Abstract
In this paper, we are concerned with the existence of the least energy sign-changing solutions for the following fractional Schrödinger–Poisson system:
where \(\lambda \in {\mathbb {R}}^{+}\) is a parameter, \(s, t\in (0, 1)\) and \(4s+2t>3\), \((-\Delta )^{s}\) stands for the fractional Laplacian. By constraint variational method and quantitative deformation lemma, we prove that the above problem has one least energy sign-changing solution. Moreover, for any \(\lambda >0\), we show that the energy of the least energy sign-changing solutions is strictly larger than two times the ground state energy. Finally, we consider \(\lambda \) as a parameter and study the convergence property of the least energy sign-changing solutions as \(\lambda \searrow 0\).
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1 Introduction
In this article, we are interested in the existence, energy property of the least energy sign-changing solution \(u_{\lambda }\) and a convergence property of \(u_{\lambda }\) as \(\lambda \searrow 0\) for the nonlinear fractional Schrödinger–Poisson system
where \(\lambda >0\) is a parameter, \(s, t\in (0, 1)\) and \(4s+2t>3\), \((-\Delta )^{s}\) stands for the fractional Laplacian and the potential V(x) satisfies the following assumptions:
- \((V_{1})\):
\(V\in C({\mathbb {R}}^{3})\) satisfies \({\inf }_{x\in {\mathbb {R}}^{3}}V(x)\ge V_{0}>0\), where \(V_{0}\) is a positive constant;
- \((V_{2})\):
There exists \(h>0\) such that \({\lim _{\vert y\vert \rightarrow \infty }}\text {meas}(\{x\in B_{h}(y): V(x)\le c\})=0\) for any \(c>0\);
where \(B_{h}(y)\) denotes an open ball of \({\mathbb {R}}^{3}\) centered at y with radius \(h>0\), and \(\text {meas}(A)\) denotes the Lebesgue measure of set A. Condition \((V_{2})\), which is weaker than the coercivity assumption: \(V(x)\rightarrow \infty \) as \(\vert x\vert \rightarrow \infty \), was originally introduced by Bartsch and Wang [1] to overcome the lack of compactness for the local elliptic equations and then was used by Pucci, Xia and Zhang [18] for the fractional Schrödinger–Kirchhoff type equations. Moreover, on the nonlinearity f, we assume that
- \((f_1)\):
\(f: {\mathbb {R}}^{3}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function and \(f(x, u)=o(\vert u\vert )\) as \(u\rightarrow 0\) for \(x\in {\mathbb {R}}^{3}\) uniformly;
- \((f_2)\):
For some \(1<p<2_{s}^{*}-1\), there exits \(C>0\) such that
$$\begin{aligned} \vert f(x, u)\vert \le C(1+\vert u\vert ^{p}), \end{aligned}$$where \(2_{s}^{*}=\frac{6}{3-2s}\);
- \((f_3)\):
\({\lim }_{\vert u\vert \rightarrow \infty }\frac{F(x, u)}{u^{4}}=+\infty \), where \( F(x, u)=\int _{0}^{u}f(x, s)\hbox {d}s\);
- \((f_4)\):
\(\frac{f(x, t)}{\vert t\vert ^{3}}\) is an increasing function of t on \({\mathbb {R}}\setminus \{0\}\) for a.e. \(x\in {\mathbb {R}}^{3}\).
When \(s=t=1\), the system (1.1) reduces to the following Schrödinger–Poisson system
This kind of system has a strong physical meaning. For instance, they appear in quantum mechanics models [4, 6] and in semiconductor theory [2, 3]. For the research of Schrödinger–Poisson system, we may refer to [9, 10, 13, 19, 23].
In recent years, there has been a great deal works dealing with the nonlinear equations or systems involving fractional Laplacian, which arise in fractional quantum mechanics [11, 12], physics and chemistry [14], obstacle problems [21], optimization and finance [8] and so on. In the remarkable work of Caffarelli and Silvestre [5], the authors express this nonlocal operator \((-\Delta )^{s}\) as a Dirichlet–Neumann map for a certain elliptic boundary value problem with local differential operators defined on the upper half space. This technique is a valid tool to deal with the equations involving fractional operators in the respects of regularity and variational methods. For some results on the fractional differential equations, we refer to [7, 16, 18, 25, 26]. Recently, using Caffarelli and Silvestre’s method in [5] and variational method, in [22], Teng studied the ground state for the fractional Schrödinger–Poisson system with critical Sobolev exponent. To the best of our knowledge, there are few papers which considered the least energy sign-changing solutions of system (1.1). In [20], Combining constraint variational methods and quantitative deformation lemma, Shuai firstly studied the least energy sign-changing solutions for a class of Kirchhoff problems in bounded domain, where a stronger condition that \(f\in C^{1}\) was assumed. In virtue of the fractional operator and Poisson equation which are included in (1.1), our problem is more complicated and difficult.
