.

# Nonlinear elliptic eigenvalue problems in cylindrical domains becoming
unbounded in one direction

Rama Rawat<EMAIL_ADDRESS>,  Haripada Roy<EMAIL_ADDRESS>and
Prosenjit Roy<EMAIL_ADDRESS>Indian Institute of Technology Kanpur,
India

###### Abstract.

The aim of this work is to characterize the asymptotic behaviour of the first
eigenfunction of the generalised $p$-Laplace operator with mixed (Dirichlet
and Neumann) boundary conditions in cylindrical domains when the length of the
cylindrical domains tends to infinity. This generalises an earlier work of
Chipot et.al. [Asymptotics of eigenstates of elliptic problems with mixed
boundary data on domains tending to infinity. Asymptot. Anal.,
85(3-4):199–227, 2013] where the linear case $p=2$ is studied. Asymptotic
behavior of all the higher eigenvalues of the linear case and the second
eigenvalues of general case (using topological degree) for such problems is
also studied.

###### Key words and phrases:

p-Laplacian; uniform elliptcity; Poincaré inequality; Krasnoselskii’s genus

###### 2023 Mathematics Subject Classification:

35P15; 35P30; 35B38

##  1\. Introduction

The asymptotic analysis of problems set in cylindrical domains which become
unbounded in one or several directions for various partial differential
equations has been carried out by several authors since the pioneering work of
Chipot [13]. In this paper, we revisit one of these problems which is set in
cylindrical domains becoming unbounded in one direction for the generalised
p-Laplacian with mixed boundary conditions. This continues the work in [13],
which concerns the asymptotic analysis of the linear case $p=2$ with mixed
(Dirichlet-Neumann) boundary conditions. We now briefly describe the setting.

Consider $\Omega_{\ell}=(-\ell,\ell)\times\omega$, where $\ell$ is a positive
real number and $\omega$ is an open bounded set in $\mathbb{R}^{n-1}$, $n\geq
2$. Any generic point $x\in\mathbb{R}^{n}$ will be denoted by
$x=(x_{1},X_{2})$, where $x_{1}\in\mathbb{R}$ and $X_{2}\in\mathbb{R}^{n-1}$.
Then for each $x_{1}\in(-\ell,\ell)$, $\omega$ is the cross-section of
$\Omega_{\ell}$ orthogonal to $x_{1}$-direction. Let $A$ be a $n\times n$
symmetric matrix of the form

$A(X_{2}):=\begin{pmatrix}a_{11}(X_{2})&A_{12}(X_{2})\\\
A_{12}^{t}(X_{2})&A_{22}(X_{2})\end{pmatrix},\ X_{2}\in\omega,$

where $A_{22}$ is an $(n-1)\times(n-1)$ matrix and assume that $A$ is
uniformly bounded and uniformly elliptic on $\mathbb{R}\times\omega$ ( precise
definitions are given in next section). For $p\geq 2$ consider the following
eigenvalue problem on $\Omega_{\ell}$ for the generalised p-Laplacian with
Dirichlet boundary condition

$\begin{cases}-\textrm{div}\left(|A(X_{2})\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}A(X_{2})\nabla
u_{\ell}\right)=\lambda_{D}(\Omega_{\ell})|u_{\ell}|^{p-2}u_{\ell}\
\textrm{in}\ \Omega_{\ell},\\\ u_{\ell}=0\ \textrm{on}\
\partial\Omega_{\ell},\end{cases}$ (1.1)

and the corresponding eigenvalue problem

$\begin{cases}-\textrm{div}\left(|A_{22}(X_{2})\nabla_{X_{2}}u\cdot\nabla_{X_{2}}u|^{\frac{p-2}{2}}A_{22}(X_{2})\nabla_{X_{2}}u\right)=\mu(\omega)|u|^{p-2}u\
\textrm{in}\ \omega,\\\ u=0\ \textrm{on}\ \partial\omega,\end{cases}$ (1.2)

defined on the cross section $\omega$, where
$\nabla_{X_{2}}=(\frac{\partial}{\partial x_{2}},\frac{\partial}{\partial
x_{3}},\cdots,\frac{\partial}{\partial x_{n}})$.

For the linear case $p=2,$ it is well known that each of the problems (1.1)
and (1.2) admit an infinite set of positive discrete eigenvalues tending to
infinity (see [22], Theorem 3.6.1). Let $\lambda_{D}^{k}(\Omega_{\ell})$ and
$\mu^{k}(\omega)$ denote the $k$-th eigenvalues of (1.1) and (1.2)
respectively in this case. In [12], Theorem 2.4, it was shown that as $\ell$
goes to infinity, the $k$-th eigenvalue $\lambda_{D}^{k}(\Omega_{\ell})$ of
(1.1) converges to the first eigenvalue $\mu^{1}(\omega)$ of (1.2) with the
optimal convergence rate of $\frac{1}{\ell^{2}}$. It was further proved that
under appropriate scaling, the eigenfunctions corresponding to the first
eigenvalues of (1.1) converge to the eigenfunction corresponding to the first
eigenvalue of (1.2) in the appropriate function space (see [12], Theorem 3.4).
In general for $p\geq 2$, the first eigenvalue of (1.1) and (1.2) can be
obtained by appropriate minimization problems (described later in (2.6) and
(2.7)), we continue to denote them by $\lambda_{D}^{1}(\Omega_{\ell})$ and
$\mu^{1}(\omega)$ respectively, the value of $p$ will be clear from the
context. In [21], the following theorem was proved.

######  Theorem 1.

_[21]_ Let $p\geq 2$. Then $\exists$ a constant $C$ depends only on $A$,
$\omega$ and $p$, such that

$\mu^{1}(\omega)\leq\lambda_{D}^{1}(\Omega_{\ell})\leq\mu^{1}(\omega)+\frac{C}{\ell}.$
(1.3)

We now describe the problem which will be our main focus in this paper.
Consider the following eigenvalue problem for the generalised p-Laplacian with
mixed (Dirichlet and Neumann) boundary conditions:

$\begin{cases}-\textrm{div}\left(|A(X_{2})\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}A(X_{2})\nabla
u_{\ell}\right)=\lambda_{M}(\Omega_{\ell})|u_{\ell}|^{p-2}u_{\ell}\
\textrm{in}\ \Omega_{\ell},\\\ u_{\ell}=0\ \textrm{on}\
\gamma_{\ell}:=(-\ell,\ell)\times\partial\omega,\\\ \big{(}A(X_{2})\nabla
u_{\ell}\big{)}\cdot\nu=0\ \textrm{on}\
\Gamma_{\ell}:=\\{-\ell,\ell\\}\times\omega,\end{cases}$ (1.4)

where $\nu$ denotes the outward unit normal to $\Gamma_{\ell}$.

In the linear case $p=2$, where the set of all eigenvalues of (1.4) is well
understood (see [26]) a similar asymptotic analysis of the first and second
eigenvalue was carried out in [13]. It was shown there that under an
additional condition, the first eigenvalue $\lambda_{M}^{1}(\Omega_{\ell})$ of
(1.4) converges to a limit as $\ell$ goes to infinity but in this case the
limit is strictly smaller than the first eigenvalue $\mu^{1}(\omega)$ of
(1.2). In particular there is a gap in the limiting behaviour of
$\lambda_{M}^{1}(\Omega_{\ell})$ and $\mu^{1}(\omega).$ This is in sharp
contrast to the limiting behavior of eigenvalue problem with Dirichlet
boundary condition (1.1) as discussed before. Further it was proved that under
a symmetry condition on the matrix $A$, the second eigenvalue
$\lambda_{M}^{2}(\Omega_{\ell})$ of (1.4) has the same limit as that of the
first eigenvalue $\lambda_{M}^{1}(\Omega_{\ell})$ of (1.4) as $\ell$ goes to
infinity.

This was followed up in [21] where it was shown that the gap phenomenon
continues to hold for the limit superior of the first eigenvalues of (1.4) for
the nonlinear case $p>2$ under a similar additional condition. To summarize
these results:

######  Theorem 2.

_[13, 21]_ Let $p\geq 2$ and let $W$ be the eigenfunction corresponding to the
eigenvalue $\mu^{1}(\omega)$ such that $\int_{\omega}|W|^{p}=1.$ Then
_(i)_ For $p=2$,

$\lim_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell})<\mu^{1}(\omega),$
(1.5)

provided $A_{12}\cdot\nabla_{X_{2}}W\not\equiv 0$ a.e. on $\omega$, otherwise
$\lambda_{M}^{1}(\Omega_{\ell})=\mu^{1}(\omega)$ for all $\ell>0$.
Furthermore, if $A$ satisfies the symmetry (S) (see Definition 1 later), then

$\lim_{\ell\rightarrow\infty}\lambda_{M}^{2}(\Omega_{\ell})=\lim_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell}).$

_(ii)_ For $p>2$, one has

$\limsup_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell})<\mu^{1}(\omega)$
(1.6)

provided $A_{12}\cdot\nabla_{X_{2}}W\not\equiv 0$ a.e. on $\omega$, otherwise
$\lambda_{M}^{1}(\Omega_{\ell})=\mu^{1}(\omega)$ for all $\ell>0$.

A detailed discussion on the condition $A_{12}\cdot\nabla_{X_{2}}W\not\equiv
0$ is done in [13] [see, Remark 4.2].

On the way to the proof of the gap phenomenon in Theorem 2 (i), several other
interesting results were obtained in [13]. For example, behavior of the first
normalised eigenfunction of (1.4) near the end (on sets like
$\Omega_{\ell}\setminus\Omega_{\ell-1}$), in the middle of the cylinder (sets
like $\Omega_{\ell/2}$) were established. Also the value
$\Lambda:=\limsup_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell})$ is
identified with an appropriate problem on the semi-infinite cylinders.

Similar questions remained unanswered for the case $p>2$, which we wish to
take up in this paper. Let $u_{\ell}$ be the normalised eigenfunction
corresponding to the first eigenvalue $\lambda_{M}^{1}(\Omega_{\ell})$ of
(1.4). In our first result, we analyse the behavior of $u_{\ell}$ as $\ell$
goes to infinity and show that as in the case $p=2$ these eigenfunctions decay
to zero in the middle of the cylinder and their mass concentrates near the
base of the cylinder. Throughout this paper, $[x]$ will denote the integer
part of $x\in\mathbb{R}.$

######  Theorem 1.1.

Assume that $p\geq 2$ and $A_{12}\cdot\nabla_{X_{2}}W\not\equiv 0$ a.e in
$\omega$. Then $\exists$ constants $\alpha\in(0,1)$ and $C_{1},C_{2}\geq 0$
depend only on $A$, $\omega$ and $p$ such that for $\ell>\ell_{0}$ we have for
every $0<r\leq\ell-1$,

$\int_{\Omega_{r}}|\nabla u_{\ell}|^{p}\leq C_{1}\alpha^{[\ell-r]},$ (1.7)

and

$\int_{\Omega_{r}}|u_{\ell}|^{p}\leq C_{2}\alpha^{[\ell-r]}.$ (1.8)

The following identity

$A\nabla(uv)\cdot\nabla(uv)-u^{2}\left(A\nabla v\cdot\nabla v\right)=A\nabla
u\cdot\nabla(uv^{2})$

plays a crucial role in proofs for the case $p=2$. For general $p$, we do not
have this kind of identity. To overcome this difficulty we use the uniformly
ellipticity condition on the matrix $A$ and the Poincaré inequality (see (2.3)
and (2.1) in Section 2). Other complication arises because of the fact, that
for $p=2$ the spaces involved are Hilbert spaces which is not the case for
general $p$. Our next result is crucial for understanding the value
$\Lambda:=\limsup_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell})$.

Consider the semi-infinite cylinders

$\Omega_{\infty}^{+}=(0,\infty)\times\omega\ \mathrm{and}\
\Omega_{\infty}^{-}=(-\infty,0)\times\omega$

with the boundaries

$\gamma_{\infty}^{+}=(0,\infty)\times\partial\omega,\ \
\gamma_{\infty}^{-}=(-\infty,0)\times\partial\omega,$

and the quantities

$\nu_{\infty}^{\pm}=\inf_{0\neq u\in
V(\Omega_{\infty}^{\pm})}\frac{\int_{\Omega_{\infty}^{\pm}}|A\nabla
u\cdot\nabla u|^{\frac{p}{2}}}{\int_{\Omega_{\infty}^{\pm}}|u|^{p}},$ (1.9)

where the spaces $V(\Omega_{\infty}^{\pm})$ are defined by

$V(\Omega_{\infty}^{\pm})=\\{u\in W^{1,p}(\Omega_{\infty}^{\pm}):u=0\
\textrm{on}\ \gamma_{\infty}^{\pm}\\}.$

Detailed description of these spaces is given in Section 2.

######  Theorem 1.2.

Assume that $p\geq 2.$ Then

$\lim_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell})=\min\left\\{\nu_{\infty}^{+},\nu_{\infty}^{-}\right\\}.$
(1.10)

We further provide a condition for which the gaps hold between
$\nu_{\infty}^{\pm}$ and $\mu^{1}(\omega)$ and another condition for which the
gaps fail. This was known for $p=2$ from [13].

######  Theorem 1.3.

_(i)_ Assume that $A_{12}\cdot\nabla_{X_{2}}W\not\equiv 0$. If

$\int_{\omega}|A_{22}\nabla_{X_{2}}W\cdot\nabla_{X_{2}}W|^{\frac{p-2}{2}}(A_{12}\cdot\nabla_{X_{2}}W)W\geq
0,$ (1.11)

then $\nu_{\infty}^{+}<\mu_{1}(\omega)$.

_(ii)_ If

$A_{12}\cdot\nabla_{X_{2}}W\leq 0\ \mathrm{a.e}\ \mathrm{in}\ \omega,$ (1.12)

then $\nu_{\infty}^{+}=\mu_{1}(\omega)$. Moreover in this case there is no
minimizer realizing $\nu_{\infty}^{+}$.

We conclude the paper with some results about the convergence of the other
eigenvalues of problem (1.4), these results may be compared with the ones in
[13], [12]. Unlike the linear case $p=2,$ there is no specific description of
the second and higher eigenvalues available for $p>2$ for the eigenvalue
problem (1.4). Although there is a description available of what we call
Krasnoselskii eigenvalues, through the concept of Krasnoselskii’s genus. Let
$\\{\beta_{k}(\Omega_{\ell})\\}$ denote these Krasnoselskii eigenvalues of
(1.4) which we instead use to analyze the asymptotic behaviour. We discuss
these eigenvalues briefly in the next section. For $p=2$ therefore, we have
two descriptions of eigenvalues of the problem (1.4), one through classical
methods using Hilbert space structure and the other one using Krasnoselskii’s
genus, and these two descriptions coincide in this case (see Lemma 6.1). For
$p>2$ it is not clear if Krasnoselskii eigenvalues exhaust the set of all
eigenvalues of (1.4). However the first Krasnoselskii eigenvalue
$\beta_{1}(\Omega_{\ell})$ coincides with the usual first eigenvalue
$\lambda_{M}^{1}(\Omega_{\ell})$ (see Corollary 2).

In the literature there are other descriptions of eigenvalues of (1.4)
available, interested readers may look at [20] for example.

######  Definition 1.

We shall say that $A$ satisfies the symmetry (S) if the set $\omega$ is
symmetric w.r.t the origin, i.e. $\omega=-\omega$ and $A(-X_{2})=A(X_{2})$.

The following theorem is an extension of Theorem 7.1 in [13] for higher
eigenvalues. Only the asymptotic behavior of second eigenvalue was considered
there.

######  Theorem 1.4.

Let $A$ satisfies the symmetry (S). Then for $p=2$ and for any $k\geq 2$

$\lim_{\ell\rightarrow\infty}\lambda_{M}^{k}(\Omega_{\ell})=\lim_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell}).$
(1.13)

######  Theorem 1.5.

Let $A$ satisfies the symmetry (S). Then for $p>2,$

$\lim_{\ell\rightarrow\infty}\beta_{2}(\Omega_{\ell})=\lim_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell}).$
(1.14)

For other related work on asymptotic behavior of problems defined on
cylindrical domains interested readers may look at [5, 7, 10, 14, 16] for
study of elliptic problems, [3, 4, 15] for parabolic problems. The behavior of
Stokes equations has been studied in [2, 9], whereas purely variational
problems have been studied in [6, 11, 24]. Recently, problems involving
nonlocal operators have also been considered, e.g., see [1, 17, 18, 19, 28].
For the first time in such literature semilinear problems are studied in [8].

