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July/August 2016 Existence and blow-up rate of large solutions of $p(x)$-Laplacian equations with large perturbation and gradient terms
Dumitru Motreanu, Qihu Zhang
Adv. Differential Equations 21(7/8): 699-734 (July/August 2016).

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.

Citation

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Dumitru Motreanu. Qihu Zhang. "Existence and blow-up rate of large solutions of $p(x)$-Laplacian equations with large perturbation and gradient terms." Adv. Differential Equations 21 (7/8) 699 - 734, July/August 2016.

Information

Published: July/August 2016
First available in Project Euclid: 3 May 2016

zbMATH: 1375.35144
MathSciNet: MR3493932

Subjects:
Primary: 35B40 , 35J25 , 35J60

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.21 • No. 7/8 • July/August 2016
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