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

摘要

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 agiven parameter. In the range $p \in [3, 10/3]$ we explicit a threshold valueof $c>0$ separating existence and non-existence of minimizers. We also derive anon-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, weextend some results of \cite{CJS} where a constrained minimization problem,associated to a quasilinear equation, is considered.