Nodal solutions of weighted indefinite problems

Abstract

This paper analyzes the structure of the set of nodal solutions, i.e., solutions changing sign, of a class of one-dimensional superlinear indefinite boundary values problems with an indefinite weight functions in front of the spectral parameter. Quite surprisingly, the associated high order eigenvalues may not be concave as is the case for the lowest one. As a consequence, in many circumstances, the nodal solutions can bifurcate from three or even four bifurcation points from the trivial solution. This paper combines analytical and numerical tools. The analysis carried out is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously the numerical study confirms and illuminates the analysis. 2020 MSC: 34B15, 34B08, 34L16.

Global bifurcation diagrams of positive solutions for a class of 1D superlinear indefinite problems*

“…To discretize (1.1) we have used two methods. To compute the small positive solutions bifurcating from u = 0 we implemented a pseudo-spectral method combining a trigonometric spectral method with collocation at equidistant points, as in Gómez-Reñasco and López-Gómez [26,28], López-Gómez, Eilbeck, Duncan and Molina-Meyer [37], López-Gómez and Molina-Meyer [38][39][40], López-Gómez, Molina-Meyer and Tellini [41], López-Gómez, Molina-Meyer and Rabinowitz [42], and Fencl and López-Gómez [23]. This gives high accuracy at a rather reasonable computational cost (see, e.g., Canuto, Hussaini, Quarteroni and Zang [12]).…”

Global bifurcation diagrams of positive solutions for a class of 1-D superlinear indefinite problems

“…To discretize (1.1) we have used two methods. To compute the small positive solutions bifurcating from u = 0 we implemented a pseudo-spectral method combining a trigonometric spectral method with collocation at equidistant points, as in R. Gómez-Reñasco and J. López-Gómez [28,31], J. López-Gómez, J. C. Eilbeck, K. Duncan and M. Molina-Meyer [42], J. López-Gómez and M. Molina-Meyer [44,45,46], J. López-Gómez, M. Molina-Meyer and A. Tellini [47], J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz [48], and M. Fencl and J. López-Gómez [25]. This gives high accuracy at a rather reasonable computational cost (see, e.g., Canuto, Hussaini, Quarteroni and Zang [13]).…”