mathcal{I}^{\prime}(u) = 0\bigr\} . $$

In view of Lemma 2.2 and \(\{u_{n}\}\) being bounded, we obtain \(u_{0} \neq 0\) and \(\mathcal{I}^{\prime}(u_{0}) = 0\). Thus M is not empty and \(0 \le m \le\mathcal{I}(u_{0})\). In the following we will prove m can be achieved in M. Suppose that \(\{ u_{n}\}\) is a sequence of nontrivial critical points of \(\mathcal{I}\) satisfying \(\mathcal{I}(u_{n}) \to m\). Similar to the proofs of (3.5), (3.6) and (3.8), we find that \(\{u_{n}\}\) is bounded in \(H^{s}(\mathbb {R}^{3})\). By \(\mathcal{I}^{\prime}(u_{n}) u_{n} = 0\), (2.8) and the Sobolev inequality, we have

$$\begin{aligned}[b] \Vert u_{n} \Vert ^{2}& = \int_{\mathbb{R}^{3}} f(u_{n}) u_{n} \,dx - \int_{\mathbb{R}^{3}} \phi_{u_{n}}^{t} u_{n}^{2} \,dx \\ &\le\frac{1}{4} \vert u_{n} \vert _{2}^{2} + C \vert u_{n} \vert _{3}^{3} \le \frac{1}{4} \Vert u_{n} \Vert ^{2} + C S_{s,3}^{-\frac{3}{2}} \Vert u_{n} \Vert ^{3}.\end{aligned} $$

From (3.9), there exists a positive \(\rho> 0\) such that

$$ \lim\inf_{n \to\infty} \Vert u_{n} \Vert \ge\rho> 0. $$

Applying the Lions lemma in [11], if \(\{u_{n}\}\) vanishes, one has \(u_{n} \to0\) in \(L^{q}(\mathbb{R}^{3})\) for all \(q \in(2, 6)\). Thus, if \(t > \frac{1}{2}\), then it follows (12) in [2] that

$$\int_{\mathbb{R}^{3}} \phi_{u_{n}}^{t} u_{n}^{2} \,dx \le S_{t}^{\frac{1}{2}} \vert u_{n} \vert _{\frac{12}{3+2t}}^{4} \to0. $$

By (3.7) and \(u_{n} \to0\) in \(L^{q}(\mathbb{R}^{3})\), we get \(\int_{\mathbb {R}^{3}} f(u_{n}) u_{n} \,dx \to0\). Combining with (3.9), we can easily deduce that \(\|u_{n}\| \to0\) in a contradiction with (3.10). Hence there exist \(r, \delta> 0\) and a sequence \(\{y_{n}\} \subset\mathbb{R}^{3}\) such that \(\lim_{n \to\infty} \sup_{y_{n} \in\mathbb{R}^{3}} \int_{B_{r}(y_{n})} |u_{n}|^{2} \ge\delta> 0\). Set \(\bar{u}_{n} = u_{n}(x+y_{n})\), then we have \(\bar{u}_{n} \rightharpoonup u \neq 0\) in \(H^{s}(\mathbb{R}^{3})\), \(\mathcal {I}(\bar{u}_{n}) \to m\) and \(\mathcal{I}^{\prime}(\bar{u}_{n}) = 0\). Thus, we get \(\mathcal{I}^{\prime}(u) = 0\) and \(\mathcal{I}(u) \ge m\). Since \(\mathcal{I}^{\prime}(u) = 0\), one has

$$ \int_{\mathbb{R}^{3}} \bigl( \bigl\vert (-\Delta)^{\frac{s}{2}} u \bigr\vert ^{2} + u^{2} \bigr) \,dx + \int_{\mathbb{R}^{3}} \phi_{u}^{t} u^{2} \,dx - \int_{\mathbb{R}^{3}} f(u) u \,dx = 0, $$

and by the Pohožaev identity [2, 12], we have

$$ \frac{3-2s}{2} \int_{\mathbb{R}^{3}} \bigl\vert (-\Delta)^{\frac {s}{2}} u \bigr\vert ^{2} \,dx + \frac{3}{2} \int_{\mathbb{R}^{3}} u^{2} \,dx + \frac{3+2t}{4} \int_{\mathbb{R}^{3}} \phi_{u}^{t} u^{2} \,dx = 3 \int_{\mathbb {R}^{3}} F(u) \,dx. $$