Now, we recall some theory of the fractional Sobolev spaces. We firstly define the homogeneous fractional Sobolev space \({\mathcal {D}}^{\alpha , 2}({\mathbb {R}}^{3})(\alpha \in (0, 1))\) as follows
which is the completion of \({\mathcal {C}}_{0}^{\infty }({\mathbb {R}}^{3})\) under the norm
The embedding \( {\mathcal {D}}^{\alpha , 2}({\mathbb {R}}^{3})\hookrightarrow L^{2_{\alpha }^{*}}\) is continuous and there exists a best constant \({\mathcal {S}}_{\alpha }>0\) such that
The fractional Sobolev space \(H^{\alpha }({\mathbb {R}}^{3})\) is defined by
endowed with the norm
In this paper, we denote the fractional Sobolev space for (1.1) by
with the norm
In the sequel, we need the following embedding lemma which is a special case of Lemma 1 in [18], so we omit its proof.
Lemma 1.1
(i) Suppose that \((V_{1})\) holds. Let \(q\in [2, 2_{s}^{*}]\), then the embeddings
are continuous, with \(\min \{1, V_{0}\}\Vert u\Vert _{H^s({\mathbb {R}}^{3})}^{2}\le \Vert u\Vert ^{2}\) for all \(u\in H\). In particular, there exists a constant \(C_{q}>0\) such that
Moreover, if \(q\in [1, 2_{s}^{*})\), then the embedding \(H\hookrightarrow \hookrightarrow L^{q}(B_{R})\) is compact for any \(R>0\).
(ii) Suppose that \((V_{1})-(V_{2})\) hold. Let \(q\in [2, 2_{s}^{*})\) be fixed and \(\{u_{n}\}_{n}\) be a bounded sequence in H, then there exists \(u\in H\cap L^{q}({\mathbb {R}}^{N})\) such that, up to a subsequence,
Assume that \(s, t\in (0, 1)\), if \(4s+2t\ge 3\), there holds \(2\le \frac{12}{3+2t} \le \frac{6}{3-2s}\) and thus \(H\hookrightarrow L^{\frac{12}{3+2t}}({\mathbb {R}}^{3})\) by Lemma 1.1. For \(u\in H\), the linear functional \({\mathcal {L}}_{u}:{\mathcal {D}}^{t, 2}({\mathbb {R}}^{3})\rightarrow {\mathbb {R}}\) is defined by
the Hölder’s inequality and (1.2) implies that
By the Lax–Milgram theorem, there exists a unique \(\phi _{u}^{t}\in {\mathcal {D}}^{t, 2}({\mathbb {R}}^{3})\) such that
that is \(\phi _{u}^{t}\) is the weak solution of
and the representation formula holds
which is called t-Riesz potential, where
In the sequel, we often omit the constant \(c_{t}\) for convenience in (1.3). The properties of the function \(\phi _{u}^{t}\) are given as follows.
Lemma 1.2
([22]) If \(4s+2t\ge 3\), then for any \(u\in H^s({\mathbb {R}}^{3})\), we have
- (1)
\(\phi _{u}^{t}: H^s({\mathbb {R}}^{3})\rightarrow {\mathcal {D}}^{t, 2}({\mathbb {R}}^{3})\) is continuous and maps bounded sets into bounded maps;
- (2)
\(\int _{{\mathbb {R}}^{3}}\phi _{u}^{t}{u}^{2}\hbox {d}x\le {\mathcal {S}}_{t}^{2}\Vert u\Vert ^{4}_{L^{\frac{12}{3+2t}}}\);
- (3)
\(\phi _{\tau u}^{t}=\tau ^{2}\phi _{u}^{t}\) for all \(\tau \in {\mathbb {R}}\), \(\phi _{u(\cdot +y)}^{t}=\phi _{u}^{t}(x+y)\);
- (4)
\(\phi _{u_{\theta }}=\theta ^{2s}(\phi _{u}^{t})_{\theta }\) for all \(\theta >0\), where \(u_{\theta }=u(\frac{\cdot }{\theta });\)
- (5)
If \(u_{n}\rightharpoonup u\) in \(H^s({\mathbb {R}}^{3}),\) then \(\phi _{u_{n}}^{t}\rightharpoonup \phi _{u}^{t}\) in \({\mathcal {D}}^{t, 2}({\mathbb {R}}^{3})\);
- (6)
If \(u_{n}\rightarrow u\) in \(H^s({\mathbb {R}}^{3}),\) then \(\phi _{u_{n}}^{t}\rightarrow \phi _{u}^{t}\) in \({\mathcal {D}}^{t, 2}({\mathbb {R}}^{3})\) and \(\int _{{\mathbb {R}}^{3}}\phi _{u_{n}}^{t}{u_{n}}^{2}\hbox {d}x\rightarrow \int _{{\mathbb {R}}^{3}}\phi _{u}^{t}{u}^{2}\hbox {d}x\).
