In this paper we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations $${\Delta \left(v\left(x\right)\right|\Delta u|}^{q-2}\Delta u)-\sum _{j=1}^n{D}_j\left[\omega \left(x\right){𝒜}_j(x,u,\nabla u)\right]=f_0\left(x\right)-\sum _{j=1}^n{D}_j{f}_j\left(x\right),\text{in}\Omega$$ in the setting of the weighted Sobolev spaces.

Let Ω be a bounded domain in Rn with n ≥ 3. In this paper we are concerned with the problem of finding u ∈ H01 (Ω) satisfying the nonlinear elliptic problemsΔu + |u|(n+2/n-2) + f(x) = 0  in Ω and u(x) = 0 on ∂Ω, andΔu + u + |u|(n+2/n-2) + f(x) = 0  in Ω and u(x) = 0 on ∂Ω, when of f ∈ L∞(Ω).

We consider the anisotropic quasilinear elliptic Dirichlet problem $$\left\{\begin{array}{cc}{\displaystyle -\sum _{i=1}^N{D}^i{a}_i(x,u,\nabla u)+{\left|u\right|}^{s\left(x\right)-1}u=f+\lambda \frac{{\left|u\right|}^{p_0\left(x\right)-2}u}{{\left|x\right|}^{p_0\left(x\right)}}}\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is an open bounded subset of ${ℝ}^N$ containing the origin. We show the existence of entropy solution for this equation where the data $f$ is assumed to be in $L^1\left(\Omega \right)$ and $\lambda$ is a positive constant.

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 $F\in {\Pi }_{i=1}^N{L}^{p^{\prime }}(\Omega ,w_i^{1-p^{\prime }})$.

The paper is dedicated to the existence of local solutions of strongly nonlinear equations in RN and the Orlicz spaces framework is used.