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

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.

Keywords

• Positive radially symmetric solution
• biharmonic equation
• uniqueness
• dependence
• cone
• mixed monotone operator