Infinite valued solutions of non-uniformly elliptic problems

International audienceWe consider a quasilinear equation with $L^1$ data and with a diffusion matrix $\bold A(x,u)$, which is not uniformly coercive with respect to $u$. Under such assumptions it is not realistic, in general, to search for a solution which is finite almost everywhere. We introduce two equivalent notions of solutions which take into account the possible values $+\infty$ and $-\infty$. Then we prove that there exists at least one such solution. Finally, we establish a uniqueness result in a class of simultaneous infinite-valued solutions