Sharp nonexistence results of prescribed L ²-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations

Abstract

In this paper, we study the existence of minimizers for

$$F(u) = \frac{1}{2} \int_{\mathbb{R}^3} |\nabla u|^{2} {\rm d}x + \frac{1}{4} \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{| u(x)|^2 | u(y)|^2}{| x-y|} {\rm d}x{\rm d}y-\frac{1}{p} \int_{\mathbb{R}^3}|u|^p {\rm d}x$$

on the constraint

$$S(c) = \{u \in H^1(\mathbb{R}^3) : \int_{\mathbb{R}^3}|u|^2 {\rm d}x = c\}$$

, where c >  0 is a given parameter. In the range ${p \in [3,\frac{10}{3}]}$ , we explicit a threshold value of c >  0 separating existence and nonexistence of minimizers. We also derive a nonexistence result of critical points of F(u) restricted to S(c) when c >  0 is sufficiently small. Finally, as a by-product of our approaches, we extend some results of Colin et al. (Nonlinearity 23(6):1353–1385, 2010) where a constrained minimization problem, associated with a quasi-linear equation, is considered.