Abstract:

In this paper, we consider the $p$-Laplacian Schrödinger-Poisson equation with $L^{2}$-norm constraint

$$-\Delta_p u+|u|^{p-2}u+\lambda u+ \left(\frac{1}{4\pi|x|}*|u|^2\right)u=|u|^{q-2} u,\and x \in \mathbb{R}^3,$$

where $2 \leq p<3$, $\frac{5 p}{3}<q<p^{*}=\frac{3p}{3-p}$, $\lambda>0$ is a Lagrange multiplier. We obtain the critical point of the corresponding functional of the problem on mass constraint by the variational method and the Mountain pass lemma, and then find a normalized solution to this equation.

Key words: Normalized solution, $p$-Laplacian equation , Schr?dinger-Poisson equation , Mountain pass lemma