1,105 research outputs found
Concentrating solutions for a fractional Kirchhoff equation with critical growth
In this paper we consider the following class of fractional Kirchhoff
equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll}
\left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u
\quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0
&\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where
is a small parameter, are constants, , is the fractional critical
exponent, is the fractional Laplacian operator, is a
positive continuous potential and is a superlinear continuous function with
subcritical growth. Using penalization techniques and variational methods, we
prove the existence of a family of positive solutions which
concentrates around a local minimum of as .Comment: arXiv admin note: text overlap with arXiv:1810.0456
Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth
We investigate the existence, multiplicity and concentration of nontrivial
solutions for the following fractional magnetic Kirchhoff equation with
critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3}
[u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u
\quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where is a
small positive parameter, are fixed constants, , is the fractional critical exponent,
is the fractional magnetic Laplacian,
is a smooth magnetic potential,
is a positive continuous potential
verifying the global condition due to Rabinowitz \cite{Rab}, and
is a subcritical nonlinearity. Due
to the presence of the magnetic field and the critical growth of the
nonlinearity, several difficulties arise in the study of our problem and a
careful analysis will be needed. The main results presented here are
established by using minimax methods, concentration compactness principle of
Lions \cite{Lions}, a fractional Kato's type inequality and the
Ljusternik-Schnirelmann theory of critical points.Comment: arXiv admin note: text overlap with arXiv:1808.0929
A multiplicity result for a fractional Kirchhoff equation in with a general nonlinearity
In this paper we deal with the following fractional Kirchhoff equation
\begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)-
u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-\Delta)^{s}u = g(u) \mbox{ in }
\mathbb{R}^{N}, \end{equation*} where , , , is a
small positive parameter and is an odd
function satisfying Berestycki-Lions type assumptions. By using minimax
arguments, we establish a multiplicity result for the above equation, provided
that is sufficiently small
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
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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