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We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation ${\sum }_{i=1}^m\frac{\partial }{\partial x_i}{a}_i(x,u,\nabla u)-c_0{\left|u\right|}^{p-2}u=f(x,u,\nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $\lambda \left(\right|u\left|\right){\sum }_{i=1}^m{a}_i(x,u,\eta ){\eta }_i\ge \nu \left(x\right){\left|\eta \right|}^p,$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$.

We obtain an existence theorem for the problem (0.1) where the coefficients $a_{ij}(x,s)$ satisfy a degenerate ellipticity condition and hypotheses weaker than the continuity with respect to the variable $s$.

Sufficient conditions are obtained so that a weak subsolution of $(0.1)$, bounded from above on the parabolic boundary of the cylinder $Q$, turns out to be bounded from above in $Q$.

We prove existence results for the Dirichlet problem associated with an elliptic semilinear second-order equation of divergence form. Degeneracy in the ellipticity condition is allowed.

We prove a generalized maximum principle for subsolutions of boundary value problems, with mixed type unilateral conditions, associated to a degenerate parabolic second-order operator in divergence form.

We study the asymptotic behaviour near infinity of the weak solutions of the Cauchy-problem.