Existence and blow-up rate of large solutions of $p(x)$-Laplacian equations with large perturbation and gradient terms

Abstract

In this paper, we investigate boundary blow-up solutions of the problem \begin{equation*} \quad \left\{ \begin{array}{l} -\Delta _{p(x)}u+f(x,u)=\rho (x,u)+K(x)|\nabla u|^{m(x)}\mbox{ in }\Omega , \\[2mm] u(x)\rightarrow +\infty \mbox{ as }d(x,\partial \Omega )\rightarrow 0, \end{array} \right. \end{equation*} where $\Delta _{p(x)}u=\mathrm{div}\,(|\nabla u|^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. Our results extend the previous work of J. García-Melián, A. Suárez [23] from the case where $p(\cdot )\equiv 2 $, without gradient term, to the case where $p(\cdot )$ is a function, with gradient term. It also extends the previous work of Y. Liang, Q.H. Zhang and C.S. Zhao [38] from the radial case in the problem to the non-radial case. The existence of boundary blow-up solutions is established and the singularity of boundary blow-up solution is also studied for several cases including when $\frac{\rho (x,u(x))}{f(x,u(x))}\rightarrow 1$ as $x\rightarrow \partial \Omega $, which means that $\rho (x,u)$ is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $d(x,\partial \Omega )$. Hence, the results of this paper are new even in the canonical case $p(\cdot )\equiv 2$. In particular, we do not have the comparison principle, because we don't make the monotone assumption of nonlinear term.