Combined effects of logarithmic and critical nonlinearities in fractional Laplacian problems

Abstract

The aim of this paper is to study the following fractional Laplacian problems with logarithmic and critical nonlinearities: \begin{align*} \begin{cases} (-\Delta)^{s}u=\lambda a(x)u\ln|u|+\mu|u|^{2_s^*-2}u \ \ & x\in\Omega,\\ u=0\ \ & x\in\mathbb{R}^N\setminus\Omega, \end{cases} \end{align*} where $s\in(0,1),\ N > 2s,$ $\lambda,\mu > 0$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $a\in L^\infty(\Omega)$ is a sign-changing function, $2_s^*$ is the critical Sobolev exponent and $(-\Delta)^{s}$ is the fractional Laplacian. When $\lambda$ and $\mu$ satisfy suitable assumptions, the existence and multiplicity of nontrivial and nonnegative continuous solutions for the above problem are obtained by applying the Nehari manifold approach.