Abstract
AbstractThis paper is devoted to study some nonlinear elliptic Neumann equations of the type{Au+g(x,u,∇u)+|u|q(⋅)-2u=f(x,u,∇u)inΩ,∑i=1Nai(x,u,∇u)⋅ni=0on∂Ω,\left\{ {\matrix{ {Au + g(x,u,\nabla u) + |u{|^{q( \cdot ) - 2}}u = f(x,u,\nabla u)} \hfill & {{\rm{in}}} \hfill & {\Omega ,} \hfill \cr {\sum\limits_{i = 1}^N {{a_i}(x,u,\nabla u) \cdot {n_i} = 0} } \hfill & {{\rm{on}}} \hfill & {\partial \Omega ,} \hfill \cr } } \right.in the anisotropic variable exponent Sobolev spaces, whereAis a Leray-Lions operator andg(x,u, ∇u),f(x,u, ∇u) are two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem.
Highlights
Let Ω be a bounded open domain of IRN (N ≥ ) with smooth boundary ∂Ω
We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem
The study of various mathematical problems with isotropic variable exponent has received considerable attention in recent years. These problems are interesting in applications and raise many di cult and interesting mathematical problems
Summary
Let Ω be a bounded open domain of IRN (N ≥ ) with smooth boundary ∂Ω. The study of various mathematical problems with isotropic variable exponent has received considerable attention in recent years. On Γd , where Au = −div (a(x, ∇u)) is a Leray-Lions type operator, and λ(x, s), Φ(x, s) are Carathéodory functions They have proved the existence and uniqueness of renormalized solutions for this problem under some growth conditions. In Ω, on ∂Ω, with f ∈ L (Ω) and g ∈ ΠiN= Lp′i (Ω) They have proved the existence and uniqueness of renormalized solution in the anisotropic Sobolev spaces. Where g(x, u, ∇u) and φ(u) are Carathéodory functions, and the data f is assumed to be in L (Ω) They have proved the existence of an entropy solution in the anisotropic variable exponent Sobolev spaces. We introduce the anisotropic variable exponent Sobolev space, used in the study of our quasilinear anisotropic elliptic problem. For more details concerning the anisotropic Sobolev spaces, we refer the reader to [6] and [11]
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