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We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $-{\sum }_{i,j=1}^n{D}_j\left(a_{ij}\left(x\right)D_{i}u\left(x\right)\right)+b\left(x\right)u\left(x\right)+div\left(\Phi \left(u\left(x\right)\right)\right)=g\left(x\right)-{\sum }_{j=1}^n{f}_j\left(x\right)$ on Ω in the setting of the space H₀(Ω).

The main result establishes that a weak solution of degenerate nonlinear  elliptic equations can be approximated by a sequence of solutions for non-degenerate nonlinear elliptic equations.

In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations $$\begin{array}{cc}\hfill \Delta \left(v\left(x\right){\left|\Delta u\right|}^{p-2}\Delta u\right)& -\sum _{j=1}^n{D}_j\left[{\omega }_1\left(x\right){𝒜}_j(x,u,\nabla u)\right]+b(x,u,\nabla u){\omega }_2\left(x\right)\hfill \\ & =f_0\left(x\right)-\sum _{j=1}^n{D}_j{f}_j\left(x\right),\mathrm{in}\Omega \hfill \end{array}$$ in the setting of the weighted Sobolev spaces.

In this article, we prove the existence of entropy solutions for the Dirichlet problem $$\left(P\right)\left\{\begin{array}{cc}-\mathrm{div}\left[\omega \right(x\left)𝒜\right(x,u,\nabla u\left)\right]=f\left(x\right)-\mathrm{div}\left(G\right),\hfill & \text{in}\Omega \hfill \\ u\left(x\right)=0,\hfill & \text{on}\partial \Omega \hfill \end{array}\right.$$ where $\Omega$ is a bounded open set of ${}^N$, $N\ge 2$, $f\in L^1\left(\Omega \right)$ and $G/\omega \in {\left[L^{p^{\text{'}}}(\Omega ,\omega )\right]}^N$.

In the paper we study the equation $Lu=f$, where $L$ is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set $\Omega$. We prove existence and uniqueness of solutions in the space $H\left(\Omega \right)$ for the Neumann problem.

In this article we establish the existence of higher order weak derivatives of weak solutions of Dirichlet problem for a class of degenerate elliptic equations.

The main result establishes that a weak solution of degenerate quasilinear elliptic equations can be approximated by a sequence of solutions for non-degenerate quasilinear elliptic equations.

In this article we are interested in the existence and uniqueness of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations $$\begin{array}{cc}\hfill \Delta \left(v\left(x\right){\left|\Delta u\right|}^{p-2}\Delta u\right)& -\sum _{j=1}^n{D}_j\left[\omega \left(x\right){𝒜}_j(x,u,\nabla u)\right]\hfill \\ \hfill =& f_0\left(x\right)-\sum _{j=1}^n{D}_j{f}_j\left(x\right),in\Omega \hfill \end{array}$$ in the setting of the weighted Sobolev spaces.

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.

The main result establishes that a weak solution of degenerate semilinear elliptic equations can be approximated by a sequence of solutions for non-degenerate semilinear elliptic equations.