Normalized solution to the nonlinear p-Laplacian equation with an L^2 constrain: mass supercritical case

In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\ge 3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -Δ_{p}u+{\vert u\vert}^{p-2}u=f(u)-μu \quad $\text{ in }$ $\mathbb{R}$^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 $-{\mathrm{\Delta }}_{p}u=div\left({\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }}^{p-2}\mathrm{\nabla }u\right)$, $2\le p<N$, $f\in C(\mathbb{R},\mathbb{R})$, $m>0$, $\mu \in \mathbb{R}$ will appear as a Lagrange multiplier and the continuous nonlinearity $f$ satisfies mass supercritical conditions. We mainly study the behavior of ground state energy $E_m$ with $m>0$ changing within a certain range and aim at extending nonlinear scalar field equation when $p=2$ and reducing the constraint condition of nonlinearity $f$.