Existence and concentration of normalized solutions for p-Laplacian equations with logarithmic nonlinearity

We investigate the existence and concentration of normalized solutions for a p-Laplacian problem with logarithmic nonlinearity of type \[ \{ {array}{ll} -ε^pΔ_p u+V(x)|u|ᵖ⁻²u=λ |u|ᵖ⁻²u+|u|ᵖ⁻²ulog|u|^p ~in~ R^N, ∫R^N|u|^pdx=a^pε^N, {array} . \] where a,ε> 0, λ∈ R is known as the Lagrange multiplier, Δₚ· =div (|∇ ·|ᵖ⁻²∇ ·) denotes the usual p-Laplacian operator with 2≤ p < N and V ∈ C⁰( RN) 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 ε→0⁺ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic p-Laplacian equations involving mass-supercritical nonlinearities.