Multiple solutions for Schr{ö}dinger-Poisson-Slater equations with critical growth

We obtain multiple solutions for the zero mass Schr{ö}dinger-Poisson-Slater equation $$-\mathrm{\Delta }u+(\frac{1}{4\pi \mathrm{\mid }x\mathrm{\mid }}\ast u^2)u=\lambda g(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{6-2}u\text{, }u\in {\mathcal{D}}^{1,2}({\mathbb{R}}^3)$$ for $\lambda \gg 1$, where $p\in (4,6)$ and $g\in L^{6\mathrm{/}(6-p)}({\mathbb{R}}^3)$. The crucial (PS)${}_c$ condition is verified using a simpler method. Similar multiplicity result is also obtained for related equation with an external potential.