We establish the existence of a normalized solution for the following quasilinear elliptic problem: \(\begin{aligned} \left\{ \begin{array}{cc} -\Delta _p u + V(x) |u|^{p-2}u = \lambda |u|^{p - 2}u + \left| u\right| ^{q-2}u\ \text {in}\ \mathbb {R}^N,\\ \int _{\mathbb {R}^N} |u|^p \textrm{d}x = m^p,\ u \in W^{1, p}(\mathbb {R}^N),\end{array}\right. \end{aligned}\) where \(N > p\) , \(1< p< p+ \frac{p^2}{N}< q < p^*\) and \(p^* = Np/(N - p)\) . Our main results deal with nonnegative potentials V such that \(\lim _{|x| \rightarrow \infty } V(x) = 0\) . Hence, we also consider the zero mass case proving that our main problem has at least one normalized solution in the \(L^p\) -supercritical case. Furthermore, using extra assumptions on the potential V, we consider some results with \(\lim _{|x| \rightarrow \infty } V(x) = L\) is allowed with \(L > 0\) and V can vanish in some open subset \(\Omega \subset \mathbb {R}^N\) .
