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Ground states for fractional Choquard equations with doubly critical exponents and magnetic fields Zhenyu Guoa, Lujuan Zhaob a School of Mathematics, Liaoning Normal University, Dalian, China b Ordos Second School Attached To Beijing Normal University, Ordos, Inner Mongolia, China
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 1, DOI: https://doi.org/10.4213/im9361 
Bibliographic databases:
    § 1. IntroductionConsider the following fractional Choquard equations with doubly critical exponents and magnetic fields:
$$
\begin{equation}
(-\Delta)_A^su+V(x)u=[I_{\alpha}\ast|u|^a]|u|^{a-2}u+[I_{\alpha}\ast|u|^b]|u|^{b-2}u \quad \text{in }\ \mathbb{R}^N,
\end{equation}
\tag{1.1}
$$
where $N\geqslant3$, $s\in(0,1)$, $\alpha\in(0,N)$, $I_{\alpha}$ is the Riesz potential defined by
$$
\begin{equation*}
I_{\alpha}=\frac{\Gamma((N-\alpha)/2)}{2^{\alpha}\pi^{N/2} \Gamma(N/2)|x|^{N-\alpha}},
\end{equation*}
\notag
$$
$a:=2^{\sharp}_{\alpha,s}=(N+\alpha)/N$ and $b:=2^*_{\alpha,s}=(N+\alpha)/(N-2s)$ are fractional lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality (see § 2 for more information on the Hardy–Littlewood–Sobolev inequality), $V\in(\mathbb{R}^N,\mathbb{R})$ is a continuous function, $u\in(\mathbb{R}^N,\mathbb{C})$ is a complex valued function, $A\in(\mathbb{R}^N,\mathbb{R}^N)$ is a magnetic potential, $(-\Delta)^s_A$ is a fractional magnetic Laplacian operator with $s\in(0,1)$. Up to normalization constants, $(-\Delta)^s_A$ can be defined on smooth complex valued functions $u \in C_{\mathrm{c}}^{\infty}(\mathbb{R}^N,\mathbb{C})$ by
$$
\begin{equation*}
(-\Delta)^s_Au(x)=\lim_{\varepsilon\to0^{+}}\int_{B_{\varepsilon}^{c}(x)}\frac{e^{-i(x-y)\cdot A((x+y)/2)}u(x)-u(y)}{|x-y|^{N+2s}}\, dy\quad \text{in } \ \mathbb{R}^N,
\end{equation*}
\notag
$$
where $B_{\varepsilon}(x)$ denotes a ball in $\mathbb{R}^N$ of radius $\varepsilon > 0$ with centre at $x\in\mathbb{R}^N$, and $B_{\varepsilon}^{c}(x)=\mathbb{R}^N\setminus B_{\varepsilon}(x)$. This nonlocal operator has been introduced by d’Avenia and Squassina [1] as a fractional extension of the magnetic pseudo-relativistic operator, or by Ichinose and Tamura [2] as a Weyl pseudo-differential operator defined by mid-point prescription. The motivation for its introduction belongs to the general theoretical framework of Lévy process. Lots of scholars have studied the Choquard equation and its nontrivial solutions. For $A=0$ and $s\to1$, Ma and Zhao [3] studied the generalized Choquard equation, in particular, they proved that its positive solution is radially symmetric and monotone decreasing. Moroz and Van Schaftingen [4] proved the existence, regularity, positivity, radial symmetry and the decaying property of ground state solutions for Choquard equation. D’Avenia, Siciliano, and Squassina [5] showed some properties of the solution of the fractional Choquard equation. In recent years, many people have paid attention to the Choquard equation with double critical exponents and studied it. Seok [6] studied the existence of nontrivial solutions for Choquard equation with double critical exponents. Su, Wang, Chen, and Liu [7] studied the multiplicity and concentration of positive solutions for the fractional Choquard equation
$$
\begin{equation*}
\varepsilon^{2s}(-\Delta)^su+V(x)u=\varepsilon^{-\alpha}(I_{\alpha}\ast F(u))F'(u),
\end{equation*}
\notag
$$
