Ground state solution for a class of magnetic equation with general convolution nonlinearity

1.   Introduction

In this article, we study the following magnetic Laplace nonlinear Choquard equation

where $\Delta_A u : = (\nabla -iA)^2u$ is the magnetic Laplace operator. Here $u: \mathbb{R}^N\rightarrow C$, $A: \mathbb{R}^N\rightarrow \mathbb{R}^N$ is a vector magnetic potential, $N\geq 3$, $F(t) = \int_{0}^{t}f(s)\text{d}s$, $V:\, \mathbb{R}^N \rightarrow \mathbb{R}$ is a scalar potential function and $I_{\alpha}$ is a Riesz potential whose order is $\alpha\, \in\, (N-2, N)$ defined by $I_{\alpha}$ = $\frac{\Gamma(\frac{N-\alpha }{2}) }{\Gamma(\frac{\alpha}{2})\pi^{\frac{N}{2}}2^\alpha |x|^{N-\alpha } }$, where $\Gamma$ is the Gamma function. $V(x): \mathbb{R}^N \rightarrow \mathbb{R}$ is a continuous, bounded potential function satisfying:

(V1) $\inf\limits_{ \mathbb{R}^N}V(x) > 0$,

(V2) there exist a constant $V_{\infty} > 0$ such that for all $x\in \mathbb{R}^N$,

We also suppose $A$ satisfies:

(A1) ${\liminf\limits_{|x| \to +\infty}} A(x) = A_{\infty}$,

(A2) $A\in L^{\upsilon }(\mathbb{R}^N, \mathbb{R}^N), \upsilon > N\geq 3$,

(AV) $|A(y)|^2+V(y) < |A_{\infty}|^2+V_{\infty}$.

Moreover, we assume that the function $f\in C^1(\mathbb{R}, \mathbb{R})$ verifies:

$(f1) \, \, f(t) = o(t^{\frac{\alpha }{N}})$ as $t\rightarrow 0,$

$(f2) \, \, \lim\limits_{|t| \rightarrow +\infty} \frac{f(t)}{t^{\frac{\alpha+2 }{N-2}}} = 0,$

$(f3) \, \, \frac{f(t)}{t}$ is increasing on $(0, +\infty)$ and decreasing on $(-\infty, 0)$.

$(f4) \, \, f(t)$ is increasing on $\mathbb{R}$.

It should be noted that there is a lot of literature on the competition phenomena for elliptic equations without magnetic potential in different situations, i.e. $A\equiv 0$. Actually, when $A\equiv 0$ it conduces to the Choquard equation. There is a huge collections of articles on the subject and some good reviews of the Choquard equation can be found in ^{[1,2,3,4,5,6,7,8,9]}.

On the other hand, there are works concerning the following Schrödinger equations with magnetic field recently:

Here $u: \Omega \rightarrow C$, $-\Delta_A u : = (-i\nabla +A)^2u$, $2 < p\leq 2^*$, where $2^{*} = \frac{2N}{N-2}$ if $N\geq 3$ and $2^{*} = \infty$ if $N = 1$ or 2. Besides, $A: \Omega \rightarrow \mathbb{R}^N$ and $V: \Omega \rightarrow \mathbb{R}$ are smooth.

To the best of our knowledge, the first paper in which problem (1.2) has been studied maybe Esteban-Lions^[10]. They have used the concentration-compactness principle and minimization arguments to prove the existence of solutions for $N = 2$ and $N = 3$. More recently, applying constrained minimization and a minimax-type argument, Arioli-Szulkin^[11] considered the equation in a magnetic filed. They established the existence of nontrivial solutions both in the critical and in the subcritical case, provided that some technical conditions relating to $A$ and $V$ were assumed. We also refer to ^[12,13] for other results related to problem (1.1) in the presence of the magnetic field when the nonlinearity has a subcritical growth. Besides, we must mention the works ^[14,15] for the critical case and also refer to the recent papers ^[16,17,18] for the study of various classes of PDEs with magnetic potential.

