For \(N\ge 3\) and \(2<p<N\) , we find normalised solutions to the equation \(\begin{aligned} -\Delta _p u+(1+V(x))|u|^{p-2}u+\lambda u&=|u|^{q-2}u\qquad \text {in }\,\, {\mathbb {R}}^{N}\\ \Vert u\Vert _2&=\rho \end{aligned}\) 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 \(q\in (p\frac{N+2}{N},\frac{Np}{N-p})\) .
