Electron. J. Diff. Equ., Vol. 2015 (2015), No. 30, pp. 1-11.

Positive radially symmetric solution for a system of quasilinear biharmonic equations in the plane

Joshua Barrow, Robert DeYeso III, Lingju Kong, Frank Petronella

Abstract:
We study the boundary value system for the two-dimensional quasilinear biharmonic equations
$$\displaylines{ \Delta (|\Delta u_i|^{p-2}\Delta u_i)=\lambda_iw_i(x)f_i(u_1,\ldots,u_m),\quad x\in B_1,\cr u_i=\Delta u_i=0,\quad x\in\partial B_1,\quad i=1,\ldots,m, }$$
where $B_1=\{x\in\mathbb{R}^2:|x|<1\}$. Under some suitable conditions on $w_i$ and $f_i$, we discuss the existence, uniqueness, and dependence of positive radially symmetric solutions on the parameters $\lambda_1,\ldots,\lambda_m$. Moreover, two sequences are constructed so that they converge uniformly to the unique solution of the problem. An application to a special problem is also presented.

Submitted September 12, 2014. Published January 30, 2015.
Math Subject Classifications: 35J48, 35J92, 31A30.
Key Words: Positive radially symmetric solution; biharmonic equation; uniqueness; dependence; cone; mixed monotone operator.

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Joshua Barrow
Department of Mathematics
Southern Adventist University
Collegedale, TN 37315, USA
email: joshuabarrow@southern.edu
Robert DeYeso III
Department of Mathematics
University of Tennessee at Martin
Martin, TN 38238, USA
email: robldeye@ut.utm.edu
Lingju Kong
Department of Mathematics
University of Tennessee at Chattanooga
Chattanooga, TN 37403, USA
email: Lingju-Kong@utc.edu
Frank Petronella
Department of Mathematics
Baylor University
Waco, TX 76798, USA
email: Frank_Petronella@baylor.edu

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