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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 $\varepsilon_0$ and a $C^{\infty}$-function $q=q(x)$ of $x\in \Bbb R$ that is $\pi_p$-periodic and even about zero, such that, for any $\varepsilon\in (0,\varepsilon_0]$, the problem $$ -(|u'|^{p-2}u')' + \varepsilon q(x)|u|^{p-2}u = \mu \, |u|^{p-2}u\quad \text{in} \ \Bbb R$$ has two eigenvalues $\mu_1^{\varepsilon}, \mu_2^{\varepsilon}$ of the Ljusternik-Schnirelmann type with $\mu_1^{\varepsilon}< \mu_2^{\varepsilon}$, and a third eigenvalue $\mu(\varepsilon)\in (\mu_1^{\varepsilon}, \mu_2^{\varepsilon})$ characterized by $$ \mu(\varepsilon)=\inf_{\gamma\in \Cal C}\max_{u\in\gamma} \bigg(\int_{-\pi_p}^{\pi_p}|u'(x)|^p\,dx+ \varepsilon \int_{-\pi_p}^{\pi_p}q(x)|u(x)|^p\,dx \bigg).$$ All eigenfunctions associated with $\mu(\varepsilon)$ have precisely two zeros in the interval $(-\pi_p,\pi_p]$. Here, $\Cal C$ is a special class of curves defined on the submanifold $\Cal S$ of the Sobolev space $$W_{\text{per}}^{1,p} = \{f \in W^{1,p}(-\pi_p,\pi_p): f(-\pi_p)= f(\pi_p) \}.$$ To ensure the existence of a {\sl local} variational eigenvalue, a special choice of the potential $q$ was motivated by the works of {\it P. A.\thinspace Binding} and {\it B. P.\thinspace 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)].

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