We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{ array}{ ll} \displaystyle -\varepsilon^p\Delta_p u+V(x)|u|^{ p-2}u=\lambda |u|^{ p-2}u+|u|^{ p-2}u\log|u|^p ~\text{ in}~\mathbb R^N,\newline \displaystyle \int_{ \mathbb R^N}|u|^pdx=a^p\varepsilon^N, \end{ array} \right. \] where $a,\varepsilon> 0$, $\lambda\in\mathbb R$ is known as the Lagrange multiplier, $\Delta_p\cdot =\text{ div} (|\nabla \cdot|^{ p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in \mathcal{ C}^0(\mathbb R^N)$ is the potential which satisfies some suitable assumptions. We prove that the number of positive solutions depends on the profile of $V$ and each solution concentrates around its corresponding global minimum point of $V$ in the semiclassical limit when $\varepsilon\to0^+$ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities.
Categories: math.AP