where $F(u)=|u|^{2^{\sharp}_{\alpha}}/2^{\sharp}_{\alpha} + |u|^{2^*_{\alpha}}/2^*_{\alpha}$. In [8], Lei and Zhang considered the existence of ground state solutions for the Choquard equations
$$
\begin{equation*}
-\Delta u+u=(|x|^{\alpha-N}\ast|u|^{p})|u|^{p-2}u+(|x|^{\alpha-N}\ast|u|^q)|u|^{q-2}u;
\end{equation*}
\notag
$$
they used the Pokhozhaev-type identity to overcome the loss of compactness caused by the doubly critical nonlinearities. Inspired by the above works, we will discuss the existence of ground states for the fractional Choquard equation with doubly critical exponents and magnetic fields. In order to determine the existence of the ground states of equation (1.1), we use the Nehari method, which transforms the problem of seeking the ground states of the equation into the problem of finding the critical point of its corresponding energy functional. In addition, we apply the Pokhozhaev identity to overcome the loss of compactness caused by the doubly critical nonlinearities. Obviously, the study of the problem will become more complex because the magnetic fractional Laplacian operator and the doubly critical nonlinearities exist at the same time. As far as we know, it seems that there is almost no work on this subject. Suppose that the potential $V\in(\mathbb{R}^N,\mathbb{R})$ is a continuous function. In order to accurately express our main results, we introduce the following assumptions: (A) $A=(A_1,\dots,A_N)\in(\mathbb{R}^N,\mathbb{R}^N)$ is continuous; (V) there exists $V_0>0$ such that $V(x)\geqslant V_0$. In what follows, we state our main results. Theorem 1.1. Assume that (A) and (V) hold. Then equation (1.1) has a ground state solution. § 2. PreliminariesLet $L^2(\mathbb{R}^N,\mathbb{C})$ be the Lebesgue space with real inner product
$$
\begin{equation*}
\langle u,v\rangle_{L^2}:=\operatorname{Re}\int_{\mathbb{R}^N}u\overline{v}\, dx,
\end{equation*}
\notag
$$
and denote by $|\,{\cdot}\,|_{q}$ the norm of $L^q(\mathbb{R}^N)$, where $\operatorname{Re} z$ is the real part of a complex number $z$. Consider the fractional magnetic critical Sobolev space
$$
\begin{equation*}
D_A^s(\mathbb{R}^N,\mathbb{C}):=\{u\in L^{2^\ast_s}(\mathbb{R}^N)\colon [u]_{D_A^s}<\infty\},
\end{equation*}
\notag
$$
where $[u]_{D_A^s}$ denotes the so-called magnetic Gagliardo semi-norm, that is,
$$
\begin{equation*}
[u]^2_{D_A^s}=\int_{\mathbb{R}^{2N}}\frac{\bigl|e^{-i(x-y)\cdot A((x+y)/2)}u(x)-u(y)\bigr|^2}{|x-y|^{N+2s}}\, dx\, dy.
\end{equation*}
\notag
$$
We consider the scalar product
$$
\begin{equation*}
\begin{aligned} \, &\langle u,v\rangle_{D_A^s} \\ &=\operatorname{Re}\int_{\mathbb{R}^{2N}}\frac{\bigl(e^{-i(x-y)\cdot A((x+y)/2)}u(x)-u(y)\bigr) \overline{\bigl(e^{-i(x-y)\cdot A((x+y)/2)}v(x)-v(y)\bigr)}}{|x-y|^{N+2s}}\, dx\, dy, \end{aligned}
\end{equation*}
\notag
$$
and the norm
$$
\begin{equation}
\|u\|^2_{D_A^s}=\langle u,u\rangle_{D_A^s}=[u]^2_{D_A^s}.
\end{equation}
\tag{2.1}
$$
We also define the fractional magnetic Hilbert space
$$
\begin{equation*}
H_A^s(\mathbb{R}^N,\mathbb{C})=\biggl\{u\in D_A^s(\mathbb{R}^N,\mathbb{C})\colon \int_{\mathbb{R}^N}|u|^2\, dx<+\infty\biggr\},
\end{equation*}
\notag
$$
and the corresponding norm
$$
\begin{equation*}
\| u\|^2_{H_A^s}=\| u\|^2_{D_A^s}+\int_{\mathbb{R}^N}|u|^2\, dx.
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
\mathcal{H}=\biggl\{u\in H_A^s(\mathbb{R}^N,\mathbb{C})\colon \int_{\mathbb{R}^N}V(x)u^2\, dx<+\infty\biggr\}.
