Abstract
In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical nonlinearity:
where \((-\Delta )_{A}^{s}\) is the fractional magnetic operator with \(0< s<1\), \(N>2s\), \(\lambda >0\), \(2_{s}^{*}=\frac{2N}{N-2s}\), \(p\geq 2_{s}^{*}\), f is a subcritical nonlinearity, and \(V \in C(\mathbb{R}^{N},\mathbb{R})\) and \(A \in C(\mathbb{R}^{N}, \mathbb{R}^{N})\) are the electric and magnetic potentials, respectively. Under some suitable conditions, by variational methods we prove that the equation has a nontrivial solution for small \(\lambda >0\). Our main contribution is related to the fact that we are able to deal with the case \(p>2_{s}^{*}\).
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1 Introduction and preliminaries
Consider the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical nonlinearity:
where \((-\Delta )_{A}^{s}\) is the fractional magnetic operator with \(0< s<1\), \(N>2s\), \(\lambda >0\), \(2_{s}^{*}=\frac{2N}{N-2s}\), \(p\geq 2_{s}^{*}\), f is a subcritical nonlinearity, and \(V \in C(\mathbb{R}^{N},\mathbb{R})\) and \(A \in C(\mathbb{R}^{N}, \mathbb{R}^{N})\) are the electric and magnetic potentials, respectively.
The fractional magnetic Laplacian is defined by
This nonlocal operator has been defined in [4] as a fractional extension (for any \(s \in (0,1)\)) of the magnetic pseudorelativistic operator or Weyl pseudodifferential operator defined with midpoint prescription [1]. As stated in [17], up to correcting the operator by the factor \((1-s)\), it follows that \((-\Delta )_{A}^{s}u\) converges to \(-(\nabla u-iA)^{2}u\) as \(s\rightarrow 1\). Thus, up to normalization, the nonlocal case can be seen as an approximation of the local one. The motivation for its introduction is described in more detail in [4, 17] and relies essentially on the Lévy–Khintchine formula for the generator of a general Lévy process.
The main driving force for the study of problem (1.1) arises in the following time-dependent Schrödinger equation when \(s=1\):
where ħ is the Planck constant, m is the particle mass, \(A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}\) is the magnetic potential, \(P:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}\) is the electric potential, ρ is the nonlinear coupling, and ψ is the wave function representing the state of the particle. This equation arises in quantum mechanics and describes the dynamics of the particle in a nonrelativistic setting [2, 15]. Clearly, the form \(\psi (x,t):=e^{-i\varpi th^{-1}}u(x)\) is a standing wave solution of (1.2) if and only if \(u(x)\) satisfies the following stationary equation:
where \(\varepsilon =\hbar \), \(V(x)=2m(P(x)-\varpi )\), and \(f=2m\rho \); see [3, 5, 7, 8]. By applying variational methods and Lyusternik–Schnirelmann theory Ambrosio and d’Avenia [1] proved the existence and multiplicity of solutions for the equation
when \(\varepsilon >0\) is small. Recently, Liang et al. [14] obtained the existence and multiplicity of solutions for the fractional Schrödinger–Kirchhoff equation
with the help of fractional version of the concentration compactness principle and variational methods. If the magnetic field \(A\equiv 0\), then the operator \((-\Delta )_{A}^{s}\) can be reduced to the fractional Laplacian operator \((-\Delta )^{s}\), which is defined as
The symbol \(\mathrm{P.V}.\) stands for the Cauchy principal value, and \(C_{N,s}\) is a dimensional constant that depends on \(N, s\), precisely given by
It is well known that the fractional Laplacian \((-\Delta )^{s}\) can be viewed as a pseudodifferential operator of symbol \(|\xi |^{2s}\), as stated in Lemma 1.1 in [6]. Simultaneously, problem (1.1) becomes the classical Schrödinger equation
Recently, there has been a lot of interest in the study of equation (1.3) and other related nonlocal problems. See, for instance, [6, 10–13, 16, 21–23] and the references therein. For more results about dealing with magnetic operators, see [9, 20]. Nonlocal problems also appear in other mathematical research fields. We refer the interested readers to [18, 19] for mathematical researches on Kirchhoff-type nonlocal equations, where Tang and Cheng [19] proposed a new approach to recover compactness for the (PS)-sequence, and Tang and Chen [18] proposed a new approach to recover compactness for the minimizing sequence.