If we substitute \(\phi _{u}^{t}\) in (1.1), it leads to the following fractional Schrödinger equation
whose solutions are the critical points of the functional \(I_{\lambda }: H\rightarrow {\mathbb {R}}\) defined by
where \(F(x, u)=\int _{0}^{u}f(x, r)\hbox {d}r\). The functional \(I_{\lambda }\in C^{1}(H, {\mathbb {R}})\) and for any \(v\in H\)
We call u a least energy sign-changing solution to problem (1.1) if u is a solution of problem (1.4) with \(u^{\pm }\ne 0\) and
where \(v^{+}=\max \{v(x), 0\}\) and \(v^{-}=\min \{v(x), 0\}\).
For problem (1.4), due to the effect of the nonlocal term \(\phi _{u}^{t}\) and \((-\Delta )^{s} u\), that is
for \(u^{\pm }\ne 0\), a straightforward computation yields that
So the methods to obtain sign-changing solutions of the local problems and to estimate the energy of the sign-changing solutions seem not suitable for our nonlocal one (1.4). In order to get a sign-changing solution for problem (1.4), we firstly try to seek a minimizer of the energy functional \(I_{\lambda }\) over the following constraint:
and then we show that the minimizer is a sign-changing solution of (1.4). To show that the minimizer of the constrained problem is a sign-changing solution, we will use the quantitative deformation lemma and degree theory.
The following are the main results of this paper.
Theorem 1.1
Let \((f_{1})-(f_{4})\) and \((V_{1})-(V_{2})\) hold. Then, for any \( \lambda >0\), problem (1.1) has a least energy sign-changing solution \(u_{\lambda }\), which has precisely two nodal domains.
Let
and
Let \(u_{\lambda }\in H\) be a sign-changing solution of problem (1.4), it is clear from (1.5) and (1.6) that \(u_{\lambda }^{\pm }\not \in {\mathcal {N}}_{\lambda }\).
Theorem 1.2
Under the assumptions of Theorem 1.1, \(c_{\lambda }>0\) is achieved and \(I_{\lambda }(u_{\lambda })>2c_{\lambda }\), where \(u_{\lambda }\) is the least energy sign-changing solution obtained in Theorem 1.1. In particular, \(c_{\lambda }>0\) is achieved either by a nonpositive or a nonnegative function.
It is clear that the energy of the sign-changing solution \(u_{\lambda }\) obtained in Theorem 1.1 depends on \(\lambda \). Furthermore, we give a convergence property of \(u_{\lambda }\) as \(\lambda \searrow 0\), which reflects some relationship between \(\lambda >0\) and \(\lambda =0\) in problem (1.4).
Theorem 1.3
If the assumptions of Theorem 1.1 hold, then for any sequence \(\{\lambda _{n}\}_{n}\) with \(\lambda _{n}\searrow 0\) as \(n\rightarrow \infty \), there exists a subsequence, still denoted by \(\{\lambda _{n}\}_{n}\), such that \(u_{\lambda _{n}}\rightarrow u_{0}\) strongly in H as \(n\rightarrow \infty \), where \(u_{0}\) is a least energy sign-changing solution of the problem
which has precisely two nodal domains.
This paper is organized as follows. In Sect. 2, we present some preliminary lemmas which are essential for this paper. In Sect. 3, we give the proofs of Theorems 1.1–1.3, respectively.
2 Some technical lemmas
We will use constraint minimization on \({\mathcal {M}}_{\lambda }\) to look for a critical point of \(I_{\lambda }\). For this, we start with this section by claiming that the set \({\mathcal {M}}_{\lambda }\) is nonempty in H.
