3,273 research outputs found
Critical fractional -Laplacian problems with possibly vanishing potentials
We obtain nontrivial solutions of a critical fractional -Laplacian
equation in the whole space and with possibly vanishing potentials. In addition
to the usual difficulty of the lack of compactness associated with problems
involving critical Sobolev exponents, the problem is further complicated by the
absence of a direct sum decomposition suitable for applying classical linking
arguments. We overcome this difficulty using a generalized linking construction
based on the -cohomological index.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1411.219
Uniqueness of radial solutions for the fractional Laplacian
We prove general uniqueness results for radial solutions of linear and
nonlinear equations involving the fractional Laplacian with for any space dimensions . By extending a monotonicity
formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear
equation in has at most one radial and
bounded solution vanishing at infinity, provided that the potential is a
radial and non-decreasing. In particular, this result implies that all radial
eigenvalues of the corresponding fractional Schr\"odinger operator
are simple. Furthermore, by combining these findings on
linear equations with topological bounds for a related problem on the upper
half-space , we show uniqueness and nondegeneracy of ground
state solutions for the nonlinear equation in for arbitrary space dimensions and all
admissible exponents . This generalizes the nondegeneracy and
uniqueness result for dimension N=1 recently obtained by the first two authors
in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves
of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.Comment: 38 pages; revised version; various typos corrected; proof of Lemma
8.1 corrected; discussion of case \kappa_* =1 in the proof of Theorem 2
corrected with new Lemma A.2; accepted for publication in Comm. Pure. Appl.
Mat
Best dispersal strategies in spatially heterogeneous environments: optimization of the principal eigenvalue for indefinite fractional Neumann problems
We study the positive principal eigenvalue of a weighted problem associated
with the Neumann spectral fractional Laplacian. This analysis is related to the
investigation of the survival threshold in population dynamics. Our main result
concerns the optimization of such threshold with respect to the fractional
order , the case corresponding to the standard Neumann
Laplacian: when the habitat is not too fragmented, the principal positive
eigenvalue can not have local minima for . As a consequence, the best
strategy for survival is either following the diffusion with (i.e.
Brownian diffusion), or with the lowest possible (i.e. diffusion allowing
long jumps), depending on the size of the domain. In addition, we show that
analogous results hold for the standard fractional Laplacian in ,
in periodic environments.Comment: Version accepted for publication. Title changed according to
referee's suggestio
A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators
We study the Dancer-Fucik spectrum of the fractional p-Laplacian operator. We
construct an unbounded sequence of decreasing curves in the spectrum using a
suitable minimax scheme. For p=2, we present a very accurate local analysis. We
construct the minimal and maximal curves of the spectrum locally near the
points where it intersects the main diagonal of the plane. We give a sufficient
condition for the region between them to be nonempty, and show that it is free
of the spectrum in the case of a simple eigenvalue. Finally we compute the
critical groups in various regions separated by these curves. We compute them
precisely in certain regions, and prove a shifting theorem that gives a
finite-dimensional reduction in certain other regions. This allows us to obtain
nontrivial solutions of perturbed problems with nonlinearities crossing a curve
of the spectrum via a comparison of the critical groups at zero and infinity.Comment: 13 pages, typos correcte
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application
We develop further the theory of symmetrization of fractional Laplacian
operators contained in recent works of two of the authors. The theory leads to
optimal estimates in the form of concentration comparison inequalities for both
elliptic and parabolic equations. In this paper we extend the theory for the
so-called \emph{restricted} fractional Laplacian defined on a bounded domain
of with zero Dirichlet conditions outside of .
As an application, we derive an original proof of the corresponding fractional
Faber-Krahn inequality. We also provide a more classical variational proof of
the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297
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