Bifurcation of positive solutions for a Neumann boundary value problem
AbstractAnalytical, approximate and numerical methods are used to study the Neumann boundary value problem− uxx + q2u = u2(1 + sin x), for 0 < x < π,subject to ux(0) = 0, ux(π) = 0,for q2 ∈ (0,∞). Asymptotic approximations to (1) are found for q2 small and q2 large. In the case where q2 is large u(x) ≈ 3qδ(x − π/2). When q2 = 0 we show that the only possible solution is u ≡ 0. However, there exist non-zero solutions for q2 > 0 as well as the trivial solution u ≡ 0. To O(q4) in the q2 small case u(x) = q2π(π + 2)−1, so that bifurcation occurs about the trivial solution branch u ≡ 0 at the first eigenvalue λ0 = 0 and in the direction of the first eigenfunction ξ0 = constant.We obtain a bifurcation diagram for (1), which confirms that there exists a positive solution for q2 ∈ (0, 10). Symmetry-breaking bifurcations and blow-up behaviour occur on certain regions of the diagram. We show that all non-trival solutions to the problem must be positive.The formal outer solution u = q2û appears to satisfy û = û2(1 + sin x), so that û ≡ 0 and û = (1 + sin x)−1 are possible limit solutions. However, in the non-trivial case ûx(0) = −1 and ûx(π) = 1; this means that û does not satisfy the boundary conditions required for a solution of (1). This behaviour usually implies that for q2 large a boundary layer exists near x = 0 (and one near x = π), which corrects the slope. However, we find no evidence for such a solution structure, and only find perturbations in the direction of a delta function about u ≡ 0. We show using the monotone convergence theorem for quadratic forms that the inverse of the operator on the left-hand side of (1) is strongly convergent as q2 → ∞. We show that strong convergence of the operator is sufficient to stop outer-layer behaviour occurring.
