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On singularly perturbed $(p, N )$-Laplace Schrödinger equation with logarithmic nonlinearity

Deepak Kumar Mahanta, Tuhina Mukherjee, Patrick Winkert

TL;DR

The paper studies a singularly perturbed $(p,N)$-Laplacian Schrödinger equation in $\mathbb{R}^N$ with a logarithmic nonlinearity and a critical exponential growth, establishing existence, multiplicity, and concentration of positive ground-state solutions. A dual penalization strategy is developed to tame the logarithmic and exponential terms, yielding a smooth variational framework on weighted Sobolev/Orlicz spaces and enabling Mountain Pass and Nehari-manifold analyses. The work shows that ground-state solutions concentrate near the minimum set $M$ of the potential $V$, with the concentration location and the number of solutions linked to the topological category $\operatorname{cat}_{M_\delta}(M)$. An autonomous-limit comparison proves that the penalized levels converge to the ground-state energy $c_0$, and a barycenter map formalizes the localization near $M$, while Lusternik–Schnirelmann theory provides quantitative multiplicity results. Overall, the paper extends variational methods to a highly nonhomogeneous, logarithmic-exponential, singularly perturbed setting with Trudinger–Moser-type nonlinearity.

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 Schrödinger 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.

On singularly perturbed $(p, N )$-Laplace Schrödinger equation with logarithmic nonlinearity

TL;DR

The paper studies a singularly perturbed -Laplacian Schrödinger equation in with a logarithmic nonlinearity and a critical exponential growth, establishing existence, multiplicity, and concentration of positive ground-state solutions. A dual penalization strategy is developed to tame the logarithmic and exponential terms, yielding a smooth variational framework on weighted Sobolev/Orlicz spaces and enabling Mountain Pass and Nehari-manifold analyses. The work shows that ground-state solutions concentrate near the minimum set of the potential , with the concentration location and the number of solutions linked to the topological category . An autonomous-limit comparison proves that the penalized levels converge to the ground-state energy , and a barycenter map formalizes the localization near , while Lusternik–Schnirelmann theory provides quantitative multiplicity results. Overall, the paper extends variational methods to a highly nonhomogeneous, logarithmic-exponential, singularly perturbed setting with Trudinger–Moser-type nonlinearity.

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 -Laplace Schrödinger equation with logarithmic nonlinearity and critical exponential nonlinearity in the sense of Trudinger-Moser in the whole Euclidean space . 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.
Paper Structure (6 sections, 36 theorems, 418 equations)

This paper contains 6 sections, 36 theorems, 418 equations.

Table of Contents

  1. Introduction
  2. Some Preliminary Results
  3. Existence of solution for the penalized problem
  4. Existence of positive solution to the main problem via Nehari manifold method
  5. Existence and concentration phenomena of positive ground state solutions
  6. Mutiplicity of solutions to the main problem via category theory

Key Result

Theorem 1.3

Let hypotheses (V1)--(V2) and (f1)--(f4) be satisfied. Then there exists $\varepsilon_0>0$ such that for any $\varepsilon\in(0,\varepsilon_0)$, problem main problem has a positive solution $v_\varepsilon$. Further, if $\eta_\varepsilon$ is the global maximum point of $v_\varepsilon$, then it holds

Theorems & Definitions (70)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3: Concentration phenomena
  • Theorem 1.4: Multiplicity of positive solutions
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 60 more