264 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
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
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
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
Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition
The purpose of this paper is to study -periodic solutions to
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in} (0,T)^{N} (P)
u(x+Te_{i})=u(x) &\mbox{for all} x \in \R^{N}, i=1, \dots, N
where , , , and is a continuous
function, -periodic in and satisfying a suitable growth assumption
weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator
can be realized as the Dirichlet to Neumann map for a
degenerate elliptic problem posed on the half-cylinder
. By using a variant of the Linking
Theorem, we show that the extended problem in admits a
nontrivial solution which is -periodic in . Moreover, by a
procedure of limit as , we also prove the existence of a
nontrivial solution to (P) with
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
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