Multiple solutions for some strongly degenerate second order elliptic equations

Article type: Research Article

Authors: Santos Junior, João R.^a | Siciliano, Gaetano^{b; *}

Affiliations: [a] Faculdade de Matemática, Instituto de Ciências Exatas e Naturais, Universidade Federal do Pará, Avenida Augusto Corrêa 01, 66075-110, Belém, PA, Brazil. E-mail: joaojunior@ufpa.br | [b] Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo, SP, Brazil. E-mail: sicilian@ime.usp.br

Correspondence: [*] Corresponding author. E-mail: sicilian@ime.usp.br.

Abstract: We consider a boundary value problem in a bounded domain involving a degenerate operator of the form L(u)=−div(a(x)∇u) and a suitable nonlinearity f. The function a vanishes on smooth 1-codimensional submanifolds of Ω where it is not allowed to be C2. By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where a vanishes.

Keywords: Elliptic equations, degenerate operators, vanishing solutions

DOI: 10.3233/ASY-211728

Journal: Asymptotic Analysis, vol. 129, no. 2, pp. 261-272, 2022