Inspired by the above works, we want to research the Eq (1.2) with general convolution term as the right-hand side, i.e. Eq (1.1). Our aim of this paper is to prove the existence of a ground state solution for problem (1.1), that is a nontrivial solution with minimal energy.

Notice that if we define

our assumptions assure that $\tilde{f}(t)$ is continuous. Therefore, Eq (1.1) can be rewritten in the form

The right-hand side of problem (1.3) generalizes the term $\big(\frac{1}{|x|^\alpha }*|u|^p\big)|u|^{p-2}u$, which was studied by Cingolani, Clapp and Secchi in ^[19]. Similar problems were also studied in ^[20,21,22]. Especially, it is worth mentioning that in ^[23], the authors obtained the ground state solution of the following Eq (1.4)

which can be rewritten in the form

They considered the "limit problem" of problem (1.4), then by the splitting lemma, they proved that (1.4) has at least a ground state solution. In our paper, we improve the growth condition of $f$ to the critical case, and generalizes the convolution term to a more general case. Most importantly, we get the ground state solution in a more straightforward way which is completely different from ^[23].

Our main result is as follows:

Theorem 1.1. If $\alpha \in (N-2, N)$, (A1), (A2), (V1), (V2), (AV) are valid, and $f\in C^1(\mathbb{R}, \mathbb{R})$ verifies (f1)-(f3), then problem(1.1) has at least a ground state solution.

Now we define $\nabla _A u = -i\nabla u-Au$ and consider the space

equipped with scalar product

Therefore

which is an equivalent norm to the norm obtained by considering $V\equiv 1$, see^[6].

Hereafter for the convenience of narration, we will use the following notations:

● $L^r(\mathbb{R}^N)(1\leq r < \infty)$ denotes the Lebesgue space in which the norm is defined as follows

● $C, C_{\varepsilon }, C_{1}, C_{2}, ...$ denote positive constants which are possibly different in different lines.

2.   Preliminaries

In this section, we will give some very important inequalities and lemmas.

Lemma 2.1. ^[10] Assume $u \in H^1_{A, V}$, then $|u| \in H^1(\mathbb{R}^N)$ and the diamagnetic inequality holds $\big|\nabla |u|(x)\big|\leq |\nabla _Au(x)|$.

Remark 2.2. It is well known that the embedding $H^1_{A, V}\hookrightarrow L^r(\mathbb{R}^N, C)$ is continuous for $r \in [1, 2^*]$.

Lemma 2.3. Assume (f1)–(f4) hold, then we have

$(1)$ $for\, \, all\, \, \varepsilon > 0, \, \, there\, \, is\, \, a\, \, C_{\varepsilon} > 0\, \, such\, \, that\, \, |f(t)|\leq \varepsilon |t|^{\frac{\alpha }{N}}+C_{\varepsilon}|t|^{{\frac{\alpha+2 }{N-2}}}\, \, and\, \, |F(t)|\leq \varepsilon |t|^{\frac{N+\alpha }{N}}+C_{\varepsilon}|t|^{\frac{N+\alpha }{N-2}},$

$(2)$ $for\, \, all\, \, \varepsilon > 0, \, \, there\, \, is\, \, a\, \, C_{\varepsilon} > 0\, \, such\, \, that\, \, for\, \, every\, \, p\in (2, 2^{*}), \, \, |F(t)|\leq \varepsilon (|t|^{\frac{N+\alpha }{N}}+|t|^{{\frac{N+\alpha }{N-2}}})+C_{\varepsilon}|t|^{\frac{p(N+\alpha) }{2N}}, and\, \, |F(t)|^{\frac{2N}{N+\alpha }}\leq \varepsilon (|t|^2+|t|^{\frac{2N }{N-2}})+C_{\varepsilon}|t|^p,$

$(3)$ $for\, \, any\, \, s\neq 0, \, sf(s) > 2F(s)\, \, and \, \, F(s) > 0.$

Proof. One can easily obtain the results by elementary calculation.

Lemma 2.4. ^[23] Let $O \subset \mathbb{R}^N$ be any open set, for $1 < p < \infty$, and $\{f_n\}$ be a bounded sequence in $L^p(O, C)$ such that $f_n(x)\rightharpoonup f(x)$ a.e., then $f_n(x)\rightharpoonup f(x)$.

