875 research outputs found
Concentration phenomena for a fractional Schr\"odinger-Kirchhoff type equation
In this paper we deal with the multiplicity and concentration of positive
solutions for the following fractional Schr\"odinger-Kirchhoff type equation
\begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}}
\iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy +
\frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2}
dx\right)[\varepsilon^{2s} (-\Delta)^{s}u+ V(x)u]= f(u) \, \mbox{in}
\mathbb{R}^{3} \end{equation*} where is a small parameter,
, is the fractional Laplacian, is a
Kirchhoff function, is a continuous positive potential and is a
superlinear continuous function with subcritical growth. By using penalization
techniques and Ljusternik-Schnirelmann theory, we investigate the relation
between the number of positive solutions with the topology of the set where the
potential attains its minimum.Comment: Mathematical Methods in the Applied Sciences (2017
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
Combined effects for non-autonomous singular biharmonic problems
We study the existence of nontrivial weak solutions for a class of
generalized -biharmonic equations with singular nonlinearity and Navier
boundary condition. The proofs combine variational and topological arguments.
The approach developed in this paper allows for the treatment of several
classes of singular biharmonic problems with variable growth arising in applied
sciences, including the capillarity equation and the mean curvature problem
- …