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

In this paper we consider the following critical Schrödinger–Bopp–Podolsky system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^2 \Delta u+ V(x)u+Q(x)\phi u=h(x,u)+K(x)\vert u \vert ^{4}u&{} \text{ in } \ \mathbb {R}^3 \\ - \Delta \phi + a^{2}\Delta ^{2} \phi = 4\pi Q(x) u^{2}&{} \text{ in } \ \mathbb {R}^3 \end{array}\right. } \end{aligned}$$

in the unknowns \(u,\phi :\mathbb {R}^{3}\rightarrow \mathbb {R}\) and where \(\varepsilon , a>0\) are parameters. The functions V, K, Q satisfy suitable assumptions as well as the nonlinearity h which is subcritical. For any fixed \(a>0\), we show existence of “small” solutions in the semiclassical limit, namely whenever \(\varepsilon \rightarrow 0\). We give also estimates of the norm of this solutions in terms of \(\varepsilon \). Moreover, we show also that fixed \(\varepsilon \) suitably small, when \(a\rightarrow 0\) the solutions found strongly converge to solutions of the Schrödinger-Poisson system.