\end{equation*}
\notag
$$
Consider the associated inner product
$$
\begin{equation*}
\langle u,v\rangle=\langle u,v\rangle_{D_A^s} +\operatorname{Re}\int_{\mathbb{R}^N}V(x)u\overline{v}\, dx,
\end{equation*}
\notag
$$
and the corresponding norm
$$
\begin{equation*}
\| u\|^2=\| u\|^2_{D_A^s}+\int_{\mathbb{R}^N}V(x)u^2\, dx,
\end{equation*}
\notag
$$
which is equivalent to the norm $\|u\|^2_{H_A^s}$. Combining Lemma 3.5 in [1] and (V), we have the injection
$$
\begin{equation*}
\mathcal{H}\hookrightarrow L^q(\mathbb{R}^N,\mathbb{C})
\end{equation*}
\notag
$$
is continuous for any $2\leqslant q\leqslant2_s^*$. Moreover, the injection is compact for any $2\leqslant q<2_s^*$, where $K\subset\mathbb{R}^N$ is a compact set and $2_s^*=2N/(N- 2s)$. We point out that the weak solution of equation (1.1) is a critical point of the energy functional
$$
\begin{equation}
\mathcal{I}_{s,A}(u)=\frac12\| u\|^2 -\frac1{2a}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx -\frac1{2b}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx
\end{equation}
\tag{2.2}
$$
for each $u\in \mathcal{H}$. Define
$$
\begin{equation*}
\mathcal{M}=\{u\in \mathcal{H}\setminus\{0\}\colon \langle \mathcal{I}'_{s,A}(u),u\rangle=0\},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\langle \mathcal{I}'_{s,A}(u),u\rangle=\| u\|^2-\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx-\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx.
\end{equation*}
\notag
$$
We also define the Pokhozhaev functional
$$
\begin{equation*}
\begin{aligned} \, \mathcal{P}(u)&=\frac{N-2s}{2}\| u\|^2_{D_A^s}+\frac{N}{2}\int_{\mathbb{R}^N}V(x)u^2\, dx-\frac{N+\alpha}{2a}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx \\ &\qquad -\frac{N+\alpha}{2b}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx. \end{aligned}
\end{equation*}
\notag
$$
Hence any critical point $u$ of $\mathcal{I}_{s,A}$ satisfies $\mathcal{P}(u)= 0$. Proposition 2.1 (the Hardy–Littlewood–Sobolev inequality [9]). Let $s,t>1$ and $0<\alpha<N$ with $1/s+(N-\alpha)/N+1/t=2$. If $f\in L^s(\mathbb{R}^N)$ and $g\in L^t(\mathbb{R}^N)$, then there exists a sharp constant $C(N,\alpha,s,t)$ independent of $f$ and $g$ such that Remark 2.1 (see [9]). If $s=t=2N/(N+\alpha)$, then Based on Proposition 2.1, the integral
$$
\begin{equation*}
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)|^q|u(y)|^q}{|x-y|^{N-\alpha}}\, dx\, dy
\end{equation*}
\notag
$$
(see [10]) is well defined if $|u|^q\in L^t(\mathbb{R}^N)$ for some $t>1$ and Thus, for any $u\in\mathcal{H}$, by the Sobolev embedding theorems, we have that is,
$$
\begin{equation*}
\frac{N+\alpha}{N}\leqslant q\leqslant\frac{N+\alpha}{N-2s}.
