517 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
On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity
We consider the fractional Schr\"{o}dinger-Kirchhoff equations with
electromagnetic fields and critical nonlinearity
where as and
is the fractional magnetic operator with , is a continuous nondecreasing
function, and are the electric and the magnetic potential,
respectively. By using the fractional version of the concentration compactness
principle and variational methods, we show that the above problem: (i) has at
least one solution provided that ; and (ii) for any
, has pairs of solutions if , where and are
sufficiently small positive numbers. Moreover, these solutions as
A fractional Kirchhoff problem involving a singular term and a critical nonlinearity
In this paper we consider the following critical nonlocal problem
\left\{\begin{array}{ll}
M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s
u = \displaystyle\frac{\lambda}{u^\gamma}+u^{2^*_s-1}&\quad\mbox{in } \Omega,\\
u>0&\quad\mbox{in } \Omega,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminus\Omega,
\end{array}\right. where is an open bounded subset of
with continuous boundary, dimension with parameter ,
is the fractional critical Sobolev exponent, is a
real parameter, exponent , models a Kirchhoff type
coefficient, while is the fractional Laplace operator. In
particular, we cover the delicate degenerate case, that is when the Kirchhoff
function is zero at zero. By combining variational methods with an
appropriate truncation argument, we provide the existence of two solutions
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