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E. Yanagida recently proved that the classical Matukuma equation with a given exponent has only one nite mass solution. We show how similar ideas can be exploited to obtain uniqueness results for other classes of equations as well as Matukuma equations with more general coe cients. One particular example covered is u + u p u = 0, with p > 1. The key ingredients of the method are energy functions and suitable transformations. We also study general boundary conditions, using an extension of a recent result by Bandle and Kwong. Yanagida's proof does not extend to solutions of Matukuma's equation satisfying other boundary conditions. We treat these with a completely di erent method of Kwong and Zhang. AMS(MOS) Subject Classi cation. Primary 34B15. Secondary 35J25, 35J65.

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In this paper a new approach based on a shooting method in a half line coupled with the technique of upper-lower solution pair is used to study the existence and nonexistence of monotone waves for one form of the delayed Fisher equation that does not have the quasimonotonicity property. A necessary and sufficient condition is provided. This new method can be extended to investigate many other nonlocal and non-monotone delayed reaction-diffusion equations.

We investigate the zeros of eigenfunctions of regular Sturm–Liouville boundary value problems with general weight functions w. In particular we are interested in the case when the set of zeros of w has positive measure. We find that in this case the first eigenfunction may have one or more zeros in the interval, in contrast to the classical case when w is positive. Necessary and sufficient conditions on w and the other coefficients are found such that the first eigenfunction has no zero.