Lemma 2.5. ^[23] Suppose that $u_n\rightharpoonup u_0$ in $H^1_{A, V}(\mathbb{R}^N, C)$, and $u_n(x)\rightarrow u_0(x)$ a.e. in $\mathbb{R}^N$, then $I_{\alpha}*F(|u_n(x)|)\rightharpoonup I_{\alpha}*F(|u_0(x)|)$ in $L^{\frac{2N}{\alpha }}(\mathbb{R}^N)$.

Corollary 2.6. Suppose that $u_n\rightharpoonup u_0$ in $H^1_{A, V}(\mathbb{R}^N, C)$, then $\mathfrak{R}e\int_{ \mathbb{R}^N}I_{\alpha}*F(|u_n|)\tilde{f}(|u_n|)u_n\overline{\varphi }\rightarrow \mathfrak{R}e\int_{ \mathbb{R}^N}I_{\alpha}*F(|u_0|)\tilde{f}(|u_0|)u_0\overline{\varphi }$ for $\varphi \in C_c^{\infty}(\mathbb{R}^N, C)$.

Lemma 2.7. (Hardy-Littlewood-Sobolev inequality ^[6]). $Let\, \, 0 < \alpha < N, \, p, q > 1\, \, and\, \, 1\leq r < s < \infty\, \, be\, \, such\, \, that$

$(1)$$For\, \, any\, \, f\in L^p(\mathbb{R}^N)\, \, and\, \, g\in L^q(\mathbb{R}^N), \, \, one \, \, has$

$(2)$$For\, \, any\, \, f\in L^r(\mathbb{R}^N)\, \, one \, \, has$

Remark 2.8. By Lemma 2.3 (1), Lemma 2.7 (1) and Sobolev imbedding theorem, we can get

3.   Variational formulation

The energy functional associated to problem (1.1) is given by:

The derivative of the energy functional $J_{A, V}(u)$ is given by

Thus,

Now, we can prove the following results.

Lemma 3.1. The functional $J_{A, V}$ possesses the mountain-pass geometry, that is

$(1)$ there exist $\rho, \delta > 0$ such that $J_{A, V} \geq \delta$ for all $\|u\Arrowvert = \rho$;

$(2)$ for any $u \in H^1_{A, V}(\mathbb{R}^N, C)\backslash \{0\}$, there exist $\tau \in (0, +\infty)$ such that $\|\tau u\Arrowvert > \rho$ and $J_{A, V}(\tau u) < 0$.

Proof. (1) By Lemma 2.7 (1) and Lemma 2.3, one can get

Thus there exist $\rho, \delta > 0$ such that $J_{A, V}\geq \delta$ for all $\|u\Arrowvert = \rho > 0$ small enough.

(2) For any fixed $u_0 \in H^1_{A, V}\backslash \{0\}$, consider the function $g_{u_0}(t): (0, +\infty)\rightarrow \mathbb{R}$ given by

then

Thus, $lng_{u_0}(t)\big|_1^{\tau \|u_0\Arrowvert_{A, V}}\geq 4lnt\big|_1^{\tau \|u_0\Arrowvert_{A, V}}$. So $\frac{g_{u_0}(\tau \|u_0\Arrowvert_{A, V})}{g_{u_0}(1)}\geq (\|u_0\Arrowvert_{A, V})^4$ which implies that $g_{u_0}(\tau \|u_0\Arrowvert_{A, V})\geq M(\|u_0\Arrowvert_{A, V})^4$ for a constant $M > 0$. Then we can get

yields that $J_{A, V}(\tau u_0) < 0$ when $\tau$ is large enough.

Hence we can define the mountain-pass level of $J_{A, V}$ :

where:$\, \, \Gamma = \{\gamma \in C([0, 1], H^1_{A, V}(\mathbb{R}^N, C)):\gamma(0) = 0, J_{A, V}(\gamma(1)) < 0\}$.