Lemma 2.1
Assume that \((f_{1})-(f_{4})\) and \((V_{1})\) hold, if \(u\in H\) with \(u^{\pm }\ne 0\), then there exists a unique pair \((\alpha _{u}, \beta _{u})\in {\mathbb {R}}_{+}\times {\mathbb {R}}_{+}\) such that \(\alpha _{u}u^{+}+\beta _{u}u^{-}\in {\mathcal {M}}_{\lambda }\).
Proof
Fixed an \(u\in H\) with \(u^{\pm }\ne 0\). We first establish the existence of \(\alpha _{u}\) and \(\beta _{u}\). Let
and
By \((f_{1})\) and \((f_{3})\), it is easy to see that \(g_{1}(\alpha , \alpha )>0\) and \(g_{2}(\alpha , \alpha )>0\) for \(\alpha >0\) small and \(g_{1}(\beta , \beta )<0\) and \(g_{2}(\beta , \beta )<0\) for \(\beta >0\) large. Thus, there exist \(0<r<R\) such that
From (2.1), (2.2) and (2.3), we have
and
By virtue of Miranda’s Theorem [15], there exists some point \((\alpha _{u}, \beta _{u})\) with \(r<\alpha _{u}, \beta _{u}<R\) such that \(g_{1}(\alpha _{u}, \beta _{u})=g_{2}(\alpha _{u}, \beta _{u})=0\). So \(\alpha _{u}u^{+}+\beta _{u}u^{-}\in {\mathcal {M}}_{\lambda }\).
Now, we prove the uniqueness of the pair \((\alpha _{u}, \beta _{u})\) and consider two cases.
Case 1\(u\in {\mathcal {M}}_{\lambda }\).
If \(u\in {\mathcal {M}}_{\lambda }\), then \(u^{+}+u^{-}=u\in {\mathcal {M}}_{\lambda }\). It means that
that is
and
We show that \((\alpha _{u}, \beta _{u})=(1, 1)\) is the unique pair of numbers such that \(\alpha _{u}u^{+}+\beta _{u}u^{-}\in {\mathcal {M}}_{\lambda }\).
Assume that \(({\tilde{\alpha }}_{u}, {\tilde{\beta }}_{u})\) is another pair of numbers such that \({\tilde{\alpha }}_{u}u^{+}+{\tilde{\beta }}_{u}u^{-}\in {\mathcal {M}}_{\lambda }\). By the definition of \({\mathcal {M}}_{\lambda }\), we have
and
Without loss of generality, we may assume that \(0<{\tilde{\alpha }}_{u}\le {\tilde{\beta }}_{u}\). Then, from (2.6), we have
Moreover, dividing the above inequality by \({\tilde{\alpha }}_{u}^{-4}\), we have
By \((f_{4})\) and (2.9), it implies that \(1\le {\tilde{\alpha }}_{u}\le {\tilde{\beta }}_{u}\). By (2.7) and the same method, we have that
It is easy to see that \({\tilde{\beta }}_{u}\le 1\). This together with \(1\le {\tilde{\alpha }}_{u}\le {\tilde{\beta }}_{u}\) shows that \({\tilde{\alpha }}_{u}={\tilde{\beta }}_{u}=1\).
Case 2\(u\not \in {\mathcal {M}}_{\lambda }\).
If \(u\not \in {\mathcal {M}}_{\lambda }\), then there exists a pair of positive numbers \((\alpha _{u}, \beta _{u})\) such that \(\alpha _{u}u^{+}+\beta _{u}u^{-}\in {\mathcal {M}}_{\lambda }\). Suppose that there exists another pair of positive numbers \((\alpha '_{u}, \beta '_{u})\) such that \(\alpha '_{u}u^{+}+\beta '_{u}u^{-}\in {\mathcal {M}}_{\lambda }\). Set \(v:=\alpha _{u}u^{+}+\beta _{u}u^{-}\) and \(v':=\alpha '_{u}u^{+}+\beta '_{u}u^{-}\), we have
Since \(v\in {\mathcal {M}}_{\lambda }\), we obtain that \(\alpha _{u}=\alpha '_{u}\) and \(\beta _{u}=\beta '_{u}\), which implies that \((\alpha _{u}, \beta _{u})\) is the unique pair of numbers such that \(\alpha _{u}u^{+}+\beta _{u}u^{-}\in {\mathcal {M}}_{\lambda }\). The proof is completed. \(\square \)
Lemma 2.2
Assume that \((f_{1})-(f_{4})\) and (V) hold for a fixed \(u\in H\) with \(u^{\pm }\ne 0\). If \(\langle I_{\lambda }'(u), u^{+}\rangle \le 0\) and \(\langle I_{\lambda }'(u), u^{-}\rangle \le 0\), then there exists a unique pair \((\alpha _{u}, \beta _{u})\in (0, 1]\times (0, 1]\) such that \(\langle I_{\lambda }'(\alpha _{u}u^{+}+ \beta _{u}u^{-} ), \alpha _{u}u^{+}\rangle =\langle I_{\lambda }'(\alpha _{u}u^{+}+ \beta _{u}u^{-}), \beta _{u}u^{-}\rangle =0\).
