Ground states for fractional nonlocal equations with logarithmic nonlinearity

Abstract

In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by $\begin{cases}\mathcal{L}_{K}u(x)+u\log|u|+|u|^{q-2}u=0, & x\in\Omega,\\ u=0, & x\in\mathbb{R}^{n}\setminus\Omega,\end{cases}$ where $2\lt q\lt 2^{*}_s$, $L_{K}$ is a non-local operator, $\Omega$ is an open bounded set of $\mathbb{R}^{n}$ with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.