Multiplicity results near the principal eigenvalue for boundary-value problems with periodic nonlinearity

Let us consider the boundary-value problem -u"(x) - lambda u(x) + g(u(x)) = asinx + (h) over tilde (x), x is an element of [0, pi], u(0) = u(pi) = 0, where g : R -> R is a continuous and T-periodic function with zero mean value, not identically zero, (lambda, a) is an element of R-2 and (h) over tilde is an element of C[0, pi] with integral(pi)(0) (h) over tilde (x)sinxdx = 0. If lambda(1) denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (lambda, a) -> (lambda(1), 0), then the number of solutions increases to infinity. The proof combines Liapunov-Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation.