Abstract
We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of R-N. The first one, of the form
-Delta(p)u = beta(u)vertical bar del u vertical bar(p) + lambda f(x) + alpha,
involves a source gradient term with natural growth, where beta is non-negative, lambda > 0, f(x) >= 0, and alpha is a non-negative 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 non-decreasing, and mu is a non-negative measure. Here beta and g can present an asymptote. The correlation gives new results of existence, non-existence, 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.