Journées des jeunes chercheurs.es du LAMA

Existence of solutions of a Dirichlet problem involving the p-Laplacian operator with weight

23 mai 2024, 15:15

50m

Salle Yvette Cauchois Bât Perrin (IHP)

Exposés

Orateur

Asma Benhamida

Description

Consider the problem $-div(\alpha (x)|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)=\lambda |u{|}^{q-2}u+|u{|}^{p^{\star }-2}u$ in a bounded domain, with
homogeneous Dirichlet boundary condition, where $\alpha (.)$ is a continuous function, $p^{\ast }$ the Sobolev critical exponent and $2\le p\le q<p^{\ast }$. We prove the existence of positive solutions which depends, among others, on the behavior of the potential $\alpha (.)$ in the neighborhood of its minima, the position of $p^2$ with respect to dimension of the space and the position of $q$ with respect to specific values.