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We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution $u$ in a singular way and on its gradient $Du$ with quadratic growth. The prototype of the problem under consideration is where $\lambda >0$, $k>0$; $f(x)\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, $f(x)\ge 0$ (and so $u\ge 0$). If , we prove the existence of a solution for both the "+" and the "-" signs, while if $k\ge 1$, we prove the existence of a solution for the "+" sign only.

Boundary value problems for linear quasi-elliptic $A+\sigma$-type operators with variable coefficients are studied in the unbounded region of ${𝐑}^n$, definited by $x_k>0$, $k=1,\mathrm{\cdots },n$; $\sigma$ means a perturbation whose behaviour is assigned at infinity and in the angular points of the domain. It is proved that the operator related to the problem has closed range and finite dimensional null space. The study is developed within a new class of dissimetric Sobolev weighted spaces.

We show an existence result for the Cauchy-Dirichlet problem in $Q_T=\mathrm{\Omega }\times (0,T)$ for parabolic equations with degenerate principal part (of porous medium type) with a lower order term having a quadratic growth with respect to the gradient. The right hand side of the equation $f$ and the initial datum $u_0$ are bounded nonnegative functions.