Proof
For \(u\in H\) with \(u^{\pm }\ne 0\), by Lemma 2.2, we know that there exist \(\alpha _{u}\) and \(\beta _{u}\) such that \(\alpha _{u}u^{+}+\beta _{u}u^{-}\in {\mathcal {M}}_{\lambda }\). Without loss of generality, suppose that \(\alpha _{u}\ge \beta _{u}>0\). Moreover, we have
Since \(\langle I_{\lambda }'(u), u^{+}\rangle \le 0\), it yields that
Combine (2.10) and (2.11), we have
If \(\alpha _{u}>1\), the left-hand side of this inequality is negative. But from \((f_{4})\), the right-hand side of this inequality is positive, so have \(\alpha _{u}\le 1\). The proof is thus complete. \(\square \)
Lemma 2.3
For a fixed \(u\in H\) with \(u^{\pm }\ne 0\), then \((\alpha _{u}, \beta _{u})\) obtained in Lemma 2.1 is the unique maximum point of the function \(\kappa :{\mathbb {R}}_{+}\times {\mathbb {R}}_{+}\rightarrow {\mathbb {R}}\) defined as \(\kappa (\alpha , \beta )=I_{\lambda }(\alpha u^{+}+\beta u^{-})\).
Proof
From the proof of Lemma 2.1, we know that \((\alpha _{u}, \beta _{u})\) is the unique critical point of \(\kappa \) in \({\mathbb {R}}_{+} \times {\mathbb {R}}_{+}\). By \((f_{3})\), we conclude that \(\kappa (\alpha , \beta )\rightarrow -\infty \) uniformly as \(\vert (\alpha , \beta )\vert \rightarrow \infty \), so it is sufficient to show that a maximum point cannot be achieved on the boundary of \(({\mathbb {R}}_{+}, {\mathbb {R}}_{+})\). If we assume that \((0, {\bar{\beta }})\) is a maximum point of \(\kappa \) with \({\bar{\beta }}\ge 0\). Then, since
is an increasing function with respect to \(\alpha \) if \(\alpha \) is small enough, the pair \((0, {\bar{\beta }})\) is not a maximum point of \(\kappa \) in \({\mathbb {R}}_{+}\times {\mathbb {R}}_{+}\). The proof is now finished. \(\square \)
By Lemma 2.1, we define the minimization problem
Lemma 2.4
Assume that \((f_{1})-(f_{4})\) and \((V_{1})-(V_{2})\) hold, then \(m_{\lambda }>0\) can be achieved for any \(\lambda >0\).
Proof
For every \(u\in {\mathcal {M}}_{\lambda }\), we have \(\langle I'_{\lambda }(u), u\rangle =0\). From \((f_{1})\), \((f_{2})\), for any \(\epsilon >0\), there exists \(C_{\epsilon }>0\) such that
By Sobolev embedding theorem, we get
Pick \(\epsilon =\frac{1}{2C_{2}}\). So there exists a constant \(\gamma >0\) such that \(\Vert u\Vert ^{2}>\gamma \).
By \((f_{4})\), we have
then
This implies that \(I_{\lambda }(u)\) is coercive in \({\mathcal {M}}_{\lambda }\) and \(m_{\lambda }\ge \frac{\gamma }{4}>0\).
