Existence and Asymptotic Behavior of Boundary Blow-Up Solutions for Weighted p ( x ) -Laplacian Equations with Exponential Nonlinearities

Abstract

This paper investigates the following $p\left(x\right)$-Laplacian equations with exponential nonlinearities: $-{\Delta }_{p\left(x\right)}u+\rho \left(x\right)e^{f(x,u)}=0$ in $\Omega$, $u\left(x\right)\to +\infty$ as $d(x,\partial \Omega )\to 0$, where $-{\Delta }_{p\left(x\right)}u=-\text{div}({|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is called $p\left(x\right)$-Laplacian, $\rho \left(x\right)\in C\left(\Omega \right)$. The asymptotic behavior of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.