On degenerate fractional Schr\"{o}dinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields

Abstract

We investigate the degenerate fractional Schr\"{o}dinger-Kirchhoff-Poisson equation in $\mathbb{R}^3$ with critical nonlinearity and electromagnetic fields $\varepsilon^{2s} M([u]_{s,A}^2)(-\Delta)_{A}^su + V(x)u + \phi u = k(x)|u|^{r-2}u + \left(\mathcal{I}_\mu*|u|^{2_s^\sharp}\right)|u|^{2_s^\sharp-2}u$ and $(-\Delta)^t\phi = u^2,$ where $\varepsilon > 0$ is a parameter, $3/4<s<1$, $0 < t < 1$, $V$ is an electric potential satisfying some suitable assumptions, $0 < k_\ast \leq k(x) \leq k^\ast$, $\mathcal{I}_\mu(x) = |x|^{3-\mu}$ with $0<\mu<3$, $2_s^\sharp =\frac{3+\mu}{3-2s},$ and $2 < r < 2_s^\sharp$. With the help of the concentration compactness principle and variational methods, together with some fine analytical tools, we establish the existence and multiplicity of solutions for the above problem when $\varepsilon \rightarrow 0$ in the degenerate cases, i.e. when the Kirchhoff term $M$ vanishes at zero.