780 research outputs found
Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity
This article concerns about the existence and multiplicity of weak solutions
for the following nonlinear doubly nonlocal problem with critical nonlinearity
in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{
\begin{split} (-\Delta)^su &= \lambda |u|^{q-2}u +
\left(\int_{\Omega}\frac{|v(y)|^{2^*_\mu}}{|x-y|^\mu}~\mathrm{d}y\right)
|u|^{2^*_\mu-2}u\; \text{in}\; \Omega
(-\Delta)^sv &= \delta |v|^{q-2}v +
\left(\int_{\Om}\frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}~\mathrm{d}y \right)
|v|^{2^*_\mu-2}v \; \text{in}\; \Omega
u &=v=0\; \text{in}\; \mb R^n\setminus\Omega, \end{split} \right.
\end{equation*} where is a smooth bounded domain in \mb R^n, , , is the well known fractional Laplacian, , is the upper critical
exponent in the Hardy-Littlewood-Sobolev inequality, and
are real parameters. We study the fibering maps
corresponding to the functional associated with and show
that minimization over suitable subsets of Nehari manifold renders the
existence of atleast two non trivial solutions of (P_{\la,\delta}) for
suitable range of \la and .Comment: 37 page
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
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