On the connection between two quasilinear elliptic problems with source terms of order 0 or 1

We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of ${\mathbb{R}}^N\mathrm{.}$ The first one, of the form $$-{\mathrm{\Delta }}_{p}u=\beta (u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+\lambda f(x)+\alpha ,$$ involves a source gradient term with natural growth, where $\beta$ is nonnegative, $\lambda >0,f(x)\geqq 0$, and $\alpha$ is a nonnegative measure. The second one, of the form $$-{\mathrm{\Delta }}_{p}v=\lambda f(x)(1+g(v){)}^{p-1}+\mu ,$$ 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.