This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by $$\begin{array}{cccc}\hfill t2& -{\mathrm{div}\left(b\right(\left|u\right|\left)\right|\nabla u|}^{p-2}{\nabla u)+d(\left|u\right|\left)\right|\nabla u|}^p=f-{\mathrm{div}\left(c\left(x\right)\right|u|}^{\alpha })\hfill & \hfill & \text{in}\Omega ,\hfill \\ & u=0\hfill & \hfill & \text{on}\partial \Omega ,\hfill \end{array}t$$ where $\Omega$ is a bounded open set of ${ℝ}^N$ ($N\ge 2$) with $1<p<N$ and $f\in L^1\left(\Omega \right),$ under some growth conditions on the function $b(·)$ and $d(·),$ where $c(·)$ is assumed to be in $L^{\frac{N}{(p-1)}}\left(\Omega \right).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

An existence theorem is proved, for a quasilinear degenerated elliptic inequality involving nonlinear operators of the form $Au+g(x,u,\nabla u)$, where $A$ is a Leray-Lions operator from $W_0^{1,p}(\Omega ,w)$ into its dual, while $g(x,s,\xi )$ is a nonlinear term which has a growth condition with respect to $\xi$ and no growth with respect to $s$, but it satisfies a sign condition on $s$, the second term belongs to $W^{-1,p^{\prime }}(\Omega ,w^{*})$.

Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where $Au:={\sum }_{i,j=1}^m{a}_{ij}\left(x\right)\left(\partial ²u\right)/\left(\partial x_i\partial x_j\right)$, $x=(x_1,...,x_m)\in G\subset {ℝ}^m$, G is a bounded domain with $C^{2+\alpha }$ (0 < α < 1) boundary, the operator A is strongly uniformly elliptic in G and u is a real $L^p\left(G̅\right)$ function. For the equation (1) we consider the Dirichlet problem with the boundary condition (2) u(x) = h(x) for x∈ ∂G. We use Chaplygin’s method [5] to prove that problem (1), (2) has at least one regular solution in a suitable class of functions. Using the method of upper and lower...