Most of the works mentioned are set in \(\mathbb{R}^{N}\), \(N>2s\), with subcritical or critical growth, and to the best of our knowledge, no results are available on the existence for problem (1.1) with supercritical exponent. In this paper, we aim at studying the existence of nontrivial solutions for critical or supercritical problem (1.1).
To reduce the statements of the main result, we introduce the following assumptions:
- (V):
\(V \in C(\mathbb{R}^{N},\mathbb{R})\), \(0< V_{0}:=\inf_{x\in \mathbb{R}^{N}}V(x)\), and \(\lim_{|x|\rightarrow +\infty }V(x)=+\infty \).
- \((f_{1})\):
\(f\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R})\), and there exists \(2< q< 2_{s}^{*}\) such that
$$ \bigl\vert f(x,t) \bigr\vert \leq C\bigl(1+ \vert t \vert ^{\frac{q-2}{2}}\bigr) $$for all \((x,t) \in \mathbb{R}^{N}\times \mathbb{R}\), where C is a positive constant.
- \((f_{2})\):
\(f(x,t)=o(1)\) as \(|t|\to 0\) uniformly in \(x \in \mathbb{R}^{N}\);
- \((f_{3})\):
\(f(x,t)t\geq \frac{q}{2}F(x,t):=\frac{q}{2}\int _{0}^{t}f(x, \tau )\,d\tau \) for all \((x,t) \in \mathbb{R}^{N}\times \mathbb{R}\);
- \((f_{4})\):
\(c_{0}:=\inf_{x\in \mathbb{R}^{N},|t|=1}F(x,t)>0\).
For a function \(u:\mathbb{R}^{N}\rightarrow \mathbb{C}\), we set
and
Then we may introduce the Hilbert space
endowed with the scalar product
and norm
where \(\mathcal{R}(z)\) is the real part of a complex number z. By Lemma 3.5 in [4] the embedding \(H_{A}^{s}(\mathbb{R}^{N}, \mathbb{C})\hookrightarrow L^{t}( \mathbb{R}^{N}, \mathbb{C})\) is continuous for any \(t \in [2,2_{s}^{*}]\), and the embedding \(H_{A}^{s}(\mathbb{R}^{N}, \mathbb{C})\hookrightarrow L_{loc}^{t}( \mathbb{R}^{N}, \mathbb{C})\) is compact for any \(t \in [1,2_{s}^{*})\). Moreover, set
with the norm
By assumption \((V)\) the embedding \(E\hookrightarrow H_{A}^{s}(\mathbb{R}^{N}, \mathbb{C})\) is continuous.
For convenience, we define the homogeneous fractional Sobolev space
which is the completion of \(C_{0}^{\infty }(\mathbb{R}^{N})\) under the norm
Define the norm on \(H^{s}(\mathbb{R}^{N})\) as follows:
Moreover, the best fractional critical Sobolev constant is given by
Our main result is the following:
Theorem 1.1
Suppose that\((V)\)and\((f_{1})\)–\((f_{4})\)are satisfied. Then there exists\(\lambda _{0}>0\)such that for each\(\lambda \in (0,\lambda _{0}]\), problem (1.1) has a nontrivial solution\(u_{\lambda }\).
As a complement of Theorem 1.1, by the Pohozaev identity we can deduce that the equation
with\(p \geq 2_{s}^{*}\)and\(\mu >0\)has no nontrivial solution for all\(\lambda >0\). Indeed, let\(u \in E\)be a weak solution of the problem. Then we have the following Pohozaev identity:
Moreover, takinguas the test function, we have
Taking into account (1.4) and (1.5), we can derive that
which implies the conclusion.
