Yasuhiro Furusho

Positive entire solutions to nonlinear biharmonic equations in the plane

Abstract Two-dimensional nonlinear biharmonic equations of the form Δ(¦Δu¦ p−2 Δu) = 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¦→∞ .

Existence of positive entire solutions for a class of quasilinear elliptic equations

D, = LT/d?r,, D,] = d’liix, ax,, 1 6 i, j< N, and i. is a real constant. By an entire solution of (1.1) we mean a function II of class C’(RN) which satisfies (1.1) in RN. In recent years, the existence of positive entire solutions of ( 1.1) has been studied under various structure hypotheses for L and f by many authors (see, e.g., [ 1-8, l&15, 17-22, 241). However, as far as we are aware, little is known about the existence of nonconstant entire solutions of Eq. ( 1.1) subject to the condition