Abstract
We prove existence of solutions for strongly nonlinear elliptic equations of the form $$ \left\{\begin{array}{c} A(u)+g(x,u,\nabla u)=f+\mbox {div}(\phi(u))\quad \textrm{in }\Omega, \\ u\equiv0\quad \partial \Omega. \end{array} \right.$$ Where $A(u)=-\mbox {div}(a(x,u,\nabla u))$ be a Leray-Lions operator defined in $D(A)\subset W^{1}_{0}L_\varphi(\Omega) \rightarrow W^{-1}_{0}L_\psi(\Omega)$, the right hand side belongs in $ L^{1}(\Omega)$, and $\phi\in C^{0}(\mathbb{R},\mathbb{R}^N)$, without assuming the $\Delta_{2}$-condition on the Musielak function.
Highlights
Benkirane abstract: We prove existence of solutions for strongly nonlinear elliptic equations of the form
Our purpose is to generalize the result [3] and we prove the existence of entropy solution of (1.1)
In the following the measurability of a function u : Ω −→ R means the Lebesgue measurability
Summary
Let φ and γ be two Musielak-orlicz functions, we say that φ dominate γ, and we write γ ≺ φ, near infinity The closure in Lφ(Ω) of the bounded measurable functions with compact support in Ω is denoted by Eφ(Ω) For u ∈ W mLφ(Ω) there functionals are a convex modular and a norm on W mLφ(Ω), respectively, and the pair W mLφ(Ω), m φ,Ω is a Banach space if φ satisfies the following condition [21] : there exist a constant c > 0 such that inf φ(x, 1) ≥ c. Let W mEφ(Ω) the space of functions u such that u and its distribution derivatives up to order m lie in Eφ(Ω), and W0mEφ(Ω) is the (norm) closure of D(Ω) in W mLφ(Ω).
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