23,826 research outputs found
New unique existence criteria for higher-order nonlinear singular fractional differential equations
In this paper, a nonlinear three-point boundary value problem of higher-order singular fractional differential equations is discussed. By applying the properties of Green function and some fixed point theorems for sum-type operator on cone, some new criteria on the existence and uniqueness of solutions are obtained. Moreover, two iterative sequences are given for uniformly approximating the positive solution, which are important for practical application. At last, we give two examples to illustrate the main results
On a diffusion model with absorption and production
We discuss the structure of radial solutions of some superlinear elliptic
equations which model diffusion phenomena when both absorption and production
are present. We focus our attention on solutions defined in R (regular) or in R
\ {0} (singular) which are infinitesimal at infinity, discussing also their
asymptotic behavior. The phenomena we find are present only if absorption and
production coexist, i.e., if the reaction term changes sign. Our results are
then generalized to include the case where Hardy potentials are considered
Positive periodic solutions of singular systems
The existence and multiplicity of positive periodic solutions for second
order non-autonomous singular dynamical systems are established with
superlinearity or sublinearity assumptions at infinity for an appropriately
chosen parameter. Our results provide a unified treatment for the problem and
significantly improve several results in the literature. The proof of our
results is based on the Krasnoselskii fixed point theorem in a cone.Comment: Journal of Differential Equations, 201
The nonlinear heat equation involving highly singular initial values and new blowup and life span results
In this paper we prove local existence of solutions to the nonlinear heat
equation with initial value , anti-symmetric with
respect to and
for where is a constant, and This gives a local
existence result with highly singular initial values.
As an application, for we establish new blowup criteria for
, including the case Moreover, if
we prove the existence of initial values
for which the resulting solution blows up in finite time
if is sufficiently small. We also construct blowing up solutions
with initial data such that has different finite
limits along different sequences . Our result extends the known
"small lambda" blow up results for new values of and a new class of
initial data.Comment: Submitte
About curvature, conformal metrics and warped products
We consider the curvature of a family of warped products of two
pseduo-Riemannian manifolds and furnished with metrics of
the form and, in particular, of the type , where are smooth
functions and is a real parameter. We obtain suitable expressions for the
Ricci tensor and scalar curvature of such products that allow us to establish
results about the existence of Einstein or constant scalar curvature structures
in these categories. If is Riemannian, the latter question involves
nonlinear elliptic partial differential equations with concave-convex
nonlinearities and singular partial differential equations of the
Lichnerowicz-York type among others.Comment: 32 pages, 3 figure
Shooting with degree theory: Analysis of some weighted poly-harmonic systems
In this paper, the author establishes the existence of positive entire
solutions to a general class of semilinear poly-harmonic systems, which
includes equations and systems of the weighted Hardy--Littlewood--Sobolev type.
The novel method used implements the classical shooting method enhanced by
topological degree theory. The key steps of the method are to first construct a
target map which aims the shooting method and the non-degeneracy conditions
guarantee the continuity of this map. With the continuity of the target map, a
topological argument is used to show the existence of zeros of the target map.
The existence of zeros of the map along with a non-existence theorem for the
corresponding Navier boundary value problem imply the existence of positive
solutions for the class of poly-harmonic systems.Comment: 19 pages, author's accepted version including corrections to a few
typographical error
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