ON THE BOUNDARY BLOW-UP SOLUTIONS OF $p(x)$-LAPLACIAN EQUATIONS WITH GRADIENT TERMS

Abstract

In this paper we investigate boundary blow-up solutions of the problem $$ \left\{ \begin{array}{l} -\triangle _{p(x)}u+f(x,u)=\rho (x,u)+K(\left\vert x\right\vert )\left\vert \nabla u\right\vert ^{\delta (\left\vert x\right\vert )}\text{ in }\Omega \text{,} \\[12pt] \text{ }u(x)\rightarrow +\infty \text{ as }d(x,\text{ }\partial \Omega )\rightarrow 0\text{,} \end{array} \right. $$ where $-\triangle _{p(x)}u=-\mbox{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. The existence of boundary blow-up solutions is proved and the singularity of boundary blow-up solution is also given for several cases including the case of $\rho (x,u)$ being a large perturbation (namely, $\frac{\rho (x,u(x))}{f(x,u(x))}\rightarrow 1$ as $x\rightarrow \partial \Omega $). In particular, we do not have the comparison principle.