\end{equation*}
\notag
$$
Therefore, $(N+\alpha)/N$ is known as the fractional lower critical exponent, and $(N+ \alpha)/(N-2s)$ is the fractional upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Lemma 2.1. For each $u\in\mathcal{H}\setminus\{0\}$ and $k>0$, the functional $\mathcal{I}_{s,A}(ku)$ has a unique critical point which corresponds to its maximum. Proof. For each $u\in\mathcal{H}\setminus\{0\}$, we have
$$
\begin{equation*}
\begin{aligned} \, \mathcal{I}_{s,A}(ku) &=\frac{k^2}{2}\biggl(\| u\|^2_{D_A^s}+\int_{\mathbb{R}^N}V(x)u^2\, dx\biggr) -\frac{k^{2a}}{2a}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx \\ &\qquad -\frac{k^{2b}}{2b}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx. \end{aligned}
\end{equation*}
\notag
$$
It is easily checked that $\mathcal{I}_{s,A}(ku)\to-\infty$ as $k\to+\infty$. By Proposition 2.1,
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx \\ &\quad\leqslant C(N,\alpha)\biggl(\int_{\mathbb{R}^N}|u|^{b \cdot 2N/(N+\alpha)}\, dx\biggr)^{(N+\alpha)/(2N)} \biggl(\int_{\mathbb{R}^N}|u|^{b \cdot 2N/(N+\alpha)}\, dx\biggr)^{(N+\alpha)/(2N)} \\ &\quad=C(N,\alpha)|u|^{2b}_{b}. \end{aligned}
\end{equation*}
\notag
$$
A similar analysis shows that $\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx\leqslant C(N,\alpha)|u|_2^{2a}$. Therefore, by (V),
$$
\begin{equation*}
\mathcal{I}_{s,A}(ku) \geqslant\frac{k^2}{2}\| u\|^2_{D_A^s} -\frac{k^{2a}}{2a}C(N,\alpha)|u|_2^{2a} -\frac{k^{2b}}{2b}C(N,\alpha)|u|^{2b}_{b}>0
\end{equation*}
\notag
$$
for sufficiently small $k>0$. Furthermore,
$$
\begin{equation*}
\begin{aligned} \, \langle I'_{s,A}(ku),u\rangle &=k\| u\|^2-k^{2a-1}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx -k^{2b-1}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx \\ &\geqslant k\| u\|^2-k^{2a-1}C(N,\alpha)|u|_2^{2a}-k^{2b-1}C(N,\alpha)|u|^{2b}_{2_s^*}>0 \end{aligned}
\end{equation*}
\notag
$$
for sufficiently small $k>0$. Thus, $\mathcal{I}_{s,A}(ku)$ has a critical point $k_0>0$ such that $I'_{s,A}(k_0u)=0$, and $I'_{s,A}(ku)>0$ as $0<k<k_0$.
In what follows, we will prove the uniqueness of the critical point of $\mathcal{I}_{s,A}(ku)$. We set for $t>0$. It can be shown that $f(0)=0$, $f(t)>0$ for sufficiently small $t>0$ and $f(t)\to-\infty$ as $t\to+\infty$. Therefore, we assume by contradiction that $f$ has at least three positive critical points $t_1$, $t_2$, $t_3$, and Hence we have
$$
\begin{equation*}
f'(t)=2a_1t-2aa_2t^{2a-1}-2ba_3t^{2b-1} =t(2a_1-2aa_2t^{2a-2}-2ba_3t^{2b-2})
\end{equation*}
\notag
$$
and Let Then Thus, $g$ has at least two positive critical points $\eta_1\in(t_1,t_2)$, $\eta_2\in(t_2,t_3)$, which satisfy However,
$$
\begin{equation*}
\begin{aligned} \, g'(t) &=-2a(2a-2)a_2t^{2a-3}-2b(2b-2)a_3t^{2b-3} \\ &=t^{2a-3}[-2a(2a-2)a_2-2b(2b-2)a_3t^{2(b-a)}]. \end{aligned}
\end{equation*}
\notag
$$
Thus, we obtain that $\eta_1=\eta_2$, which is contradictory.
Therefore, $\mathcal{I}_{s,A}(ku)$ has a unique critical point which corresponds to its maximum. Lemma is proved. Remark 2.2. If $u$ is a critical point of $\mathcal{I}_{s,A}$, then the maximum of $\mathcal{I}_{s,A}(ku)$ should be achieved at $k=1$. Define
$$
\begin{equation*}
\mathcal{N}_{\rho}=\{u\in\mathcal{H}\colon \| u\|=\rho\}.