This paper is organized as follow: In Section 2 we briefly review the notion
of Krasnoselskii’s genus and related facts, underlying function spaces,
uniform Poincaré inequality, properties of the matrix $A$ and other relevant
known results. Section 3 contains the proof of Theorem 1.1. In Section 4 we
prove Theorem 1.2, which identifies the limit of
$\lambda_{M}^{1}(\Omega_{\ell})$ as $\ell$ goes to infinity. In Section 5 gap
phenomenon on semi infinite cylinder (Theorem 1.3) is studied. In Section 6 we
conclude the paper with the convergence results (Theorem 1.4 and 1.5)
concerning the $k$-th eigenvalue for $p=2$ and the second Krasnoselskii’s
eigenvalue for $p>2$ of the problem (1.4).

##  2\. Preliminaries

Let $V$ be a Banach space. Define the collection

$\mathcal{A}=\left\\{A\subset V:A\ \mathrm{is}\ \mathrm{closed}\ \mathrm{and}\
A=-A\right\\}$

of all closed and symmetric subsets of $V$. Let
$\gamma:\mathcal{A}\rightarrow\mathbb{N}\cup\\{0,\infty\\}$ be defined by

$\gamma(A)=\begin{cases}\mathrm{inf}\left\\{m:\exists\,h:A\rightarrow\mathbb{R}^{m}\setminus\\{0\\},h\
\mathrm{is}\ \mathrm{continuous}\ \mathrm{and}\ h(-u)=-h(u)\right\\}\\\
\infty,\ \mathrm{if}\ \\{\cdots\\}=\phi,\ \mathrm{in}\
\mathrm{particular\,if}\ A\ni 0,\end{cases}$

for $A\neq\phi$ and $\gamma(\phi)=0$. $\gamma(A)$ is called the
Krasnoselskii’s genus of the set $A$. We summarise some important properties
of the Krasnoselskii’s genus in the following :

######  Proposition 2.1.

_(i)_ $\gamma$ is a monotone sub-additive continuous map and $\gamma(A)$ is
finite for any compact symmetric subset of $V\setminus\\{0\\}$.
_(ii)_ Let $\Omega$ be any bounded symmetric neighborhood of the origin in
$\mathbb{R}^{m}$. Then $\gamma(\partial\Omega)=m$.
_(iii)_ Let $A$ be a compact symmetric subset of a Hilbert space $V$ with the
inner product $(\cdot,\cdot)_{V}$ and let $\gamma(A)=m<\infty$. Then $A$
contains at least $m$ mutually orthogonal vectors, i.e. $\exists
u_{1},u_{2},\cdots,u_{m}\in A$ such that $(u_{j},u_{k})_{V}=0$ for $j\neq k$.

Suppose $E$ is an even functional on $V$ i.e., $E(-u)=E(u)$, $u\in V$. A
sequence $(u_{m})$ is called a Palais-Smale sequence for $E$, if
$|E(u_{m})|\leq c$ uniformly on $m$, while $\lVert DE(u_{m})\rVert\rightarrow
0$ as $m\rightarrow\infty$, where $DE(u)$ is the Fréchet derivative of the
functional $E$ at $u$. $E$ is said to satisfy the Palais-Smale (P.S) condition
if any Palais-Smale sequence has a strongly convergent subsequence. The
following theorem assures the existence of the critical points and
characterizes the critical values of $E$.

######  Theorem 2.2.

_(i)_ Suppose $E$ is an even functional on a complete symmetric
$C^{1,1}$-manifold $M\subset V\setminus\\{0\\}$ of some Banach space $V$. Also
suppose $E$ satisfies (P.S) condition and is bounded from below on $M$. Let
$\hat{\gamma}(M)=\mathrm{sup}\left\\{\gamma(K):K\subset M\ \mathrm{compact}\
\mathrm{and}\ \mathrm{symmetric}\right\\}.$ Then the functional $E$ possesses
at least $\hat{\gamma}(M)\leq\infty$ pairs of critical points.
_(ii)_ Let $\mathcal{A}$ be defined as above. For any $k\leq\hat{\gamma}(M)$
consider the family

$\mathcal{F}_{k}=\left\\{A\in\mathcal{A}:A\subset M,\gamma(A)\geq k\right\\}.$

If

$\beta_{k}=\underset{A\in\mathcal{F}_{k}}{\mathrm{inf}}\,\underset{u\in
A}{\mathrm{sup}}\,E(u)$

is finite, then $\beta_{k}$ is a critical value of $E$.

We refer to [27] (Chapter II, Section 5) for more details about the
Krasnoselskii’s genus and the proofs of Proposition 2.1 and Theorem 2.2.

For an open set $\Omega$ in $\mathbb{R}^{n}$ and $1<p<\infty$,
$W^{1,p}(\Omega)$ is the usual Sobolev space defined by

$W^{1,p}(\Omega)=\\{v\in L^{p}(\Omega):\partial_{x_{i}}v\in
L^{p}(\Omega),i=1,2\cdots,n\\},$

with the norm

$\lVert v\rVert_{W^{1,p}(\Omega)}=\Big{\\{}\int_{\Omega}(|v|^{p}+|\nabla
v|^{p})\Big{\\}}^{1/p}.$

As before, let $\Omega_{\ell}=(-\ell,\ell)\times\omega$ on $\mathbb{R}^{n}$,
where $\ell>0$ and $\omega$ is an open bounded set in $\mathbb{R}^{n-1}$ and
we define the space

$W_{0}^{1,p}(\Omega_{\ell})=\\{v\in W^{1,p}(\Omega_{\ell}):v=0\ \textrm{on}\
\partial\Omega_{\ell}\\}.$

We denote the boundary of $\Omega_{\ell}$ by $\gamma_{\ell}\cup\Gamma_{\ell}$,
where $\gamma_{\ell}=(-\ell,\ell)\times\partial\omega$ and
$\Gamma_{\ell}=\\{-\ell,\ell\\}\times\omega$. To analyze the eigenvalues and
eigenfunctions of the problem (1.4), we define a suitable space

$V(\Omega_{\ell})=\\{v\in W^{1,p}(\Omega_{\ell}):v=0\ \textrm{on}\
\gamma_{\ell}\\}.$

The boundary conditions are considered in the sense of traces. The following
version of Poincaré inequality is used in this work.

######  Lemma 2.1.

_(Poincaré inequality)_ Let $\Omega=U\times\omega$, where $U$ is any open
subset of $\mathbb{R}$, $\omega$ is an open bounded set in $\mathbb{R}^{n-1}$.
Let $u\in V(\Omega)$, where $V(\Omega)=\\{u\in W^{1,p}(\Omega):u=0$ on
$U\times\partial\omega\\}$. Then we have

$\lVert u\rVert_{L^{p}(\Omega)}\leq C_{p}\lVert\nabla u\rVert_{L^{p}(\Omega)}$
(2.1)

where $C_{p}>0$ is a constant depends only on $p$ and $\omega$.

Using Lemma 2.1 it can be easily verified that $\lVert
v\rVert^{p}_{p,\Omega_{\ell}}=\int_{\Omega_{\ell}}|\nabla v|^{p}$ is a norm on
$V(\Omega_{\ell})$ and $V(\Omega_{\ell})$ is a Banach space with respect to
this norm.

We consider the $n\times n$ symmetric matrix $A$ of the form

$A(X_{2})=\begin{pmatrix}a_{11}(X_{2})&A_{12}(X_{2})\\\
A_{12}^{t}(X_{2})&A_{22}(X_{2})\end{pmatrix}$

where $a_{11}(X_{2})\in\mathbb{R},$ $A_{12}(X_{2})$ is a $1\times(n-1)$ matrix
and $A_{22}(X_{2})$ is a $(n-1)\times(n-1)$ matrix. The components of
$A(X_{2})$ are assumed to be bounded measurable functions on $\omega$ and we
assume that $A$ is an uniformly bounded and uniformly positive definite matrix
i.e., there exists positve constants $C$ and $\lambda$ such that

$\lVert A(X_{2})\rVert\leq C\ \textrm{a.e}\ X_{2}\in\omega,$ (2.2)

and

$A(X_{2})\xi\cdot\xi\geq\lambda|\xi|^{2}\ \textrm{a.e}\ X_{2}\in\omega,\ \
\forall\xi\in\mathbb{R}^{n}.$ (2.3)

Here $\lVert\cdot\rVert$ denotes the norm of matrices, $|.|$ denotes the
euclidean norm and $X\cdot Y$ denotes the usual inner product of two vectors
$X$ and $Y$ in $\mathbb{R}^{n}$.

To obtain the Krasnoselskii eigenvalues of (1.4), we define a closed symmetric
$C^{1,1}$-manifold $M_{\ell}$ of the Banach space $V(\Omega_{\ell})$ as

$M_{\ell}=\left\\{u\in
V(\Omega_{\ell}):\int_{\Omega_{\ell}}|u|^{p}=1\right\\},$

and an even functional $E_{\ell}$ on $M_{\ell}$ by

$E_{\ell}(u)=\int_{\Omega_{\ell}}|A\nabla u\cdot\nabla u|^{\frac{p}{2}}.$

Then $E_{\ell}$ satisfies (P.S) condition, as shown in [20]. For
$k\in\mathbb{N}$, let $\\{v_{1},v_{2},\cdots,v_{k}\\}$ be a linearly
independent set in $M_{\ell}$ and let
$K=\left\\{v=\sum_{i=1}^{k}\alpha_{i}v_{i}:\alpha_{i}\in\mathbb{R},1\leq i\leq
k\ \mathrm{and}\ \int_{\Omega_{\ell}}|v|^{p}=1\right\\}$. Then $K$ is a
compact and symmetric subset of a $k-$dimensional subspace of $L^{p}$ and is
homeomorphic to $S^{k-1},$ the unit sphere in $\mathbb{R}^{k}.$ Therefore by
Proposition 2.1 (ii), $\gamma(K)=k$ and hence $\hat{\gamma}(M_{\ell})=\infty.$
Theorem 2.2 (i) assures that there are infinitely many critical values in both
the cases $p=2$ and $p>2$. For $p=2$, these critical values of (1.4) coincide
with the usual eigenvalues (as discussed earlier) and therefore we continue to
denote them by $\lambda_{M}^{k}(\Omega_{\ell})$ and for $p>2$, we call these
critical values as Krasnoselskii eigenvalues of the problem (1.4) and denote
them by $\beta_{k}(\Omega_{\ell})$.

Multiplying the equations (1.1) and (1.2) by functions from the corresponding
underlying spaces and then integrating by parts, we formulate the
corresponding weak version of the Dirichlet problems (1.1) and (1.2)
respectively as follows.

$\begin{cases}u\in W_{0}^{1,p}(\Omega_{\ell}),\\\ \int_{\Omega_{\ell}}|A\nabla
u\cdot\nabla u|^{\frac{p-2}{2}}A\nabla u\cdot\nabla
v=\lambda_{D}(\Omega_{\ell})\int_{\Omega_{\ell}}|u|^{p-2}uv,\ \forall v\in
W_{0}^{1,p}(\Omega_{\ell}).\end{cases}$ (2.4) $\begin{cases}u\in
W_{0}^{1,p}(\omega),\\\
\int_{\omega}|A_{22}\nabla_{X_{2}}u\cdot\nabla_{X_{2}}u|^{\frac{p-2}{2}}A\nabla_{X_{2}}u\cdot\nabla_{X_{2}}v=\mu(\omega)\int_{\omega}|u|^{p-2}uv,\
\forall v\in W_{0}^{1,p}(\omega).\end{cases}$ (2.5)

The following characterization of the first eigenvalues
$\lambda_{D}^{1}(\Omega_{\ell})$ of (2.4) and $\mu^{1}(\omega)$ of (2.5) is
well known:

$\lambda_{D}^{1}(\Omega_{\ell})=\inf_{u\in W_{0}^{1,p}(\Omega_{\ell}),u\neq
0}\frac{\int_{\Omega_{\ell}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}}{\int_{\Omega_{\ell}}|u|^{p}}.$ (2.6)
$\mu^{1}(\omega)=\inf_{u\in W_{0}^{1,p}(\omega),u\neq
0}\frac{\int_{\omega}|A_{22}\nabla_{X_{2}}u\cdot\nabla_{X_{2}}u|^{\frac{p}{2}}}{\int_{\omega}|u|^{p}}=\frac{\int_{\omega}|A_{22}\nabla_{X_{2}}W\cdot\nabla_{X_{2}}W|^{\frac{p}{2}}}{\int_{\omega}|W|^{p}}.$
(2.7)

It is also a known fact (see [23, 25]) that both
$\lambda_{D}^{1}(\Omega_{\ell})$ and $\mu^{1}(\omega)$ are simple and the
corresponding eigenfunctions do not change sign. Here $W$ is the first
normalised non-negative eigenfunction of (2.5), considered earlier in Theorem
2.

The corresponding weak form of the mixed boundary value problem (1.4) is the
following:

$\begin{cases}u\in V(\Omega_{\ell}),\\\ \int_{\Omega_{\ell}}|A\nabla
u\cdot\nabla u|^{\frac{p-2}{2}}A\nabla u\cdot\nabla
v=\lambda_{M}(\Omega_{\ell})\int_{\Omega_{\ell}}|u|^{p-2}uv,\ \forall v\in
V(\Omega_{\ell})\end{cases}$ (2.8)

and the first eigenvalue is given in terms of the Rayleigh quotient as

$\lambda_{M}^{1}(\Omega_{\ell})=\inf_{u\in V(\Omega_{\ell}),u\neq
0}\frac{\int_{\Omega_{\ell}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}}{\int_{\Omega_{\ell}}|u|^{p}}$ (2.9)

In this case as well, the eigenvalue $\lambda_{M}^{1}(\Omega_{\ell})$ is
simple and the corresponding eigenfunction $u_{\ell}$ have constant sign.

##  3\. Main result on the asymptotic behavior of the first eigenfunction

###### Proof of Theorem 1.1.

Let $0<\ell^{\prime}\leq\ell-1$. Define the function
$\rho_{\ell^{\prime}}=\rho_{\ell^{\prime}}(x_{1})$ by

$\rho_{\ell^{\prime}}(x_{1})=\begin{cases}1&\text{if
$|x_{1}|\leq\ell^{\prime},$}\\\ \ell^{\prime}+1-|x_{1}|&\text{if
$|x_{1}|\in(\ell^{\prime},\ell^{\prime}+1),$}\\\ 0&\text{if
$|x_{1}|\geq\ell^{\prime}+1.$}\end{cases}$

Clearly, $\rho_{\ell^{\prime}}u_{\ell}\in W_{0}^{1,p}(\Omega_{\ell})$. From
(2.6) and using the fact that $A$ is symmetric, we get

$\lambda_{D}^{1}(\Omega_{\ell})\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}\leq\int_{\Omega_{\ell}}|A\nabla(\rho_{\ell^{\prime}}u_{\ell})\cdot\nabla(\rho_{\ell^{\prime}}u_{\ell})|^{\frac{p}{2}}\\\
=\int_{\Omega_{\ell}}|\rho_{\ell^{\prime}}^{2}A\nabla u_{\ell}\cdot\nabla
u_{\ell}+2\rho_{\ell^{\prime}}u_{\ell}A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}+u_{\ell}^{2}A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}\\\
=\int_{\Omega_{\ell}}|\rho_{\ell^{\prime}}^{2}A\nabla u_{\ell}\cdot\nabla
u_{\ell}+A\nabla(\rho_{\ell^{\prime}}u_{\ell}^{2})\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}.$

On R.H.S we use the inequality $(a+b)^{q}\leq a^{q}+q2^{q-1}(b^{q}+a^{q-1}b)$
for $a,b\geq 0$, $q\geq 1$, which gives

$\lambda_{D}^{1}(\Omega_{\ell})\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}\leq\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla u_{\ell}|^{\frac{p}{2}}\rho_{\ell^{\prime}}^{p}\\\
+\frac{p}{2}C_{0}\int_{\Omega_{\ell}}\left(|A\nabla(\rho_{\ell^{\prime}}u_{\ell}^{2})\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}+|A\nabla
u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla(\rho_{\ell^{\prime}}u_{\ell}^{2})\cdot\nabla\rho_{\ell^{\prime}}|\right),$
(3.1)

where $C_{0}=2^{\frac{p-2}{2}}$. We estimate the integrals on R.H.S
separately.

