2,057 research outputs found
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity
This paper deals with the existence and the asymptotic behavior of
non-negative solutions for a class of stationary Kirchhoff problems driven by a
fractional integro-differential operator and involving a
critical nonlinearity. The main feature, as well as the main difficulty, of the
analysis is the fact that the Kirchhoff function can be zero at zero, that
is the problem is degenerate. The adopted techniques are variational and the
main theorems extend in several directions previous results recently appeared
in the literature
Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness
We study the following Kirchhoff equation A
special feature of this paper is that the nonlinearity and the potential
are indefinite, hence sign-changing. Under some appropriate assumptions on
and , we prove the existence of two different solutions of the equation
via the Ekeland variational principle and Mountain Pass Theorem
Infinitely many periodic solutions for a class of fractional Kirchhoff problems
We prove the existence of infinitely many nontrivial weak periodic solutions
for a class of fractional Kirchhoff problems driven by a relativistic
Schr\"odinger operator with periodic boundary conditions and involving
different types of nonlinearities
Nonlinear equations involving the square root of the Laplacian
In this paper we discuss the existence and non-existence of weak solutions to
parametric fractional equations involving the square root of the Laplacian
in a smooth bounded domain ()
and with zero Dirichlet boundary conditions. Namely, our simple model is the
following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda
f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega.
\end{array}\right. \end{equation*} The existence of at least two non-trivial
-bounded weak solutions is established for large value of the
parameter requiring that the nonlinear term is continuous,
superlinear at zero and sublinear at infinity. Our approach is based on
variational arguments and a suitable variant of the Caffarelli-Silvestre
extension method
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