\end{equation*}
\notag
$$
Lemma 2.2. There exist $\delta,\rho>0$ such that $\mathcal{I}_{s,A}(u)\geqslant\delta$ for any $u\in\mathcal{N}_{\rho}$. Proof. Suppose that $\| u\|=\rho>0$, which is small enough. According to Proposition 2.1 and Sobolev inequality, we have
$$
\begin{equation}
\begin{aligned} \, \mathcal{I}_{s,A}(u) &=\frac12\| u\|^2-\frac1{2a}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx -\frac1{2b}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx \nonumber \\ &\geqslant\frac12\| u\|^2-C_1\| u\|^{2a}-C_2\| u\|^{2b} \geqslant\frac12\rho^2-C_1\rho^{2a}-C_2\rho^{2b}>0. \end{aligned}
\end{equation}
\tag{2.4}
$$
Let $\delta=\frac12\rho^2-C_1\rho^{2a}-C_2\rho^{2b}$. Then, we have $\mathcal{I}_{s,A}(u)\geqslant\delta>0$. Lemma is proved. Proof. We set
$$
\begin{equation}
\begin{aligned} \, \mathcal{J}(u) &=\langle I'_{s,A}(u),u\rangle-\frac{2\alpha}{N(2s+\alpha)}\mathcal{P}(u) \nonumber \\ &=\frac{2s}{2s+\alpha}\biggl(a\| u\|^2_{D_A^s}+\int_{\mathbb{R}^N}V(x)u^2\, dx -\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx \nonumber \\ &\qquad -a\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx\biggr). \end{aligned}
\end{equation}
\tag{2.5}
$$
We assume by contradiction that there exists $u_*\in\mathcal{M}$ such that $\mathcal{J}'(u_*)=0$. Then, we have
$$
\begin{equation}
\begin{aligned} \, &a\| u_*\|^2_{D_A^s}+\int_{\mathbb{R}^N}V(x)u_*^2\, dx \nonumber \\ &\qquad=a\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_*|^a]|u_*|^a\, dx +ab\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_*|^b]|u_*|^b\, dx \end{aligned}
\end{equation}
\tag{2.6}
$$
and
$$
\begin{equation}
\begin{aligned} \, &a\| u_*\|^2_{D_A^s}+\int_{\mathbb{R}^N}V(x)u_*^2\, dx \nonumber \\ &\qquad=\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_*|^a]|u_*|^a\, dx +a\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_*|^b]|u_*|^b\, dx. \end{aligned}
\end{equation}
\tag{2.7}
$$
Then, by (2.6) and (2.7), we obtain that
$$
\begin{equation*}
(a-1)\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_*|^a]|u_*|^a\, dx +a(b-1)\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_*|^b]|u_*|^b\, dx=0,
\end{equation*}
\notag
$$
which is contradictory. Therefore, $\mathcal{J}'(u_*)\neq0$. This proves Lemma 2.1. Proof. For each $u\in\mathcal{M}$, we have $\mathcal{P}(u)=0$. Now by (2.5)
$$
\begin{equation*}
\begin{aligned} \, \mathcal{I}_{s,A}(u) &=\frac12\biggl(\| u\|^2_{D_A^s}+\int_{\mathbb{R}^N}V(x)u^2\, dx\biggr) -\frac1{2b}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx \\ &\qquad -\frac1{2a}\biggl(a\| u\|^2_{D_A^s}+\int_{\mathbb{R}^N}V(x)u^2\, dx -a\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx\biggr) \\ &=\biggl(\frac12-\frac1{2a}\biggr)\int_{\mathbb{R}^N}V(x)u^2\, dx +\biggl(\frac12-\frac1{2b}\biggr)\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx>0. \end{aligned}
\end{equation*}
\notag
$$
Therefore, we have $\inf_{\mathcal{M}}\mathcal{I}_{s,A}>0$, proving the lemma. Lemma 2.5. There exists $\delta_{\ast}>0$ such that and, for every compact subset $W\subset\mathcal{N}_1$, there exists $C_{W}>0$ such that Proof. For each $u\in\mathcal{N}_1$, by Lemma 2.1 there exists a unique $k_u>0$ such that $k_uu\in\mathcal{M}$. Hence which means that, for every $u\in\mathcal{N}_1$, there exists $\delta_{\ast}>0$ such that $k_u\geqslant\delta_{\ast}$.
