Infinitely many solutions at a resonance ∗
نویسندگان
چکیده
We use bifurcation theory to show the existence of infinitely many solutions at the first eigenvalue for a class of Dirichlet problems in one dimension. It has been observed that complexity of the solution curve for the boundary value problem u + λf(u) = 0 for 0 < x < L, u(0) = u(L) = 0 (1) seems to mirror that of the nonlinearity f(u), see e.g. P. Korman, Y. Li and T. Ouyang [6]. Namely, if f(u) is convex (f = e is a prominent example), f(u) has at most one critical point, and correspondingly the solution curve of (1) admits only one turn. Similarly for many functions with two inflection points (like cubics, or modified Gelfand’s equation) one can show that solution curve admits exactly two turns, see [6] and also P. Korman and Y. Li [5]. It is natural to ask: will solution curve admit infinitely many turns if f(u) changes concavity infinitely many times. It has been known for a while that this indeed might happen, see e.g. R. Schaaf and K. Schmitt [8], D. Costa, H. Jeggle, R. Schaaf and K. Schmitt [2], H. Kielhofer and S. Maier [4]. Recent contributions include Y. Cheng [1] and S.-H. Wang [10]. This note was stimulated by an example in Y. Cheng [1], who has shown that for f(u) = u+sin √ u the problem (1) admits infinitely many solutions at λ = λ1, where λ1 denotes as usual the principal eigenvalue of −u′′ on (0, L), λ1 = π L2 . The proof in [1] used the quadrature method. In this note we use bifurcation theory to obtain a similar result. The bifurcation approach gives a clear understanding of the solution curve, and opens a way to considering higher dimensions. The papers [8] and [2] also used bifurcation approach, and they considered more general equations, as well as the PDE case in two dimensions. Our approach is different, we work with a smooth curve of solutions (rather than a continuum of solutions bifurcating from ∗Mathematics Subject Classifications: 34B15.
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تاریخ انتشار 2000