Let \(\{u_{n}\}_n\subset {\mathcal {M}}_{\lambda }\) be such that \( I_{\lambda }(u_{n})\rightarrow m_{\lambda }\). Then \(\{u_{n}\}_n\) is bounded in H by (2.14). Using Lemma 1.1, up to a subsequence, we have
Moreover, the conditions \((f_{1})\), \((f_{2})\) and Lemma 1.1 imply that
Since \(u_{n}\in {\mathcal {M}}_{\lambda }\), we have \(\langle I'_{\lambda }(u_{n}), u_{n}^{\pm }\rangle =0\), that is
and
Similar to (2.12) and (2.13), we also have \(\Vert u_{n}^{\pm }\Vert ^{2}\ge \delta \) for all \(n\in N\), where \(\delta \) is a constant.
Since \(u_{n}\in {\mathcal {M}}_{\lambda }\), by (2.17) and (2.18) again, we have
Using the boundedness of \(\{u_{n}\}_n\), there is \(C_{2}>0\) such that
Choosing \(\epsilon =\delta /(2C_{2})\), we get
where \({\bar{C}}\) is a positive constant, in fact, \({\bar{C}}=C_{\frac{\delta }{2C_{2}}}\).
By (2.19) and Lemma 1.1 (ii), we get
Thus, \(u_{\lambda }^{\pm }\ne 0\). From Lemma 2.1, there exists \(\alpha _{u_{\lambda }}\), \(\beta _{u_{\lambda }}>0\) such that
Now, we show that \(\alpha _{u_{\lambda }}\), \(\beta _{u_{\lambda }}\le 1\). By (2.15), (2.17), the weak semicontinuity of norm, Fatou’s Lemma and Lemma 1.2, we have
From (2.20) and Lemma 2.2, we have \(\alpha _{u_{\lambda }}\le 1\). Similarly, \(\beta _{u_{\lambda }}\le 1\). The condition \((f_{4})\) implies that \(H(u):=uf(x, u)-4F(x, u)\) is a nonnegative function, increasing in \(\vert u\vert \), so we have
We then conclude that \(\alpha _{u_{\lambda }}=\beta _{u_{\lambda }}=1\). Thus, \({\bar{u}}_{\lambda }=u_{\lambda }\) and \(I_{\lambda }(u_{\lambda })=m_{\lambda }\). \(\square \)
3 Proof of main results
In this section, we are devoted to proving our main results.
Proof of Theorem 1.1
We firstly prove that the minimizer \(u_{\lambda }\) for the minimization problem is indeed a sign-changing solution of problem (1.4), that is, \(I'_{\lambda }(u_{\lambda })=0\). For it, we will use the quantitative deformation lemma.
It is clear that \(I'_{\lambda }(u_{\lambda })u_{\lambda }^{+}=0=I'_{\lambda }(u_{\lambda })u_{\lambda }^{-}\). From Lemma 2.2, for any \((\alpha , \beta )\in {\mathbb {R}}_{+}\times {\mathbb {R}}_{+}\) and \((\alpha , \beta )\ne (1, 1)\),
If \(I'_{\lambda }(u_{\lambda })\ne 0\), then there exist \(\delta >0\) and \(\kappa >0\) such that
Let \(D:=(\frac{1}{2}, \frac{3}{2})\times (\frac{1}{2}, \frac{3}{2})\) and \(g(\alpha , \beta ):=\alpha u_{\lambda }^{+}+\beta u_{\lambda }^{-}\). From Lemma 2.3, we also have
For \(\epsilon :=\min \{(m_{\lambda }-{\bar{m}}_{\lambda })/2, \kappa \delta /8\}\) and \(S:=B(u_{\lambda }, \delta )\), there is a deformation \(\eta \) such that
- (a):
\(\eta (1, u)=u\) if \(u\not \in I_{\lambda }^{-1}([m_{\lambda }-2\epsilon , m_{\lambda }+2\epsilon ])\cap S_{2\delta }\);
- (b):
\(\eta (1, I_{\lambda }^{m_{\lambda }+\epsilon }\cap S)\subset I_{\lambda }^{m_{\lambda }-\epsilon }\);
- (c):
\(I_{\lambda }(\eta (1, u)))\le I_{\lambda }(u)\) for all \(u\in H\).
See [24] for more details. It is clear that
Now we prove that \(\eta (1, g(D))\cap {\mathcal {M}}_{\lambda }\ne \emptyset \) which contradicts the definition of \(m_{\lambda }\). Let us define \(h(\alpha , \beta )=\eta (1, g(\alpha , \beta )))\) and
Lemma 2.1 and the degree theory imply that \(\deg (\Psi _{0}, D, 0)=1\). It follows from that \(g=h\) on \(\partial D\). Consequently, we obtain
Thus, \(\Psi _{1}(\alpha _{0}, \beta _{0})=0\) for some \((\alpha _{0}, \beta _{0})\in D\), so that
which is a contradiction. From this, \(u_{\lambda }\) is a critical point of \(I_{\lambda }\), and moreover, it is a sign-changing solution for problem (1.4).
