Nonlinear elliptic anisotropic problem with Fourier boundary condition

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Résumé

The authors obtain an existence and uniqueness result of entropy solution for nonlinear anisotropic elliptic equation with Fourier boundary condition. Precisely
{−∑Ni=1Diai(x,Diu)+|u|pM(x)−2u=f∑Ni=1ai(x,Diu)νi+λu=gin Ω,on ∂Ω,
where Ω⊂ℝN, N≥3 is a bounded domain with smooth boundary, f∈L1(Ω), g∈L1(∂Ω) and λ>0.
The authors start with proving the existence and uniqueness of a weak solution when the data are bounded and deduce the main result of existence and uniqueness of entropy solution

Mots-clés

generalized Lebesgue-Sobolev spaces; anisotropic Sobolev spaces; weak solution; entropy solution; Fourier boundary condition; Marcinkiewicz spaces