2 Proof of Theorem 1.1
It is well known that a weak solution of problem (1.1) is a critical point of the following functional:
Clearly, we cannot apply variational methods directly because the functional \(I_{\lambda }\) is not well defined on E unless \(p=2_{s}^{*}\). To overcome this difficulty, we define the function
where \(M>0\). Then \(\phi \in C(\mathbb{R},\mathbb{R})\), \(\phi (t)t\geq q\varPhi (t):=q\int _{0}^{t}\phi (\tau )\,d\tau \geq 0\), and \(|\phi (t)|\leq M^{p-q}|t|^{q-1}\) for all \(t \in \mathbb{R}\). Set \(h_{\lambda }(x,t)=\lambda \phi (t)+f(x,|t|^{2})t\) for \((x,t) \in \mathbb{R}^{N} \times \mathbb{R}\). Then \(h_{\lambda }(x,t)\) admits the following properties:
- \((h_{1})\):
\(h_{\lambda }\in C(\mathbb{R}^{N} \times \mathbb{R}, \mathbb{R})\), and \(|h_{\lambda }(x,t)|\leq \lambda M^{p-q}|t|^{q-1}+C(|t|+|t|^{q-1})\) for all \((x,t) \in \mathbb{R}^{N} \times \mathbb{R}\).
- \((h_{2})\):
\(h_{\lambda }(x,t)t\geq qH_{\lambda }(x,t):=q\int _{0}^{t}h_{\lambda }(x,\tau )\,d\tau \geq 0\) for all \((x,t) \in \mathbb{R}^{N} \times \mathbb{R}\).
- \((h_{3})\):
\(\inf_{x\in \mathbb{R}^{N},|t|=1}H_{\lambda }(x,t) \geq \frac{c_{0}}{2}>0\).
Let
By \((h_{1})\)–\((h_{3})\), \((V)\), and the mountain pass theorem, using a standard argument, we easily see that the equation
has a nontrivial solution \(u_{\lambda }\in E\) with \(J_{\lambda }^{\prime }(u_{\lambda })=0\) and \(J_{\lambda }(u_{\lambda })=c_{\lambda }:=\inf_{\gamma \in \varGamma _{\lambda }}\sup_{t \in [0,1]}J_{\lambda }(\gamma (t))\), where
We further set
and
Then \(\varGamma \subset \varGamma _{\lambda }\) and \(c_{\lambda }\leq c\).
Lemma 2.1
The solution\(u_{\lambda }\)satisfies\(\| u_{\lambda }\|^{2}\leq \frac{2q}{q-2}c_{\lambda }\), and there exists a constant\(A>0\)independent onλsuch that\(\| u_{\lambda }\|^{2}\leq A\).
Proof
By \((h_{2})\) we know that
which means that \(\| u_{\lambda }\|^{2}\leq \frac{2q}{q-2}c_{\lambda }\leq \frac{2q}{q-2}c:=A>0\). This completes the proof. □
Lemma 2.2
There exist two constants\(B, D>0\)independent onλsuch that\(\Vert \vert u_{\lambda } \vert \Vert _{\infty }\leq B(1+\lambda )^{D}\).
Proof
For any \(L>0\) and \(\beta >1\), set \(\gamma (a)=aa_{L}^{2(\beta -1)}, a \in \mathbb{R}\), where \(a_{L}:=\min \{|a|,L\}\). Since γ is an increasing function, we have
Let \(\varPhi (t)=\frac{|t|^{2}}{2}\) and \(\varGamma (t)=\int _{0}^{t} (\gamma ^{\prime }(\tau ) )^{ \frac{1}{2}}\,d\tau \) for \(t\geq 0\). Then if \(a>b\), then we have
If \(a\leq b\), then we can use a similar argument to obtain the conclusion. It follows that
for all \(a, b \in \mathbb{R}\), which implies that
Choosing \(u_{\lambda }u_{\lambda,L}^{2(\beta -1)}\) as a test function, where \(u_{\lambda,L}:=\min \{|u_{\lambda }|,L\}\), we obtain
Note that
Consequently, by (2.1) we have
For any \(\varepsilon >0\), by \((f_{1})\)–\((f_{2})\) and properties of ϕ, there exists \(C_{\varepsilon }>0\) such that
and
for all \((x,t) \in \mathbb{R}^{N} \times \mathbb{R}\). Thereby, for fixed \(\lambda >0\) and small \(\varepsilon >0\), we have
for all \((x,t) \in \mathbb{R}^{N} \times \mathbb{R}\). Simultaneously, \(\varGamma (|u_{\lambda }|)\geq \frac{1}{\beta }|u_{\lambda }| u_{\lambda,L}^{ \beta -1}\), and
Therefore, taking into account (2.2)–(2.4) and condition \((V)\), we can see that
which implies that
Setting \(w_{\lambda,L}=|u_{\lambda }| u_{\lambda,L}^{\beta -1}\), by the Hölder inequality we can derive that
where \(\alpha _{s}^{*}=\frac{22_{s}^{*}}{2_{s}^{*}-(q-2)} \in (2,2_{s}^{*})\).