Assume on the contrary that, for every $n\in\mathbb{N}$, there exists $\{u_n\}\subset W\subset\mathcal{N}_1$ such that $k_{u_n}\to+\infty$ as $n\to\infty$. By compactness of $W$, we can suppose that there exists $u\in W$ such that $u_n\to u$ as $n\to\infty$. Assume that
$$
\begin{equation*}
Q(u):=\frac1{2b}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^b]|u|^b\, dx, \qquad u \in \mathcal{H},
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
G(u):=\frac1{2a}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u|^a]|u|^a\, dx, \qquad u \in \mathcal{H}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\begin{aligned} \, Q(k_{u_n}u_n) &=\frac1{2b}k^{2b}_{u_n}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n|^b]|u_n|^b\, dx \nonumber \\ &=k^{2b}_{u_n}\| u_n\|^{2b}\frac1{2b\| u_n\|^{2b}} \int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n|^b]|u_n|^b\, dx \nonumber \\ &=k^{2b}_{u_n}\| u_n\|^{2b}\frac1{2b}\int_{\mathbb{R}^N}\biggl[I_{\alpha}\ast\biggl|\frac{u_n}{\| u_n\|}\biggr|^b\biggr]\biggl|\frac{u_n}{\| u_n\|}\biggr|^b\, dx =k^{2b}_{u_n}\| u_n\|^{2b}Q\biggl(\frac{u_n}{\| u_n\|}\biggr). \end{aligned}
\end{equation}
\tag{2.8}
$$
A similar analysis shows that
$$
\begin{equation}
G(k_{u_n}u_n)=k^{2a}_{u_n}\| u_n\|^{2a}G\biggl(\frac{u_n}{\| u_n\|}\biggr).
\end{equation}
\tag{2.9}
$$
By (2.8) and (2.9), we have
$$
\begin{equation*}
\begin{aligned} \, \mathcal{I}_{s,A}(k_{u_n}u_n) &=\frac12k_{u_n}^2\| u_n\|^2-G(k_{u_n}u_n)-Q(k_{u_n}u_n) \\ &=\frac12k_{u_n}^2\| u_n\|^2-k^{2a}_{u_n}\| u_n\|^{2a}G\biggl(\frac{u_n}{\|u_n\|}\biggr) -k^{2b}_{u_n}\| u_n\|^{2b}Q\biggl(\frac{u_n}{\| u_n\|}\biggr) \\ &=\frac12k_{u_n}^2-k^{2a}_{u_n}G(u_n)-k^{2b}_{u_n}Q(u_n)\to-\infty \quad \text{as }\ n\to\infty. \end{aligned}
\end{equation*}
\notag
$$
However, since $k_{u_n}u_n\in\mathcal{M}$, we have
$$
\begin{equation*}
\begin{aligned} \, &\lim_{n\to\infty}\mathcal{I}_{s,A}(k_{u_n}u_n) \\ &=\lim_{n\to\infty}\biggl[\frac12k_{u_n}^2 \biggl(\|u_n\|^2_{D_A^s}+\int_{\mathbb{R}^N}V(x)u_n^2\, dx\biggr)- \frac{k^{2b}_{u_n}}{2b}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n|^b]|u_n|^b\, dx\biggr] \\ &\quad -\lim_{n\to\infty}\frac1{2a}\biggl(ak_{u_n}^2\|u_n\|^2_{D_A^s} +k_{u_n}^2\int_{\mathbb{R}^N}V(x)u_n^2\, dx -ak^{2b}_{u_n}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n|^b]|u_n|^b\, dx\biggr) \\ &=\lim_{n\to\infty}\biggl[\biggl(\frac12-\frac1{2a}\biggr)k_{u_n}^2\int_{\mathbb{R}^N} \! V(x)u_n^2\, dx \,{+}\,\biggl(\frac12-\frac1{2b}\biggr)k^{2b}_{u_n} \int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n|^b]|u_n|^b\, dx\biggr]{>}\,0. \end{aligned}
\end{equation*}
\notag
$$
This contradiction proves the lemma. Define the mapping $\mu\colon \mathcal{N}_1\to\mathcal{M}$ by $\mu(w):=k_ww$, where $k_w$ is as in Lemma 2.1. Lemma 2.6 (see [11]). The mapping $\mu$ is a homeomorphism between $\mathcal{N}_1$ and $\mathcal{M}$, and the inverse mapping of $\mu$ is given by $\mu^{-1}(u)=u/\|u\|$. Consider the functional $\varUpsilon\in(\mathcal{N}_1,\mathbb{R})$ defined by The following result holds. Lemma 2.7 (see [11]). The following holds. (i) For every $n\in\mathbb{N}$, if $\{w_n\}$ is a Palais–Smale sequence for $\varUpsilon$, then $\{\mu(w_n)\}$ is a Palais–Smale sequence for $\mathcal{I}_{s,A}$. Conversely, if $\{u_n\}\subset\mathcal{M}$ is a bounded Palais–Smale sequence for $\mathcal{I}_{s,A}$, then $\{\mu^{-1}(u_n)\}$ is a Palais–Smale sequence for $\varUpsilon$. (ii) $w\in\mathcal{N}_1$ is a critical point of $\varUpsilon$ if and only if $\mu(w)$ is a nontrivial critical point of $\mathcal{I}_{s,A}$. In addition, the corresponding values of $\varUpsilon$ and $\mathcal{I}_{s,A}$ are equal, and $\inf_{\mathcal{N}_1}\varUpsilon=\inf_{\mathcal{M}}\mathcal{I}_{s,A}$. (iii) The minimum of $\mathcal{I}_{s,A}$ on $\mathcal{M}$ is the ground state solution of equation (1.1). § 3. Proof of Theorem 1.1In this section, we will prove Theorem 1.1. Proof of Theorem 1.1. Let $\{w_n\}_{n\in\mathbb{N}}\subset\mathcal{N}_1$ be a minimizing sequence for $\varUpsilon$. By (ii) of Lemma 2.7,
$$
\begin{equation}
\varUpsilon(w_n)\to c=\inf_{\mathcal{N}_1}\varUpsilon\quad \text{as }\ n\to\infty.
