Normalised solutions for $p$-Laplacian equations with $L^p$-supercritical growth

For $N\ge 3$ and $2<p<N$, we find normalised solutions to the equation \begin{align*} -\Delta_p u+(1+V(x))|u|^{p-2}u+\lambda u&=|u|^{q-2}u\qquad\text{in $\mathbb{R}^N$}\\ \|u\|_2&=\rho \end{align*} in the mass supercritical and Sobolev subcritical case, that is $q\in(p\frac{N+2}{N},\frac{Np}{N-p})$, at least if $\rho>0$ is small enough. The function $V\in L^{N/p}(\mathbb{R}^N)$, which plays the role of potential, is assumed to be non-positive and vanishing at infinity. Moreover, we will prove the compactness of the embedding of the space of radial functions $W^{1,p}_{rad}(\mathbb{R}^N)\subset L^q(\mathbb{R}^N)$ for $p\in(1,N)$ and $q\in(p\frac{N+2}{N},\frac{Np}{N-p})$.