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Existence of ground state solutions for an asymptotically 2-linear fractional Schrödinger–Poisson system
Boundary Value Problems volume 2020, Article number: 5 (2020)
Abstract
In this paper, we investigate the following fractional Schrödinger–Poisson system:
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:
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:
- \((f_{1})\):
\(f \in C(\mathbb{R}, \mathbb{R})\), \(\lim_{\tau\to0} \frac {f(\tau)}{\tau} = 0\);
- \((f_{2})\):
- $$\lim_{ \vert \tau \vert \to\infty} \frac{f(\tau)}{ \vert \tau \vert ^{2}} = \mu \quad \text{with } \sqrt{ \frac{54}{\pi} C(3, s)^{-1} S^{2} \biggl(\frac{32 \pi}{3} \biggr)^{\frac{5}{3}}} < \mu< + \infty, $$
where the constant S and the function \(C(3, s)\) will be specified in Sect. 2;
- \((f_{3})\):
- $$f(\tau) \tau \ge3 F(\tau) \ge0, \quad\forall\tau \in\mathbb {R}, \text{where } F(\tau) = \int_{0}^{\tau} f(s) \,ds. $$
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:
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:
Yin, Wu and Tang [5] proved the existence of ground state solutions of (1.2) by using
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 [6–8].
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.
endowed with the norm
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:
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:
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:
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
Hence, if u is a critical point of \(\mathcal{I}\), then \((u, \phi _{u}^{t})\) is a solution of (1.1).
Set
Hence \(u_{R} \in H^{s}(\mathbb{R}^{3})\). By Proposition 3.4 in [9], we have
where \(\mathcal{F}\) is the usual Fourier transform in \(\mathbb{R}^{3}\), and
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
If \(t > \frac{1}{2}\), then we have by Lemma 2.3 in [2]
where
Remark 2.1
If \(t = 1\), then the above inequalities modifies to the following inequalities:
where
From [5], we have
and
Lemma 2.1
If\((f_{1})\)and\((f_{2})\)hold, then
- (i)
there exists a\(v \in H^{s}(\mathbb{R}^{3}) \setminus\{0\}\)such that\(\mathcal{I}(v) \le0\);
- (ii)
\(c := \inf_{\gamma\in\varGamma} \max_{t \in[0, 1]} \mathcal {I}(\gamma(t)) > 0\), where
$$\varGamma= \bigl\{ \gamma\in C\bigl([0, 1], H^{s}\bigl( \mathbb{R}^{3}\bigr)\bigr) : \gamma(0) = 0, \gamma(1) = v\bigr\} . $$
Proof
Set \(R = \frac{8 \pi\mu_{0}}{3 S^{2} (\frac{32 \pi}{3} )^{\frac {5}{3}}}\), where
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
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
From (2.8) and by using the Sobolev inequality, we obtain
where
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.
Proof
For convenience, we introduce a functional on \(H^{s}(\mathbb{R}^{3})\) as follows:
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:
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:
where the family of paths is denoted by
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
From Theorem 2.9 of [11], (3.3) and setting \(u_{n} = \varOmega(\lambda_{n}, w_{n})\), one has
By (3.4) and \((f_{3})\), we derive that
which implies that \(\{u_{n}\}\) is bounded in \(L^{2}(\mathbb{R}^{3})\). According to (3.4), we obtain
Combining \((f_{1})\) and \((f_{2})\), we have
By means of (3.7), the interpolation and Sobolev inequalities, we get
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 \(H^{s}(\mathbb{R}^{3})\).
Define
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
From (3.9), there exists a positive \(\rho> 0\) such that
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
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
and by the Pohožaev identity [2, 12], we have
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
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.
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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.
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This work is supported by the Natural Science Foundation of China (11571136).
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Yang, D., Bai, C. Existence of ground state solutions for an asymptotically 2-linear fractional Schrödinger–Poisson system. Bound Value Probl 2020, 5 (2020). https://doi.org/10.1186/s13661-019-01314-2
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DOI: https://doi.org/10.1186/s13661-019-01314-2