Uncountably Infinite Solutions To A Class Of Nonlinear Singular Two-Point Eigenvalue Boundary Value Problems

Abstract

We proved the existence of unique weak solutions in weighted Sobolev spaces, up to each of an arbitrarily selectable parameter m ∈ (0, a], for the nonlinear singular second order two-point boundary value problems
u''(r) + a/r u'(r) + g(u(r)) = h(r), a ≥ 1, r ∈ (0, 1)
u(0) = u(1) = 0,
where h ∈ L 2 [(0, 1), rm], for 1 < m ≤ a, and g : R → R is differentiable, with 0 6≡ |g 0| ≤ γ = constant. Our solutions are uncountably infinite, since the possible choices of the parameter m ∈ (1, a] are uncountably infinite