  The p-Laplace equation
  $$ \n \cdot (|\n u|^n \n u)=0 \whereA n>0, $$ in a bounded domain $\O \subset
\re^2$, with inhomogeneous Dirichlet conditions on the smooth boundary $\p \O$
is considered. In addition, there is a finite collection of curves
  $$\Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \O, \quad \{on which we assume
homogeneous Dirichlet boundary conditions} \quad u=0, $$ modeling a multiple
crack formation, focusing at the origin $0 \in \O$. This makes the above
quasilinear elliptic problem overdetermined. Possible types of the behaviour of
solution $u(x,y)$ at the tip 0 of such admissible multiple cracks, being a
"singularity" point, are described, on the basis of blow-up scaling techniques
and a "nonlinear eigenvalue problem". Typical types of admissible cracks are
shown to be governed by nodal sets of a countable family of nonlinear
eigenfunctions, which are obtained via branching from harmonic polynomials that
occur for $n=0$. Using a combination of analytic and numerical methods,
saddle-node bifurcations in $n$ are shown to occur for those nonlinear
eigenvalues/eigenfunctions.
