A REMARK ON THE UNIQUENESS OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS WITH DOUBLE-POWER NONLINEARITIES

作者：Kawano Shinji^*

来源：Differential and Integral Equations, 2011, 24(3-4): 201-207.

We consider the uniqueness of positive solutions to
{Delta u + f(u) = 0 in R(n),
lim(|x|-->infinity) u(x) = 0,
where f(u) = -omega u + u(p) - u(2p-1) with omega > 0 and p > 1.
It is known that for fixed p > 1, a positive solution exists if and only if omega < omega(p) := p/(p+1)(2). We deduce the uniqueness in the case where omega is close to omega(p), from the argument in the classical paper by Peletier and Serrin [9], thereby recovering a part of the uniqueness result of Ouyang and Shi [8] for all omega is an element of (0, omega(p)).
In the appendix we consider the more general nonlinearity
f(u) = -omega u + u(p) - u(q), omega > 0, q > p > 1
and discuss the existence and uniqueness conditions. There we prove the fact that f having positive part is equivalent to f remaining negative, where (f) over tilde (u) := (uf' (u))' f (u) u f'(u)(2).

• 出版日期2011-4