A nonlinear Dirichlet problem involving terms with logarithmic growth

Article type: Research Article

Authors: Polito, Andrea^{; *}

Affiliations: Sapienza Università di Roma, Italy

Correspondence: [*] Corresponding author. E-mail: andrea.polito@sbai.uniroma1.it.

Abstract: We study existence and regularity of weak solutions for a class of boundary value problems, whose form is −div(log(1+|∇u|)|∇u|m(x)∇u)+u|∇u|log(1+|∇u|)=f(x),in Ωu=0,on ∂Ω where both the principal part and the lower order term have a logarithmic growth with respect to the gradient of the solutions. We prove that the solutions, due to the regularizing effect given by the lower order term, belong to the Orlicz–Sobolev space generated by the function slog(1+|s|) even for L1(Ω) data.

Keywords: Nonlinear partial differential equations, Orlicz–Sobolev spaces, Logarithmic growth

DOI: 10.3233/ASY-221773

Journal: Asymptotic Analysis, vol. 131, no. 2, pp. 233-254, 2023