\end{equation}
\tag{3.1}
$$
For $n\in\mathbb{N}$, set $u_n=\mu(w_n)\in\mathcal{M}$. By the Ekeland variational principle, we can assume that $\varUpsilon'(w_n)\to0$ in the dual $\mathcal{H}^*$ of $\mathcal{H}$. By (i), (ii) of Lemma 2.7 and (3.1), we have
$$
\begin{equation}
\mathcal{I}_{s,A}(u_n)=\varUpsilon(\mu^{-1}(u_n))=\varUpsilon(w_n)\to c=\inf_{\mathcal{M}}\mathcal{I}_{s,A}
\end{equation}
\tag{3.2}
$$
and
$$
\begin{equation}
\mathcal{I}'_{s,A}(u_n)\to0\quad\text{in }\ \mathcal{H}^*
\end{equation}
\tag{3.3}
$$
as $n\to\infty$. Furthermore, $\mathcal{P}(u_n)=0$. Then, by (V), we have
$$
\begin{equation*}
\begin{aligned} \, c+o(1)>\mathcal{I}_{s,A}(u_n) &=\mathcal{I}_{s,A}(u_n)-\frac1l{N+\alpha}\mathcal{P}(u_n) \\ &\geqslant\biggl(\frac12-\frac{N-2s}{2(N+\alpha)}\biggr) \|u_n\|_{D_A^s}^2 +\biggl(\frac12-\frac{N}{2(N+\alpha)}\biggr)V_0\int_{\mathbb{R}^N}u_n^2\, dx, \end{aligned}
\end{equation*}
\notag
$$
which means that $\{u_n\}_{n\in\mathbb{N}}$ is bounded in $\mathcal{H}$. Hence, in the sense of subsequence, we have
$$
\begin{equation*}
\begin{aligned} \, u_n &\rightharpoonup u_0\quad \text{in }\ \mathcal{H}, \\ u_n &\to u_0\quad \text{in }\ L^q(K,\mathbb{C})\text{ for }2\leqslant q<2_s^*, \\ u_n(x) &\to u_0(x)\quad \text{a. e. on }\ \mathbb{R}^N, \end{aligned}
\end{equation*}
\notag
$$
where $K\subset\mathbb{R}^N$ is a compact set. Since $\mathcal{I}'_{s,A}(u_n)\to0$ as $n\to\infty$, we have
$$
\begin{equation*}
\begin{aligned} \, 0=\langle \mathcal{I}'_{s,A}(u_0),v\rangle &=\langle u_0,v\rangle -\operatorname{Re}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_0|^a]|u_0|^{a-1}\overline{v}\, dx \\ &\qquad -\operatorname{Re}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_0|^b]|u_0|^{b-1}\overline{v}\, dx \end{aligned}
\end{equation*}
\notag
$$
for each $v\in\mathcal{H}$. Thus,
$$
\begin{equation*}
\mathcal{I}'_{s,A}(u_0)=0 \quad\text{and} \quad \mathcal{P}(u_0)=0.