Now we recall the Nehari manifold

Let$\, \, c_{\alpha } = \inf\limits_{u \in \mathcal {N}_{\alpha } }J_{A, V}(u),$ Moreover by the similar argument as Chapter 4^[24], we have the following characterization

Remark 3.2. If we set $\Phi (t) = \frac{1}{2}\|tu\Arrowvert^2_{A, V}-\frac{1}{2}\int_{ \mathbb{R}^N}(I_{\alpha}*F(|tu|))F(|tu|)\text{d}x$, the proof of Lemma 3.1 assures that $\Phi (t) > 0$ for $t$ small enough, and $\Phi (t) < 0$ for $t$ large enough. Besides $g'_{u}(t) > 0$ if $t > 0$, we can get that $\max\limits_{t \geq 0}\Phi (t)$ is achieved at a unique $t_u > 0$. Furthermore, $\Phi' (t_u) = 0$ implies that $t_u u \in \mathcal {N}_{\alpha }$ and the map $u\rightarrow t_u (u\neq 0)$ is continuous.

4.   Ground state solution for problem (1.1)

In this section, we prove the main theorem.

Proof of Theorem 1.1. Let $\{u_n\}$ be minimizing sequence given as a consequence of Lemma 3.1 i.e. $\{u_n\} \subset H^1_{A, V}$ such that $J_{A, V}'(u_n)\rightarrow 0$, $J_{A, V}(u_n)\rightarrow c$, where $c = c_{\alpha } = \inf\limits_{u \in \mathcal {N}_{\alpha } }J_{A, V}(u) = c^{*} = \inf\limits_{u \in H^1_{A, V}(\mathbb{R}^N, C)\backslash \{0\} }\max\limits_{t \geq 0}J_{A, V}(tu)$. Then we have

Consequence, $\{u_n\}$ is bounded. Then by standard methods we can get the convergence of $\{u_n\}$.

Next, let $\delta : = \limsup\limits_{n\rightarrow \infty}\sup\limits_{y\in \mathbb{R}^N}\int_{B_1 (y)}|u_n|^2\text{d}x$. We claim $\delta > 0$. On the contrary, by Lions' concentration compactness principle, we have $u_n\rightarrow 0$ in $L^p (\mathbb{R}^N)$ for $2 < p < 2^*$. By Lemma 2.3(2), for any $\varepsilon > 0$ there exist a constant $C_{\varepsilon } > 0$ such that

Note that $\varepsilon$ is arbitrary, we get

Combining with $J_{A, V}'(u_n)\rightarrow 0$, we can get

which implies that

Then we have $\int_{ \mathbb{R}^N}[|\nabla_A u_n|^2+V(x)|u_n|^2]\text{d}x\rightarrow 0$, which implies $u_n\rightarrow 0$ in $H^1_{A, V}$. We deduce that $c_\alpha = 0$, which contradicts to the fact that $c_{\alpha } > 0$. Hence $\delta > 0$ and there exist $\{y_n\} \subset \mathbb{R}^N$ such that $\int_{B_1(y_n)}|u_n|^p\text{d}x\geq \frac{\delta }{2} > 0$. We set $v_n(x) = u_n(x+y_n)$, then $\|u_n\Arrowvert = \|v_n\Arrowvert, \, \, \int_{B_1(0)}|v_n|^p\text{d}x > \frac{\delta }{2}$ and $J_{A, V}(v_n)\rightarrow c_{\alpha } = c, \, \, J'_{A, V}(v_n)\rightarrow 0$. Thus there exist a $v_0\neq 0$ such that

Then for any $\varphi \in C_0^{\infty}(\mathbb{R}^N)$ we have $0 = \langle J_{A, V}'(v_n), \varphi \rangle+o(1) = \langle J_{A, V}'(v_0), \varphi \rangle$, which means $v_0$ is a solition of Eq (1.1).

On the other hand, combining with the Fatou Lemma, we can obtain

At the same time, we know $c_{\alpha }\leq J_{A, V}(v_0)$ by the definition of $c_{\alpha }$. Then we can deduce that $v_0$ is a ground state solution of Eq (1.1).

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 11771198, 11361042, 11901276).

Conflict of interest

There is no conflict of interest.