EXISTENCE OF POSITIVE NONRADIAL SOLUTIONS FOR NONLINEAR ELLIPTIC-EQUATIONS IN ANNULAR DOMAINS

Abstract

We study the existence of positive nonradial solutions of equation DELTA-u + f(u) = 0 in OMEGA(a) , u = 0 on partial derivative-OMEGA(a) , where OMEGA(a) = {x is-an-element-of R(n) : a < x < 1} is an annulus in R(n) , n greater-than-or-equal-to 2 , and f is positive and superlinear at both 0 and infinity . We use a bifurcation method to show that there is a nonradial bifurcation with mode k at a(k) is-an-element-of (0, 1) for any positive integer k if f is subcritical and for large k if f is supercritical. When f is subcritical, then a Nehari-type variational method can be used to prove that there exists a* is-an-element-of (0, 1) such that for any a is-an-element-of (a* , 1) , the equation has a nonradial solution on OMEGA(a) .