Taiwanese Journal of Mathematics


Yuan Liang, Qihu Zhang, and Chunshan Zhao

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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.

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Taiwanese J. Math., Volume 18, Number 2 (2014), 599-632.

First available in Project Euclid: 10 July 2017

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Primary: 35J25: Boundary value problems for second-order elliptic equations 35B40: Asymptotic behavior of solutions 35J60: Nonlinear elliptic equations

$p(x)$-Laplacian sub-solution super-solution boundary blow-up solution singularity


Liang, Yuan; Zhang, Qihu; Zhao, Chunshan. ON THE BOUNDARY BLOW-UP SOLUTIONS OF $p(x)$-LAPLACIAN EQUATIONS WITH GRADIENT TERMS. Taiwanese J. Math. 18 (2014), no. 2, 599--632. doi:10.11650/tjm.18.2014.3327. https://projecteuclid.org/euclid.twjm/1499706404

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