W01,p(Ω)×W01,p(Ω) versus C01(Ω)×C01(Ω) local minimizers

Article type: Research Article

Authors: Da Silva, João Pablo P.^{; *}

Affiliations: Universidade Federal do Pará, Departamento de Matemática, Brazil

Correspondence: [*] Corresponding author. E-mail: jpabloufpa@gmail.com.

Abstract: In this work, we consider a functional I:W01,p(Ω)×W01,p(Ω)→R of the form I(u,v)=1p∫Ω(|∇u|p+|∇v|p)dx−∫ΩH(x,u(x),v(x))dx where Ω⊂RN is a smooth bounded domain, max{|∂sH(x,s,t)|,|∂tH(x,s,t)|}⩽C(1+|s|σ1−1+|t|σ2−1) a.e. x∈Ω, for some C>0, ∀t,s∈R, p<σi⩽p∗:=Np/(N−p), i=1,2, and 1<p<N. We prove that a local minimum in the topology of C01(Ω)×C01(Ω) is a local minimum in the topology of W01,p(Ω)×W01,p(Ω). An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.

Keywords: Local minimization, p-Laplacian system, Sub-super solution method

DOI: 10.3233/ASY-241911

Journal: Asymptotic Analysis, vol. 140, no. 1-2, pp. 59-76, 2024