Existence Results for a Class of the Quasilinear Elliptic Equations with the Logarithmic Nonlinearity

Abstract

In this paper, the nonlinear quasilinear elliptic problem with the logarithmic nonlinearity $-\mathrm{div}\left({\left|\nabla u\right|}^{p-2}\nabla u\right)=a\left(x\right){\phi }_p\left(u\right)\log \left|u\right|+h\left(x\right){\psi }_p\left(u\right)$ in $\Omega \subset R^n$ was studied. By means of a double perturbation argument and Nehari manifold, the authors obtain the existence results.