### Pseudo-monotonicity and degenerate elliptic operators of second order.

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In this article we define Musielak−Orlicz−Sobolev spaces on arbitrary metric spaces with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a pointwise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds, and that the Lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application,...

We study a general class of nonlinear elliptic problems associated with the differential inclusion $\beta \left(u\right)-div\left(a\right(x,Du)+F(u\left)\right)\ni f$ in Ω where $f\in {L}^{\infty}\left(\Omega \right)$. The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general ${L}^{\infty}$-data.

In this paper, we shall be concerned with the existence result of the Degenerated unilateral problem associated to the equation of the type $Au+g(x,u,\nabla u)=f-\mathrm{div}F,$ where $A$ is a Leray-Lions operator and $g$ is a Carathéodory function having natural growth with respect to $\left|\nabla u\right|$ and satisfying the sign condition. The second term is such that, $f\in {L}^{1}\left(\Omega \right)$ and $F\in {\Pi}_{i=1}^{N}{L}^{{p}^{\prime}}(\Omega ,{w}_{i}^{1-{p}^{\prime}})$.

In this paper we study the existence of solutions for quasilinear degenerated elliptic operators A(u) + g(x,u,∇u) = f, where A is a Leray-Lions operator from W (Ω,ω) into its dual, while g(x,s,ξ) is a nonlinear term which has a growth condition with respect to ξ and no growth with respect to s, but it satisfies a sign condition on s. The right hand side f is assumed to belong either to W(Ω,ω*) or to L(Ω).

An existence theorem is proved, for a quasilinear degenerated elliptic inequality involving nonlinear operators of the form $Au+g(x,u,\nabla u)$, where $A$ is a Leray-Lions operator from ${W}_{0}^{1,p}(\Omega ,w)$ into its dual, while $g(x,s,\xi )$ is a nonlinear term which has a growth condition with respect to $\xi $ and no growth with respect to $s$, but it satisfies a sign condition on $s$, the second term belongs to ${W}^{-1,{p}^{\prime}}(\Omega ,{w}^{*})$.

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