Positive entire solutions to nonlinear biharmonic equations in the plane

Two-dimensional nonlinear biharmonic equations of the form Delta(\Delta u\(p-2)Delta u) = f(\x\,u), x is an element of R-2, (*) are considered, where p>1 is a constant and f:(R) over bar(+) x R+ --> R+ is a continuous function. It can be shown that any positive radially symmetric solution is unbounded and grows at least as fast as positive constant multiples of \x\(2)(log\x\)(1/(p-1)) as as \x\ --> infinity In this paper sharp conditions on f are presented under which (*) has infinitely many positive symmetric entire solutions which are asymptotic to positive constant multiples of \x\(2)(log\x\)(1/(p-1)) as \x\ --> infinity. (C) 1998 Elsevier Science B.V. All rights reserved.