Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient

Abstract

Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of ${\mathbb{R}}^N.$ The first one, of the form $-{\Delta }_{p}u=\beta (u)|\nabla u{|}^p+\lambda f(x),$ where $\beta$ is nonnegative, involves a gradient term with natural growth. The second one, of the form $-{\Delta }_{p}v=\lambda f(x)(1+g(v){)}^{p-1}$ where $g$ is nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems.