Positive entire solutions to nonlinear biharmonic equations in the plane

Two-dimensional nonlinear biharmonic equations of the form Δ(¦Δu¦p−2Δu) = f(¦x¦,u), x ∈ R2 are considered, where p>1 is a constant and f: R+ × 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 ¦x¦→∞. 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¦→∞.