$\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p}{2}}\rho_{\ell^{\prime}}^{p}\\\
=\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}\left(\rho_{\ell^{\prime}}^{p}A\nabla
u_{\ell}\cdot\nabla u_{\ell}+p\rho_{\ell^{\prime}}^{p-1}u_{\ell}A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}-p\rho_{\ell^{\prime}}^{p-1}u_{\ell}A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}\right)\\\
=\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}A\nabla
u_{\ell}\cdot\nabla(\rho_{\ell^{\prime}}^{p}u_{\ell})-p\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p-1}u_{\ell}|A\nabla
u_{\ell}\cdot\nabla u_{\ell}|^{\frac{p-2}{2}}A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}$

Using the fact that $u_{\ell}$ is the first eigenfunction of (2.8) and taking
modulus in the last integral, we get

$\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p}{2}}\rho_{\ell^{\prime}}^{p}\leq\lambda_{M}^{1}(\Omega_{\ell})\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}+p\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p-1}|A\nabla
u_{\ell}\cdot\nabla u_{\ell}|^{\frac{p-2}{2}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}||u_{\ell}|.$ (3.2)

Now we estimate the second integral on R.H.S of (3.1).

$\int_{\Omega_{\ell}}|A\nabla(\rho_{\ell^{\prime}}u_{\ell}^{2})\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}=\int_{\Omega_{\ell}}|2\rho_{\ell^{\prime}}u_{\ell}A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}+u_{\ell}^{2}A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}\\\
\leq 2^{\frac{p}{2}}C_{0}\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}\rho_{\ell^{\prime}}^{\frac{p}{2}}|u_{\ell}|^{\frac{p}{2}}+C_{0}\int_{\Omega_{\ell}}|A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}|u_{\ell}|^{p}.$
(3.3)

Finally the last integral on R.H.S of (3.1) gives

$\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla(\rho_{\ell^{\prime}}u_{\ell}^{2})\cdot\nabla\rho_{\ell^{\prime}}|\\\
=\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|2\rho_{\ell^{\prime}}u_{\ell}A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}+u_{\ell}^{2}A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|\\\
\leq 2\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}||u_{\ell}|+\int_{\Omega_{\ell}}u_{\ell}^{2}|A\nabla
u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|.$
(3.4)

Combining (3.1), (3.2), (3.3) and (3.4) and using
$0\leq\rho_{\ell^{\prime}}\leq 1$ we obtain

$\left(\lambda_{D}^{1}(\Omega_{\ell})-\lambda_{M}^{1}(\Omega_{\ell})\right)\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}\leq
p\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}||u_{\ell}|\,+\\\
pC_{0}^{3}\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}|u_{\ell}|^{\frac{p}{2}}\,+\frac{p}{2}C_{0}^{2}\int_{\Omega_{\ell}}|A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}|u_{\ell}|^{p}\,+\\\
pC_{0}\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}||u_{\ell}|\,+\frac{p}{2}C_{0}\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|u_{\ell}^{2}$
(3.5) $=p(C_{1}I_{1}+C_{2}I_{2}+C_{3}I_{3}+C_{4}I_{4}),$

where

$I_{1}=\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}||u_{\ell}|,$
$I_{2}=\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}|u_{\ell}|^{\frac{p}{2}},$
$I_{3}=\int_{\Omega_{\ell}}|A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}|u_{\ell}|^{p},$
$I_{4}=\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p-2}{2}}|A\nabla\rho_{\ell^{\prime}}\cdot\nabla\rho_{\ell^{\prime}}|u_{\ell}^{2}$

and $C_{1}=(1+C_{0})$, $C_{2}=C_{0}^{3}$, $C_{3}=\frac{1}{2}C_{0}^{2}$ and
$C_{4}=\frac{1}{2}C_{0}$.

Let
$D_{\ell^{\prime}}=\Omega_{\ell^{\prime}+1}\setminus\Omega_{\ell^{\prime}}$.
Note that $|\nabla\rho_{\ell^{\prime}}|=\chi_{D_{\ell^{\prime}}}$. We now
estimate the integrals $I_{1}$, $I_{2}$, $I_{3}$ and $I_{4}$ respectively.
Estimate for $I_{1}$: Using Cauchy-Schwarz inequality, we obtain

$I_{1}\leq\int_{\Omega_{\ell}}|A\nabla u_{\ell}|^{\frac{p}{2}}|\nabla
u_{\ell}|^{\frac{p-2}{2}}|\nabla\rho_{\ell^{\prime}}||u_{\ell}|\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p-1}|u_{\ell}|.$

By the Hölder’s inequality, and then by the Poincaré inequality (2.1), we
obtain

$I_{1}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}\left(\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}\right)^{\frac{p-1}{p}}\left(\int_{D_{\ell^{\prime}}}|u_{\ell}|^{p}\right)^{\frac{1}{p}}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}C_{p}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}.$ (3.6)

Estimate for $I_{2}$: Cauchy-Schwarz inequality gives

$I_{2}\leq\int_{\Omega_{\ell}}|A\nabla
u_{\ell}|^{\frac{p}{2}}|\nabla\rho_{\ell^{\prime}}|^{\frac{p}{2}}|u_{\ell}|^{\frac{p}{2}}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{\frac{p}{2}}|u_{\ell}|^{\frac{p}{2}}.$

Using (2.2) and then the Hölder’s inequality and the Poincaré inequality
(2.1), we obtain

$I_{2}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}\left(\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}\right)^{\frac{1}{2}}\left(\int_{D_{\ell^{\prime}}}|u_{\ell}|^{p}\right)^{\frac{1}{2}}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}C_{p}^{\frac{p}{2}}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}.$ (3.7)

Estimate for $I_{3}$: Using the Poincaré inequality (2.1), we estimate the the
integral

$I_{3}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}\int_{D_{\ell^{\prime}}}|u_{\ell}|^{p}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}C_{p}^{p}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}.$ (3.8)

Estimate for $I_{4}$: Similarly we estimate the integral $I_{4}$ as follows.

$I_{4}\leq\lVert A\rVert_{\infty}^{\frac{p}{2}}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p-2}|u_{\ell}|^{2}.$

Again we use here Hölder’s inequality, and then the Poincaré inequality (2.1)
to obtain

$I_{4}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}\left(\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}\right)^{\frac{p-2}{p}}\left(\int_{D_{\ell^{\prime}}}|u_{\ell}|^{p}\right)^{\frac{2}{p}}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}C_{p}^{2}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}.$ (3.9)

Substituting (3.6), (3.7), (3.8) and (3.9) in (3.5), we get

$\left(\lambda_{D}^{1}(\Omega_{\ell})-\lambda_{M}^{1}(\Omega_{\ell})\right)\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}\leq
p\lVert
A\rVert_{\infty}^{\frac{p}{2}}C^{\prime}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}$ (3.10)

where
$C^{\prime}=\left(C_{1}C_{p}+C_{2}C_{p}^{\frac{p}{2}}+C_{3}C_{p}^{p}+C_{4}C_{p}^{2}\right)$.
Notice that, (1.6) together with (1.3) gives a $\beta^{\prime}>0$ such that
for $\ell>\ell_{0}$ we have
$\lambda_{D}^{1}(\Omega_{\ell})-\lambda_{M}^{1}(\Omega_{\ell})>\beta^{\prime}$
and (3.10) gives

$\beta^{\prime}\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}\leq
p\lVert
A\rVert_{\infty}^{\frac{p}{2}}C^{\prime}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}.$ (3.11)

Since $\rho_{\ell^{\prime}}^{p}u_{\ell}\in V(\Omega_{\ell})$, from (2.8) we
get

$\lambda_{M}^{1}(\Omega_{\ell})\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}=\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla u_{\ell}|^{\frac{p-2}{2}}A\nabla
u_{\ell}\cdot\nabla(\rho_{\ell^{\prime}}^{p}u_{\ell})\\\
=\int_{\Omega_{\ell}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p}{2}}\rho_{\ell^{\prime}}^{p}+p\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p-1}u_{\ell}|A\nabla
u_{\ell}\cdot\nabla u_{\ell}|^{\frac{p-2}{2}}A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}.$

Using the inequality $a-|b|\leq a+b$ for $a,b\in\mathbb{R}$, we obtain

$\lambda_{M}^{1}(\Omega_{\ell})\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}\geq\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p}{2}}\rho_{\ell^{\prime}}^{p}-p\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla u_{\ell}|^{\frac{p-2}{2}}|A\nabla
u_{\ell}\cdot\nabla\rho_{\ell^{\prime}}|\rho_{\ell^{\prime}}^{p-1}|u_{\ell}|.$

Using (2.3) and $\rho_{\ell^{\prime}}=1$ in $\Omega_{\ell^{\prime}}$ in the
first integral, $0\leq\rho_{\ell^{\prime}}\leq 1$ and
$|\nabla\rho_{\ell^{\prime}}|=\chi_{D_{\ell^{\prime}}}$ and then the Hölder’s
inequality in the second integral of the R.H.S, we get

$\lambda_{M}^{1}(\Omega_{\ell})\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}\geq\lambda^{\frac{p}{2}}\int_{\Omega_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}-p\lVert
A\rVert_{\infty}^{\frac{p}{2}}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p-1}|u_{\ell}|\\\
\geq\lambda^{\frac{p}{2}}\int_{\Omega_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}-p\lVert
A\rVert_{\infty}^{\frac{p}{2}}\left(\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}\right)^{\frac{p-1}{p}}\left(\int_{D_{\ell^{\prime}}}|u_{\ell}|^{p}\right)^{\frac{1}{p}}.$

Finally the Poincaré inequality (2.1) on the last integral gives

$\lambda_{M}^{1}(\Omega_{\ell})\int_{\Omega_{\ell}}\rho_{\ell^{\prime}}^{p}|u_{\ell}|^{p}\geq\lambda^{\frac{p}{2}}\int_{\Omega_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}-p\lVert
A\rVert_{\infty}^{\frac{p}{2}}C_{p}\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}.$ (3.12)

Combining (3.12) with (3.11), we get

$\beta^{\prime}\lambda^{\frac{p}{2}}\int_{\Omega_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}\leq p\lVert
A\rVert_{\infty}^{\frac{p}{2}}\left(\lambda_{M}^{1}(\Omega_{\ell})C^{\prime}+\beta^{\prime}C_{p}\right)\int_{D_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p},$

which gives

$\beta\int_{\Omega_{\ell^{\prime}}}|\nabla u_{\ell}|^{p}\leq
C\int_{D_{\ell^{\prime}}}|\nabla u_{\ell}|^{p},$ (3.13)

where $\beta=\beta^{\prime}\lambda^{\frac{p}{2}}$, and $C=p\lVert
A\rVert_{\infty}^{\frac{p}{2}}(\mu^{1}(\omega)C^{\prime}+\beta^{\prime}C_{p})$.
Here we use the fact that $\lambda_{M}^{1}(\Omega_{\ell})\leq\mu^{1}(\omega)$,
which follows from (2.9) by using $u(x)=W(X_{2})\in V(\Omega_{\ell})$ as a
test function. Therefore, from (3.13) we deduce that

$(\beta+C)\int_{\Omega_{\ell^{\prime}}}|\nabla u_{\ell}|^{p}\leq
C\int_{\Omega_{\ell^{\prime}+1}}|\nabla u_{\ell}|^{p},$

and we have

$\int_{\Omega_{\ell^{\prime}}}|\nabla
u_{\ell}|^{p}\leq\alpha\int_{\Omega_{\ell^{\prime}+1}}|\nabla u_{\ell}|^{p},$

where $\alpha=\frac{C}{\beta+C}<1$. Applying this procedure successively for
$\ell^{\prime}=r,r+1,\cdots,r+[\ell-r]-1$, we get

$\int_{\Omega_{r}}|\nabla
u_{\ell}|^{p}\leq\alpha^{[\ell-r]}\int_{\Omega_{\ell}}|\nabla
u_{\ell}|^{p}\leq\frac{\alpha^{[\ell-r]}}{\lambda^{\frac{p}{2}}}\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p}{2}}=\frac{\lambda_{M}^{1}(\Omega_{\ell})}{\lambda^{\frac{p}{2}}}\alpha^{[\ell-r]}\leq\frac{\mu^{1}(\omega)}{\lambda^{\frac{p}{2}}}\alpha^{[\ell-r]},$
(3.14)

which proves (1.7). To prove (1.8), we use the Poincaré inequality (2.1) and
(3.14) and we get

$\int_{\Omega_{r}}|u_{\ell}|^{p}\leq C_{p}^{p}\int_{\Omega_{r}}|\nabla
u_{\ell}|^{p}\leq\frac{\mu^{1}(\omega)C_{p}^{p}}{\lambda^{\frac{p}{2}}}\alpha^{[\ell-r]}.$
(3.15)

(3.14) and (3.15) together completes the proof. ∎

An immediate corollary of this theorem is the following, which gives the
concentration of the masses near the ends of the cylinder.

######  Corollary 1.

If $A_{12}\cdot\nabla_{X_{2}}W\not\equiv 0$ a.e. in $\omega$, then for large
$\ell$ and for any $0<r\leq\ell-1$, we have

$\int_{\Omega_{\ell}\setminus\Omega_{r}}|\nabla u_{\ell}|^{p}\geq
1-C_{1}\alpha^{[\ell-r]}\quad
and\quad\int_{\Omega_{\ell}\setminus\Omega_{r}}|u_{\ell}|^{p}\geq
1-C_{2}\alpha^{[\ell-r]}.$ (3.16)

##  4\.  Connection with the problem on semi-infinite cylinder

Although, unlike the first Dirichlet eigenvalues it is not known whether the
function $\ell\rightarrow\lambda_{M}^{1}(\Omega_{\ell})$ is monotone or not,
we show that the limit of $\lambda_{M}^{1}(\Omega_{\ell})$ exists as $\ell$
goes to infinity (Theorem 1.2). To identify the limit we introduce the
variational problems (1.9) on semi-infinite cylinders $\Omega_{\infty}^{\pm}$.
In next lemma, we find the possible range of the quantities
$\nu_{\infty}^{\pm}$ defined in (1.9) and then (in Lemma 4.2) we identify
$\nu_{\infty}^{\pm}$ with the limits of specific minimization problems
(defined later in (4.4)) defined on the half-cylinders $\Omega_{\ell}^{\pm}$,
which we will use to prove Theorem 1.2.

######  Remark 1.

If $A$ has the symmetry (S), then it is easy to check that
$\nu_{\infty}^{+}=\nu_{\infty}^{-}$, since $v(-x_{1},-X_{2})\in
V(\Omega_{\infty}^{-})$ for any $v(x_{1},X_{2})\in V(\Omega_{\infty}^{+})$ and
vice versa, and clearly both give the same value in (1.9). Otherwise, we use
$\nu_{\infty}^{-}=\tilde{\nu}_{\infty}^{+}$, where $\tilde{\nu}_{\infty}^{+}$
is defined similarly as $\nu_{\infty}^{+}$, but in place of $A(X_{2})$ we put

$\tilde{A}(X_{2})=\begin{pmatrix}a_{11}(X_{2})&-A_{12}(X_{2})\\\
-A_{12}^{t}(X_{2})&A_{22}(X_{2})\end{pmatrix}.$

######  Lemma 4.1.

We have

$\lambda C_{p}^{p}\leq\nu_{\infty}^{\pm}\leq\mu^{1}(\omega).$ (4.1)

where $\lambda$ is the constant given in (2.3) and $C_{p}$ is the constant of
Poincaré inequality (2.1).

###### Proof.

Since $\tilde{A}_{22}(X_{2})=A_{22}(X_{2})$ as shown in Remark 1, and
$\mu^{1}(\omega)$ depends only on $A_{22}(X_{2})$, it is enough to prove (4.1)
for $\nu_{\infty}^{+}$. We get the lower bound $\nu_{\infty}^{+}\geq\lambda
C_{p}^{p}$ by using the uniform ellipticity (2.3) of $A$ and the Poincaré
inequality (2.1). To show the upper bound
$\nu_{\infty}^{+}\leq\mu^{1}(\omega)$, we first fix $\epsilon>0$. For
$x=(x_{1},X_{2})\in\Omega_{\infty}^{+}$ define the function $v_{\epsilon}$ in
$V(\Omega_{\infty}^{+})$ as

$v_{\epsilon}(x)=e^{-\epsilon x_{1}}W(X_{2}).$

We have

$\int_{\Omega_{\infty}^{+}}|A\nabla v_{\epsilon}\cdot\nabla
v_{\epsilon}|^{\frac{p}{2}}=\int_{\Omega_{\infty}^{+}}\left|e^{-2\epsilon
x_{1}}\left(\epsilon^{2}a_{11}W^{2}-2\epsilon(A_{12}\cdot\nabla_{X_{2}}W)W+A_{22}\nabla_{X_{2}}W\cdot\nabla_{X_{2}}W\right)\right|^{\frac{p}{2}}\\\
\leq\left(\int_{0}^{\infty}e^{-p\epsilon
x_{1}}\right)\int_{\omega}\left(|A_{22}\nabla_{X_{2}}W\cdot\nabla_{X_{2}}W|+\epsilon|\epsilon
a_{11}W^{2}-2(A_{12}\cdot\nabla_{X_{2}}W)W|\right)^{\frac{p}{2}}.$

Splitting the integral on $\omega$ by using the inequality $(a+b)^{q}\leq
a^{q}+q2^{q-1}(a^{q-1}b+b^{q})$ for $a,b\geq 0$, $q\geq 1$, and then using
(2.7) and the fact that $W$ is normalised we get

$\int_{\Omega_{\infty}^{+}}|A\nabla v_{\epsilon}\cdot\nabla
v_{\epsilon}|^{\frac{p}{2}}\leq\left(\int_{0}^{\infty}e^{-p\epsilon
x_{1}}\right)\Big{[}\mu^{1}(\omega)+\epsilon
C\int_{\omega}|A_{22}\nabla_{X_{2}}W\cdot\nabla_{X_{2}}W|^{\frac{p-2}{2}}\\\
\times|\epsilon
a_{11}W^{2}-2(A_{12}\cdot\nabla_{X_{2}}W)W|+\epsilon^{\frac{p}{2}}C\int_{\omega}|\epsilon
a_{11}W^{2}-2(A_{12}\cdot\nabla_{X_{2}}W)W|^{\frac{p}{2}}\Big{]}.$ (4.2)

As

$\int_{\Omega_{\infty}^{+}}|v_{\epsilon}|^{p}=\int_{0}^{\infty}e^{-p\epsilon
x_{1}},$ (4.3)

substituting (4.2), (4.3) in (1.9), we get

$\nu_{\infty}^{+}\leq\mu^{1}(\omega)+\epsilon
C\int_{\omega}|A_{22}\nabla_{X_{2}}W_{1}\cdot\nabla_{X_{2}}W_{1}|^{\frac{p-2}{2}}|a_{11}\epsilon
W_{1}^{2}-2(A_{12}\cdot\nabla_{X_{2}}W_{1})W_{1}|\\\
+\epsilon^{\frac{p}{2}}C\int_{\omega}|a_{11}\epsilon
W_{1}^{2}-2(A_{12}\cdot\nabla_{X_{2}}W_{1})W_{1}|^{\frac{p}{2}}.$

Passing to the limit $\epsilon\rightarrow 0$ we get the result. ∎

We now consider the minimization problems

$\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{\pm})=\inf\left\\{\int_{\Omega_{\ell}^{\pm}}|A\nabla
u\cdot\nabla u|^{\frac{p}{2}}:u\in
W^{1,p}(\Omega_{\ell}^{\pm}),\int_{\Omega_{\ell}^{\pm}}|u|^{p}=1,u=0\
\textrm{on}\ \gamma_{\ell}^{\pm}\cup\Gamma_{\ell}^{\pm}\right\\}.$ (4.4)

in the half cylinders

$\Omega_{\ell}^{+}=(0,\ell)\times\omega\ \mathrm{and}\
\Omega_{\ell}^{-}=(-\ell,0)\times\omega,$ (4.5)

where the boundaries are defined by

$\Gamma_{\ell}^{\pm}=\\{\pm\ell\\}\times\omega,\
\gamma_{\ell}^{+}=(0,\ell)\times\partial\omega\ \textrm{and}\
\gamma_{\ell}^{-}=(-\ell,0)\times\partial\omega.$ (4.6)

In the following lemma, we identify the values $\nu_{\infty}^{\pm}$ with the
limits (as $\ell$ goes to infinity) of the quantities defined in (4.4).

######  Remark 2.

Since for any fixed $\ell>0$ the domains are bounded, it is a well known fact
that the infimum in (4.4) is attained. We denote by $\tilde{u}_{\ell}^{\pm}$,
the positive normalized minimizers corresponding to
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{\pm})$.

######  Lemma 4.2.

We have

$\nu_{\infty}^{\pm}=\lim_{\ell\rightarrow\infty}\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{\pm}).$
(4.7)

###### Proof.

It is enough to prove (4.7) only for
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})$, proof for
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{-})$ is similar. If
$\ell_{1}<\ell_{2}$, then any admissible function in (4.4) for
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell_{1}}^{+})$ can be extended to an
admissible function for $\tilde{\lambda}_{M}^{1}(\Omega_{\ell_{2}}^{+})$ by
setting it zero on $\Omega_{\ell_{2}}^{+}\setminus\Omega_{\ell_{1}}^{+}$,
which shows that
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell_{1}}^{+})\geq\tilde{\lambda}_{M}^{1}(\Omega_{\ell_{2}}^{+}).$
Therefore, monotonicity of the function
$\ell\mapsto\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})$ implies that
$\lim_{\ell\rightarrow\infty}\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})$
exists.
A similar argument shows that
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})\geq\nu_{\infty}^{+}$, $\forall$
$\ell>0$.

The following space

$V_{s}(\Omega_{\infty}^{+})=\left\\{u\in C^{\infty}(\Omega_{\infty}^{+})\cap
V(\Omega_{\infty}^{+}):\exists M=M(u)>0\ \textrm{such that}\ u=0\ \textrm{on}\
(M,\infty)\times\omega\right\\}$ (4.8)

is clearly dense in $V(\Omega_{\infty}^{+})$. Let $\tilde{u}=\frac{u}{\lVert
u\rVert_{L^{p}}}$, where $u\in V(\Omega_{\infty}^{+})\setminus\\{0\\}$. Then
$\exists\\{v_{n}\\}\in V_{s}(\Omega_{\infty}^{+})\setminus\\{0\\}$ with
supp$(v_{n})\subset V(\Omega_{\ell_{n}}^{+})$ such that
$v_{n}\rightarrow\tilde{u}$ in $V(\Omega_{\infty}^{+})$ as
$n\rightarrow\infty$. In particular $\\{v_{n}\\}$ is bounded in
$V(\Omega_{\infty}^{+})$ and $\lVert
v_{n}\rVert_{L^{p}(\Omega_{\infty}^{+})}\rightarrow 1$.

Define $\tilde{v_{n}}=\frac{v_{n}}{\lVert v_{n}\rVert_{L^{p}}}$ and let
$\epsilon>0$. Then for all $n\geq n_{0}$,

$\lVert\tilde{u}-\tilde{v_{n}}\rVert_{W^{1,p}(\Omega_{\infty}^{+})}\leq\lVert\tilde{u}-v_{n}\rVert_{W^{1,p}(\Omega_{\infty}^{+})}+\lVert
v_{n}-\frac{v_{n}}{\lVert
v_{n}\rVert_{L^{p}}}\rVert_{W^{1,p}(\Omega_{\infty}^{+})}\\\
=\lVert\tilde{u}-v_{n}\rVert_{W^{1,p}(\Omega_{\infty}^{+})}+\left|1-\frac{1}{\lVert
v_{n}\rVert_{L^{p}}}\right|\lVert
v_{n}\rVert_{W^{1,p}(\Omega_{\infty}^{+})}<\epsilon.$ (4.9)

Using the inequality $|a^{p}-b^{p}|\leq
2p|a-b|\mathrm{max}\left\\{a^{p-1},b^{p-1}\right\\}$ for $a,b\geq 0$ and
$p\geq 1$ and then Holder’s inequality we obtain

$I_{n}=\left|\int_{\Omega_{\infty}^{+}}\left(|A\nabla\tilde{u}\cdot\nabla\tilde{u}|^{\frac{p}{2}}-|A\nabla\tilde{v_{n}}\cdot\nabla\tilde{v_{n}}|^{\frac{p}{2}}\right)\right|\\\
\leq
p\int_{\Omega_{\infty}^{+}}|A\nabla\tilde{u}\cdot\nabla\tilde{u}-A\nabla\tilde{v_{n}}\cdot\nabla\tilde{v_{n}}|\mathrm{max}\left\\{|A\nabla\tilde{u}\cdot\nabla\tilde{u}|^{\frac{p-2}{2}},|A\nabla\tilde{v_{n}}\cdot\nabla\tilde{v_{n}}|^{\frac{p-2}{2}}\right\\}\\\
\leq
p\left(\int_{\Omega_{\infty}^{+}}|A\nabla\tilde{u}\cdot\nabla\tilde{u}-A\nabla\tilde{v_{n}}\cdot\nabla\tilde{v_{n}}|^{\frac{p}{2}}\right)^{\frac{2}{p}}\left(\int_{\Omega_{\infty}^{+}}\mathrm{max}\left\\{|A\nabla\tilde{u}\cdot\nabla\tilde{u}|^{\frac{p}{2}},|A\nabla\tilde{v_{n}}\cdot\nabla\tilde{v_{n}}|^{\frac{p}{2}}\right\\}\right)^{\frac{p-2}{p}}$
(4.10)

The second integral in R.H.S of (4.10) is clearly bounded. Using the identity
$A\nabla u\cdot\nabla u-A\nabla v\cdot\nabla
v=A\nabla(u-v)\cdot\nabla(u-v)+2A\nabla(u-v)\cdot\nabla v$ in (4.10), we get

$I_{n}\leq
pC_{1}\left(\int_{\Omega_{\infty}^{+}}|A\nabla(\tilde{u}-\tilde{v_{n}})\cdot\nabla(\tilde{u}-\tilde{v_{n}})+2A\nabla(\tilde{u}-\tilde{v_{n}})\cdot\nabla\tilde{v_{n}}|^{\frac{p}{2}}\right)^{\frac{2}{p}}\\\
\leq
pC_{1}\left(2^{\frac{p-2}{2}}\int_{\Omega_{\infty}^{+}}|A\nabla(\tilde{u}-\tilde{v_{n}})\cdot\nabla(\tilde{u}-\tilde{v_{n}})|^{\frac{p}{2}}+2^{p-1}\int_{\Omega_{\infty}^{+}}|A\nabla(\tilde{u}-\tilde{v_{n}})\cdot\nabla\tilde{v_{n}}|^{\frac{p}{2}}\right)^{\frac{2}{p}}$
(4.11)

Using (4.9), we estimate the first integral of (4.11) as

$\int_{\Omega_{\infty}^{+}}|A\nabla(\tilde{u}-\tilde{v_{n}})\cdot\nabla(\tilde{u}-\tilde{v_{n}})|^{\frac{p}{2}}\leq\lVert
A\rVert_{\infty}^{\frac{p}{2}}\epsilon^{p}.$ (4.12)

Hölder’s inequality together with (4.9) gives

$\int_{\Omega_{\infty}^{+}}|A\nabla(\tilde{u}-\tilde{v_{n}})\cdot\nabla\tilde{v_{n}}|^{\frac{p}{2}}\leq
C_{2}\lVert A\rVert_{\infty}^{\frac{p}{2}}\epsilon^{\frac{p}{2}}.$ (4.13)

Combining (4.12), (4.13) with (4.11) we get for $n\geq n_{0}$,

$I_{n}=\left|\frac{\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}}{\int_{\Omega_{\infty}^{+}}|u|^{p}}-\frac{\int_{\Omega_{\infty}^{+}}|A\nabla
v_{n}\cdot\nabla
v_{n}|^{\frac{p}{2}}}{\int_{\Omega_{\infty}^{+}}|v_{n}|^{p}}\right|\leq
C\epsilon.$ (4.14)

This proves (4.7).

∎

###### Proof of Theorem 1.2.

(i) First we show that

$\limsup_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell})\leq\min\left\\{\nu_{\infty}^{+},\nu_{\infty}^{-}\right\\}.$
(4.15)

Without loss of generality we may assume that
$\nu_{\infty}^{+}=\min\\{\nu_{\infty}^{+},\nu_{\infty}^{-}\\}.$ For any
$\ell>0$ the function $\tilde{u}_{\ell}^{+}(x_{1}-\frac{\ell}{2},X_{2})\in
V(\Omega_{\ell/2})$, where $\tilde{u}_{\ell}^{+}(x_{1},X_{2})$ is the
minimizer for $\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})$ (see Remark 2), is
a suitable test function in (2.9) and we have
$\lambda_{M}^{1}(\Omega_{\ell/2})\leq\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})$.
On the other hand, for any $\epsilon>0$, Lemma 4.2 gives an
$\ell_{\epsilon}>0$ such that
$\tilde{\lambda}_{M}^{1}(\Omega_{2\ell_{\epsilon}}^{+})<\nu_{\infty}^{+}+\epsilon$.
Monotonicity of $\ell\mapsto\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})$ gives
$\sup_{\ell\geq\ell_{\epsilon}}\lambda_{M}^{1}(\Omega_{\ell})\leq\nu_{\infty}^{+}+\epsilon$
and (4.15) holds.

(ii) For the reverse inequality first we consider the case $A_{12}\cdot\nabla
W\not\equiv 0$ a.e. in $\omega$. Let $\rho(x_{1})$ be the function defined by

$\rho(x_{1})=\begin{cases}0&\text{if $x_{1}\leq-1,$}\\\ 1+x_{1}&\text{if
$x_{1}\in(-1,0),$}\\\ 1&\text{if $x_{1}\geq 0.$}\end{cases}$

Define $w_{\ell+1}(x_{1},X_{2})=\rho(x_{1}+\ell)u_{\ell}(x_{1}+\ell,X_{2})$ on
$\Omega_{\ell+1}^{-}$, where $u_{\ell}$ is the minimizer in (2.9). It is easy
to see that $w_{\ell+1}$ is a suitable test function for defining
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell+1}^{-})$ in (4.4). Note that
$|\nabla\rho|=1$ in $(-1,0)\times\omega$. Hence

$\int_{\Omega_{\ell+1}^{-}}|A\nabla w_{\ell+1}\cdot\nabla
w_{\ell+1}|^{\frac{p}{2}}\leq\int_{\Omega_{\ell}^{+}}|A\nabla
u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p}{2}}+\int_{(-1,0)\times\omega}|A\nabla(\rho
u_{\ell})\cdot\nabla(\rho u_{\ell})|^{\frac{p}{2}}\\\
\leq\int_{\Omega_{\ell}^{+}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p}{2}}+2^{p-1}\lVert
A\rVert_{\infty}^{\frac{p}{2}}\int_{(-1,0)\times\omega}\left(|u_{\ell}|^{p}+|\nabla
u_{\ell}|^{p}\right).$ (4.16)

Substituting (1.7), (1.8) in (4.16), we get a constant $C>0$ such that

$\int_{\Omega_{\ell+1}^{-}}|A\nabla w_{\ell+1}\cdot\nabla
w_{\ell+1}|^{\frac{p}{2}}\leq\int_{\Omega_{\ell}^{+}}|A\nabla
u_{\ell}\cdot\nabla u_{\ell}|^{\frac{p}{2}}+C\alpha^{[\ell-1]}.$ (4.17)

We denote by $N_{\ell}^{\pm}$ and $D_{\ell}^{\pm}$ the quantities

$N_{\ell}^{\pm}=\int_{\Omega_{\ell}^{\pm}}|A\nabla u_{\ell}\cdot\nabla
u_{\ell}|^{\frac{p}{2}},\quad
D_{\ell}^{\pm}=\int_{\Omega_{\ell}^{\pm}}|u_{\ell}|^{p}.$

Clearly

$N_{\ell}^{+}+N_{\ell}^{-}=\lambda_{M}^{1}(\Omega_{\ell})\quad and\quad
D_{\ell}^{+}+D_{\ell}^{-}=1.$ (4.18)

By the same computation for $\Omega_{\ell+1}^{+}$ and then using (4.18) we get

$\tilde{\lambda}_{M}^{1}(\Omega_{\ell+1}^{-})\leq\frac{N_{\ell}^{+}+C\alpha^{[\ell-1]}}{D_{\ell}^{+}}\
\mathrm{and}\
\tilde{\lambda}_{M}^{1}(\Omega_{\ell+1}^{+})\leq\frac{N_{\ell}^{-}+C\alpha^{[\ell-1]}}{D_{\ell}^{-}}.$
(4.19)

By (4.19) and (4.18), we have

$\min\\{\tilde{\lambda}_{M}^{1}(\Omega_{\ell+1}^{-}),\tilde{\lambda}_{M}^{1}(\Omega_{\ell+1}^{+})\\}\leq
D_{\ell}^{+}\tilde{\lambda}_{M}^{1}(\Omega_{\ell+1}^{-})+D_{\ell}^{-}\tilde{\lambda}_{M}^{1}(\Omega_{\ell+1}^{+})\leq\lambda_{M}^{1}(\Omega_{\ell})+2C\alpha^{[\ell-1]}.$
(4.20)

Passing to the limit as $\ell\rightarrow\infty$ on both sides, and using Lemma
4.2 it follows that

$\min\\{\nu_{\infty}^{+},\nu_{\infty}^{-}\\}\leq\liminf_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell}),$
(4.21)

which together with (4.15) proves the theorem in this case.

(iii) We now consider the case $A_{12}\cdot\nabla W\equiv 0$. In this case it
is already known from Theorem 2 that
$\lambda_{M}^{1}(\Omega_{\ell})=\mu^{1}(\omega)$ for all $\ell>0$. On the
other hand we see in Theorem 1.3 (ii) that
$\nu_{\infty}^{\pm}=\mu^{1}(\omega)$ in this case. This completes the proof. ∎

In the next proposition we show that the first eigenfunction for the problem
(1.4) on $\Omega_{\ell}$ decays to $0$ on that semi-infinite cylinder which
corresponds to the strictly smaller of the two values $\nu_{\infty}^{+}$ and
$\nu_{\infty}^{-}.$

######  Proposition 4.1.

If $\nu_{\infty}^{+}<\nu_{\infty}^{-}$, then
$\lim_{\ell\rightarrow\infty}\int_{\Omega_{\ell}^{+}}|\nabla
u_{\ell}|^{p}+|u_{\ell}|^{p}=0$.

###### Proof.

Taking the limit $\ell\rightarrow\infty$ in (4.20) and using (4.7) and (1.10),
we obtain

$\left(\limsup_{\ell\rightarrow\infty}D_{\ell}^{+}\right)\nu_{\infty}^{-}+\left(1-\limsup_{\ell\rightarrow\infty}D_{\ell}^{+}\right)\nu_{\infty}^{+}\leq\nu_{\infty}^{+}$

which gives

$\limsup_{\ell\rightarrow\infty}D_{\ell}^{+}(\nu_{\infty}^{-}-\nu_{\infty}^{+})\leq
0,$

and we must have $\limsup_{\ell\rightarrow\infty}D_{\ell}^{+}=0$. Again by
(4.19) we have,

$\frac{N_{\ell}^{+}}{D_{\ell}^{-}}+\tilde{\lambda}_{M}^{1}(\Omega_{\ell+1}^{+})-\frac{C\alpha^{[\ell-1]}}{D_{\ell}^{-}}\leq\frac{N_{\ell}^{+}+N_{\ell}^{-}}{D_{\ell}^{-}}\\\
=\frac{N_{\ell}^{+}+N_{\ell}^{-}}{D_{\ell}^{+}+D_{\ell}^{-}}+\frac{(N_{\ell}^{+}+N_{\ell}^{-})D_{\ell}^{+}}{(D_{\ell}^{+}+D_{\ell}^{-})D_{\ell}^{-}}=\lambda_{M}^{1}(\Omega_{\ell})+\frac{D_{\ell}^{+}}{D_{\ell}^{-}}(N_{\ell}^{+}+N_{\ell}^{-}).$
(4.22)

Since $\lim_{\ell\rightarrow\infty}D_{\ell}^{-}=1$, passing to the limit
$\ell\rightarrow\infty$ in (4.22) and using (4.7) and (1.10), we get
$\lim_{\ell\rightarrow\infty}N_{\ell}^{+}=0$. Finally
$\lim_{\ell\rightarrow\infty}\int_{\Omega_{\ell}^{+}}|\nabla u_{\ell}|^{p}=0$
follows using (2.3). ∎

##  5\. Gap phenomenon on semi-infinite cylinder

In this section we revisit the problem (1.9) on the semi-infinite cylinders.
We first investigate when the infimum in (1.9) is attained and follow it up
with Theorem 1.3 where we investigate when the gap between
$\nu_{\infty}^{\pm}$ and $\mu^{1}(\omega)$ holds and when does not hold.

######  Proposition 5.1.

If

$\nu_{\infty}^{+}<\mu^{1}(\omega),$ (5.1)

then $\nu_{\infty}^{+}$ is attained. The minimizer $\tilde{u}^{+}$ in (1.9) is
unique upto multiplication by a constant, has constant sign and satisfies

$\begin{cases}-\mathrm{div}(|A\nabla\tilde{u}^{+}\cdot\nabla\tilde{u}^{+}|^{\frac{p-2}{2}}A\nabla\tilde{u}^{+})=\nu_{\infty}^{+}|\tilde{u}^{+}|^{p-2}\tilde{u}^{+}\
\mathrm{in}\ \Omega_{\infty}^{+},\\\ \tilde{u}^{+}=0\ \mathrm{on}\
\gamma_{\infty}^{+},\\\ (A\nabla\tilde{u}^{+})\cdot\nu=0\ \mathrm{on}\
\\{0\\}\times\omega.\end{cases}$ (5.2)

###### Proof.

For any $\ell>0$ the function $\tilde{u}_{\ell}^{+}$ (defined in Remark 2) can
be extended by zero to a function in $W^{1,p}(\Omega_{\infty}^{+})$. Since the
sequence $\\{\tilde{u}_{\ell}^{+}\\}$ is bounded in
$W^{1,p}(\Omega_{\infty}^{+})$, there is a subsequence
$\\{\tilde{u}_{\ell_{k}}^{+}\\}$ which converges weakly to a limit
$\tilde{u}^{+}\in W^{1,p}(\Omega_{\infty}^{+})$. Definition of
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell_{k}}^{+})$ gives the following identity:

$\int_{\Omega_{\infty}^{+}}|A\nabla\tilde{u}_{\ell_{k}}^{+}\cdot\nabla\tilde{u}_{\ell_{k}}^{+}|^{\frac{p}{2}}=\tilde{\lambda}_{M}^{1}(\Omega_{\ell_{k}}^{+})\int_{\Omega_{\infty}^{+}}|\tilde{u}_{\ell_{k}}^{+}|^{p}.$
(5.3)

We claim that

$\int_{\Omega_{\infty}^{+}}|A\nabla\tilde{u}^{+}\cdot\nabla\tilde{u}^{+}|^{\frac{p}{2}}=\nu_{\infty}^{+}\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}|^{p}.$
(5.4)

Clearly $\tilde{u}^{+}\in V(\Omega_{\infty}^{+})$. From (1.9) we have

$\int_{\Omega_{\infty}^{+}}|A\nabla\tilde{u}^{+}\cdot\nabla\tilde{u}^{+}|^{\frac{p}{2}}\geq\nu_{\infty}^{+}\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}|^{p}.$
(5.5)

By Fatou’s lemma and by (5.3), we get

$\int_{\Omega_{\infty}^{+}}|A\nabla\tilde{u}^{+}\cdot\nabla\tilde{u}^{+}|^{\frac{p}{2}}\leq\liminf_{k\rightarrow\infty}\int_{\Omega_{\infty}^{+}}|A\nabla\tilde{u}_{\ell_{k}}^{+}\cdot\nabla\tilde{u}_{\ell_{k}}^{+}|^{\frac{p}{2}}=\nu_{\infty}^{+}\left(\liminf_{k\rightarrow\infty}\int_{\Omega_{\infty}^{+}}|\tilde{u}_{\ell_{k}}^{+}|^{p}\right)=\nu_{\infty}^{+}.$
(5.6)

To prove our claim, it is enough to show that
$\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}|^{p}=1$. In addition we will show
that $\tilde{u}_{\ell}^{+}$ decays to zero for large $x_{1}$, and that implies
concentration near $x_{1}=0$, using the same technique as in the proof of
Theorem 1.1.

Let $\ell$ and $\ell^{\prime}$ be such that $0<\ell^{\prime}\leq\ell-1$.
Define $\tilde{\rho}_{\ell^{\prime}}=\tilde{\rho}_{\ell^{\prime}}(x_{1})$ by

$\tilde{\rho}_{\ell^{\prime}}(x_{1})=\begin{cases}0&\text{$x_{1}\leq\ell^{\prime}$}\\\
x_{1}-\ell^{\prime}&\text{$x_{1}\in(\ell^{\prime},\ell^{\prime}+1)$}\\\
1&\text{$x_{1}\geq\ell^{\prime}+1.$}\end{cases}$

Clearly $\tilde{u}_{\ell}^{+}\tilde{\rho}_{\ell^{\prime}}$ satisfies Dirichlet
boundary condition on $\partial\Omega_{\ell}^{+}.$ Translating this function
in $\Omega_{\ell/2}$ and using definition of
$\lambda_{D}^{1}(\Omega_{\ell/2})$ and
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})$ respectively, we get

$\lambda_{D}^{1}(\Omega_{\ell/2})\int_{\Omega_{\ell}^{+}}|\tilde{u}_{\ell}^{+}|^{p}\tilde{\rho}_{\ell^{\prime}}^{p}\leq\int_{\Omega_{\ell}^{+}}|A\nabla(\tilde{u}_{\ell}^{+}\tilde{\rho}_{\ell^{\prime}})\cdot\nabla(\tilde{u}_{\ell}^{+}\tilde{\rho}_{\ell^{\prime}})|^{\frac{p}{2}}$

and

$\int_{\Omega_{\ell}^{+}}|A\nabla\tilde{u}_{\ell}^{+}\cdot\nabla\tilde{u}_{\ell}^{+}|^{\frac{p-2}{2}}A\nabla\tilde{u}_{\ell}^{+}\cdot\nabla(\tilde{\rho}_{\ell^{\prime}}^{p}\tilde{u}_{\ell}^{+})=\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})\int_{\Omega_{\ell}^{+}}|\tilde{u}_{\ell}^{+}|^{p}\tilde{\rho}_{\ell^{\prime}}^{p}.$

Let
$D_{\ell^{\prime}}^{+}=\Omega_{\ell^{\prime}+1}^{+}\setminus\Omega_{\ell^{\prime}}^{+}$.
An analogous calculation to derive (3.10) in the proof of Theorem 1.1 gives

$\left(\lambda_{D}^{1}(\Omega_{\ell/2})-\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})\right)\int_{\Omega_{\ell}^{+}}|\tilde{u}_{\ell}^{+}|^{p}\tilde{\rho}_{\ell^{\prime}}^{p}\leq
p\lVert
A\rVert_{\infty}^{\frac{p}{2}}C^{\prime}\int_{D_{\ell^{\prime}}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}.$
(5.7)

By (1.3), (4.7) and the assumption (5.1), there exists a
$\tilde{\beta}^{\prime}>0$ such that for $\ell>\ell_{0}$ we have
$\lambda_{D}^{1}(\Omega_{\ell/2})-\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})\geq\tilde{\beta}^{\prime}$,
which implies

$\tilde{\beta}^{\prime}\int_{\Omega_{\ell}^{+}}|\tilde{u}_{\ell}^{+}|^{p}\tilde{\rho}_{\ell^{\prime}}^{p}\leq
p\lVert
A\rVert_{\infty}^{\frac{p}{2}}C^{\prime}\int_{D_{\ell^{\prime}}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}.$
(5.8)

Again by an analogous computation as in derivation of the (3.13) in the proof
of Theorem 1.1, and by the properties of $\tilde{\rho}_{\ell^{\prime}}$, we
obtain

$\tilde{\beta}^{\prime}\lambda^{\frac{p}{2}}\int_{\Omega_{\ell}^{+}\setminus\Omega_{\ell^{\prime}+1}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}\leq
p\lVert
A\rVert_{\infty}^{\frac{p}{2}}\left(\lambda_{M}^{1}(\Omega_{\ell})C^{\prime}+\tilde{\beta}^{\prime}C_{p}\right)\int_{D_{\ell^{\prime}}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p},$

which gives

$\tilde{\beta}\int_{\Omega_{\ell}^{+}\setminus\Omega_{\ell^{\prime}+1}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}\leq\tilde{C}\int_{\Omega_{\ell^{\prime}+1}^{+}\setminus\Omega_{\ell^{\prime}}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p},$
(5.9)

where $\tilde{\beta}=\tilde{\beta}^{\prime}\lambda^{\frac{p}{2}}$ and
$\tilde{C}=p\lVert
A\rVert_{\infty}^{\frac{p}{2}}\left(\mu^{1}({\omega})C^{\prime}+\tilde{\beta}^{\prime}C_{p}\right)$.
From (5.9) we get

$(\tilde{\beta}+\tilde{C})\int_{\Omega_{\ell}^{+}\setminus\Omega_{\ell^{\prime}+1}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}\leq\tilde{C}\int_{\Omega_{\ell}^{+}\setminus\Omega_{\ell^{\prime}}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p},$

which implies

$\int_{\Omega_{\ell}^{+}\setminus\Omega_{\ell^{\prime}+1}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}\leq\tilde{\alpha}\int_{\Omega_{\ell}^{+}\setminus\Omega_{\ell^{\prime}}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p},$
(5.10)

where $\tilde{\alpha}=\frac{\tilde{C}}{\tilde{\beta}+\tilde{C}}<1$. Fix $r>1$.
Applying this procedure successively for $\ell^{\prime}=r-1,r-2,\cdots,r-[r]$
and then using (2.3), we obtain

$\int_{\Omega_{\ell}^{+}\setminus\Omega_{r}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}\leq\tilde{\alpha}^{[r]}\int_{\Omega_{\ell}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}\\\
\leq\frac{\tilde{\alpha}^{[r]}}{\lambda^{\frac{p}{2}}}\int_{\Omega_{\ell}^{+}}|A\nabla\tilde{u}_{\ell}^{+}\cdot\nabla\tilde{u}_{\ell}^{+}|^{\frac{p}{2}}=\frac{\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})}{\lambda^{\frac{p}{2}}}\tilde{\alpha}^{[r]}\leq\frac{\mu^{1}(\omega)}{\lambda^{\frac{p}{2}}}\tilde{\alpha}^{[r]}.$
(5.11)

By Poincaré inequality (2.1) and (5.11), we have

$\int_{\Omega_{\ell}^{+}\setminus\Omega_{r}^{+}}|\tilde{u}_{\ell}^{+}|^{p}\leq
C_{p}^{p}\int_{\Omega_{l}^{+}\setminus\Omega_{r}^{+}}|\nabla\tilde{u}_{\ell}^{+}|^{p}\leq\frac{\mu^{1}(\omega)C_{p}^{p}}{\lambda^{\frac{p}{2}}}\tilde{\alpha}^{[r]}=C\tilde{\alpha}^{[r]},\
\forall\ell>r,$

which implies

$\int_{\Omega_{r}^{+}}|\tilde{u}_{\ell}^{+}|^{p}\geq 1-C\tilde{\alpha}^{[r]}.$
(5.12)

Since $\tilde{u}_{\ell_{k}}^{+}\rightarrow\tilde{u}^{+}$ strongly in
$L^{p}(\Omega^{+}_{r})$, we deduce from (5.12) that

$\int_{\Omega_{r}^{+}}|\tilde{u}^{+}|^{p}\geq 1-C\tilde{\alpha}^{[r]}.$

Finally, limit $r\rightarrow\infty$ gives

$\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}|^{p}=1.$ (5.13)

Since $\tilde{u}^{+}\not\equiv 0$, we can conclude that it is a minimizer
realizing $\nu_{\infty}^{+}$ in (1.9). It is a standard fact that the
minimizer is unique upto multiplication by a constant and $\tilde{u}^{+}$ has
constant sign follows from the fact that $\tilde{u}_{\ell_{k}}^{+}$ are non-
negative.

