Abstract :
In this paper, a semi-linear elliptic equation with Dirichlet boundary condition in a cylindrical domain R × Ω is considered, where Ω is a bounded domain in Rn with n = 1, 2. Let μ1 and μ2 be the first two eigenvalues for the negative Laplacian in Ω. The existence of periodic solutions for a bifurcation parameter λ between μ1 and μ2 is obtained. When λ is near μ2, a local branch of nontrivial solutions which bifurcates from the zero solution is found with an assumption of nondegeneracy. The first order approximations are the decaying solutions while the solutions have small oscillations at infinity. The amplitude of the oscillations is small beyond all orders. The cases for λ > μ2 are deferred to a subsequent study.
