Existence and concentration of positive solutions for a fractional
Schr\”odinger logarithmic equation
Abstract
In this paper, we study the existence and concentration of positive
solutions for the following fractional Schr\”odinger
logarithmic equation: \begin{equation*}
\left\{ \begin{aligned}
& \varepsilon^{2s}
(-\Delta)^{s} u+V(x)u =u\log
u^2,\ x\in
\mathbb{R}^N,\\
&u\in H^s(\mathbb{R}^N),
\end{aligned} \right.
\end{equation*} where $\varepsilon
> 0$ is a small parameter, $N>2s,$ $s
\in ( 0 ,1), (-\Delta)^{s}$ is the
fractional Laplacian, the potential $V$ is a continuous function
having a global minimum. Using variational method to modify the
nonlinearity with the sum of a $C^1 $ functional and a convex lower
semicontinuous functional, we prove the existence of positive solutions
and concentration around of a minimum point of $V$ when
$\varepsilon$ tends to zero.