Multiplicity of solutions for a class of quasilinear problems involving the 1-Laplacian operator with critical growth

Author(s)

Alves, Claudianor O. ^[1] ; Ourraoui, Anass ^[2] ; Pimenta, Marcos T. O. ^[3]

Affiliation

^[1] Univ Fed Campina Grande, Unidade Acad Matemat, BR-58429970 Campina Grande, Paraiba - Brazil

^[2] Univ Mohamed I, Dept Math, FSO, Oujda - Morocco

^[3] Univ Estadual Paulista Unesp, Fac Ciencias & Tecnol, Dept Matemat & Comp, BR-19060900 Presidente Prudente, SP - Brazil

• Document type: Journal article
• Source: Journal of Differential Equations; v. 308, p. 545-574, JAN 25 2022.
• Web of Science Citations: 0

Abstract

The aim of this paper is to establish two results about multiplicity of solutions to problems involving the 1-Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem [-Delta(u)(1) + xi u/vertical bar u vertical bar = lambda vertical bar u vertical bar(q-2)u + vertical bar u vertical bar(1{*})-2u, in Omega, u=0, on partial derivative Omega, where Omega is a smooth bounded domain in R-N, N >= 2 and xi is an element of [0, 1]. Moreover, lambda > 0, q is an element of (1, 1{*}) and 1{*} = N/N-1. The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that xi = 1, Omega= [x is an element of RN : r < |x| < r + 1], N > 2, N not equal 3 and r > 0. In the second one, is a smooth bounded domain, xi = 0, and the multiplicity of solutions is proved through an abstract result which involves genus theory for functionals which are sum of a C1 functional with a convex lower semicontinuous functional. (C) 2021 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 19/14330-9 - Variational and nonvariational elliptic problems involving the 1-Laplacian operator
• Grantee: Marcos Tadeu de Oliveira Pimenta
• Support Opportunities: Regular Research Grants