Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations

We study the boundary behaviour of the solutions of (E) $-\Delta_p u+|\nabla u|^q=0$inadomain$\Omega \subset $\mathbb{R}$^N$,when$N\geq p > q >p-1$. We show the existence of a critical exponent $q_{\ast }<p$ such that if $p-1 < q < q_*$thereexistpositivesolutionsof(E)withanisolatedsingularityon$\partial\Omega$ and that these solutions belong to two different classes of singular solutions. If $q_{\ast }\le q<p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the two types of singular solutions that we have constructed.