66 research outputs found
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
Infinitely many solutions for the p-fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity
In this paper, we consider the fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity. By means of the concentration-compactness principle in fractional Sobolev space and the Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions, which tend to zero for suitable positive parameters
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
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
Multiplicity and concentration results for a fractional Schr\"odinger-Poisson type equation with magnetic field
This paper is devoted to the study of fractional Schr\"odinger-Poisson type
equations with magnetic field of the type \begin{equation*}
\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u
\quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where is a
parameter, are such that ,
is a smooth magnetic potential,
is the fractional magnetic Laplacian,
is a continuous electric potential and
is a subcritical nonlinear term.
Using variational methods, we obtain the existence, multiplicity and
concentration of nontrivial solutions for small enough
Solutions for a nonhomogeneous p&q-Laplacian problem via variational methods and sub-supersolution technique
In this paper it is obtained, through variational methods and sub-supersolution arguments, existence and multiplicity of solutions for a nonhomogeneous problem which arise in several branches of science such as chemical reactions, biophysics and plasma physics. Under a general hypothesis it is proved an existence result and multiple solutions are obtained by considering an additional natural condition
Existence of infinitely many radial and non-radial solutions for quasilinear Schrödinger equations with general nonlinearity
In this paper, we prove the existence of many solutions for the following quasilinear Schrödinger equation
\begin{equation*}
-\Delta u - u\Delta(|u|^2) + V(|x|)u = f(|x|,u),\qquad x \in \mathbb{R}^N.
\end{equation*}
Under some generalized assumptions on , we obtain infinitely many radial solutions for , many non-radial solutions for and , and a non radial solution for . Our results generalize and extend some existing results
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