On singularly perturbed $(p, N )$-Laplace Schr\"{o}dinger equation with logarithmic nonlinearity

TL;DR

This paper investigates the existence, multiplicity, and concentration of solutions for a complex $(p, N)$-Laplace Schr"odinger equation with logarithmic and critical exponential nonlinearities, using advanced variational methods.

Contribution

It introduces new analytical techniques to establish solution multiplicity linked to the potential's topology in the context of a singularly perturbed $(p, N)$-Laplace Schr"odinger equation.

Findings

Existence of multiple positive solutions.

Concentration behavior of solutions near potential minima.

Application of Lusternik-Schnirelmann theory to relate solutions to topology.

Abstract

This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a $(p, N)$-Laplace Schr\"{o}dinger equation with logarithmic nonlinearity and critical exponential nonlinearity in the sense of Trudinger-Moser in the whole Euclidean space $\mathbb{R}^N$. Through the use of smooth variational methods, penalization techniques, and the application of the Lusternik-Schnirelmann category theory, we establish a connection between the number of positive solutions and the topological properties of the set in which the potential function achieves its minimum values.