In this paper, we study the existence of ground state solutions to the following $p$-Laplacian equation with an $L^2$ constraint
$$\begin{equation*} \begin{cases} -\Delta_{p}u+{\vert u\vert}^{p-2}u=f(u)-\mu u, x\in\mathbb{R}^N,\\ {\Vert u\Vert}^2_{L^2(\mathbb{R}^N)}=m,\\ u\in W^{1,p}(\mathbb{R}^N)\cap L^2(\mathbb{R}^N), \end{cases} \end{equation*}$$
where $N\geq3$, $2\leq p<N$, $m>0$, the nonlinearity $f\in C(\mathbb{R},\mathbb{R})$ satisfies the mass supercritical conditions and $\mu\in\mathbb{R}$ appears as a Lagrange multiplier. We also reduce the conditions for the nonlinearity $f$ and analyse the behavior of the ground state energy $E_m$ for $m>0$ which generalize the results for the nonlinear scalar field equation with $p=2$.
Q1 （JIF=1.2； JCI=1.15）