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Multiple critical points theorems for a class of nonsmooth functionals and applications to problems driven by 1-Laplacian and discontinuous nonlinearities

Ismael Sandro da Silva, Marcos T. Oliveira Pimenta, Pedro Fellype Pontes

Abstract

In this paper, we present a novel approach to investigate the existence of multiple critical points for a class of nonsmooth functionals. This method provides a robust framework to analyze the existence of solutions for problems involving the $1$-Laplacian operator with discontinuous nonlinearities. Our results contribute to advancing the study of nonsmooth variational problems, by establishing new nonsmooth multiple critical point theorems.

Multiple critical points theorems for a class of nonsmooth functionals and applications to problems driven by 1-Laplacian and discontinuous nonlinearities

Abstract

In this paper, we present a novel approach to investigate the existence of multiple critical points for a class of nonsmooth functionals. This method provides a robust framework to analyze the existence of solutions for problems involving the -Laplacian operator with discontinuous nonlinearities. Our results contribute to advancing the study of nonsmooth variational problems, by establishing new nonsmooth multiple critical point theorems.
Paper Structure (8 sections, 28 theorems, 205 equations)

This paper contains 8 sections, 28 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. On nonsmooth analysis
  3. Abstract theorems involving the $G$-index theory
  4. Tools of the $G$-index theory
  5. Minimax theorems related with the $G$-index theory
  6. Applications to quasilinear problems involving the 1-Laplacian operator
  7. A problem with a discontinuous nonlinearity with linear-subcritical growth
  8. A critical perturbation of a discontinuous nonlinearity

Key Result

Theorem 1.1

There exists $\lambda_0>0$ such that $I_{\lambda,a}$ has a sequence of critical points $(u_{\lambda,a}^{(n)})_{n \in \mathbb{N}}$ for any $\lambda \in (0,\lambda_0)$. Consequently, the problem $(P_{\lambda,a})$ has infinitely many solutions for $\lambda \in (0,\lambda_0)$. Moreover, there exists $a_ for any $a \in (0,a_0)$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 1
  • Lemma 2.4
  • proof
  • Definition 3.1: $G$-index
  • Proposition 3.2
  • ...and 44 more