On the Generalization of the Courant Nodal Domain Theorem

Abstract

In this paper we consider the analogue of the Courant nodal domain theorem for the nonlinear eigenvalue problem for the p-Laplacian. In particular we prove that if u ln is an eigenfunction associated with the nth variational eigenvalue, l n , then u ln has at most 2n − 2 nodal domains. Also, if u ln has n+k nodal domains, then there is another eigenfunction with at most n − k nodal domains. © 2002 Elsevier Science (USA)

Multi-dimensional Morse Index Theorems and a symplectic view of elliptic boundary value problems

“…There is a long history of work aimed at relating this index to the nodal structure of the associated eigenfunction. This goes back, at least to Courant and Hilbert [3], and was further developed in [9] and [11]. The interesting paper of Johnson, see [13], can also be seen in this light.…”

Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue

“…Therefore, for example, if i = 2 and Ω is convex, in Corollary 2.1, we prove that the solutions have exactly two nodal domains and the nodal line touches the boundary of Ω. For completeness, we also recall a version of Courant's nodal theorem for the p-Laplacian found in [15], and that in the case of sublinear elliptic problems in R N , in [2] the authors prove the existence of radial compactly supported solutions with any given number of nodes.…”

Eigenvalues of the p(x)-Laplacian Steklov problem

“…In order to solve the eigenvalue problem (3.2), in general, the constrained variational method is employed (see [4,7,13,14,29,35,36]). In this paper, we take F as an objective functional and G as a constraint functional.…”

“…For example, see [2][3][4]6,9,10,13,14,24,25,28,29,32,34] and references therein. The investigations mainly have relied on variational methods and deduce the existence of a principal eigenvalue, which is the smallest of all possible eigenvalues, as a consequence of minimization results of appropriate functionals.…”