Special Session 138: 

Ground states of elliptic equations with competition between power and gradient terms

Marie-Francoise BIDAUT-VERON
University Francois Rabelais, Tours
France
Co-Author(s):    Pr Marta Garcia-Huidobro, Pr Laurent Veron
Abstract:
Here we consider the nonnegative solutions of equations in a punctured ball $B(0,R)\setminus\left\{ 0\right\} \subset\mathbb{R}^{N}$ or in $\mathbb{R}^{N},$ of type \begin{equation} -\Delta u=u^{p}+M|\nabla u|^{q}\label{two}% \end{equation} where $p,q>1$ and $M\in\mathbb{R}$. We give new a priori estimates on the solutions and their gradient, and Liouville type results. We use Bernstein technique and Osserman`s or Gidas-Spruck`s type methods. The most interesting case is $q=2p/(p+1)$, where the equation is invariant by scaling. In the radial case, we give a precise description of all the solutions, improving the known results. The situation appears to be quite complicated in the case $M