Critical Schrödinger–Bopp–Podolsky systems: solutions in the semiclassical limit

In this paper we consider the following critical Schrödinger–Bopp–Podolsky system \(\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^2 \Delta u+ V(x)u+Q(x)\phi u=h(x,u)+K(x)\vert u \vert ^{4}u&{} \text{ in } \ \mathbb {R}^3 \\ - \Delta \phi + a^{2}\Delta ^{2} \phi = 4\pi Q(x) u^{2}&{} \text{ in } \ \mathbb {R}^3 \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-{ϵ}^2\mathrm{\Delta }u+V(x)u+Q(x)\varphi u=h(x,u)+K(x){|u|}^{4}u\hfill & \text{in}{\mathbb{R}}^3\hfill \\ -\mathrm{\Delta }\varphi +a^2{\mathrm{\Delta }}^2\varphi =4\pi Q(x)u^2\hfill & \text{in}{\mathbb{R}}^3\hfill \end{array}\right)\end{array}$$ in the unknowns \(u,\phi :\mathbb {R}^{3}\rightarrow \mathbb {R}\) $u,\varphi :{\mathbb{R}}^3\to \mathbb{R}$ and where \(\varepsilon , a>0\) $\epsilon ,a>0$ are parameters. The functions V, K, Q satisfy suitable assumptions as well as the nonlinearity h which is subcritical. For any fixed \(a>0\) $a>0$ , we show existence of “small” solutions in the semiclassical limit, namely whenever \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ . We give also estimates of the norm of this solutions in terms of \(\varepsilon \) $\epsilon$ . Moreover, we show also that fixed \(\varepsilon \) $\epsilon$ suitably small, when \(a\rightarrow 0\) $a\to 0$ the solutions found strongly converge to solutions of the Schrödinger-Poisson system.