9,013 research outputs found
A nodal domain theorem and a higher-order Cheeger inequality for the graph -Laplacian
We consider the nonlinear graph -Laplacian and its set of eigenvalues and
associated eigenfunctions of this operator defined by a variational principle.
We prove a nodal domain theorem for the graph -Laplacian for any .
While for the bounds on the number of weak and strong nodal domains are
the same as for the linear graph Laplacian (), the behavior changes for
. We show that the bounds are tight for as the bounds are
attained by the eigenfunctions of the graph -Laplacian on two graphs.
Finally, using the properties of the nodal domains, we prove a higher-order
Cheeger inequality for the graph -Laplacian for . If the eigenfunction
associated to the -th variational eigenvalue of the graph -Laplacian has
exactly strong nodal domains, then the higher order Cheeger inequality
becomes tight as
A p-Laplacian supercritical Neumann problem
For , we consider the quasilinear equation
in the unit ball of , with homogeneous Neumann boundary
conditions. The assumptions on are very mild and allow the nonlinearity to
be possibly supercritical in the sense of Sobolev embeddings. We prove the
existence of a nonconstant, positive, radially nondecreasing solution via
variational methods. In the case , we detect the asymptotic
behavior of these solutions as .Comment: 34 pages, 1 figur
Weak perturbations of the p-Laplacian
We consider the p-Laplacian in R^d perturbed by a weakly coupled potential.
We calculate the asymptotic expansions of the lowest eigenvalue of such an
operator in the weak coupling limit separately for p>d and p=d and discuss the
connection with Sobolev interpolation inequalities.Comment: 20 page
Limit of p-Laplacian Obstacle problems
In this paper we study asymptotic behavior of solutions of obstacle problems
for Laplacians as For the one-dimensional case and for the
radial case, we give an explicit expression of the limit. In the n-dimensional
case, we provide sufficient conditions to assure the uniform convergence of
whole family of the solutions of obstacle problems either for data that
change sign in or for data (that do not change sign in )
possibly vanishing in a set of positive measure
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