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 {[ -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, ]. where a,ε> 0, λ∈ℝ is known as the Lagrange multiplier, Δ_p· =div (|∇·|^p-2∇·) denotes the usual p-Laplacian operator with 2≤ p < N and V ∈𝒞^0(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 ε→0^+ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic p-Laplacian equations involving mass-supercritical nonlinearities.