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We prove the existence of an entropy solution for a class of nonlinear anisotropic elliptic unilateral problem associated to the following equation $$\begin{aligned} -\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) -\sum _{i=1}^{N}\partial _{i}\phi _{i}( u)=\mu , \end{aligned}$$where the right hand side $$\mu $$ belongs to $$L^{1}(\Omega )+ W^{-1, \vec {p'}}(\Omega )$$. The operator $$-\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) $$ is a Leray–Lions anisotropic operator and $$\phi _{i} \in C^{0}({\mathbb {R}}, {\mathbb {R}})$$.

In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form-∑i=1N∂iai(x,u,∇u)-∑i=1N∂iφi(u)=f,$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$where the right hand sidefbelongs toL1(Ω). The operator-∑i=1N∂iai(x,u,∇u)$- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $is a Leray-Lions anisotropic operator andϕi∈C0(ℝ,ℝ).

In this article we define the weighted variable exponent-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 point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that 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 prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.

In this paper, we establish the existence of a renormalized solutions of anisotropic parabolic operators of the type $\frac{\partial b(x,u)}{\partial t}-\sum\limits_{i=1}^{N}\partial_{i}a_{i}(x, t, u, \nabla u)=f$ and $b(x, u)(t\ =\ 0) = b(x, u_{0})$ , where the right hand side $f$ ’ belongs to $L^{1}(Q_{T})$ and $b(x, u_{0})$ belongs to $L^{1}(\Omega)$ . The function $b(x, u)$ is unbounded on u, the operator $-\sum_{i=1}^{N}\partial_{i}a_{i}(x, t, u, \nabla u)$ is a Leray-Lions operator.

In this work, we shall be concerned with the existence of weak solutions of anisotropic elliptic operators Au +\sum_{i=1}^{N}g_{i}(x, u, \nabla u)+\sum_{i=1}^{N}H_{i}(x, \nabla u)=f-\sum_{i=1}^{N} \frac{\partial }{\partial x_{i}}k_{i}Au+∑ i=1 N ​ g i ​ (x,u,∇u)+∑ i=1 N ​ H i ​ (x,∇u)=f−∑ i=1 N ​ ∂x i ​ ∂ ​ k i ​ where the right hand side ff belongs to L^{p^{'}_{\infty}}(\Omega)L p ∞ ′ ​ (Ω) and k_{i}k i ​ belongs to L^{p_{i}^{'}}(\Omega)L p i ′ ​ (Ω) for i=1,...,Ni=1,...,N and AA is a Leray-Lions operator. The critical growth condition on g_{i}g i ​ is the respect to \nabla u∇u and no growth condition with respect to u, while the function H_{i}H i ​ grows as |\nabla u|^{p_{i}-1}∣∇u∣ p i ​ −1 .

In this paper, we study the existence of solutions for strongly nonlinear degenerated elliptic unilateral problems of the form \(A(u)+ g(x,u,\nabla u)+ H(x,\nabla u)=f\), where A is a Leray–Lions operator acting from \(W_0^{1,p}(\Omega ,w)\) to its dual. On the nonlinear term \(g(x,s,\xi )\), we assume growth condition on \(\xi \) and a sign condition on s, while the convection term \(H(x,\xi )\) is only growing at most as \(|\xi |^{p-1}\). The right-hand side f belongs to \(L^1(\Omega )\).

We study the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form ∂ ⁡ b ⁢ ( x , u ) ∂ ⁡ t - div ⁡ ( a ⁢ ( x , t , u , ∇ ⁡ u ) ) + g ⁢ ( x , t , u , ∇ ⁡ u ) + H ⁢ ( x , t , ∇ ⁡ u ) = μ in ⁢ Ω × ( 0 , T ) , ${\partial b(x,u)\over\partial t}-\operatorname{div}(a(x,t,u,\nabla u))+g(x,t,u% ,\nabla u)+H(x,t,\nabla u)=\mu\quad\text{in }\Omega\times(0,T),$ where the right-hand side belongs to L 1 ⁢ ( Q T ) + L p ′ ⁢ ( 0 , T ; W - 1 , p ′ ⁢ ( Ω ) ) ${L^{1}(Q_{T})+L^{p^{\prime}}(0,T;W^{-1,p^{\prime}}(\Omega))}$ and b ⁢ ( x , u ) ${b(x,u)}$ is unbounded function of u, - div ⁡ ( a ⁢ ( x , t , u , ∇ ⁡ u ) ) ${{-}\operatorname{div}(a(x,t,u,\nabla u))}$ is a Leray–Lions type operator with growth | ∇ ⁡ u | p - 1 ${|\nabla u|^{p-1}}$ in ∇ ⁡ u ${\nabla u}$ . The critical growth condition on g is with respect to ∇ ⁡ u ${\nabla u}$ and there is no growth condition with respect to u, while the function H ⁢ ( x , t , ∇ ⁡ u ) ${H(x,t,\nabla u)}$ grows as | ∇ ⁡ u | p - 1 ${|\nabla u|^{p-1}}$ .