To show that $\tilde{u}^{+}$ is a weak solution of (5.2), for any fixed
$\phi\in V(\Omega_{\infty}^{+})$ and $t\in\mathbb{R}$ we define the function

$J_{\phi}(t)=\frac{\int_{\Omega_{\infty}^{+}}|A\nabla(\tilde{u}^{+}+t\phi)\cdot\nabla(\tilde{u}^{+}+t\phi)|^{\frac{p}{2}}}{\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}+t\phi|^{p}}.$

The derivative of $J_{\phi}$ with respect to $t$ is

$J_{\phi}^{\prime}(t)=\frac{\frac{p}{2}\int_{\Omega_{\infty}^{+}}|A\nabla(\tilde{u}^{+}+t\phi)\cdot\nabla(\tilde{u}^{+}+t\phi)|^{\frac{p-2}{2}}(2A\nabla\tilde{u}^{+}\cdot\nabla\phi+2tA\nabla\phi\cdot\nabla\phi)}{\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}+t\phi|^{p}}\\\
-\frac{\frac{p}{2}\left(\int_{\Omega_{\infty}^{+}}|A\nabla(\tilde{u}^{+}+t\phi)\cdot\nabla(\tilde{u}^{+}+t\phi)|^{\frac{p}{2}}\right)\left(\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}+t\phi|^{p-2}(2\tilde{u}^{+}\phi+2t\phi^{2})\right)}{\left(\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}+t\phi|^{p}\right)^{2}}.$

Since $\tilde{u}^{+}$ is a minimizer, we must have $J_{\phi}^{\prime}(0)=0$.
Substituting this and using (5.4) we get

$\int_{\Omega_{\infty}^{+}}|A\nabla\tilde{u}^{+}\cdot\nabla\tilde{u}^{+}|^{\frac{p-2}{2}}A\nabla\tilde{u}^{+}\cdot\nabla\phi=\nu_{\infty}^{+}\int_{\Omega_{\infty}^{+}}|\tilde{u}^{+}|^{p-2}u\phi.$
(5.14)

This completes the proof. ∎

###### Proof of Theorem 1.3.

(i) Assume that $A_{12}\cdot\nabla_{X_{2}}W\not\equiv 0$ and (1.11) holds. For
simplicity of notation, we use $\nabla W=\nabla_{X_{2}}W$ in this proof. For
any $\ell>0$ we define the set
$T_{\ell}=\\{x\in\omega:\mathrm{dist}(x,\partial\omega)\leq\ell\\}$. Let
$\beta\in(0,1)$ is fixed and $\rho_{\ell}$ be an approximation of the
characteristic function of $\omega$, i.e. $\rho_{\ell}\rightarrow 1$ pointwise
as $\ell\rightarrow 0$ and $\rho_{\ell}$ satisfies the following properties:

$\rho_{\ell}\in C_{c}^{\infty}(\omega),\ 0\leq\rho_{\ell}\leq 1,\
\rho_{\ell}=1\ \mathrm{in}\ \omega\setminus T_{\ell},\
|\nabla\rho_{\ell}|\leq\frac{1}{\ell^{\beta}}\ \mathrm{in}\ T_{\ell}.$ (5.15)

Define the function

$v_{\ell}^{\epsilon}(x_{1},X_{2})=W(X_{2})-x_{1}\rho_{\ell}(X_{2})G_{\epsilon}(X_{2})$

on $\Omega_{\ell}$, where $\\{G_{\epsilon}\\}\subset C_{c}^{\infty}(\omega)$
satisfies

$\lim_{\epsilon\rightarrow
0}G_{\epsilon}(X_{2})=\frac{A_{12}(X_{2})\cdot\nabla
W(X_{2})}{a_{11}(X_{2})}\quad\textrm{in}\ L^{p}(\omega).$

We claim that

$\inf_{\epsilon>0}\lim_{\ell\rightarrow
0}\frac{\int_{\Omega_{\ell}^{-}}|A\nabla v_{\ell}^{\epsilon}\cdot\nabla
v_{\ell}^{\epsilon}|^{\frac{p}{2}}}{\int_{\Omega_{\ell}^{-}}|v_{\ell}^{\epsilon}|^{p}}\leq\int_{\omega}\left(A_{22}\nabla
W\cdot\nabla W-\frac{A_{12}\cdot\nabla W}{a_{11}}\right)^{\frac{p}{2}}.$
(5.16)

Since

$a_{11}G_{\epsilon}^{2}-2(A_{12}\cdot\nabla W)G_{\epsilon}+A_{22}\nabla
W\cdot\nabla W\rightarrow A_{22}\nabla W\cdot\nabla W-\frac{A_{12}\cdot\nabla
W}{a_{11}}$

in $L^{p}(\omega)$ as $\epsilon\rightarrow 0$, and L.H.S is always positive by
uniform ellipticity of the matrix $A$, the function on the R.H.S of (5.16) is
always non-negative. The following inequality holds for any two vectors $a,$
$b$ and $q\geq 1$ :

$|b|^{q}\geq|a|^{q}+q\langle|a|^{q-2}a,b-a\rangle.$

Using this we obtain

$\int_{\Omega_{\ell}^{-}}|v_{\ell}^{\epsilon}|^{p}=\int_{\Omega_{\ell}^{-}}|W(X_{2})-x_{1}\rho_{\ell}(X_{2})G_{\epsilon}(X_{2})|^{p}\\\
\geq\int_{\Omega_{\ell}^{-}}|W(X_{2})|^{p}-p\int_{\Omega_{\ell}^{-}}|W(X_{2})|^{p-1}x_{1}\rho_{\ell}(X_{2})G_{\epsilon}(X_{2})=\ell+p\frac{\ell^{2}}{2}\int_{\omega}|W|^{p-1}\rho_{\ell}G_{\epsilon}.$
(5.17)

Now

$\int_{\Omega_{\ell}^{-}}|A\nabla v_{\ell}^{\epsilon}\cdot\nabla
v_{\ell}^{\epsilon}|^{\frac{p}{2}}=\int_{\Omega_{\ell}^{-}}|a_{11}(\partial_{x_{1}}v_{\ell}^{\epsilon})^{2}+2(A_{12}\cdot\nabla_{X_{2}}v_{\ell}^{\epsilon})\partial_{x_{1}}v_{\ell}^{\epsilon}+A_{22}\nabla_{X_{2}}v_{\ell}^{\epsilon}\cdot\nabla_{X_{2}}v_{\ell}^{\epsilon}|^{\frac{p}{2}}\\\
=\int_{\Omega_{\ell}^{-}}|a_{11}\rho_{\ell}^{2}G_{\epsilon}^{2}-2\rho_{\ell}G_{\epsilon}(A_{12}\cdot\nabla
W-x_{1}(G_{\epsilon}A_{12}\cdot\nabla\rho_{\ell}+\rho_{\ell}A_{12}\cdot\nabla
G_{\epsilon}))\\\ +\left(A_{22}\nabla
W-x_{1}(G_{\epsilon}A_{22}\nabla\rho_{\ell}+\rho_{\ell}A_{22}\nabla
G_{\epsilon})\right)\cdot\left(\nabla
W-x_{1}(G_{\epsilon}\nabla\rho_{\ell}+\rho_{\ell}\nabla
G_{\epsilon})\right)|^{\frac{p}{2}}.$ (5.18)

Applying Minkowski’s inequality in (5.18), we get

$\int_{\Omega_{\ell}^{-}}|A\nabla v_{\ell}^{\epsilon}\cdot\nabla
v_{\ell}^{\epsilon}|^{\frac{p}{2}}\leq({I_{1}^{\epsilon}}^{\frac{2}{p}}+{I_{2}^{\epsilon}}^{\frac{2}{p}}+{I_{3}^{\epsilon}}^{\frac{2}{p}})^{\frac{p}{2}},$
(5.19)

where the integrals

$I_{1}^{\epsilon}=\int_{\Omega_{\ell}^{-}}|a_{11}G_{\epsilon}^{2}-2(A_{12}\cdot\nabla
W)G_{\epsilon}+A_{22}\nabla W\cdot\nabla
W|^{\frac{p}{2}}=\ell\int_{\omega}|a_{11}G_{\epsilon}^{2}-2(A_{12}\cdot\nabla
W)G_{\epsilon}+A_{22}\nabla W\cdot\nabla W|^{\frac{p}{2}},$ (5.20)
$I_{2}^{\epsilon}=\int_{\Omega_{\ell}^{-}}|a_{11}G_{\epsilon}^{2}(\rho_{\ell}^{2}-1)+2(A_{12}\cdot\nabla
W)G_{\epsilon}(1-\rho_{\ell})|^{\frac{p}{2}}=\ell K_{\ell}^{\epsilon},$ (5.21)

where

$K_{\ell}^{\epsilon}=\int_{\omega}|a_{11}G_{\epsilon}^{2}(\rho_{\ell}^{2}-1)+2(A_{12}\cdot\nabla
W)G_{\epsilon}(1-\rho_{\ell})|^{\frac{p}{2}}$

and

$I_{3}^{\epsilon}=\int_{\Omega_{\ell}^{-}}|2x_{1}H_{\ell}^{\epsilon}(X_{2})+x_{1}^{2}F_{\ell}^{\epsilon}(X_{2})|^{\frac{p}{2}},$
(5.22)

where

$H_{\ell}^{\epsilon}(X_{2})=\rho_{\ell}G_{\epsilon}^{2}(A_{12}\cdot\nabla\rho_{\ell})+\rho_{\ell}^{2}G_{\epsilon}(A_{12}\cdot\nabla
G_{\epsilon})-G_{\epsilon}(A_{22}\nabla
W\cdot\nabla\rho_{\ell})-\rho_{\ell}(A_{22}\nabla W\cdot\nabla G_{\epsilon})$

and

$F_{\ell}^{\epsilon}(X_{2})=(G_{\epsilon}A_{22}\nabla\rho_{\ell}+\rho_{\ell}A_{22}\nabla
G_{\epsilon})\cdot(G_{\epsilon}\nabla\rho_{\ell}+\rho_{\ell}\nabla
G_{\epsilon}).$

Since $\rho_{\ell}\rightarrow 1$ pointwise as $\ell\rightarrow 0$,
$K_{\ell}^{\epsilon}$ converges to $0$ as $\ell\rightarrow 0$ by dominated
convergence theorem. For any fix $\epsilon>0$, (5.15) gives

$|H_{\ell}^{\epsilon}|\leq C_{1}+\frac{C_{2}}{\ell^{\beta}}\ \mathrm{and}\ \
|F_{\ell}^{\epsilon}|\leq C_{3}+\frac{C_{4}}{\ell^{2\beta}},$ (5.23)

where the constants $C_{i}>0$ ($1\leq i\leq 4$) are independent of $\ell$.
Applying Minkowski’s inequality in (5.22), we obtain

$I_{3}^{\epsilon}\leq\left(\left(\int_{\Omega_{\ell}^{-}}|2x_{1}H_{\ell}^{\epsilon}(X_{2})|^{\frac{p}{2}}\right)^{\frac{2}{p}}+\left(\int_{\Omega_{\ell}^{-}}|x_{1}^{2}F_{\ell}^{\epsilon}(X_{2})|^{\frac{p}{2}}\right)^{\frac{2}{p}}\right)^{\frac{p}{2}}\\\
\leq\left(2\ell\left(\int_{\Omega_{\ell}^{-}}|H_{\ell}^{\epsilon}(X_{2})|^{\frac{p}{2}}\right)^{\frac{2}{p}}+\ell^{2}\left(\int_{\Omega_{\ell}^{-}}|F_{\ell}^{\epsilon}(X_{2})|^{\frac{p}{2}}\right)^{\frac{2}{p}}\right)^{\frac{p}{2}}\\\
=\ell\left(2\ell\left(\int_{\omega}|H_{\ell}^{\epsilon}(X_{2})|^{\frac{p}{2}}\right)^{\frac{2}{p}}+\ell^{2}\left(\int_{\omega}|F_{\ell}^{\epsilon}(X_{2})|^{\frac{p}{2}}\right)^{\frac{2}{p}}\right)^{\frac{p}{2}}$

By (5.23), we get

$I_{3}^{\epsilon}\leq\ell\left[C_{1}\ell+C_{2}\ell^{1-\beta}+C_{3}\ell^{2}+C_{4}\ell^{2-2\beta}\right]^{\frac{p}{2}}=\ell
C(\ell)^{\frac{p}{2}},$ (5.24)

where $C(\ell)\rightarrow 0$ as $\ell\rightarrow 0$. Using the estimates
(5.20), (5.21) and (5.24) in (5.19), we obtain

$\int_{\Omega_{\ell}^{-}}|A\nabla v_{\ell}^{\epsilon}\cdot\nabla
v_{\ell}^{\epsilon}|^{\frac{p}{2}}\leq\ell\left[\left(\int_{\omega}|a_{11}G_{\epsilon}^{2}-2(A_{12}\cdot\nabla
W)G_{\epsilon}+A_{22}\nabla W\cdot\nabla
W|^{\frac{p}{2}}\right)^{\frac{2}{p}}+(K_{\ell}^{\epsilon})^{\frac{2}{p}}+C(\ell)\right]^{\frac{p}{2}}.$
(5.25)

Hence we have by (5.17) and (5.25),

$\frac{\int_{\Omega_{\ell}^{-}}|A\nabla v_{\ell}^{\epsilon}\cdot\nabla
v_{\ell}^{\epsilon}|^{\frac{p}{2}}}{\int_{\Omega_{\ell}^{-}}|v_{\ell}^{\epsilon}|^{p}}\leq\frac{\left[\left(\int_{\omega}|a_{11}G_{\epsilon}^{2}-2(A_{12}\cdot\nabla
W)G_{\epsilon}+A_{22}\nabla W\cdot\nabla
W|^{\frac{p}{2}}\right)^{\frac{2}{p}}+(K_{\ell}^{\epsilon})^{\frac{2}{p}}+C(\ell)\right]^{\frac{p}{2}}}{1+\frac{p}{2}\ell\int_{\omega}|W|^{p-1}\rho_{\ell}G_{\epsilon}}$
(5.26)

Sending the limits $\ell\rightarrow 0$ and $\epsilon\rightarrow 0$ in (5.26),
we prove our claim. In particular we have

$\inf_{\epsilon>0}\lim_{\ell\rightarrow
0}\frac{\int_{\Omega_{\ell}^{-}}|A\nabla v_{\ell}^{\epsilon}\cdot\nabla
v_{\ell}^{\epsilon}|^{\frac{p}{2}}}{\int_{\Omega_{\ell}^{-}}|v_{\ell}^{\epsilon}|^{p}}<\mu^{1}(\omega).$

Hence we can find some $\ell_{1}$ and $\epsilon_{1}$ for which

$-\gamma_{1}=\int_{\Omega_{\ell_{1}}^{-}}|A\nabla
v_{\ell_{1}}^{\epsilon_{1}}\cdot\nabla
v_{\ell_{1}}^{\epsilon_{1}}|^{\frac{p}{2}}-\mu^{1}(\omega)\int_{\Omega_{\ell_{1}}^{-}}|v_{\ell_{1}}^{\epsilon_{1}}|^{p}<0.$
(5.27)

For any $\alpha>0$ we define the following function in
$V(\Omega_{\infty}^{+})$:

$z_{\alpha}(x_{1},X_{2})=\begin{cases}v_{\ell_{1}}^{\epsilon_{1}}(x_{1}-\ell_{1},X_{2})&\text{$x_{1}\in[0,\ell_{1})$}\\\
W(X_{2})e^{-\alpha(x_{1}-\ell_{1})}&\text{$x_{1}\in[\ell_{1},\infty)$}.\end{cases}$

We have

$\int_{\Omega_{\infty}^{+}}|z_{\alpha}|^{p}=\int_{\Omega_{\ell_{1}}^{-}}|v_{\ell_{1}}^{\epsilon_{1}}|^{p}+\Big{(}\int_{0}^{\infty}e^{-p\alpha
x_{1}}\Big{)}\int_{\omega}|W|^{p}=\int_{\Omega_{\ell_{1}}^{-}}|v_{\ell_{1}}^{\epsilon_{1}}|^{p}+\frac{1}{p\alpha},$
(5.28)

and

$\int_{\Omega_{\infty}^{+}}|A\nabla z_{\alpha}\cdot\nabla
z_{\alpha}|^{\frac{p}{2}}=\int_{\Omega_{\ell_{1}}^{-}}|A\nabla
v_{\ell_{1}}^{\epsilon_{1}}\cdot\nabla
v_{\ell_{1}}^{\epsilon_{1}}|^{\frac{p}{2}}\\\
+\frac{1}{p\alpha}\int_{\omega}|\alpha^{2}a_{11}W^{2}-2\alpha(A_{12}\cdot\nabla
W)W+A_{22}\nabla W\cdot\nabla W|^{\frac{p}{2}}.$ (5.29)

