273 research outputs found
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
Multiple solutions for a fractional -Laplacian equation with sign-changing potential
We use a variant of the fountain Theorem to prove the existence of infinitely
many weak solutions for the following fractional p-Laplace equation
(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u) in R^N, where ,,, is the fractional -Laplace operator, the
nonlinearity f is -superlinear at infinity and the potential V(x) is allowed
to be sign-changing
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
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
Periodic solutions for critical fractional problems
We deal with the existence of -periodic solutions to the following
non-local critical problem \begin{equation*} \left\{\begin{array}{ll}
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in}
(-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N},
\quad i=1, \dots, N, \end{array} \right. \end{equation*} where , , , is the fractional critical
Sobolev exponent, is a positive continuous function, and is a
superlinear -periodic (in ) continuous function with subcritical
growth. When , the existence of a nonconstant periodic solution is
obtained by applying the Linking Theorem, after transforming the above
non-local problem into a degenerate elliptic problem in the half-cylinder
, with a nonlinear Neumann boundary
condition, through a suitable variant of the extension method in periodic
setting. We also consider the case by using a careful procedure of limit.
As far as we know, all these results are new.Comment: Calculus of Variations and Partial Differential Equations (2018
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
Ground states for a fractional scalar field problem with critical growth
We prove the existence of a ground state solution for the following
fractional scalar field equation in
where , is the fractional Laplacian, and
is an odd function satisfying the
critical growth assumption
Mountain pass solutions for the fractional Berestycki-Lions problem
We investigate the existence of least energy solutions and infinitely many
solutions for the following nonlinear fractional equation (-\Delta)^{s} u =
g(u) \mbox{ in } \mathbb{R}^{N}, where , ,
is the fractional Laplacian and is an
odd function satisfying Berestycki-Lions type
assumptions. The proof is based on the symmetric mountain pass approach
developed by Hirata, Ikoma and Tanaka in \cite{HIT}. Moreover, by combining the
mountain pass approach and an approximation argument, we also prove the
existence of a positive radially symmetric solution for the above problem when
satisfies suitable growth conditions which make our problem fall in the so
called "zero mass" case
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