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

Haydar Abdelhamid (1) , Marie-Françoise Bidaut-Véron (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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 $-\Delta_{p}u=\beta(u)\left\vert \nabla u\right\vert ^{p}+\lambda f(x),$ 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.

Mots clés: Quasilinear elliptic equations; renormalized solutions; extremal solutions; measure data

Domaines: Equations aux dérivées partielles [math.AP]

Haydar Abdelhamid, Marie-Françoise Bidaut-Véron. Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2008, 346, pp.1251-1256. ⟨hal-00326563v2⟩