Monotonicity properties for the variational Dirichlet eigenvalues of the p-Laplace operator

Let D ≥ 2 be an integer. For each open and bounded set Ω ⊂ R D and each integer k ≥ 1 we denote by R k ( Ω ) the largest number r > 0 for which there exists k disjoint open balls in Ω of radius r . Next, for each open, bounded, convex set Ω ⊂ R D with smooth boundary and each real number p ∈ ( 1 , ∞ ) we denote by { λ k ( p ; Ω ) } k ≥ 1 the sequence of eigenvalues of the p -Laplace operator subject to the homogeneous Dirichlet boundary conditions, given by the Ljusternik-Schnirelman theory. For each integer k ≥ 1 we show that there exists M k ∈ [ ( k e ) − 1 , 1 ] such that for any open, bounded, convex set Ω ⊂ R D with smooth boundary for which R k ( Ω ) is less than or equal to M k , the k-th eigenvalue of the p -Laplacian on Ω, λ k ( p ; Ω ) , is an increasing function of p on ( 1 , ∞ ) . Moreover, there exists N k ≥ M k such that for any real number s ∈ ( N k , ∞ ) ∖ { 1 } there exists an open, bounded, convex set Ω ⊂ R D with smooth boundary which has R k ( Ω ) equal to s such that λ k ( p ; Ω ) is not a monotone function of p ∈ ( 1 , ∞ ) .