By Lemma 2.1 we have
Now we observe that if \(|u_{\lambda }|^{\beta }\in L^{\alpha _{s}^{*}}(\mathbb{R}^{N})\), then from the definition of \(\{u_{\lambda, L}\}\), the inequality \(u_{\lambda,L}\leq |u_{\lambda }|\), and (2.5) we obtain
Passing to the limit in (2.6) as \(L\rightarrow +\infty \), by the Fatou lemma we deduce that
whenever \(|u_{\lambda }|^{\beta \alpha _{s}^{*}} \in L^{1}(\mathbb{R}^{N})\).
Now set \(\beta:=\frac{2_{s}^{*}}{\alpha _{s}^{*}}>1\). Since \(|u_{\lambda }| \in L^{2_{s}^{*}}(\mathbb{R}^{N})\), the inequality holds for this choice of β. Then, since \(\beta ^{2}\alpha _{s}^{*}=\beta 2_{s}^{*}\), it follows that (2.7) holds with β replaced by \(\beta ^{2}\). Consequently,
Iterating this process and recalling that \(\beta \alpha _{s}^{*}=2_{s}^{*}\), we conclude that for every \(m \in \mathbb{N}\),
Set \(d_{m}=\sum_{i=1}^{m}\frac{1}{\beta ^{i}}\) and \(e_{m}=\sum_{i=1}^{m}\frac{i}{\beta ^{i}}\). Then \(d_{m}\rightarrow \sigma _{1}>0\) and \(e_{m}\rightarrow \sigma _{2}>0\) as \(m\rightarrow \infty \). Then, taking the limit in (2.8) as \(m\rightarrow +\infty \), by Lemma 2.1 we have
where \(B:=C^{\sigma _{1}}\beta ^{\sigma _{2}}C>0\) and \(D:=\frac{\sigma _{1}}{2}\). This completes the proof. □
Proof of Theorem 1.1
By Lemma 2.2, for large \(M>0\), we can choose small \(\lambda _{0}>0\) such that \(\Vert \vert u_{\lambda } \vert \Vert _{L^{\infty }}\leq B(1+\lambda )^{D}\leq M\) for all \(\lambda \in (0,\lambda _{0}]\). Consequently, \(u_{\lambda }\) is a nontrivial solution of (1.1) with \(\lambda \in (0,\lambda _{0}]\). This completes the proof. □
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We would like to thank the referees for their valuable comments and suggestions.
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This work was supported in part by the National Natural Science Foundation of China (11801153, 11501403, 11601145, 11701322, 11901514), the Honghe University Doctoral Research Programs (XJ17B11, XJ17B12), the Yunnan Province Applied Basic Research for Youths (2018FD085), the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013), the Yunnan Province Applied Basic Research for General Project (2019FB001), Technology Innovation Team of University in Yunnan Province, and the Project funded by China Postdoctoral Science Foundation (2019M652790).
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QL and JZ conceived of the idea of this manuscript and wrote the manuscript. KT and WW discussed about some estimation and checked the calculations. All authors read and approved the final manuscript.
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Li, Q., Teng, K., Wang, W. et al. Existence of nontrivial solutions for fractional Schrödinger equations with electromagnetic fields and critical or supercritical nonlinearity. Bound Value Probl 2020, 112 (2020). https://doi.org/10.1186/s13661-020-01409-1
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DOI: https://doi.org/10.1186/s13661-020-01409-1