"Eigenvalues of the (p, q, r)-Laplacian with a parametric boundary condition"

"Consider in a bounded domain $\Omega \subset \mathbb{R}^N$, $N\ge 2$, with smooth boundary $\partial \Omega$ the following nonlinear eigenvalue problem \begin{equation*} \left\{\begin{array}{l} -\sum_{\alpha \in \{ p,q,r\}}\rho_{\alpha}\Delta_{\alpha}u=\lambda a(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega,\\[1mm] \big(\sum_{\alpha \in \{p,q,r\}}\rho_{\alpha}\mid \nabla u\mid ^{\alpha-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) \mid u\mid ^{r-2}u ~ \mbox{ on} ~ \partial \Omega, \end{array}\right. \end{equation*} where $p, q, r\in (1, +\infty),~q0.$ Such a triple-phase problem is motivated by some models arising in mathematical physics. If $r \not\in (q, p),$ we determine a positive number $\lambda_r$ such that the set of eigenvalues of the above problem is precisely $\{ 0\} \cup (\lambda_r, +\infty )$. On the other hand, in the complementary case $r \in (q, p)$ with $r < q(N-1)/(N-q)$ if $q