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

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...

A priori estimates and strong solvability results in Sobolev space $W^{2,p}\left(\Omega \right)$, $1<p<\infty$ are proved for the regular oblique derivative problem $$\left\{\begin{array}{c}\sum _{i,j=1}^n{a}^{ij}\left(x\right)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}=f\left(x\right)\text{a.e.}\Omega \hfill \\ \frac{\partial u}{\partial \ell }+\sigma \left(x\right)u=\varphi \left(x\right)\text{on}\partial \Omega \hfill \end{array}\right.$$ when the principal coefficients $a^{ij}$ are $VMO\cap L^{\infty }$ functions.

In this paper we prove a regularity result for very weak solutions of equations of the type $-divA(x,u,Du)=B(x,u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x,u,Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$\left\{\begin{array}{cc}-divA(x,u,Du)=B(x,u,Du)\hfill & \text{in}\Omega ,u-u_o\in W^{1,r}(\Omega ,{ℝ}^m).\hfill \end{array}\right.$$