Now we prove that \(u_{\lambda }\) has exactly two nodal domains. By contradiction, we assume that \(u_{\lambda }\) has at least three nodal domains \(\Omega _{1}\), \(\Omega _{2}\), \(\Omega _{3}\). Without loss generality, we may assume that \(u_{\lambda }\ge 0\) a.e. in \(\Omega _{1}\), \(u_{\lambda }\le 0\) a.e. in \(\Omega _{2}\). Set
where
So \(\text {supp}(u_{\lambda _{1}})\cap \text {supp}(u_{\lambda _{2}})=\emptyset \), \({u_{\lambda _{i}}}\ne 0\) and \(\langle I'(u_{\lambda }), {u_{\lambda _{i}}}\rangle =0\) for \(i=1, 2, 3\). Assume that \(v:=u_{\lambda _{1}}+u_{\lambda _{2}}\), then \(v^{+}=u_{\lambda _{1}}\) and \(v^{-}=u_{\lambda _{2}}\), i.e., \(v^{\pm }\ne 0\). By Lemma 2.1, there is a unique pair \((\alpha _{v}, \beta _{v})\) of positive numbers such that
so we have
From \(\langle I'(u_{\lambda }), {u_{\lambda _{i}}}\rangle =0\) for \(i=1, 2, 3\), we have
By Lemma 2.2, we know that \((\alpha _{v}, \beta _{v})\in (0, 1]\times (0, 1]\). Since
From \((f_{4})\), we have
which is impossible, so \(u_{\lambda }\) has exactly two nodal domains. \(\square \)
Proof of Theorem 1.2
Similar to the proof of Lemma 2.4, for each \(\lambda >0\), we can get a \(v_{\lambda }\in {\mathcal {N}}_{\lambda }\) such that \(I_{\lambda }(v_{\lambda })=c_{\lambda }>0\), where \({\mathcal {N}}_{\lambda }\) and \(c_{\lambda }\) are defined by (1.5) and (1.6), respectively. Moreover, the critical points of \(I_{\lambda }\) on \({\mathcal {N}}_{\lambda }\) are the critical points of \(I_{\lambda }\) in H. Thus, \(v_{\lambda }\) is a ground state solution of problem (1.4).
From Theorem 1.1, we know that problem (1.4) has a least energy sign-changing solution \(u_{\lambda }\) which changes sign only once. Suppose that \(u_{\lambda }=u_{\lambda }^{+}+u_{\lambda }^{-}\). As the proof of Step 1 in Lemma 2.1, there is a unique \(\alpha _{u_{\lambda }^{+}}>0\) such that
Similarly, there exists a unique \(\beta _{u_{\lambda }^{-}}>0\), such that
Moreover, Lemma 2.2 implies that \(\alpha _{u_{\lambda }^{+}}, \beta _{u_{\lambda }^{-}}\in (0, 1]\). Therefore, by Lemma 2.3, we obtain that
that is \(I_{\lambda }(u_{\lambda }) \ge 2c_{\lambda }\). It follows that \(c_{\lambda }>0\) which cannot be achieved by a sign-changing function. This completes the proof. \(\square \)
Now, we prove Theorem 1.3. In the following, we regard \(\lambda >0\) as a parameter in problem (1.4). We shall study the convergence property of \(u_{\lambda }\) as \(\lambda \searrow 0\).
Proof of Theorem 1.3
For any \(\lambda >0\), let \(u_{\lambda }\in H\) be the least energy sign-changing solution of problem (1.1) obtained in Theorem 1.1, which has exactly two nodal domains.
Step 1 We show that, for any sequence \(\{\lambda _{n}\}_n\) with \(\lambda _{n} \searrow 0\) as \(n\rightarrow \infty \), \(\{{u_{\lambda _{n}}}\}_n\) is bounded in H.