Using (1.9) and then (5.28) and (5.29), we obtain

$\nu_{\infty}^{+}-\mu^{1}(\omega)\leq\frac{\int_{\Omega_{\ell_{1}}^{-}}|A\nabla
v_{\ell_{1}}^{\epsilon_{1}}\cdot\nabla
v_{\ell_{1}}^{\epsilon_{1}}|^{\frac{p}{2}}\\\
+\frac{1}{p\alpha}\int_{\omega}|\alpha^{2}a_{11}W^{2}-2\alpha(A_{12}\cdot\nabla
W)W+A_{22}\nabla W\cdot\nabla
W|^{\frac{p}{2}}}{\int_{\Omega_{\ell_{1}}^{-}}|v_{\ell_{1}}^{\epsilon_{1}}|^{p}+\frac{1}{p\alpha}}-\mu^{1}(\omega).$
(5.30)

Substituting the value $\gamma_{1}$ defined in (5.27), we get

$\nu_{\infty}^{+}-\mu^{1}(\omega)\leq\frac{-\gamma_{1}+\frac{1}{p\alpha}\int_{\omega}|\alpha^{2}a_{11}W^{2}-2\alpha(A_{12}\cdot\nabla
W)W+A_{22}\nabla W\cdot\nabla
W|^{\frac{p}{2}}-\frac{\mu^{1}(\omega)}{p\alpha}}{\int_{\Omega_{\ell_{1}}^{-}}|v_{\ell_{1}}^{\epsilon_{1}}|^{p}+\frac{1}{p\alpha}}.$
(5.31)

Now we approximate the integral

$I_{\alpha}=\int_{\omega}|\alpha^{2}a_{11}W^{2}-2\alpha(A_{12}\cdot\nabla
W)W+A_{22}\nabla W\cdot\nabla W|^{\frac{p}{2}}.$

Using the inequality $(a+b)^{q}\leq a^{q}+qa^{q-1}b+Ca^{q-2}b^{2}$ where
$|b|<a$, $q\geq 1$ and $C$ is some positive constant, for small $\alpha,$ we
have

$I_{\alpha}\leq\int_{\omega}|A_{22}\nabla W\cdot\nabla
W|^{\frac{p}{2}}+\frac{p}{2}\int_{\omega}|A_{22}\nabla W\cdot\nabla
W|^{\frac{p-2}{2}}\left(\alpha^{2}a_{11}W^{2}-2\alpha(A_{12}\cdot\nabla
W)W\right)\\\ +C\int_{\omega}|A_{22}\nabla W\cdot\nabla
W|^{\frac{p-4}{2}}\left(\alpha^{2}a_{11}W^{2}-2\alpha(A_{12}\cdot\nabla
W)W\right)^{2}\\\ =\mu^{1}(\omega)+\alpha^{2}\frac{p}{2}a_{11}I_{1}-\alpha
pI_{2}+\alpha^{2}CI_{3},$ (5.32)

where

$I_{1}=\int_{\omega}|A_{22}\nabla W\cdot\nabla W|^{\frac{p-2}{2}}W^{2},$
$I_{2}=\int_{\omega}|A_{22}\nabla W\cdot\nabla
W|^{\frac{p-2}{2}}(A_{12}\cdot\nabla W)W$

and

$I_{3}=\int_{\omega}|A_{22}\nabla W\cdot\nabla W|^{\frac{p-4}{2}}\left(\alpha
a_{11}W^{2}-2(A_{12}\cdot\nabla W)W\right)^{2}.$

Using (5.32) in (5.31), we get

$\nu_{\infty}^{+}-\mu^{1}(\omega)\leq\frac{-\gamma_{1}+\frac{\alpha}{2}a_{11}I_{1}-I_{2}+\frac{\alpha}{p}CI_{3}}{\int_{\Omega_{\ell_{1}}^{-}}|v_{\ell_{1}}^{\epsilon_{1}}|^{p}+\frac{1}{p\alpha}}.$
(5.33)

Since $\gamma_{1}>0$, and $I_{2}\geq 0$ by (1.11), choose $\alpha$ small
enough so that RHS becomes negative and this completes the proof.

(ii) It is known that the space $V_{s}(\Omega_{\infty}^{+})$ defined in (4.8)
is dense in $V(\Omega_{\infty}^{+})$. The following subspace

$V_{s}^{0}(\Omega_{\infty}^{+})=\\{u\in
V_{s}(\Omega_{\infty}^{+}):\exists\delta=\delta(u)>0\ \mathrm{s.t}\ u(x)=0\
\mathrm{for}\ x\ \mathrm{with}\
\mathrm{dist}(x,\gamma_{\infty}^{+})\leq\delta\\}$

is also dense in $V(\Omega_{\infty}^{+})$. By regularity of the eigenfunction,
we know that $W$ is continuous and positive in $\omega$ (see [23]). Picone’s
identity says that for $u\geq 0$ and $v>0$, we have

$R(u,v)=|A\nabla u\cdot\nabla u|^{\frac{p}{2}}-|A\nabla v\cdot\nabla
v|^{\frac{p-2}{2}}A\nabla v\cdot\nabla\left(\frac{u^{p}}{v^{p-1}}\right)\geq
0.$ (5.34)

Let $u\in V_{s}^{0}(\Omega_{\infty}^{+})$ be such that $u\geq 0$ and let
$v=W$. With these $u$ and $v$, integrating (5.34) over $\Omega_{\infty}^{+}$
and applying integration by parts, we get

$0\leq\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}-\int_{\Omega_{\infty}^{+}}|A\nabla W\cdot\nabla
W|^{\frac{p-2}{2}}A\nabla W\cdot\nabla\left(\frac{u^{p}}{W^{p-1}}\right)\\\
=\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}+\int_{\Omega_{\infty}^{+}}\mathrm{div}\left(|A\nabla
W\cdot\nabla W|^{\frac{p-2}{2}}A\nabla W\right)\frac{u^{p}}{W^{p-1}}\\\
-\int_{\\{0\\}\times\omega}\left(|A\nabla W\cdot\nabla
W|^{\frac{p-2}{2}}A\nabla W\cdot\nu\right)\frac{u^{p}}{W^{p-1}}\\\
=\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}-\mu^{1}(\omega)\int_{\Omega_{\infty}^{+}}u^{p}+\int_{\omega}|A_{22}\nabla
W\cdot\nabla W|^{\frac{p-2}{2}}\left(A_{12}\cdot\nabla
W\right)\frac{u^{p}(0,X_{2})}{W^{p-1}(0,X_{2})}.$ (5.35)

Here we have used $-\mathrm{div}(|A\nabla W\cdot\nabla
W|^{\frac{p-2}{2}}A\nabla W)=\mu^{1}(\omega)W^{p-1}$, which is true in view of
(1.2). The last integral is less than or equal to $0$ by the assumption
(1.12), we have

$0\leq\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}-\mu^{1}(\omega)\int_{\Omega_{\infty}^{+}}u^{p}.$ (5.36)

Now let $u\in V_{s}^{0}(\Omega_{\infty}^{+})$ and $u=u^{+}-u^{-}$, where
$u^{+}(x)=$max$\\{u(x),0\\}$ and $u^{-}(x)=$min$\\{0,u(x)\\}$. Clearly $0\leq
u^{\pm}\in V_{s}^{0}(\Omega_{\infty}^{+})$ and $|u|=u^{+}+u^{-}$. Since
$u^{+}$ and $u^{-}$ have disjoint supports, it is easy to check that

$\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}=\int_{\Omega_{\infty}^{+}}|A\nabla u^{+}\cdot\nabla
u^{+}|^{\frac{p}{2}}+\int_{\Omega_{\infty}^{+}}|A\nabla u^{-}\cdot\nabla
u^{-}|^{\frac{p}{2}}$ (5.37)

and

$\int_{\Omega_{\infty}^{+}}|u|^{p}=\int_{\Omega_{\infty}^{+}}\left(u^{+}\right)^{p}+\int_{\Omega_{\infty}^{+}}\left(u^{-}\right)^{p}.$
(5.38)

Hence from (5.36), (5.37) and (5.38) we obtain, for every $u\in
V_{s}^{0}(\Omega_{\infty}^{+})$

$\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}\geq\mu^{1}(\omega)\int_{\Omega_{\infty}^{+}}u^{p},\
\mathrm{i.e.}\ \ \frac{\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}}{\int_{\Omega_{\infty}^{+}}u^{p}}\geq\mu^{1}(\omega).$ (5.39)

By the density of the $V_{s}^{0}(\Omega_{\infty}^{+})$ in
$V(\Omega_{\infty}^{+})$, inequality (5.39) holds for every $u\in
V(\Omega_{\infty}^{+})$ and (1.9) gives $\nu_{\infty}^{+}\geq\mu^{1}(\omega)$.
The reverse inequality follows from Lemma 4.1 and we conclude
$\nu_{\infty}^{+}=\mu^{1}(\omega)$.

Now, if $u$ be a minimizer realizing $\nu_{\infty}^{+}$, from (5.35) we must
have

$\int_{\Omega_{\infty}^{+}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}=\int_{\Omega_{\infty}^{+}}|A\nabla W\cdot\nabla
W|^{\frac{p-2}{2}}A\nabla W\cdot\nabla\left(\frac{u^{p}}{W^{p-1}}\right).$

This is true only when $u=cW$ for some constant $c$. But this is a
contradiction since $W\not\in V(\Omega_{\infty}^{+})$. This completes the
proof. ∎

######  Remark 3.

It is clear from Remark 1 that Theorem 1.3 (i) and (ii) follows for
$\nu_{\infty}^{-}$ if we replace the inequalities (1.11) and (1.12) by
$\int_{\omega}|A_{22}\nabla W\cdot\nabla W|^{\frac{p-2}{2}}(A_{12}\cdot\nabla
W)W\leq 0$ and $A_{12}\cdot\nabla W\geq 0$ respectively. In particular, if
$A_{12}\cdot\nabla W\not\equiv 0$ and $\int_{\omega}|A_{22}\nabla W\cdot\nabla
W|^{\frac{p-2}{2}}(A_{12}\cdot\nabla W)W=0$, then
$\nu_{\infty}^{+}<\mu^{1}(\omega)$ and $\nu_{\infty}^{-}<\mu^{1}(\omega)$.
Hence, if $A$ satisfies the symmetry (S) defined in Definition 1 and
$A_{12}\cdot\nabla W\not\equiv 0$, then it follows that
$\nu_{\infty}^{\pm}<\mu^{1}(\omega)$. Another consequence of Theorem 1.3 is
that if $A_{12}\cdot\nabla W\not\equiv 0$, at least one of the
$\nu_{\infty}^{+}$ and $\nu_{\infty}^{-}$ is strictly less than
$\mu^{1}(\omega)$, in particular
$\min\\{\nu_{\infty}^{+},\nu_{\infty}^{-}\\}<\mu^{1}(\omega)$, which already
follows from Theorem 2 (ii) and Theorem 1.2.

In the next theorem we describe the behaviour of the eigenfunctions
$\\{u_{\ell}\\}$ of mixed boundary value problem (1.4) at the ends of the
cylinder. Here we have shown two possible cases may happen: concentration near
one of the ends of the cylinder, or near both ends. Let $\tilde{u}^{\pm}$ be
the unique positive normalized minimizer in (1.9), whenever it exists. For
each $\ell>0$ we define the functions

$\tilde{v}_{\ell}^{+}(x_{1},X_{2})=u_{\ell}(x_{1}-\ell,X_{2})\quad
on\quad\Omega_{\ell}^{+},$
$\tilde{v}_{\ell}^{-}(x_{1},X_{2})=u_{\ell}(x_{1}+\ell,X_{2})\quad
on\quad\Omega_{\ell}^{-}.$

######  Theorem 5.2.

_(i)_ If $\nu_{\infty}^{+}<\nu_{\infty}^{-}$, then for every $r>0$,

$\tilde{v}_{\ell}^{+}\longrightarrow\tilde{u}^{+}\ \ in\ \
W^{1,p}(\Omega_{r}^{+})\ \ and\ \ \tilde{v}_{\ell}^{-}\longrightarrow 0\ \ in\
\ W^{1,p}(\Omega_{r}^{-}).$

_(ii)_ If $A$ satisfies the symmetry (S) defined in Definition 1 and (5.2)
holds, then we have
$\tilde{v}_{\ell}^{+}(x_{1},X_{2})=\tilde{v}_{\ell}^{-}(-x_{1},-X_{2})$ and
for every $r>0$,

$\tilde{v}_{\ell}^{+}\longrightarrow\tilde{u}^{+}\ \ in\ \
W^{1,p}(\Omega_{r}^{+})\ \ and\ \
\tilde{v}_{\ell}^{-}\longrightarrow\tilde{u}^{-}\ \ in\ \
W^{1,p}(\Omega_{r}^{-}).$

###### Proof.

(i) The convergence of $\\{\tilde{v}_{\ell}^{-}\\}$ to $0$ in
$W^{1,p}(\Omega_{r}^{-})$ for any $r>0$ follows from Proposition 4.1. It
remains to prove the convergence for $\\{\tilde{v}_{\ell}^{+}\\}$. Let
$\\{\tilde{v}_{\ell_{k}}^{+}\\}\subset\\{\tilde{v}_{\ell}^{+}\\}$ be any
sequence. Since $\\{\tilde{v}_{\ell_{k}}^{+}\\}$ is bounded in
$W^{1,p}(\Omega_{r}^{+})$ for any $r>0$, using a diagonal argument in
$\\{\tilde{v}_{\ell_{k}}^{+}\\}$, we get a subsequence, we also denote it by
$\\{\tilde{v}_{\ell_{k}}^{+}\\}$, which converges weakly in
$W^{1,p}(\Omega_{r}^{+})$ and strongly in $L^{p}(\Omega_{r}^{+})$ to a
function $v^{+}\in W^{1,p}(\Omega_{\infty}^{+})$. By (1.8) we have

$\int_{\Omega_{r}^{+}}|\tilde{v}_{\ell}^{+}|^{p}=\int_{\Omega_{\ell}^{-}\setminus\Omega_{\ell-r}}|u_{\ell}|^{p}=1-\int_{\Omega_{\ell-r}^{-}}|u_{\ell}|^{p}-\int_{\Omega_{\ell}+}|u_{\ell}|^{p}\geq
1-C\alpha^{[r]}-\int_{\Omega_{\ell}+}|u_{\ell}|^{p}.$ (5.40)

Clearly the integral on the right hand side goes to $0$ as
$\ell\rightarrow\infty$ by Proposition 4.1. Setting $\ell=\ell_{k}$ in (5.40)
and taking the limit $\ell_{k}\rightarrow\infty$, we get

$\int_{\Omega_{r}^{+}}|v^{+}|^{p}\geq 1-C\alpha^{[r]}$ (5.41)

We conclude that $\int_{\Omega_{\infty}^{+}}|v^{+}|^{p}=1$ by taking $r$ to
infinity. Now by (1.10) and Fatou’s lemma, we have

$\nu_{\infty}^{+}=\lim_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell})=\lim_{\ell\rightarrow\infty}\int_{\Omega_{\ell}}|A\nabla
u_{\ell}\cdot\nabla u_{\ell}|^{\frac{p}{2}}\\\
\geq\limsup_{k\rightarrow\infty}\int_{\Omega_{r}^{+}}|A\nabla\tilde{v}_{\ell_{k}}^{+}\cdot\nabla\tilde{v}_{\ell_{k}}^{+}|^{\frac{p}{2}}\geq\int_{\Omega_{r}^{+}}|A\nabla
v^{+}\cdot\nabla v^{+}|^{\frac{p}{2}}.$ (5.42)

Hence from (5.41) and (5.42) we deduce that
$\int_{\Omega_{\infty}^{+}}|A\nabla v^{+}\cdot\nabla
v^{+}|^{\frac{p}{2}}\leq\nu_{\infty}^{+}$. The reverse inequality follows from
(1.9) and we conclude that $v^{+}$ is a normalised minimizer realizing
$\nu_{\infty}^{+}$. Since the minimizer is unique upto multiply by a constant,
it necessarily coincides with $\tilde{u}^{+}$. Now we define the following
function on $(0,\infty)$,

$f(r)=\limsup_{k\rightarrow\infty}\int_{\Omega_{r}^{+}}|A\nabla\tilde{v}_{\ell_{k}}^{+}\cdot\nabla\tilde{v}_{\ell_{k}}^{+}|^{\frac{p}{2}}-\int_{\Omega_{r}^{+}}|A\nabla
v^{+}\cdot\nabla v^{+}|^{\frac{p}{2}}.$

From (5.42) it follows that $f$ is non-negative and non-decreasing. On the
other hand, $f(r)\rightarrow 0$ as $r\rightarrow\infty$. Hence $f$ must be
identically zero on $(0,\infty)$. This implies $\tilde{v}_{\ell_{k}}^{+}$
converges to $\tilde{u}^{+}$ strongly in $W^{1,p}(\Omega_{r}^{+})$ for any
$r>0$. The convergence for the whole family $\\{\tilde{v}_{\ell}^{+}\\}$
follows from the uniqueness of the all possible limits.

