\documentclass[10pt]{studiamnew} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amssymb} \sloppy \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{solution}[theorem]{Solution} \newtheorem{notation}[theorem]{Notation} \renewcommand{\theequation}{\thesection.\arabic{equation}} \numberwithin{equation}{section} \begin{document} % \setcounter{page}{1} \setcounter{firstpage}{1} \setcounter{lastpage}{4} \renewcommand{\currentvolume}{??} \renewcommand{\currentyear}{??} \renewcommand{\currentissue}{??} % \title{On eigenvalue problems governed by the $(p,q)$-Laplacian} \author{Lumini\c{t}a Barbu} \address{"Ovidius" University, \\ Faculty of Mathematics and Computer Science\\ 124 Mamaia Blvd,\\ 900527 Constan\c{t}a, \\Romania} \email{lbarbu@univ-ovidius.ro} \author{Gheorghe Moro\c{s}anu} \address{``Babe\c{s}-Bolyai'' University, \\ Faculty of Mathematics and Computer Science\\ 1 Mihail Kog\u{a}lniceanu Street,\\ 400084 Cluj-Napoca,\\ Romania} \email{morosanu@math.ubbcluj.ro} \subjclass{35J60, 35J92, 35P30} \keywords{eigenvalue problem, $(p,q)$-Laplacian, Sobolev space, Nehari manifold, Variational methods, Lagrange multipliers} \begin{abstract} This is a survey on recent results, mostly of the authors, regarding eigenvalue problems governed by the $(p,q)-$Laplacian and related open problems. \end{abstract} \maketitle \section{ Introduction}\label{se:1} Let $\Omega \subset \mathbb{R}^N,~N\geq 2$, be a bounded domain with smooth boundary $\partial\Omega$. For $\theta \in (1, \infty)$, consider in $\Omega$ the $\theta$-Laplace operator $\Delta_\theta u=\, \mbox{div}~ (\mid\nabla u\mid^{\theta-2}\nabla u)$. Obviously, $\Delta_2$ is the classic Laplacian $\Delta$. \noindent There are many applications involving such kind of operators, including the so called two phase problems. For example, the operator $\big(\Delta + c\Delta_{\theta}\big),~c>0, \ \theta \in (1, \infty)$, has applications in Born-Infeld theory for electrostatic fields (see Bonheure, Colasuonno \& Fortunato \cite{[BCF]}, Fortunato, Orsina \& Pisani \cite{FOP}). We also refer to Benci et al. \cite{[BAF]} and Benci, Fortunato \& Pisani \cite{[BFP]} for more general applications to quantum physics. Two phase equations arise also in other parts of mathematical physics as reaction diffusion equations (see Cherfils \& Il'yasov \cite{CI}) and nonlinear elasticity theory (see Marcellini \cite{Mp} and Zhikov \cite{[Z]}). In fact, the literature related to this subject is vast and daily increasing. \vskip5pt\noindent For $p,~q\in (1, \infty)$, define $\mathcal{A}_{pq}:=\Delta_p +\Delta_q,$ which is usually called $(p,q)-$Laplacian. We assumes that $p\neq q$, because for $p=q$ $\mathcal{A}_{pq} = 2\Delta_p$ and this case is not relevant for our discussion here. Notice that the operator introduced above $\big(\Delta + c\Delta_{\theta}\big)$ with $c=1$ is a $(2,\theta )-$Laplacian. The restriction to the case $c=1$ is not important. \vskip5pt In what follows we recall some facts concerning the classic eigenvalue problem for $-\Delta_p$, $p\in (1,\infty)$, under the Dirichlet boundary condition \begin{equation}\label{eq:1.1} \left\{\begin{array}{l} -\Delta_p u=\lambda \mid u\mid ^{p-2}u\ \ \mbox{in} ~ \Omega,\\[1mm] u=0~ \mbox{on} ~ \partial\Omega. \end{array}\right. \end{equation} A real number $\lambda$ is called an {\it eigenvalue} of problem \eqref{eq:1.1} if this problem admits a nontrivial weak solution, i.e. there exists $u_\lambda\in W_0^{1,p}(\Omega)\setminus \{0\}$ such that \begin{equation}\label{eq:1.2} \int_\Omega \mid \nabla u_\lambda\mid ^{p-2}\nabla u_\lambda \cdot \nabla w~dx=\lambda\int_\Omega \mid u_\lambda\mid ^{p-2} u_\lambda w~dx~\forall~w\in W_0^{1,p}(\Omega). \end{equation} The nontrivial solutions $u_\lambda$ of problem \eqref{eq:1.1} are called {\it eigenfunctions} corresponding to the eigenvalue $\lambda$, and $(\lambda , u_{\lambda} )$ are called {\it eigenpairs} of problem \eqref{eq:1.1}. A standard method to show the existence of an increasing sequence of eigenvalues for problem \eqref{eq:1.1}, \begin{equation}\label{eq:1.3} 0<\lambda_1^D<\lambda_2^D \leq \lambda_3^D \leq \cdots \rightarrow \infty, \end{equation} relies on the Ljusternik-Schnirelmann principle and on the concept of Krasnosel'sk\u{\i}i genus. There are also other methods to prove the existence of such a sequence (see Garc\'{\i}a-Azorero \& Peral \cite{GP}, Dr\'{a}bek \& Robinson \cite{DR}). It is still not known whether this sequence includes all eigenvalues of problem \eqref{eq:1.1}, except for the well-known particular case $p=2$. On the other hand, it is well-known that $-\Delta_p$ with the Dirichlet boundary condition admits a lowest positive eigenvalue $\lambda_1$ (called {\it principal eigenvalue}), which is simple, and there exists a corresponding eigenfunction which is positive in $\Omega$ (see Lindqvist \cite{L}, L\^{e} \cite{Le} and the references therein). Note also that the properties of the next lowest eigenvalue $\lambda_2$ have been investigated by Anane \& Tsouli in \cite{AT}, who proved that $\lambda_2$ has a variational characterization similar to that corresponding to the linear case $p=2.$ Similar situations can be reported in the case of Neumann, Robin or Steklov boundary conditions. \section{Eigenvalue problems governed by the $(p,q)-$Laplacian}\label{se:2} In this section we shall present some recent results on eigenvalue problems involving the $(p,q)-$Laplacian with various boundary conditions. More precisely, these results contain information regarding the corresponding eigenvalue sets. \noindent As seen below, the fact that the differential operator $\mathcal{A}_{pq}$ is \emph{non-homogeneous} (i.e., $p \neq q$) implies that the eigenvalue sets are intervals or contain intervals. \bigskip Throughout this section we will assume that $p, q\in (1, \infty),~p\neq q$, and introduce the following notations: \begin{equation}\label{eq:2.1def} \begin{split} W&:=W^{1,\max\{p,q\}}(\Omega),\\ \frac{\partial u}{\partial\nu_{pq}}&:=\big(\mid \nabla u\mid ^{p-2}+\mid \nabla u\mid ^{q-2}\big)\frac{\partial u}{\partial\nu}, \end{split} \end{equation} where $\nu$ is the outward unit normal to $\partial\Omega$. \subsection{The case of Dirichlet, Neumann, Robin or Steklov boundary conditions}\label{se:02} Let us begin with the case of the \emph{Dirichlet boundary condition}. Specifically, we consider the problem \begin{equation}\label{eq:1.4} \left\{\begin{array}{l} -\mathcal{A}_{pq} u=\lambda \mid u\mid ^{p-2}u\ \ \mbox{in} ~ \Omega,\\[1mm] u=0~ \mbox{on} ~ \partial\Omega. \end{array}\right. \end{equation} The definitions of eigenvalues, eigenfunctions and eigenpairs for