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.
Citation
Mingqi Xiang. Binlin Zhang. "Combined effects of logarithmic and critical nonlinearities in fractional Laplacian problems." Adv. Differential Equations 26 (7/8) 363 - 396, July/August 2021. https://doi.org/10.57262/ade026-0708-363
Information