3,478 research outputs found
Existence and concentration results for some fractional Schr\"odinger equations in with magnetic fields
We consider some nonlinear fractional Schr\"odinger equations with magnetic
field and involving continuous nonlinearities having subcritical, critical or
supercritical growth. Under a local condition on the potential, we use minimax
methods to investigate the existence and concentration of nontrivial weak
solutions.Comment: arXiv admin note: text overlap with arXiv:1807.0744
Nonlinear fractional magnetic Schr\"odinger equation: existence and multiplicity
In this paper we focus our attention on the following nonlinear fractional
Schr\"odinger equation with magnetic field \begin{equation*}
\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{
in } \mathbb{R}^{N}, \end{equation*} where is a parameter,
, , is the fractional magnetic
Laplacian, and
are continuous potentials and
is a subcritical nonlinearity. By
applying variational methods and Ljusternick-Schnirelmann theory, we prove
existence and multiplicity of solutions for small.Comment: 23 page
Concentration phenomena for critical fractional Schr\"odinger systems
In this paper we study the existence, multiplicity and concentration behavior
of solutions for the following critical fractional Schr\"odinger system
\begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}
(-\Delta)^{s}u+V(x) u=Q_{u}(u, v)+\frac{1}{2^{*}_{s}}K_{u}(u, v) &\mbox{ in }
\mathbb{R}^{N}\varepsilon^{2s} (-\Delta)^{s}u+W(x) v=Q_{v}(u,
v)+\frac{1}{2^{*}_{s}}K_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} u, v>0 &\mbox{ in
} \R^{N}, \end{array} \right. \end{equation*} where is a
parameter, , , is the fractional Laplacian
operator, and
are positive H\"older continuous
potentials, and are homogeneous -functions having subcritical
and critical growth respectively. We relate the number of solutions with the
topology of the set where the potentials and attain their minimum
values. The proofs rely on the Ljusternik-Schnirelmann theory and variational
methods.Comment: arXiv admin note: text overlap with arXiv:1704.0060
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 results for some nonlinear Schr\"odinger equations with the fractional -Laplacian
We consider a class of parametric Schr\"odinger equations driven by the
fractional -Laplacian operator and involving continuous positive potentials
and nonlinearities with subcritical or critical growth. By using variational
methods and Ljusternik-Schnirelmann theory, we study the existence,
multiplicity and concentration of positive solutions for small values of the
parameter
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
Concentration phenomena for a fractional Choquard equation with magnetic field
We consider the following nonlinear fractional Choquard equation
\varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u =
\varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u
\mbox{ in } \mathbb{R}^{N}, where is a parameter, , , , is the fractional magnetic
Laplacian, is a smooth magnetic
potential, is a positive potential
with a local minimum and is a continuous nonlinearity with subcritical
growth. By using variational methods we prove the existence and concentration
of nontrivial solutions for small enough.Comment: arXiv admin note: text overlap with arXiv:1801.0019
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