May  2010, 9(3): 819-837. doi: 10.3934/cpaa.2010.9.819

Layered solutions in $R^2$ for a class of $p$-Laplace equations

1. 

College of Mathematics and Econometrics, Hunan University, Changsha, China

Received  March 2009 Revised  September 2009 Published  January 2009

This paper studies the entire solutions of a class of $p$-Laplace equation

- div$(|\nabla u|^{p-2}\nabla u)+a(x)W^'(u(x,y))=0, (x,y)\in R^2$

in the case $p>2$. where $a:\mathbf{R}\rightarrow \mathbf{R_{+}}$ is a periodic, positive function and $W:\mathbf{R}\rightarrow \mathbf{R}$ is a non-negative $C^{2}$ function. We look for the entire solutions of the above equation with asymptotic conditions $u(x,y)\rightarrow \pm 1 $ as $x\rightarrow\pm\infty$ uniformly with respect to $y\in \mathbf{R}$. Via variational methods we find layered solutions which depend on both x and y, i.e., solutions which do not exhibit one dimensional symmetries.

Citation: Zheng Zhou. Layered solutions in $R^2$ for a class of $p$-Laplace equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 819-837. doi: 10.3934/cpaa.2010.9.819
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