Choose a nonzero function \(\varphi \in C_{0}^{\infty }({\mathbb {R}}^{3})\) with \(\varphi ^{\pm }\ne 0\). By \((f_{3})\) and \((f_{4})\), for any \(\lambda \in [0, 1]\), there exists a pair \((\theta _{1}, \theta _{2})\in ({\mathbb {R}}_{+}\times {\mathbb {R}}_{+})\), which does not depend on \(\lambda \), such that
Then, in view of Lemmas 2.1 and Lemma 2.2, for any \(\lambda \in [0, 1]\), there is a unique pair \((\alpha _{\varphi }(\lambda ), \beta _{\varphi }(\lambda ) )\in (0, 1]\times (0, 1]\) such that \({\bar{\varphi }}:=\alpha _{\varphi }(\lambda )\theta _{1}\varphi ^{+}+\beta _{\varphi }(\lambda )\theta _{2}\varphi ^{-}\in {\mathcal {M}}_{\lambda }\). Thus, for all \(\lambda \in [0, 1]\), we have
Moreover, for n large enough, we obtain
So \(\{{u_{\lambda }}_{n}\}_n\) is bounded in H.
Step 2 The problem has a sign-changing solution \(u_{0}\).
By step 1 and Lemma 1.1, there exists a subsequence of \(\{\lambda _{n}\}_n\), up to a subsequence and \(u_{0}\in H\) such that
Since \(u_{\lambda _{n}}\) is the least energy sign-changing solution of (1.4) with \(\lambda =\lambda _{n}\), then we have
for all \(v\in C_{0}^{\infty }({\mathbb {R}}^{3})\). From (3.1), we get that
for all \(v\in C_{0}^{\infty }({\mathbb {R}}^{3})\). So \(u_{0}\) is a weak solution of (1.7). From a similar argument of the proof in Lemma 2.4, we know that \(u_{0}^{\pm }\ne 0\).
Step 3 The problem (1.7) has a least energy sign-changing solution \(v_{0}\), and there is a unique pair \((\alpha _{\lambda _{n}}, \beta _{\lambda _{n}} )\in {\mathbb {R}}^{+}\times {\mathbb {R}}^{+}\) such that \(\alpha _{\lambda _{n}}{v_{0}}^{+}+ \beta _{\lambda _{n}}{v_{0}}^{-} \in {\mathcal {M}}_{\lambda }\). Moreover, \((\alpha _{\lambda _{n}}, \beta _{\lambda _{n}})\rightarrow (1, 1)\) as \(n\rightarrow \infty \).
Via a similar argument in the proof of Theorem 1.1, there is a least energy sign-changing solution \(v_{0}\) for problem (1.7) with two nodal domain, so we have
and
By Lemma 2.1, there exits an unique pair of \((\alpha _{\lambda _{n}}, \beta _{\lambda _{n}} )\) such that \(\alpha _{\lambda _{n}}{v_{0}}^{+}+ \beta _{\lambda _{n}}{v_{0}}^{-} \in {\mathcal {M}}_{\lambda }\). So we have
and
From \((f_{3})\) and \(\lambda _{n}\rightarrow 0\) as \(n\rightarrow \infty \), we get that the sequences \(\{\alpha _{\lambda _{n}}\}\) and \(\{\beta _{\lambda _{n}}\}\) are bounded. Assume that \(\alpha _{\lambda _{n}}\rightarrow \alpha _{0}\) and \(\beta _{\lambda _{n}}\rightarrow \beta _{0}\) as \(n\rightarrow \infty \). From (2.16), (3.4) and (3.5), we have
and
Moreover, by \((f_{3})\) and \((f_{4})\), we know that \(\frac{f(x, s)}{\vert s\vert ^{3}}\) is nondecreasing in \(\vert s\vert \). So from (3.2), (3.3), (3.6), (3.7), we obtain that \((\alpha _{0}, \beta _{0})=(1, 1)\).
Now, we complete the proof of Theorem 1.3. We only need to show that \(u_{0}\) obtained in step 2 is a least energy sign-changing solution of problem (1.7). By Lemma 2.3, we have
This show that \(u_{0}\) is a least energy sign-changing solution of problem (1.7) which has precisely two nodal domains. We complete the proof of Theorem 1.3. \(\square \)
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Acknowledgements
Chao Ji is supported by Shanghai Natural Science Foundation (18ZR1409100), NSFC (No. 11301181) and China Postdoctoral Science Foundation.
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Ji, C. Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger–Poisson system in \({\mathbb {R}}^{3}\). Annali di Matematica 198, 1563–1579 (2019). https://doi.org/10.1007/s10231-019-00831-2
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DOI: https://doi.org/10.1007/s10231-019-00831-2
Keywords
- Fractional Schrödinger–Poisson system
- Sign-changing solutions
- Constraint variational method
- Quantitative deformation lemma