On the logarithmic p-Laplacian Schrödinger equation with multi-well potential: Multi-peak positive solutions

This paper investigates the existence and multiplicity of multi-peak positive solutions for a class of [Formula: see text]-Laplacian Schrödinger equations with logarithmic non-linearity in [Formula: see text]: [Formula: see text] Here, [Formula: see text] is a small parameter and the potential [Formula: see text] has multiple wells. The problem is characterized by the combined difficulties of the non-homogeneous [Formula: see text]-Laplacian and a logarithmic term which prevents standard variational approaches. By combining a penalization method with variational techniques, we prove that for any non-empty set of potential wells, there exists a family of positive solutions. These solutions exhibit multiple peaks which concentrate at the minima of the potential as [Formula: see text]. This establishes the existence of at least [Formula: see text] distinct multi-peak solutions, where [Formula: see text] is the number of wells, providing the first systematic result on the construction of such solutions for this class of quasilinear logarithmic problems.