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Concentration Phenomena for $(p,N)$-Laplace Equation Under Discontinuous Nonlinearities and Penalization Method

Ankit, Giovany M. Figueiredo, Abhishek Sarkar

Abstract

In this paper, we investigate the existence and concentration of solutions to a $(p,N)$-Laplace equation in $\mathbb{R}^N$ involving a discontinuous nonlinearity and critical exponential growth. To establish the existence of solutions, we employ a penalization technique in the sense of Del Pino and Felmer adapted to a locally Lipschitz functional. Furthermore, by combining variational methods with Moser-type iteration techniques, we obtain the concentration behavior of the solutions. Our results contribute to the study of nonlinear elliptic problems with irregular nonlinearities and critical growth phenomena.

Concentration Phenomena for $(p,N)$-Laplace Equation Under Discontinuous Nonlinearities and Penalization Method

Abstract

In this paper, we investigate the existence and concentration of solutions to a -Laplace equation in involving a discontinuous nonlinearity and critical exponential growth. To establish the existence of solutions, we employ a penalization technique in the sense of Del Pino and Felmer adapted to a locally Lipschitz functional. Furthermore, by combining variational methods with Moser-type iteration techniques, we obtain the concentration behavior of the solutions. Our results contribute to the study of nonlinear elliptic problems with irregular nonlinearities and critical growth phenomena.
Paper Structure (12 sections, 17 theorems, 234 equations)

This paper contains 12 sections, 17 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Auxiliary Problem
  4. Mountain Pass Geometry of the Energy Functional
  5. Compactness of the Energy Functional--Cerami Condition
  6. Existence of Critical Point of the Energy Functional Corresponding to the Auxiliary Problem
  7. Autonomous Problem
  8. Existence of Critical Point
  9. Relationship Between Both Mountain Pass Levels
  10. Compactness Result
  11. Moser Iteration Argument
  12. Proof of Main Theorem

Key Result

Theorem 1.4

(Existence and Concentration) Assume that conditions V1-V2 and f1-f5 are fulfilled. Then, there exists $\tilde{\epsilon}$,$\tilde{\beta}$ such that for all $\epsilon\in(0,\tilde{\epsilon})$ and $\beta\in(0,\tilde{\beta})$, problem main problem possesses a weak solution $u_{\epsilon,\beta}\in \mathbf

Theorems & Definitions (38)

  • Remark 1.1
  • Example 1.2
  • Definition 1.3: Weak Solution
  • Theorem 1.4
  • Definition 2.1: Generalized Directional Derivative
  • Definition 2.2: Cerami Condition--Non Smooth Version Stuart-2011
  • Definition 2.3: Orlicz space
  • Definition 2.4
  • Definition 2.5: Complementary/conjugate of a function
  • Lemma 2.1: Moser-Trudinger Inequality do-1997
  • ...and 28 more