9,140 research outputs found
The p-Laplace equation in domains with multiple crack section via pencil operators
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 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 . Using a combination of analytic and numerical methods,
saddle-node bifurcations in are shown to occur for those nonlinear
eigenvalues/eigenfunctions.Comment: arXiv admin note: substantial text overlap with arXiv:1310.065
Double-phase problems with reaction of arbitrary growth
We consider a parametric nonlinear nonhomogeneous elliptic equation, driven
by the sum of two differential operators having different structure. The
associated energy functional has unbalanced growth and we do not impose any
global growth conditions to the reaction term, whose behavior is prescribed
only near the origin. Using truncation and comparison techniques and Morse
theory, we show that the problem has multiple solutions in the case of high
perturbations. We also show that if a symmetry condition is imposed to the
reaction term, then we can generate a sequence of distinct nodal solutions with
smaller and smaller energies
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