According to (3.11)–(3.12), one has \(\mathcal{J}(u) = 0\). As \(\mathcal {I}^{\prime}(\bar{u}_{n}) = 0\), \(\mathcal{I}^{\prime}(u) = 0\) and by Fatou’s lemma we have

$$\begin{aligned} m &= \lim_{n \to\infty} \biggl(\frac{2-(s+t)}{5-2t} \int_{\mathbb {R}^{3}} \bigl\vert (-\Delta)^{\frac{s}{2}} \bar{u}_{n} \bigr\vert ^{2} \,dx + \frac{2-t}{5-2t} \int_{\mathbb{R}^{3}} \bar{u}_{n}^{2} \,dx \\ &\quad + \frac{2}{5-2t} \int_{\mathbb{R}^{3}} \bigl[f(\bar{u}_{n}) \bar{u}_{n} - 3 F(\bar{u}_{n})\bigr] \,dx \biggr) \\ &\ge\frac{2-(s+t)}{5-2t} \int_{\mathbb{R}^{3}} \bigl\vert (-\Delta)^{\frac{s}{2}} u \bigr\vert ^{2} \,dx + \frac{2-t}{5-2t} \int_{\mathbb{R}^{3}} u^{2} \,dx \\ &\quad+ \frac{2}{5-2t} \int_{\mathbb{R}^{3}} \bigl[f(u) u - 3 F(u)\bigr] \,dx \\ &= \mathcal{I}(u) - \frac{1}{5-2t} \mathcal{J}(u) = \mathcal{I}(u) \ge m.\end{aligned} $$

Hence \(\mathcal{I}(u) = m\). The proof is complete. □

Remark 3.1

If \(s = t =1\), Our main result Theorem 3.1 reduces to Theorem 1.1 in [5]. On the other hand, Theorem 3.1 in this paper relaxes the condition of super-quadratic nonlinearity in [1] to being asymptotically 2-linear.


  1. Gao, Z., Tang, X., Chen, S.: Ground state solutions for a class of nonlinear fractional Schrödinger–Poisson systems with super-quadratic nonlinearity. Chaos Solitons Fractals 105, 189–194 (2017)

    Article  MathSciNet  Google Scholar 

  2. Teng, K.M.: Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent. J. Differ. Equ. 261, 3061–3106 (2016)

    Article  Google Scholar 

  3. He, Y., Jing, L.: Existence and multiplicity of non-trivial solutions for the fractional Schrödinger–Poisson system with superlinear terms. Bound. Value Probl. 2019, Article ID 25 (2019)

    Article  Google Scholar 

  4. Wang, D.B., Ma, Y.M., Guan, W.: Least energy sign-changing solutions for the fractional Schrödinger–Poisson systems in \(\mathbb{R}^{3}\). Bound. Value Probl. 2019, Article ID 4 (2019)

    Article  Google Scholar 

  5. Yin, L.F., Wu, X.P., Tang, C.L.: Ground state solutions for an asymptotically 2-linear Schrödinger–Poisson system. Appl. Math. Lett. 87, 7–12 (2019)

    Article  MathSciNet  Google Scholar 

  6. Azzollini, A., Pomponio, A.: A note on the ground state solutions for the nonlinear Schrödinger–Maxwell equations. Boll. Unione Mat. Ital. (9) 2(1), 93–104 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Azzollini, A., d’Avenia, P., Pompomio, A.: On the Schrödinger–Maxwell equations under the effiect of a general nonlinear term. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27(2), 779–791 (2010)

    Article  MathSciNet  Google Scholar 

  8. Sun, J.J., Ma, S.W.: Ground state solutions for some Schrödinger–Poisson systems with periodic potentials. J. Differ. Equ. 260(3), 2119–2149 (2016)

    Article  Google Scholar 

  9. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  10. Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28(10), 1633–1659 (1997)

    Article  MathSciNet  Google Scholar 

  11. Willem, M.: Minimax Theorems. Progr. Nonlinear Differential Equations Appl., vol. 24. Birkhäuser, Boston (1996)

    Book  Google Scholar 

  12. Teng, K.M.: Ground state solutions for the non-linear fractional Schrödinger–Poisson system. Appl. Anal. 98, 1959–1996 (2019)

    Article  MathSciNet  Google Scholar 

Download references


The authors would like to thank the referees for their careful reading of the manuscript and for helpful suggestions which improved the quality of the paper.

Availability of data and materials

Not applicable.


This work is supported by the Natural Science Foundation of China (11571136).

Author information

Authors and Affiliations



The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Chuanzhi Bai.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information


Not applicable.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, D., Bai, C. Existence of ground state solutions for an asymptotically 2-linear fractional Schrödinger–Poisson system. Bound Value Probl 2020, 5 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: