» 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 $p, q, r\in (1, +\infty),~q<p,~r\not\in \{p, q\};$ $\rho_p, \rho_q, \rho_r$ are positive constants; $\Delta_{\alpha}$ is the usual $\alpha$ -Laplacian, i.e., $\Delta_\alpha u=\, \mbox{div} \, (|\nabla u|^{\alpha-2}\nabla u)$ ; $\nu$ is the unit outward normal to $\partial \Omega$ ; $a\in L^{\infty}(\Omega),$ $b\in L^{\infty}(\partial\Omega)$ {are given nonnegative functions satisfying} $\int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0.$ 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 $q<N$ , we prove that
there exist two positive constants $\lambda_*<\lambda^*$ such that any $\lambda\in \{0\}\cup [\lambda^*, \infty)$ is an eigenvalue of the above problem, while the set $(-\infty, 0)\cup (0, \lambda_*)$ contains no eigenvalue $\lambda$ of the problem.