On degenerate fractional Schrödinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields

Abstract

We investigate the degenerate fractional Schrödinger-Kirchhoff-Poisson equation in ${\mathbb{R}}^3$ with critical nonlinearity and electromagnetic fields ${\epsilon }^{2s}M([u{]}_{s,A}^2)(-\Delta {)}_A^{s}u+V(x)u+\varphi u=k(x)|u{|}^{r-2}u+\left({\mathcal{I}}_{\mu }\ast |u{|}^{2_s^{♯}}\right)|u{|}^{2_s^{♯}-2}u$ and $(-\Delta {)}^t\varphi =u^2,$ where $\epsilon >0$ is a parameter, $3/4<s<1$, $0<t<1$, $V$ is an electric potential satisfying some suitable assumptions, $0<k_{\ast }\le k(x)\le k^{\ast }$, ${\mathcal{I}}_{\mu }(x)=|x{|}^{3-\mu }$ with $0<\mu <3$, $2_s^{♯}=\frac{3+\mu }{3-2s},$ and $2<r<2_s^{♯}$. 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 $\epsilon \to 0$ in the degenerate cases, i.e. when the Kirchhoff term $M$ vanishes at zero.

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