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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 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 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 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 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.

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