1,042 research outputs found
Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian
We study the regularity of stable solutions to the problem where
. Our main result establishes an bound for stable and
radially decreasing solutions to this problem in dimensions . In particular, this estimate holds for all
in dimensions . It applies to all nonlinearities .
For such parameters and , our result leads to the regularity of the
extremal solution when is replaced by with . This
is a widely studied question for , which is still largely open in the
nonradial case both for and
Boundedness of stable solutions to semilinear elliptic equations: a survey
This article is a survey on boundedness results for stable solutions to
semilinear elliptic problems. For these solutions, we present the currently
known estimates that hold for all nonlinearities. Such estimates
are known to hold up to dimension 4. They are expected to be true also in
dimensions 5 to 9, but this is still an open problem which has only been
established in the radial case
Boundedness of stable solutions to semilinear elliptic equations: a survey
This article is a survey on boundedness results for stable solutions to
semilinear elliptic problems. For these solutions, we present the currently
known estimates that hold for all nonlinearities. Such estimates
are known to hold up to dimension 4. They are expected to be true also in
dimensions 5 to 9, but this is still an open problem which has only been
established in the radial case
Regularity of radial extremal solutions for some non local semilinear equations
We investigate stable solutions of elliptic equations of the type
\begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad
{\mbox{ in }} \\ u&= 0 \qquad{\mbox{ on ,}}\end{aligned}\right . \end{equation*} where , ,
and is any smooth positive superlinear function. The
operator stands for the fractional Laplacian, a
pseudo-differential operator of order . According to the value of
, we study the existence and regularity of weak solutions
Mass and Asymptotics associated to Fractional Hardy-Schr\"odinger Operators in Critical Regimes
We consider linear and non-linear boundary value problems associated to the
fractional Hardy-Schr\"odinger operator on domains of
containing the singularity , where and , the latter being the best constant in the
fractional Hardy inequality on . We tackle the existence of
least-energy solutions for the borderline boundary value problem
on
, where and is the critical fractional
Sobolev exponent. We show that if is below a certain threshold
, then such solutions exist for all , the latter being the first eigenvalue of
. On the other hand, for , we prove existence of such solutions only for those
in for which the domain
has a positive {\it fractional Hardy-Schr\"odinger mass} . This latter notion is introduced by way of an invariant of
the linear equation on
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