Variational properties of the first curve of the Fu\v{c}\'{\i}k spectrum for elliptic operators

DOI10.1007/S00526-015-0920-4arXiv2205.01496MaRDI QIDQ6398133

Publication date: 3 May 2022

Abstract: In this paper we present a new variational characteriztion of the first nontrival curve of the Fuv{c}'{i}k spectrum for elliptic operators with Dirichlet boundary conditions. Moreover, we describe the asymptotic behaviour and some properties of this curve and of the corresponding eigenfunctions. In particular, this new characterization allows us to compare the first curve of the Fuv{c}'{i}k spectrum with the infinitely many curves we obtained in previous works (see R. Molle, D. Passaseo, New properties of the Fuv{c}'{i}k spectrum. C. R. Math. Acad. Sci. Paris 351 (2013), no. 17/18, 681--685 and R. Molle, D. Passaseo, Infinitely many new curves of the Fuv{c}'{i}k spectrum. Ann. I. H. Poincar'e - AN (2014), http://dx.doi.org/10.1016/j.anihpc.2014.05.007): for example, we show that these curves are all asymptotic to the same lines as the first curve, but they are all distinct from such a curve.

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs (35P30) Variational methods for second-order elliptic equations (35J20) Semilinear elliptic equations (35J61) Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian (35J91)

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