problem \eqref{eq:1.4} are similar to those corresponding to problem \eqref{eq:1.1}, the only differences being the following: the left hand side of equation \eqref{eq:1.2} is replaced by $\int_\Omega \Big(\mid \nabla u_\lambda\mid ^{p-2}+\mid \nabla u_\lambda\mid ^{q-2}\Big)\nabla u_\lambda \cdot \nabla w~dx$, and the Sobolev space in which the weak solution is sought is now $W_0^{1, \max\{p, q\}}(\Omega)$. The existence of eigenvalues for this problem in the case when the right hand side of equation \eqref{eq:1.4}$_1$ is of the form $\lambda m_p(x) \mid u\mid ^{p-2}u~ \mbox{in} ~ \Omega,$ where $m_p \in L^\infty (\Omega)$ such that the Lebesgue measure of $\{x\in \Omega;~m_p(x)>0 \}$ is positive, was studied by Tanaka in \cite{T}. Using the Mountain Pass Theorem, Tanaka was able to obtain the full eigenvalue set (\cite[Theorem 1, Theorem 2]{T}). In the particular case $m_p \equiv 1$, Tanaka's result is the following: \begin{theorem} If $p, q \in (1, \infty),~p \neq q,$ then the set of eigenvalues of problem \eqref{eq:1.4} is precisely $(\lambda_1^D, \infty),$ where $\lambda_1^D$ denotes the first eigenvalue of the negative Dirichlet $p-$Laplacian, more exactly \begin{equation}\label{eq:00} {\lambda}_1^D:=\inf \Biggl\{\frac{\int_\Omega \mid \nabla u\mid^pdx}{\int_\Omega \mid u\mid^pdx},~u\in W_0^{1, p}(\Omega)\Biggr\}. \end{equation} \end{theorem} Notice that the eigenvalue set of $-\mathcal{A}_{pq}$ with Dirichlet boundary condition has been completely determined, being an interval independent of $q$. \bigskip Next, let us consider the case of a generalized \emph{Neumann boundary condition}. More precisely, consider the eigenvalue problem \begin{equation}\label{eq:2.1} \left\{\begin{array}{l} -\mathcal{A}_{pq} u=\lambda \mid u\mid ^{q-2}u\ \ \mbox{in} ~ \Omega,\\[1mm] \frac{\partial u}{\partial\nu_{pq}}=0 ~ \mbox{on} ~ \partial\Omega. \end{array}\right. \end{equation} The solution $u$ of problem \eqref{eq:2.1} is understood in a weak sense, as an element of the Sobolev space $W$ satisfying equation \eqref{eq:2.1}$_1$ in the sense of distributions and \eqref{eq:2.1}$_2$ in the sense of traces. The scalar $\lambda\in \mathbb{R}$ is an eigenvalue of problem \eqref{eq:2.1} if there exists $u_\lambda\in W\setminus \{0\}$ such that for all $w\in W$ we have \begin{equation}\label{eq:2.2} \int_\Omega \Big(\mid \nabla u_\lambda\mid ^{p-2}+\mid \nabla u_\lambda\mid ^{q-2}\Big)\nabla u_\lambda \cdot \nabla w~dx=\lambda\int_\Omega \mid u_\lambda\mid ^{q-2} u_\lambda w~dx. \end{equation} Problem \eqref{eq:2.1} was investigated by Mih\u{a}ilescu \cite[Theorem 1.1]{MMih} (for $q=2,~p\in (2, \infty$)), F\u{a}rc\u{a}\c{s}eanu, Mih\u{a}ilescu \& Stancu-Dumitru \cite[Theorem 1.1]{FMD} (for $q=2,~p\in (1, 2)$), Mih\u{a}ilescu \& Moro\c{s}anu \cite[Theorem 1.1]{MM} (for $q\in (2, \infty), p\in (1, \infty),~p\neq q$) and Barbu \& Moro\c{s}anu \cite[Theorem 1]{[BM20]} (for $q\in (1, 2),~p\in (1, \infty),~p\neq q).$ To investigate such a problem, one can use techniques based on minimization arguments, which will be briefly described in what follows. To begin with, let us choose $w=u_\lambda$ in \eqref{eq:2.2}. Clearly, we see that the eigenvalues of problem \eqref{eq:2.1} cannot be negative. It is also obvious that $\lambda_0=0$ is an eigenvalue of this problem with the corresponding eigenfunctions given by the nonzero constant functions. Now, if we assume that $\lambda>0$ is an eigenvalue of problem \eqref{eq:2.1} and choose $w\equiv 1$ in \eqref{eq:2.2} we obtain that every eigenfunction $u_{\lambda}$ corresponding to $\lambda$ necessarily belong to the set \begin{equation}\label{eq:2.4} \mathcal{C}_{Ne}:=\Big\{ u\in W;~\int_\Omega \mid u\mid ^{q-2} u ~dx=0\Big\}. \end{equation} This is a symmetric cone. Moreover, $\mathcal{C}_{Ne}$ is a weakly closed subset of $W$ and $\mathcal{C}_{Ne}\setminus \{0\}\neq \emptyset$ (see \cite[Section 2]{BM}). Next, we shall briefly describe the method we can use to solve the eigenvalue problem \eqref{eq:2.1}. For $\lambda>0$ consider the $C^1$ functional $\mathcal{J}_{\lambda}: W \rightarrow \mathbb{R},$ defined as \begin{equation}\label{eq:2.5} \mathcal{J}_\lambda(u)=\frac{1}{p}\int_\Omega \mid \nabla u\mid ^{p}~dx+\frac{1}{q}\int_\Omega\mid \nabla u\mid ^{q}~dx- \frac{\lambda}{q}\int_\Omega \mid u\mid ^{q}~dx. \end{equation} This functional is often called the \emph{energy functional} associated to problem \eqref{eq:2.1}. Clearly, $\lambda$ is an eigenvalue of problem \eqref{eq:2.1} if and only if there exists a critical point $u_\lambda\in W\setminus\{0\}$ of $\mathcal{J}_{\lambda}$, i. e. $\mathcal{J}'_{\lambda}(u_\lambda)=0$. Define \begin{equation}\label{eq:2.6} \widetilde{\lambda}^{Ne}:=\underset{w\in\mathcal{C}_{Ne}\setminus\{0\}}{\inf }~\frac{\int_\Omega\mid\nabla w\mid^q~dx}{\int_{\Omega}\mid w\mid^q~dx}. \end{equation} Since $\widetilde{\lambda}^{Ne}=\lambda^{Ne_q}_{1}$ for $q>p$ and $\widetilde{\lambda}^{Ne}\geq\lambda^{Ne_q}_{1}$ for $q0$ (we have denoted by $\lambda^{Ne_q}_{1}$ the first positive eigenvalue of the negative Neumann $q-$Laplace operator). Also, one can easily check that there is no eigenvalue of problem \eqref{eq:2.1} in the set $(-\infty, \widetilde{\lambda}^{Ne}] \setminus \{0\}.$ So, from now on we shall consider that $\lambda$ is arbitrary but fixed in the interval $(\widetilde{\lambda}^{Ne}, \infty).$ We distinguish two cases related to $p$ and $q$:\\ \textbf{Case 1:} $1\widetilde{\lambda}^{Ne},$ the functional $\mathcal{J}_\lambda$ is coercive on $\mathcal{C}_{Ne}\subset W=W^{1,p}(\Omega)$, i.e., \[ \underset{\parallel u\parallel_{W^{1,p}(\Omega)}\rightarrow\infty, u\in\mathcal{C}_{Ne}}{\lim}\mathcal{J}_\lambda(u)=\infty. \] In particular, there exists $u_{*}\in \mathcal{C}_{Ne}\setminus \{0\}$ where $\mathcal{J}_{\lambda}$ attains its minimal value over $\mathcal{{C}}_{Ne},$ \[J_{\lambda}(u_{*})= \underset{w\in\mathcal{{C}}_{Ne}\setminus \{0\}}{\inf }{\mathcal{{J}}_{\lambda} (w) }\neq 0\] (see \cite[Lemma 6]{[BM20]}).\\ \noindent \textbf{Case 2:} $10~\forall~u\in \mathcal{N}_\lambda. \end{equation*} Moreover, any possible eigenfunction corresponding to $\lambda$ belongs to $\mathcal{N}_\lambda$. In addition, since $\lambda>\widetilde{\lambda}^{Ne},$ we can easily check that $\mathcal{N}_\lambda \neq \emptyset.$ In this case we have the following result (see \cite[Case 2, Steps 1-4]{BM} and \cite[Lemma 6]{[BM20]}):\\ If $1\widetilde{\lambda}^{Ne},$ then there exists $u_{*}\in \mathcal{N}_{\lambda}$ where $\mathcal{J}_{\lambda}$ attains its minimal value over $\mathcal{{N}}_{\lambda},$ \[ m_{\lambda}:= \underset{w\in\mathcal{{N}}_{\lambda}}{\inf }{\mathcal{{J}}_{\lambda} (w) }>0.\] Using the above preliminary results and applying the Lagrange Multipliers Rule in the case $q\geq 2$ and, respectively, an approximation technique in the case $1< q < 2,$ one can show that in fact the minimizer $u_*$ of functional $\mathcal{{J}}_{\lambda}$ over $\mathcal{C}_{Ne}$ if $qp,$ is a global minimizer of $\mathcal{{J}}_{\lambda}$ over the whole $W,$ i.e. $u_*$ is an eigenfunction of problem \eqref{eq:2.1} corresponding to the eigenvalue $\lambda>\widetilde{\lambda}^{Ne}.$ Thus, we have the following important result which provides the full spectrum of the eigenvalue problem \eqref{eq:2.1}: \begin{theorem}\label{te:N} Assume that $p, q\in (1, \infty),~p\neq q.$ Then the set of eigenvalues of problem \eqref{eq:2.1} is precisely $\{0\} \cup(\widetilde{\lambda}^{Ne}, \infty)$, where $\widetilde{\lambda}^{Ne}$ is the positive constant defined by \eqref{eq:2.6}. \end{theorem} Now, consider the eigenvalue problem for the \emph{Steklov $(p,q)-$Laplacian}, namely \begin{equation}\label{eq:2.1S} \left\{\begin{array}{l} \mathcal{A}_{pq} u=0\ \ \mbox{in} ~ \Omega,\\[1mm] \frac{\partial u}{\partial\nu_{pq}}=\lambda \mid u\mid ^{q-2}u ~ \mbox{on} ~ \partial\Omega. \end{array}\right. \end{equation} Using an approach similar to that used before for the Neumann $(p,q)-$Laplacian, one can determine the full spectrum of the eigenvalue problem \eqref{eq:2.1S}. More exactly, if we denote \begin{equation}\label{eq:2.4S} \mathcal{C}_S:=\Big\{ u\in W;~\int_{\partial\Omega} \mid u_\lambda\mid ^{q-2} u_\lambda ~d\sigma=0\Big\}, \end{equation} \begin{equation}\label{eq:2.6S} \widetilde{\lambda}^S:=\underset{w\in\mathcal{C}_S\setminus\{0\}}{\inf }~\frac{\int_\Omega\mid\nabla w\mid^q~dx}{\int_{\partial\Omega}\mid w\mid^q~d\sigma}, \end{equation} we have the following result \begin{theorem}\label{te:S} Assume that $p, q\in (1, \infty),~p\neq q.$ Then the set of eigenvalues of problem \eqref{eq:2.1S} is precisely $\{0\} \cup(\widetilde{\lambda}_S, \infty)$, where $\widetilde{\lambda}_S$ is the positive constant defined by \eqref{eq:2.6S}. \end{theorem} This theorem was proved by Costea \& Moro\c{s}anu \cite[Theorem 3.1]{CM} in the case $p\in (1, \infty),~q\in [2, \infty),~p\neq q$ and later by Barbu \& Moro\c{s}anu \cite[Theorem 1]{[BM20]} in the case $p\in (1, \infty),~q\in (1, 2),~p\neq q.$ Next, we pay attention to equation \eqref{eq:2.1}$_1$ with a generalized \emph{Robin boundary condition}. More precisely, we consider the following eigenvalue problem \begin{equation}\label{eq:2.1R} \left\{\begin{array}{l} -\mathcal{A}_{pq} u= \lambda \mid u\mid ^{q-2}u\ \ \mbox{in} ~ \Omega,\\[1mm] \frac{\partial u}{\partial\nu_{pq}}+\beta \mid u\mid ^{q-2}u=0 ~ \mbox{on} ~ \partial\Omega, \end{array}\right. \end{equation} where $\beta$ is a positive constant. The eigenvalue problem \eqref{eq:2.1R} was studied by Gyulov \& Moro\c{s}anu \cite{GM}, who found an interval of eigenvalues for this problem. \noindent In order to state the main result in \cite{GM}, we define \begin{equation}\label{eq:2.6R} \begin{split} \widetilde{\lambda}^R&:=\underset{w\in W\setminus\{0\}}{\inf }~\frac{\int_\Omega\mid\nabla w\mid^q~dx+\beta\int_{\partial\Omega}\mid\nabla w\mid^q~d\sigma}{\int_{\Omega}\mid w\mid^q~dx},\\ \lambda_0&:=\beta\frac{\mid\partial\Omega\mid_{N-1}}{\mid\Omega\mid_N}, \end{split} \end{equation} where $\mid\cdot\mid_{N}$ and $\mid\cdot\mid_{N-1}$ denote the Lebesgue measures of the two sets. Obviously, the constant $\widetilde{\lambda}_R$ coincides with the first eigenvalue of the Robin $q-$Laplace operator (see L\^{e} \cite{Le}) in the case $q>p$ and is greater than or equal to that if $qN.$ The constant $\beta$ is replaced by a function $\beta\in W^{1, \infty}(\partial \Omega),~\beta \geq 0,$ $\beta \not\equiv 0$ such that \begin{equation}\label{eq:2.7} \int_\Omega \zeta~dx + \int_{\partial\Omega} \beta~d\sigma>0. \end{equation} By arguing as in \cite{GM}, the authors obtain a result similar to Theorem~\ref{te:R} (see \cite[Theorem 1]{PVV}). Finally, let us consider the Steklov like eigenvalue problem \begin{equation}\label{eq:2.1R2} \left\{\begin{array}{l} -\mathcal{A}_{pq} u+\rho_1(x) \mid u\mid ^{p-2}u+\rho_2(x) \mid u\mid ^{q-2}u=0,~ \, x\in \Omega,\\[1mm] \frac{\partial u}{\partial\nu_{pq}}+\gamma_1(x)\mid u\mid ^{p-2}u+\gamma_2(x)\mid u\mid ^{q-2}u=\lambda \mid u\mid ^{q-2}u, ~ \, x\in \partial \Omega. \end{array}\right. \end{equation} Assume that the following hypotheses are fulfilled:\\ \noindent $(h_{\rho_1\gamma_1}) \ \ \rho_1\in L^{\infty}(\Omega)$ and $\gamma_1\in L^{\infty}(\partial \Omega),$ $\rho_1,~\gamma_1$ are nonnegative functions such that \begin{equation}\label{eq:2.2R2} \int_\Omega \rho_1~dx+\int _{\partial\Omega} \gamma_1~d\sigma >0; \end{equation} \noindent $(h_{\rho_2\gamma_2}) \ \ \rho_2\in L^{\infty}(\Omega),$ $\gamma_2\in L^{\infty}(\partial \Omega)$ and $\rho_2$ is a nonnegative function. \vskip12pt\noindent It is worth pointing out that the potential function $\gamma_2$ is allowed to be sign changing. As usual, a scalar $\lambda\in \mathbb{R}$ is said to be an eigenvalue of the problem \eqref{eq:2.1R2} if there exists $u_\lambda\in W \setminus \{0\}$ such that for all $w\in W$ \begin{equation}\label{eq:2.3R2} \begin{split} \int_\Omega &\big(\mid \nabla u_\lambda\mid ^{p-2}+\mid \nabla u_\lambda\mid ^{q-2}\big)\nabla u_\lambda \cdot \nabla w~dx\\ &+\int_\Omega \big( \rho_1\mid u_\lambda\mid ^{p-2} +\rho_2\mid u_\lambda\mid ^{q-2}\big) u_\lambda w~dx\\ &+\int_{\partial\Omega} \big(\gamma_1 \mid u_\lambda\mid ^{p-2} +\gamma_2 \mid u_\lambda\mid ^{q-2}\big)u_\lambda w~d\sigma =\lambda \int_{\partial\Omega} \mid u_\lambda\mid ^{q-2} u_\lambda w~d\sigma. \end{split} \end{equation} The function $u_\lambda$ is called an eigenfunction of the problem \eqref{eq:2.1R2} (corresponding to the eigenvalue $\lambda$). Define \begin{equation}\label{eq:3.6R} \widetilde{\lambda}^{SR}:=\underset{w\in W\setminus \{0\}}{\inf }~\frac{\int_\Omega \big(\mid \nabla w\mid ^{q}+\rho_2\mid w\mid ^{q}\big)dx+\int_{\partial\Omega} \gamma_2 \mid w\mid ^{q}d\sigma}{\int_{\partial\Omega} \mid w\mid ^{q}d\sigma}. \end{equation} Problem \eqref{eq:2.1R2} was studied by Barbu \& Moro\c{s}anu \cite{BM22}. Let us recall the main result on its eigenvalue set: \begin{theorem}\label{te:R2} (\cite[Theorem 1]{BM22}) Assume that $p, q\in (1, \infty),~p\neq q$ and assumptions $(h_{\rho_i\gamma_i}), ~i=1,2$, are fulfilled. Then the set of eigenvalues of problem \eqref{eq:2.1R2} is precisely $(\widetilde{\lambda}^{SR}, \infty).$ \end{theorem} Note that if $\gamma_1 \equiv 0$ and $\gamma_2 \equiv ~\mbox{const.