\end{equation*}
\notag
$$
Combining this with Lemma 2.4, we find that $u_0\neq0$ and $u_0\in\mathcal{M}$. By Lemma 2.4 in [4] and Lemma 1.32 in [12], we have
$$
\begin{equation}
\lim_{n\to\infty}\biggl(\int_{\mathbb{R}^N}|u_n-u_0|^2\, dx-\int_{\mathbb{R}^N}|u_n|^2\, dx\biggr) =\int_{\mathbb{R}^N}|u_0|^2\, dx,
\end{equation}
\tag{3.4}
$$
$$
\begin{equation}
\lim_{n\to\infty}\bigl(\| u_n-u_0\|_{D_A^s}^2-\| u_n\|^2_{D_A^s}\bigr)=\| u_0\|_{D_A^s}^2,
\end{equation}
\tag{3.5}
$$
$$
\begin{equation}
\begin{split} &\lim_{n\to\infty}\biggl(\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n-u_0|^b]|u_n-u_0|^b\, dx -\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n|^b]|u_n|^b\, dx\biggr) \\ &\qquad=\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_0|^b]|u_0|^b\, dx \end{split}
\end{equation}
\tag{3.6}
$$
and
$$
\begin{equation}
\begin{aligned} \, &\lim_{n\to\infty}\biggl(\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n-u_0|^a]|u_n-u_0|^a\, dx -\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n|^a]|u_n|^a\, dx\biggr) \nonumber \\ &\qquad=\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_0|^a]|u_0|^a\, dx. \end{aligned}
\end{equation}
\tag{3.7}
$$
Since $\mathcal{P}(u_0)=0$ and using (3.4)–(3.7), we have
$$
\begin{equation*}
\lim_{n\to\infty}\mathcal{P}(u_n-u_0)=\lim_{n\to\infty}\mathcal{P}(u_n)=0.
\end{equation*}
\notag
$$
In addition, $\langle\mathcal{I}'_{s,A}(u_0),u_0\rangle=0$ and $\langle\mathcal{I}'_{s,A}(u_n),u_n\rangle=o(1)$, and hence
$$
\begin{equation}
\langle \mathcal{I}'_{s,A}(u_n-u_0),u_n-u_0\rangle=o(1).
\end{equation}
\tag{3.8}
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{P}(u_n-u_0)-\frac{N-2s}{2}\langle \mathcal{I}'_{s,A}(u_n-u_0),u_n-u_0\rangle \\ &\qquad=s\biggl(\int_{\mathbb{R}^N}V(x)|u_n-u_0|^2\, dx -\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n-u_0|^a]|u_n-u_0|^a\, dx\biggr)=o(1), \end{aligned}
\end{equation*}
\notag
$$
that is,
$$
\begin{equation}
\int_{\mathbb{R}^N}V(x)|u_n-u_0|^2\, dx=\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n-u_0|^a]|u_n-u_0|^a\, dx+o(1).
\end{equation}
\tag{3.9}
$$
Now by (3.8) and (3.9) we have
$$
\begin{equation*}
\begin{aligned} \, \mathcal{I}_{s,A}(u_n) &=\mathcal{I}_{s,A}(u_n-u_0)+\mathcal{I}_{s,A}(u_0)+o(1) \\ &=\frac{a-1}{2a}\| u_n-u_0\|^2_{D_A^s}+\frac{a-1}{2a}\int_{\mathbb{R}^N}V(x)|u_n-u_0|^2\, dx \\ &\qquad -\frac{a-b}{2ab}\int_{\mathbb{R}^N}[I_{\alpha}\ast|u_n-u_0|^b]|u_n-u_0|^b\, dx +\mathcal{I}_{s,A}(u_0)+o(1) \\ &=\frac{a-1}{2a}\| u_n-u_0\|^2_{D_A^s}+\frac{b-1}{2b}\int_{\mathbb{R}^N}V(x)|u_n-u_0|^2\, dx +\mathcal{I}_{s,A}(u_0)+o(1) \\ &\geqslant\mathcal{I}_{s,A}(u_0)+o(1). \end{aligned}
\end{equation*}
\notag
$$
Hence $\mathcal{I}_{s,A}(u_0)\leqslant c$. Since $\mathcal{I}_{s,A}(u_0)\geqslant c$, we find that $\mathcal{I}_{s,A}(u_0)=c$. This means that $u_0$ is a ground state solution of equation (1.1). Theorem 1.1 is proved.
Citation in format AMSBIB
\Bibitem{GuoZha24} |
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