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The aim of this work is to give a generalization of the theorem of H. Brezis and F.E. Browder in the anisotropic Sobolev space $$W^{1,{\vec{p}}}(\Omega ).$$ By introducing capacity and using a Hedberg-type’s approximation in $$W^{1,{\vec{p}}}(\Omega ),$$ we prove our main result whenever $$\Omega$$ is an open bounded set of in $$\mathbb {R}^{N}(N\ge 2)$$ satisfying the segment properties.

This paper is concerned with the study of the existence results to the obstacle problem associated with the equation having degenerate coercivity, whose prototype is given by: $$\begin{aligned} \left\{ \begin{array}{ll} -\hbox { div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^{p} = f(x,u) &{}\quad \hbox {in}\,\Omega ,\\ u = 0 &{}\quad \hbox {on}\,\partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a bounded open set of $$\mathbb {R}^N$$ ( $$N\ge 2$$ ), with $$1

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 will prove the existence of bounded solutions for the nonlinear elliptic equations - div(a(x,u,\nabla u)) = g(x,u,\nabla u) -divf,−div(a(x,u,∇u))=g(x,u,∇u)−divf, in the setting of the weighted Sobolev space W^{1,p}(\Omega,w)W 1,p (Ω,w) where aa, gg are Caratheodory functions which satisfy some conditions and ff satisfies suitable summability assumption.

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 a general class of nonlinear anisotrpic elliptic problems associated with the differential inclusion β(u) - div(a(x, D(u) + F(u)) ∋ f in Ω, where f ∈ L∞(Ω). A vector field a(. , .) is a Caratheodory function. Using trunction techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for L∞-data.

In this paper, we study in the anisotropic Sobolev spaces with variable exponent W01, p(.)(Ω), the existence result of unilateral problem associated to the equations of the form, -∑i=1N ∂i ai(x, u, ∇ u) = g, where the right hand side f belongs to L1(Ω).

In this paper we study the existence of a renormalized solution for the nonlinear p(x)--elliptic problem in the Weighted--Variable--Exponent Soblev spaces, of the form: - div (a(x, u, ∇ u)) + H(x, u, ∇ u) = f ∈ Ω, where the right-hand side f belong to L1(Ω) and H(x, s, ξ) is the nonlinear term satisfying some growth condition, but no sign condition on s.

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 ${|\nabla u|^{p-1}}$ .

In this article, we study the existence of renormalized solution for the nonlinear $p(x)$-parabolic problem of the form:\\ $\begin{cases} \frac{\partial b(x,u)}{\partial t} - div (a(x,t,u,\nabla u)) + H(x,t,u,\nabla u) = f - divF $ \; in $Q= \Omega\times(0,T)\\$ $b(x,u)\mid_{t=0} = b(x,u_{0})$ \; in $\Omega\\$ $ u = 0 $ \quad on $\partial\Omega\times(0,T)\\$ $\end{cases}$ with $ f $ $ \in L^{1} (Q),$\; $b(x,u_{0}) \in L^{1} (\Omega)$ and $ F \in (L^{P'(.)}(Q))^{N}. $\\ The main contribution of our work is to prove the existence of renormalized solution of the variable exponent Soblev spaces, and we suppose that\;$ H(x,t,u,\nabla u)$\; is the non linear term satisfying some growth condition but no sig

In this article, we study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion b(u)−div(a(x,Du)+F(u))∋ f in W where f ∈ L 1 (W). A vector field a(.,.) is a Caratheodory function. Using truncation techniques and the generalized monotonicity method in the framework of weighted variable exponent Sobolev spaces, we prove existence of renormalized solutions for general L 1 -data. AMS subject classifications: 35J15, 35J70, 35J85

An existence result of a entropy unilateral solution for a class of strongly nonlinear parabolic equations in Orlicz spaces is proved. No growth assumption is made on the nonlinearities.