Monotonicity properties for the variational Dirichlet eigenvalues of the \(p\)-Laplace operator (Q2161495)

Let \(D\in\mathbb{N}\) and let \(\mathbb{P}^D\) denote the set of all open, bounded, convex sets with smooth boundary in \(\mathbb{R}^D\). This paper investigates some monotonicity properties of \(\lambda_k(p;\Omega)\) with respect to \(p\in (1,\infty)\), where \(\Omega\in \mathbb{P}^D\) and \(\{\lambda_k(p;\Omega)\}_{k\in\mathbb{N}}\) is the sequence of eigenvalues of \(-\Delta_p\) on \(W_0^{1,p}(\Omega)\) obtained by the Ljusternik-Schnirelman theory. When \(\Omega=(a,b)\subset\mathbb{R}^1\), the monotonicity of the function \(p\mapsto \lambda_k(p;(a,b))\) is deduced from Theorem 1.1 of [\textit{R. Kajikiya} et al., Electron. J. Differ. Equ. 2017, Paper No. 107, 37 p. (2017; Zbl 1370.34038)]. The case \(D\geq 2\) is more delicate, but for this case the monotonicity of the function \(p\mapsto \lambda_1(p;\Omega)\) was also discussed in [\textit{V. Bobkov} and \textit{M. Tanaka}, Calc. Var. Partial Differ. Equ. 54, No. 3, 3277--3301 (2015; Zbl 1328.35052)] and [\textit{M. Bocea} and \textit{M. Mihăilescu}, Adv. Calc. Var. 14, No. 1, 147--152 (2021; Zbl 1458.35294)]. The current work provides some monotonicity properties of higher eigenvalues with \(D\geq 2\). For \(\Omega\in\mathbb{P}^D\) and \(k\in \mathbb{N}\), define \[R_k(\Omega):=\sup\{r>0:\text{ there exists } k \text{ disjoint open balls in } \Omega \text{ of radius } r\}.\] Then, the main results are roughly stated as follows: (i) For each \(k\in\mathbb{N}\), there exists \(M_k\in [(ke)^{-1},1]\) such that for any \(\Omega\in\mathbb{P}^D\) with \(R_k(\Omega)\leq M_k\), the function \(p\mapsto \lambda_k(p;\Omega)\) is increasing on \((1,\infty)\). (ii) For each \(k\in\mathbb{N}\), there exists \(N_k\geq M_k\) such that for any real number \(s\in (N_k,\infty)\setminus\{1\}\) there exists \(\Omega\in\mathbb{P}^D\) with \(R_k(\Omega)= s\) such that the function \(p\mapsto \lambda_k(p;\Omega)\) is not monotone on \((1,\infty)\). (iii) For any ball \(B_R\subset\mathbb{R}^D\) of radius \(R>1\), the function \(p\mapsto \lambda_1(p;\Omega)\) is not monotone on \((1,\infty)\).