}~>0$, then we have a Steklov-Robin boundary condition. The arguments we have used in the mentioned paper can easily be adapted to the following eigenvalue problem \begin{equation}\label{eq:1.3R3} \left\{\begin{array}{l} -\mathcal{A}_{pq} u+\rho_1(x) \mid u\mid ^{p-2}u+\rho_2(x) \mid u\mid ^{q-2}u=\lambda \mid u\mid ^{q-2}u,~ \, x\in \Omega,\\[1mm] \frac{\partial u}{\partial\nu_{pq}}+\gamma_1(x)\mid u\mid ^{p-2}u+\gamma_2(x)\mid u\mid ^{q-2}u=0, ~ \, x\in \partial \Omega, \end{array}\right. \end{equation} under similar assumptions for the functions $\rho_i,~\gamma_i,~ i=1,2$. While in the previous works \cite{GM} and \cite{PVV} only subsets of the corresponding spectra were found, in this case the presence of the potential functions $\rho_i,~\gamma_i$ satisfying assumptions $(h_{\rho_i\gamma_i}),~i=1,2$, allows the full description of the spectrum. \subsection{The case of parametric boundary conditions}\label{se:3} Consider the following eigenvalue problem \begin{equation}\label{eq:3.1} \left\{\begin{array}{l} -\mathcal{A}_{pq} u=\lambda \alpha(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega,\\[1mm] \frac{\partial u}{\partial\nu_{pq}}=\lambda \beta(x) \mid u\mid ^{r-2}u ~ \mbox{ on} ~ \partial \Omega, \end{array}\right. \end{equation} under the following hypotheses $(h_{pqr})~~ p, q, r\in (1, \infty),~ p\neq q$;\\ $(h_{\alpha\beta}) ~~ \alpha\in L^{\infty}(\Omega)$ and $b\in L^{\infty}(\partial \Omega)$ are given nonnegative functions satisfying \begin{equation}\label{eq:3.2} \int_\Omega \alpha~dx+\int_{\partial\Omega} \beta~d\sigma >0. \end{equation} Such eigenvalue problems were discussed for the first time by Von Below \& Fran\c{c}ois \cite{VF} (see also Fran\c{c}ois \cite{F}) who considered the linear eigenvalue problem \begin{equation*} \left\{\begin{array}{l} -\Delta u=\lambda u\ \ \mbox{ in} ~ \Omega,\\[1mm] \frac{\partial u}{\partial\nu}=\lambda \beta u ~ \mbox{ on} ~ \partial \Omega. \end{array}\right. \end{equation*} They call it a \emph{dynamical eigenvalue problem} since it can be derived from the study of the heat equation with dynamical boundary conditions. Also, the motivation behind problem \eqref{eq:3.1} comes from the study of a double phase parabolic equation (see Arora \& Shmarev \cite{AS}, Huang \cite{H}, Marcellini \cite{Mp} and the references therein) under a dynamical boundary condition. The existence theory for such parabolic problems relies on the spectral theory of associated elliptic problems with the parameter $\lambda$ both in the equation and the boundary condition. The eigenvalues and eigenfunctions of problem \eqref{eq:3.1} can be defined as before. \noindent All eigenfunctions of problem \eqref{eq:3.1} belong to the set \begin{equation}\label{eq:1.5} \mathcal{C}_r:=\Big\{ u\in W;~\int_\Omega \alpha\mid u\mid ^{r-2} u ~dx+\int_{\partial\Omega} \beta \mid u\mid ^{r-2} u ~d\sigma=0\Big\}. \end{equation} In the case $r=q$, define \begin{equation}\label{eq:3.6} \widetilde{\lambda}:=\underset{w\in\mathcal{C}_q\setminus\{0\}}{\inf }~\frac{\int_\Omega\mid\nabla w\mid^q~dx}{\int_{\Omega}\alpha\mid w\mid^q~dx+ \int_{\partial\Omega}\beta\mid w\mid^q~d\sigma}. \end{equation} If $r\neq q$ we assume, without any loss of generality, that $10$ and the set of eigenvalues of problem \eqref{eq:3.1} (with $r=q$) is precisely $\{0\} \cup (\widetilde{\lambda}, \infty)$, where $\widetilde\lambda$ is the constant defined by \eqref{eq:3.6}. \end{theorem} Note that problem \eqref{eq:3.1} in the case $q=2$ and $p\in (1, \infty), ~p\neq 2,$ had been previously studied by Abreu \& Madeira\cite{AM}. In the case $r \not\in \{p, q\}$, we have the following result: \begin{theorem}\label{te:3.2}(\cite[Theorem 1.1]{BM_21CJM}, \cite[Theorem 1]{BM_DI}) Suppose that assumption $(h_{\alpha\beta})$ holds. \\ $(a)$~If either $(10, $$ instead of $\mathcal{A}_{pqr}$, and with $$ \frac{\partial u}{\partial\nu_\mathcal{B}}:=\Big(\sum\limits_{\alpha\in\{p, q, r\}} \rho_{\alpha}\mid \nabla u\mid ^{\alpha-2}\Big)\frac{\partial u}{\partial\nu},~\rho_p=1, $$ instead of $\frac{\partial u}{\partial\nu_{pqr}}$ (see \cite[Section 4]{BM_22CJM]}). Under assumption $(h_{pqr})^{\prime},$ the appropriate Sobolev space for problem \eqref{eq:4.1} is $\widetilde{W}:=W^{1,\max\{p,r\}}(\Omega).$ One can define the eigenvalues of problem \eqref{eq:4.1} as follows:\\ $\lambda\in \mathbb{R}$ is an eigenvalue of problem \eqref{eq:4.1} if there exists $u_\lambda\in \widetilde{W} \setminus \{0\}$ such that \begin{equation}\label{eq:4.3} \begin{split} \int_\Omega &\Big(\mid \nabla u_\lambda\mid ^{p-2}+\mid \nabla u_\lambda\mid ^{q-2}+\mid \nabla u_\lambda\mid ^{r-2}\Big)\nabla u_\lambda \cdot \nabla w~dx \\ &=\lambda\Big(\int_\Omega a\mid u_\lambda\mid ^{r-2} u_\lambda w~dx+\int_{\partial\Omega} b \mid u_\lambda\mid ^{r-2} u_\lambda w~d\sigma\Big)~\forall~w\in \widetilde{W}. \end{split} \end{equation} If $u_\lambda$ is an eigenfunction corresponding to a positive eigenvalue $\lambda$ then necessarily $u_\lambda$ belongs to the set \begin{equation}\label{eq:4.5} \mathcal{C}:=\Big\{ u\in \widetilde{W};~\int_\Omega \alpha\mid u\mid ^{r-2} u ~dx+\int_{\partial\Omega} \beta \mid u\mid ^{r-2} u ~d\sigma=0\Big\}. \end{equation} Let us introduce the notations \begin{equation}\label{eq:notations} \begin{split} K_\alpha(u)&:=\int_\Omega \mid \nabla u\mid ^{\alpha}dx,~\alpha\in\{p, q, r\},\\ k_r(u)&:=\int_\Omega \alpha\mid u\mid ^{r}dx+\int_{\partial\Omega} \beta \mid u\mid ^{r}d\sigma~\forall~u \in W,\\ \mathcal{Z}&:=\{v\in W;~k_r(v)=0\}. \end{split} \end{equation} \noindent Define \begin{equation}\label{eq:4.6} \Lambda_r:=\underset{v\in\mathcal{C}\setminus \mathcal{Z}}{\inf }~\frac{K_r(v)}{k_r(v)}. \end{equation} For $r\in (q, p)$ denote \begin{equation}\label{eq:4.7} \begin{split} \Lambda_*&:=\underset{v\in \mathcal{C}\setminus \mathcal{Z} }{\inf} \Biggl(\Gamma\frac{K_p(v)^{1-\gamma}K_q(v)^\gamma}{k_{r}(v)}+\frac{K_r(v)}{k_{r}(v)}\Biggr),\\ \Lambda^*&:=\underset{v\in \mathcal{C}\setminus \mathcal{Z} }{\inf} \Biggl(\Gamma\frac{r}{p^{1-\gamma}q^\gamma}\frac{K_p(v)^{1-\gamma}K_q(v)^\gamma}{k_{r}(v)}+\frac{K_r(v)}{k_{r}(v)}\Biggr),\\ \gamma&:=\frac{p-r}{p-q},~~\Gamma:=\frac{p-q}{(r-q)^{1-\gamma}(p-r)^\gamma}. \end{split} \end{equation} The main result concerning problem \eqref{eq:4.1} is the following: \begin{theorem}\label{te:pqr}(see \cite[Theorems 1.1 and 1.2]{BM_22CJM]}) Assume that $(h_{pqr}^\prime)$ and $(h_{\alpha\beta})$ above are fulfilled. If $r\not \in (q,p),$ then $\Lambda_r>0$ and the set of eigenvalues of problem \eqref{eq:4.1} is precisely $\{0\} \cup (\Lambda_r, \infty)$, where $\Lambda_r$ is the constant defined by \eqref{eq:4.6}. Otherwise, if $r\in (q,p)$, and $r < q(N-1)/(N-q)$ if $q