Sharp non-existence results of prescribed L^2-norm solutions for some class of Schr\"odinger-Poisson and quasilinear equations

In this paper we study the existence of minimizers for $$F(u)=\text{1}\mathrm{/}2{\int }_{{\mathbb{R}}^3}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}dx+1\mathrm{/}4{\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}\frac{\mathrm{\mid }u(x){\mathrm{\mid }}^2\mathrm{\mid }u(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y\mathrm{\mid }}dxdy-\frac{1}{p}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }u{\mathrm{\mid }}^{p}dx$$ on the constraint $$S(c)=\{u\in H^1({\mathbb{R}}^3):{\int }_{{\mathbb{R}}^3}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=c\},$$ where $c>0$ is a given parameter. In the range $p\in [3,10\mathrm{/}3]$ we explicit a threshold value of $c>0$ separating existence and non-existence of minimizers. We also derive a non-existence result of critical points of $F(u)$ restricted to $S(c)$ when $c>0$ is sufficiently small. Finally, as a byproduct of our approaches, we extend some results of \cite{CJS} where a constrained minimization problem, associated to a quasilinear equation, is considered.