Sharp non-existence results of prescribed L²-norm solutions for some class of Schr\"odinger-Poisson and quasilinear equations

In this paper we study the existence of minimizers for $$ F(u) = \1/2\int_{\R^3} |\nabla u|^2 dx + 1/4\int_{\R^3}\int_{\R^3}\frac{| u(x) |^2| u(y) |^2}{| x-y |}dxdy-\frac{1}{p}\int_{\R^3}| u |^p dx$$ on the constraint $$S(c) = \{u \in H^1(\R^3) : \int_{\R^3}|u|^2 dx = c \},$$ where $c>0$ is a given parameter. In the range $p \in [3, 10/3]$ 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.