  We establish a precise connection between two elliptic quasilinear problems
with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of
the form \[ -\Delta_{p}u=\beta(u)| \nabla u| ^{p}+\lambda f(x)+\alpha, \]
involves a source gradient term with natural growth, where $\beta$ is
nonnegative, $\lambda>0,f(x)\geqq0$, and $\alpha$ is a nonnegative measure. The
second one, of the form \[ -\Delta_{p}v=\lambda f(x)(1+g(v))^{p-1}+\mu, \]
presents a source term of order $0, $where $g$ is nondecreasing, and $\mu$ is a
nonnegative measure. Here $\beta$ and $g$ can present an asymptote. The
correlation gives new results of existence, nonexistence, regularity and
multiplicity of the solutions for the two problems, without or with measures.
New informations on the extremal solutions are given when $g$ is superlinear.
