For certain functions f, positive
in (0,∞) and continuous in [0,∞), the partial differential equation Δx = x−xf(x2)
has spherically symmetric solutions xn(t),n = 1,2,⋯ , which vanish at zero, infinity
and n − 1 distinct values in (0,∞). This and similar existence theorems for the
ordinary differential equation ÿ−y + yF(y2,t) = 0 are proved by way of variational
problems and the solutions are thus characterized by associated “eigenvalues”. The
asymptotic behavior of these eigenvalues is studied and some numerical data on the
solutions is furnished for special cases of the above equations which are of interest in
nuclear physics.
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