Semiclassical States of Fractional Choquard Equations with Exponential Critical Growth

Abstract

This paper is concerned with the following one-dimensional fractional Choquard equation

$$\begin{aligned} \varepsilon (-\Delta )^{1/2}u+V(x)u=\varepsilon ^{\mu }\left( I_{\mu }*F(u)\right) f(u), \ \ x\in \mathbb {R},\end{aligned}$$

where $(-\Delta )^{1/2}$ denotes the 1/2-Laplacian opertor, $I_{\mu }$ is the Riesz potential with $0<\mu <1$, V is a continuous real function satisfying some mild assumptions and $f\in {\mathcal {C}}(\mathbb {R},\mathbb {R})$ is a nonlinearity with exponential critical growth. The present paper has three typical features. Firstly, using a weaker assumption on f, we establish the energy inequality to recover the compactness. Second, without the strictly monotone condition and by establishing some new tricks, we obtain the existence of ground state solutions when $\varepsilon$ is small enough. Finally, we take advantage of some refined analysis techniques to get over the difficulty carried by the nonlocality of the 1/2-Laplacian and prove the concentration of the ground state solutions when $\varepsilon \rightarrow 0$. Our results extend and complement the results of Alves et al. (J Differ Equ 261:1933–1972, 2016) and Clemente et al. (Z Angew Math Phys 72:16, 2021].

Appendix

To supplement the proof of Lemma 17, in this section we collect some preliminary facts. First of all, we denote the upper half-space in $\mathbb {R}^{2}$ by $\mathbb {R}_{+}^{2}=\{(x,y)\in \mathbb {R}^{2}:y>0\}$. Define the following:

$$\begin{aligned} \Vert v\Vert ^{\varepsilon }&:=\left( \int _{\mathbb {R}_{+}^{2}}|\nabla v(x,y)|^{2}\mathrm {d}x\mathrm {d}y +\int _{\mathbb {R}}V(\varepsilon x)|v(x,0)|^{2}\mathrm {d}x\right) ^{1/2}. \end{aligned}$$

The space $X^{1}(\mathbb {R}_{+}^{2})$ is given by the completion of $C_{0}^{\infty }(\overline{\mathbb {R}_{+}^{2}})$ with the norm $\Vert \cdot \Vert ^{\varepsilon }$, that is

$$\begin{aligned} X^{1}(\overline{\mathbb {R}_{+}^{2}})&:=\overline{C_{0}^{\infty }(\overline{\mathbb {R}_{+}^{2}})}^{\Vert \cdot \Vert ^{\varepsilon }}. \end{aligned}$$

Moreover, we denote by $\Vert \cdot \Vert$ the usual norm in $X^{1}(\mathbb {R}_{+}^{2})$, that is

$$\begin{aligned} \Vert v\Vert =\left( \int _{\mathbb {R}_{+}^{2}}|\nabla v(x,y)|^{2}\mathrm {d}x\mathrm {d}y+\int _{\mathbb {R}}|v(x,0)|^{2}\mathrm {d}x\right) ^{\frac{1}{2}}. \end{aligned}$$

Note that the potential V is bounded from above and below, it is easy to verify that $\Vert \cdot \Vert ^{\varepsilon }$ and $\Vert \cdot \Vert$ are equivalent norms in $X^{1}(\mathbb {R}_{+}^{2})$. Using the above definition, we see that if $v\in X^{1}(\mathbb {R}_{+}^{2})$, then $u(x)=v(x,0)$ belongs to $H^{1/2}(\mathbb {R})$ and

$$\begin{aligned} \Vert v\Vert =\Vert u\Vert _{H^{1/2}}. \end{aligned}$$

Since $H^{1/2}(\mathbb {R})$ is continuously embedding into $L^{q}(\mathbb {R})$ for all $q\ge 2$, c.f. [14, Theorem 6.9], it follows that $X^{1}(\mathbb {R}_{+}^{2})$ is also continuously embedded into $L^{q}(\mathbb {R})$ for all $q\ge 2$. Moreover, the embeddings

$$\begin{aligned} X^{1}(\mathbb {R}_{+}^{2})\hookrightarrow L^{q}(A) \end{aligned}$$

are compact for any bounded measurable set $A\subset \mathbb {R}$. See [18, Proposition 3.6] also [15, Remark 2.1].

Next, we will give a brief introduction with the technique used in the proof of Lemma 17, using the change of variable $u(x)=v(\varepsilon x)$, it is easy to know that Problem (4) is equivalent to the problem

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{1/2}u+V(\varepsilon z)u=[I_{\mu }*F(u)]f(u), &{} \text {in} \ \mathbb {R},\\ u\in H^{1/2}(\mathbb {R}), \quad u>0, &{} \text {on} \ \mathbb {R}. \end{array} \right. \end{aligned}$$

(47)

Hereafter, we will use the method due to Caffarelli and Silvestre in [8] to seek the solution of (47), more exactly, due to Frank and Lenzmann [18] for a whole line. In the seminal above papers, were developed a local interpretation of the fractional Laplacian given in $\mathbb {R}$ by considering a Dirichlet to Neumann type operator in the domain $\mathbb {R}_{+}^{2}=\{(x,t)\in \mathbb {R}^{2}:t>0\}$. For $u\in H^{1/2}(\mathbb {R})$, the solution $w\in X^{1}(\mathbb {R}_{+}^{2})$ of

$$\begin{aligned} \left\{ \begin{array}{ll} -\text {div}(\nabla w)=0, &{} \text {in} \ \mathbb {R}_{+}^{2},\\ w=u, &{} \text {on} \ \mathbb {R}\times \{0\} \end{array} \right. \end{aligned}$$

(48)

is called $1/2-$harmonic extension $w=E_{1/2}(u)$ of u and it is proved in [8] that

$$\begin{aligned} \lim _{y\rightarrow 0^{+}}\frac{\partial w}{\partial y}(x,y)=-(-\Delta )^{1/2}u(x). \end{aligned}$$

To get a solution for the nonlocal Problem (47), we can study the existence of solutions for the local problem defined on the upper half plane

$$\begin{aligned} \left\{ \begin{array}{ll} -\text {div}(\nabla w)=0, &{} \text {in} \ \mathbb {R}_{+}^{2},\\ -\frac{\partial w}{\partial v}=-V(\varepsilon x)w+[I_{\mu }*F(w)]f(w), &{} \text {on} \ \mathbb {R}\times \{0\}, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \frac{\partial w}{\partial v}=\lim _{y\rightarrow 0^{+}}\frac{\partial w}{\partial y}(x,y). \end{aligned}$$

We finally emphasize that u is the solution of (47), if and only if, $u=v(x,0)$ for all $x\in \mathbb {R}$, where v is some critical point of the corresponding energy functional of (48).