3,069 research outputs found
Caffarelli-Kohn-Nirenberg type equations of fourth order with the critical exponent and Rellich potential
We study the existence/nonexistence of positive solution of when is a bounded domain and ,
, and
. We prove the nonexistence result when
is an open subset of which is star shaped with respect
to the origin. We also study the existence of positive solution in
when is a bounded domain with non trivial topology and ,
, for certain and . Different behavior of PS sequences have been obtained depending on
or .Comment: 21 page
Virtual characters on L-functions
In this expository note we show the inception and development of the
Heilbronn characters and their application to the holomorphy of quotients of
Artin L-functions. Further we use arithmetic Heilbronn characters introduced by
Wong, to deal with holomorphy of quotients of certain L-functions, e.g,,
L-functions associated to CM elliptic curves. Furthermore we use the
supercharacter theory introduced by Diaconis and Isaacs to study Artin
L-functions associated to such characters. We conclude the note surveying about
various other unconditional approaches taken based on character theory of
finite groups
Infinitely many sign changing solutions of an elliptic problem involving critical Sobolev and Hardy-Sobolev exponent
We study the existence and multiplicity of sign changing solutions of the
following equation where is a
bounded domain in , , all the principal curvatures of
at are negative and $\mu\geq 0, \ \ a>0, \ \ N\geq 7, \ \
0<t<2, \ \ 2^{\star}=\frac{2N}{N-2}2^{\star}(t)=\frac{2(N-t)}{N-2}$
A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities
We study the existence and non-existence of nontrivial weak solution of where , , ,
and . Using Pohozaev
type of identity, we prove the non-existence result when . On the
other hand when the equation has multiple critical nonlinearities i.e.
and , we establish the existence of
nontrivial solution using the Mountain-Pass theorem by Ambrosetti and
Rabinowitz and the variational methods.Comment: 12 page
Entire solutions for a class of elliptic equations involving -biharmonic operator and Rellich potentials
We study existence, multiplicity and qualitative properties of entire
solutions for a noncompact problem related to p-biharmonic type equations with
weights. More precisely, we deal with the following family of equations where , , , and is smaller than the Rellich
constant.Comment: 12 page
Semilinear elliptic PDE's with biharmonic operator and a singular potential
We study the existence/nonexistence of positive solution to the problem of
the type: \begin{equation}\tag{} \begin{cases} \Delta^2u-\mu
a(x)u=f(u)+\lambda b(x)\quad\textrm{in ,}\\ u>0 \quad\textrm{in
,}\\ u=0=\Delta u \quad\textrm{on ,} \end{cases}
\end{equation} where is a smooth bounded domain in ,
, are nonnegaive functions satisfying certain hypothesis
which we will specify later. are positive constants. Under some
suitable conditions on functions and the constant , we show that
there exists such that when ,
() admits a solution in
and for , it does not have any solution in
. Moreover as
, minimal positive solution of ()
converges in to a solution of
(). We also prove that there exists
such that and for ,
the above problem () does not have any solution even in the
distributional sense/very weak sense and there is complete {\it blow-up}. Under
an additional integrability condition on , we establish the uniqueness of
positive solution of () in .Comment: 21 page
Multiplicity results for fractional elliptic equations involving critical nonlinearities
In this paper we prove the existence of infinitely many nontrivial solutions
for the class of fractional elliptic equations involving
concave-critical nonlinearities in bounded domains in . Further,
when the nonlinearity is of convex-critical type, we establish the multiplicity
of nonnegative solutions using variational methods. In particular, we show the
existence of at least nonnegative solutions.Comment: 36 page
Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities
In this paper we prove the existence of infinitely many nontrivial solutions
of the following equations driven by a nonlocal integro-differential operator
with concave-convex nonlinearities and homogeneous Dirichlet boundary
conditions \begin{eqnarray*}
\mathcal{L}_{K} u + \mu\, |u|^{q-1}u + \lambda\,|u|^{p-1}u &=& 0
\quad\text{in}\quad \Omega, \\[2mm]
u&=&0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega,
\end{eqnarray*} where is a smooth bounded domain in ,
, , . Moreover, when
reduces to the fractional laplacian operator ,
, ,
, we find such that for any , there
exists at least one sign changing solution.Comment: 32 pages. Proof of Claim 4 in Theorem 4.1 has been modified in this
versio
Semilinear elliptic equations admitting similarity transformations
In this paper we study the equation in a smooth bounded domain
where , and is a
non-decreasing function which satisfies Keller-Osserman condition. We introduce
a condition on which implies that the equation is subcritical, i.e. the
corresponding boundary value problem is well posed with respect to data given
by finite measures. Under additional assumptions on we show that this
condition is necessary as well as sufficient. We also discuss b.v. problems
with data given by positive unbounded measures. Our results extend results of
\cite{MV1} treating equations of the form with
, .Comment: 30 page
Sign changing solutions of p-fractional equations with concave-convex nonlinearities
In this article we study the existence of sign changing solution of the
following p-fractional problem with concave-critical nonlinearities:
\begin{eqnarray*}
(-\Delta)^s_pu &=& \mu |u|^{q-1}u + |u|^{p^*_s-2}u \quad\mbox{in}\quad
\Omega, u&=&0\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{eqnarray*}
where and are fixed parameters, ,
and . is an open, bounded
domain in with smooth boundary with .Comment: 28 pages. arXiv admin note: text overlap with arXiv:1603.0555
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