Nodal solutions of weighted indefinite problems
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Journal of Evolution Equations
Nodal solutions of weighted indefinite problems M. Fencl and J. López- Gómez
This paper is dedicated to M. Hieber at the occasion of his 60th birthday mit Wertschätzung und Freundschaft 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 value problems with 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.
1. Introduction In this paper, we analyze the nodal solutions of the one-dimensional nonlinear weighted boundary value problem −u − μu = λm(x)u − a(x)u 2 in (0, 1), (1.1) u(0) = u(1) = 0, where a, m ∈ C[0, 1] are functions that change sign in (0, 1) and λ, μ ∈ R are regarded as bifurcation parameters. More precisely, λ is the primary parameter and μ the secondary one. All the numerical experiments carried out in this paper have been implemented in the special case when π ⎧ ⎨ −0.2 πsin 0.2 (0.2− x) if 0 ≤ x ≤ 0.2, (1.2) a(x) := sin 0.6 (x − 0.2) if 0.2 < x ≤ 0.8, π ⎩ (x − 0.8) if 0.8 < x ≤ 1, −0.2 sin 0.2 Mathematics Subject Classification: 34B15, 34B08, 34L16 Keywords: Superlinear indefinite problems, Weighted problems, Positive solutions, Nodal solutions, Eigencurves, Concavity, Bifurcation, Global components, Path-following, Pseudo-spectral methods, Finitedifference scheme. Partially supported by the Research Grant PGC2018-097104-B-I00 of the Spanish Ministry of Science, Innovation and Universities, and the Institute of Inter-disciplinary Mathematics (IMI) of Complutense University. M. Fencl has been supported by the Project SGS-2019-010 of the University of West Bohemia, the Project 18-03253S of the Grant Agency of the Czech Republic and the Project LO1506 of the Czech Ministry of Education, Youth and Sport.
M. Fencl and J. López- Gómez
J. Evol. Equ.
Figure 1. Graph of the weight function a(x)
because this is the weight function a(x) considered by López-Gómez and MolinaMeyer [29] to compute the global bifurcation diagrams of positive solutions. Figure 1 shows a plot of this function. In this paper, we pay a very special attention to the particular, but very interesting, case when m(x) = sin( jπ x) for some integer j ≥ 2. To the best of our knowledge, this is the first paper where the problem of the existence and the structure of nodal solutions of a weighted superlinear indefinite problem is addressed when m(x) changes sign. The existence results for large solutions of Mawhin et al. [40] required m ≡ 1, as well as the results of López-Gómez et
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