NEGATIVE EIGENVALUES FOR A NONLINEAR DIFFERENTIAL-EQUATION ON R

This paper considers a nonlinear eigenvalue problem involving a second-order ordinary differential equation on R with variable coefficients. One of the coefficients may be negative on some subsets of R and both may be unbounded. In case that the coefficients are positive constants this problem has been studied by H. Berestycki and P.-L. Lions [Arch. Rational Mech. Anal., 82 (1983), pp. 313-345]. The paper shows that the problem has a negative eigenvalue and a positive classical solution decaying exponentially at infinity. Moreover, in some special cases the existence of bifurcation is proved. The main tools are direct methods from the calculus of variations, some comparison techniques, and Lebesgue's theorem on monotone functions.