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On variational eigenvalues of the p-Laplacian which are not of Ljusternik-Schnirelmann type. (English) Zbl 1195.47041

From the abstract: “We demonstrate the fact that the famous Ljusternik-Schnirelmann characterization of some eigenvalues of nonlinear elliptic problems (by a minimax formula) has a global variational character. Indeed, we show that, for some homogeneous quasilinear elliptic eigenvalue problems, there are variational eigenvalues other than those of the Ljusternik-Schnirelmann type. In contrast, these eigenvalues have a local variational character. Such a phenomenon does not occur in typical linear elliptic eigenvalue problems.”

The main result of this paper is as follows:

There exist a positive number ε 0 and a C -function q=q(x) of x that is π p -periodic and even about zero, such that, for any ε(0,ε 0 ], the problem

-(|u ' | p-2 u ' ) ' +εq(x)|u| p-2 u=μ|u| p-2 uin

has two eigenvalues μ 1 ε ,μ 2 ε of the Ljusternik-Schnirelmann type with μ 1 ε <μ 2 ε , and a third eigenvalue μ(ε)(μ 1 ε ,μ 2 ε ) characterized by

μ(ε)=inf γ𝒞 max uγ -π p π p | u ' (x)| p d x + ε -π p π p q (x) |u(x)| p d x·

All eigenfunctions associated with μ(ε) have precisely two zeros in the interval (-π p ,π p ]. Here, 𝒞 is a special class of curves defined on the submanifold 𝒮 of the Sobolev space

W per 1,p ={fW 1,p (-π p ,π p ):f(-π p )=f(π p )}·

To ensure the existence of a local variational eigenvalue, a special choice of the potential q was motivated by the works of P. A. Binding and B. P. Rynne [J. Differ. Equations 235, No. 1, 199–218 (2007; Zbl 1218.34100); J. Differ. Equations 244, No. 1, 24–39 (2008: Zbl 1136.35061)].

MSC:
47J10Nonlinear spectral theory, nonlinear eigenvalue problems
47J30Variational methods (nonlinear operator equations)
49R05Variational methods for eigenvalues of operators
34L30Nonlinear ordinary differential operators