(ii) If $A$ satisfies the symmetry (S), it is easy to check that
$u_{\ell}(x_{1},X_{2})=u_{\ell}(-x_{1},-X_{2})$. Using this, a similar
computation gives the required result for this case. ∎

##  6\. Convergence of eigenvalues using Krasnoselskii genus

In this section we prove the convergence results for higher eigenvalues
(Theorem 1.4 and Theorem 1.5) using the Krasnoselskii genus. The following
lemma shows that for $p=2$, the $k$-th Krasnoselskii eigenvalue
$\beta_{k}(\Omega_{\ell})$ of problem (1.4) is same as the $k$-th eigenvalue
$\lambda_{M}^{k}(\Omega_{\ell})$. This fact may be known, but for the sake of
completeness we include the proof here.

######  Lemma 6.1.

Let for any $k\in\mathbb{N}$, $\beta_{k}(\Omega_{\ell})$ be defined as in
Theorem _2.2_ with the closed and symmetric $C^{1,1}$-manifold

$M_{\ell}=\left\\{u\in V(\Omega_{\ell}):\int_{\Omega_{\ell}}u^{2}=1\right\\}$

of the Banach space $V(\Omega_{\ell})$ and with the even functional

$E_{\ell}(u)=\int_{\Omega_{\ell}}A\nabla u\cdot\nabla u$

on $M_{\ell}$. Then $\beta_{k}(\Omega_{\ell})=\lambda_{M}^{k}(\Omega_{\ell})$,
where $\lambda_{M}^{k}(\Omega_{\ell})$ is the $k$-th eigenvalue of the problem
(1.4) for $p=2$.

###### Proof.

Let $\left\\{u_{\ell}^{1},u_{\ell}^{2},\cdots u_{\ell}^{k}\right\\}$ be the
first $k$ normalised eigenfunctions of the problem (1.4). It is well known
that $u_{\ell}^{i}$’s are mutually orthogonal and

$\lambda_{M}^{k}(\Omega_{\ell})=\underset{\underset{||u||_{L^{2}}=1}{u\in\mathrm{sp}\\{u_{\ell}^{1},\cdots
u_{\ell}^{k}\\}}}{\mathrm{max}}\,E_{\ell}(u)=\underset{\underset{\mathrm{dim}V=k}{V\subset
V(\Omega_{\ell})}}{\mathrm{min}}\,\underset{\underset{||u||_{L^{2}}=1}{u\in
V}}{\mathrm{max}}\,E_{\ell}(u).$ (6.1)

Consider the subspace
$V_{0}=\mathrm{sp}\left\\{u_{\ell}^{1},u_{\ell}^{2},\cdots,u_{\ell}^{k}\right\\}$
of $V(\Omega_{\ell})$. Define a closed symmetric subset $A_{0}$ of $M_{\ell}$
as

$A_{0}=V_{0}\cap
M_{\ell}=\left\\{\sum_{i=1}^{k}\alpha_{i}u_{\ell}^{i}:\int_{\Omega_{\ell}}\left(\sum_{i=1}^{k}\alpha_{i}u_{\ell}^{i}\right)^{2}=1\right\\}=\left\\{\sum_{i=1}^{k}\alpha_{i}u_{\ell}^{i}:\sum_{i=1}^{k}\alpha_{i}^{2}=1\right\\}.$

Clearly, $V_{0}\cong\mathbb{R}^{k}$ and $A_{0}$ is homeomorphic to the unit
sphere in $\mathbb{R}^{k},$ and therefore Proposition 2.1 gives
$\gamma(A_{0})=k$. Now, definition of $\beta_{k}(\Omega_{\ell})$ gives

$\beta_{k}(\Omega_{\ell})\leq\underset{u\in
A_{0}}{\mathrm{sup}}\,E_{\ell}(u)=\underset{\underset{||u||_{L^{2}}=1}{u\in\mathrm{sp}\\{u_{\ell}^{1},\cdots
u_{\ell}^{k}\\}}}{\mathrm{sup}}\,E_{\ell}(u)=\lambda_{M}^{k}(\Omega_{\ell}).$

For the reverse inequality, first we fix $\epsilon>0$. $\exists$
$A_{0}\in\mathcal{F}_{k}$ such that $\underset{u\in
A_{0}}{\mathrm{sup}}E_{\ell}(u)<\beta_{k}(\Omega_{\ell})+\epsilon$. Clearly by
(2.2) and (2.3), $(u,v)_{A}=\int_{\Omega_{\ell}}A\nabla u\cdot\nabla v$ is an
equivalent inner product on $V(\Omega_{\ell})$. By Propotion 2.1, let
$\\{w_{1},w_{2},\cdots w_{k}\\}$ be a set of orthogonal (w.r.t
$(\cdot,\cdot)_{A}$) vectors in $A_{0}$.
Let $V_{0}=\mathrm{sp}\left\\{w_{1},w_{2},\cdots w_{k}\right\\}$. For any
$u=\alpha_{1}w_{1}+\cdots+\alpha_{k}w_{k}\in V_{0}$ with
$||u||_{L^{2}(\Omega_{\ell})}=1$, we have

$E_{\ell}(u)=\int_{\Omega_{\ell}}A\nabla u\cdot\nabla
u=\int_{\Omega_{\ell}}A\nabla(\sum_{i=1}^{k}\alpha_{i}w_{i})\cdot\nabla(\sum_{i=1}^{k}\alpha_{i}w_{i})=\sum_{i=1}^{k}\alpha_{i}^{2}E_{\ell}(w_{i})\leq\underset{1\leq
i\leq k}{\mathrm{max}}E_{\ell}(w_{i}).$ (6.2)

Hence from (6.2)

$\lambda_{M}^{k}(\Omega_{\ell})\leq\underset{\underset{||u||_{L^{2}}=1}{u\in
V_{0}}}{\mathrm{max}}E_{\ell}(u)\leq\underset{1\leq i\leq
k}{\mathrm{max}}\,E_{\ell}(w_{i})\leq\underset{u\in
A_{0}}{\mathrm{sup}}\,E_{\ell}(u)<\beta_{k}(\Omega_{\ell})+\epsilon.$ (6.3)

Since $\epsilon$ is arbitrary, this completes the proof. ∎

######  Corollary 2.

For $p>2,$ we have $\beta_{1}(\Omega_{\ell})=\lambda_{M}^{1}(\Omega_{\ell})$.

###### Proof of Theorem 1.4.

First we consider the even case, i.e. the $2k$-th eigenvalue of the problem
(1.4). Since $A$ satisfies the symmetry (S),
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+})=\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{-})$,
holds $\forall\,\ell>0$ and Remark 1, Lemma 4.2 and Theorem 1.2 together gives

$\lim_{\ell\rightarrow\infty}\lambda_{M}^{1}(\Omega_{\ell})=\lim_{\ell\rightarrow\infty}\tilde{\lambda}_{M}^{1}(\Omega_{\ell}^{+}).$
(6.4)

Again we have
$\lambda_{M}^{1}(\Omega_{\ell})\leq\tilde{\lambda}_{M}^{1}(\Omega_{\ell_{1}}^{+})$,
$\forall\ell_{1}\leq\ell$ by considering the test function $u\in
V(\Omega_{\ell})$ in (2.9) as
$u(x_{1},X_{2})=\tilde{u}_{\ell_{1}}^{+}(x_{1}+\ell,X_{2})$ on
$\Omega_{\ell}^{-}\setminus\Omega_{\ell-\ell_{1}}^{-}$ and $0$ on
$\Omega_{\ell-\ell_{1}}^{-}\cup\Omega_{\ell}^{+}$, where
$\tilde{u}_{\ell_{1}}^{+}$ is the eigenfunction corresponding to
$\tilde{\lambda}_{M}^{1}(\Omega_{\ell_{1}}^{+})$ (see Remark 2). Let
$\epsilon>0$. Then $\exists$ $\ell_{0}>0$ such that for $1\leq i\leq k$

$\tilde{\lambda}_{M}^{1}(\Omega_{\frac{i}{k}\ell}^{+})<\lambda_{M}^{1}(\Omega_{\ell})+\epsilon,\quad\forall\
\frac{\ell}{k}>\ell_{0}.$ (6.5)

For $1\leq i\leq k$ and $\ell\geq\ell_{0}$, we define the following functions
on $\Omega_{\ell}$:

$h_{i}^{-}(x_{1},X_{2})=\begin{cases}\tilde{u}_{\frac{i}{k}\ell}^{+}(x_{1}+\ell,X_{2})&\text{on
$\Omega_{\ell}^{-}\setminus\Omega_{(1-\frac{i}{k})\ell}^{-}$}\\\
0&\text{otherwise,}\end{cases}$ (6.6)
$h_{i}^{+}(x_{1},X_{2})=\begin{cases}\tilde{u}_{\frac{i}{k}\ell}^{-}(x_{1}-\ell,X_{2})&\text{on
$\Omega_{\ell}^{+}\setminus\Omega_{(1-\frac{i}{k})\ell}^{+}$}\\\
0&\text{otherwise,}\end{cases}$ (6.7)

where $\tilde{u}_{\frac{i}{k}\ell}^{\pm}$ are defined in Remark 2 for $p=2$.
Let $u_{2i-1}=h_{i}^{-}$ and $u_{2i}=h_{i}^{+}$, where $1\leq i\leq k$ and we
define the subspace
$V_{0}=\mathrm{sp}\left\\{u_{1},u_{2},\cdots,u_{2k}\right\\}$ of
$V(\Omega_{\ell})$. Consider the closed symmetric subset $A_{0}=V_{0}\cap
M_{\ell}$, where $M_{\ell}$ is the $C^{1,1}$-manifold defined in Lemma 6.1.
Clearly $\gamma(A_{0})=2k$. Now for any $u=\sum_{j=1}^{2k}\alpha_{j}u_{j}\in
A_{0}$,

$\int_{\Omega_{\ell}}A\nabla u\cdot\nabla
u=\sum_{j=1}^{2k}\alpha_{j}^{2}\int_{\Omega_{\ell}}A\nabla u_{j}\cdot\nabla
u_{j}+2\sum_{1\leq i<j\leq 2k}\alpha_{i}\alpha_{j}\int_{\Omega_{\ell}}A\nabla
u_{i}\cdot\nabla u_{j}.$ (6.8)

For $1\leq j\leq 2k$, we have

$\int_{\Omega_{\ell}}A\nabla u_{j}\cdot\nabla
u_{j}=\begin{cases}\tilde{\lambda}_{M}^{1}(\Omega_{\frac{j+1}{2k}\ell}^{+})\int_{\Omega_{\ell}}u_{j}^{2}&\text{if
$j$ is odd,}\\\
\tilde{\lambda}_{M}^{1}(\Omega_{\frac{j}{2k}\ell}^{+})\int_{\Omega_{\ell}}u_{j}^{2}&\text{if
$j$ is even,}\end{cases}$ (6.9)

and for $1\leq i<j\leq 2k$,

$\int_{\Omega_{\ell}}A\nabla u_{i}\cdot\nabla
u_{j}=\begin{cases}\tilde{\lambda}_{M}^{1}(\Omega_{\frac{j+1}{2k}\ell}^{+})\int_{\Omega_{\ell}}u_{i}u_{j}&\text{if
$j$ is odd,}\\\
\tilde{\lambda}_{M}^{1}(\Omega_{\frac{j}{2k}\ell}^{+})\int_{\Omega_{\ell}}u_{i}u_{j}&\text{if
$j$ is even.}\end{cases}$ (6.10)

Using (6.9), (6.10) and (6.5) in (6.8) we get

$\int_{\Omega_{\ell}}A\nabla u\cdot\nabla
u\leq\left(\lambda_{M}^{1}(\Omega_{\ell})+\epsilon\right)\left[\sum_{j=1}^{2k}\alpha_{j}^{2}\int_{\Omega_{\ell}}u_{j}^{2}+2\sum_{1\leq
i<j\leq 2k}\alpha_{i}\alpha_{j}\int_{\Omega_{\ell}}u_{i}u_{j}\right]\\\
=\left(\lambda_{M}^{1}(\Omega_{\ell})+\epsilon\right)\int_{\Omega_{\ell}}u^{2}=\lambda_{M}^{1}(\Omega_{\ell})+\epsilon.$
(6.11)

Finally, from the definition of $\beta_{2k}(\Omega_{\ell})$ we obtain

$\lambda_{M}^{1}(\Omega_{\ell})\leq\lambda_{M}^{2k}(\Omega_{\ell})=\beta_{2k}(\Omega_{\ell})\leq\underset{u\in
A_{0}}{\mathrm{sup}}\int_{\Omega_{\ell}}A\nabla u\cdot\nabla
u\leq\lambda_{M}^{1}(\Omega_{\ell})+\epsilon.$

Since $\epsilon$ is arbitrary, (1.13) is true for $k$ is even. Convergence of
odd eigenvalues follows because

$\lambda_{M}^{2k-2}(\Omega_{\ell})\leq\lambda_{M}^{2k-1}(\Omega_{\ell})\leq\lambda_{M}^{2k}(\Omega_{\ell}).$

∎

###### Proof of Theorem 1.5.

Let $h_{1}^{\pm}$ is defined as in (6.6) and (6.7) with $k=1$. Define the
closed and symmetric subset $A_{0}=V_{0}\cap M_{\ell}$ of genus $2$, where
$V_{0}=\mathrm{sp}\\{h_{1}^{-},h_{1}^{+}\\}$. Since $h_{1}^{-}$ and
$h_{1}^{+}$ have disjoint support, for any $u=\alpha h_{1}^{-}+\beta
h_{1}^{+}\in A_{0}$, we have

$\int_{\Omega_{\ell}}|A\nabla u\cdot\nabla
u|^{\frac{p}{2}}=\alpha^{p}\int_{\Omega_{\ell}}|A\nabla h_{1}^{-}\cdot\nabla
h_{1}^{-}|^{\frac{p}{2}}+\beta^{p}\int_{\Omega_{\ell}}|A\nabla
h_{1}^{+}\cdot\nabla h_{1}^{+}|^{\frac{p}{2}}.$

An analogous calculation as in the proof of Theorem 1.4 gives the rest of the
proof. ∎

Acknowledgement: The third author would like to thank the facilities provided
by IMSc. Part of this work was carried out when the third author was visiting
IMSc. Research of third author is funded by Core Research grant under project
number CRG/2022/007867. The third author would like to thank Prof. Itai
Shafrir and Prof. Michel Chipot for several interesting discussion on the
subject. In fact, the key ideas of the proof of Theorem 1.2 and Theorem 1.3
are due to Prof. Shafrir.

Note: This work is a part of doctoral thesis work of the second author. He
acknowledges the support provided by IIT Kanpur, India and MHRD, Government of
India (GATE fellowship).

## References

  * [1] Claudianor O. Alves, Francisco Julio S. A. Corrˆea, and Michel Chipot. On a class of intermediate local-nonlocal elliptic problems, Topol. Methods Nonlinear Anal., 49(2), 497–509, 2017.
  * [2] M. Chipot and S. Mardare. On correctors for the Stokes problem in cylinders. In Nonlinear phenomena with energy dissipation, volume 29 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 37–52. Gakkotosho, Tokyo, 2008.
  * [3] M. Chipot and A. Rougirel. On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions. Discrete Contin. Dyn. Syst. Ser. B, 1(3), 319–338, 2001.
  * [4] M. Chipot and A. Rougirel. Remarks on the asymptotic behaviour of the solution to parabolic problems in domains becoming unbounded. In Proceedings of the Third World Congress of Nonlinear Analysts, Part 1 (Catania, 2000), volume 47, pages 3–11, 2001.
  * [5] M. Chipot and A. Rougirel. On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded. Commun. Contemp. Math., 4(1), 15–44, 2002.
  * [6] Michel Chipot. On the asymptotic behaviour of some problems of the calculus of variations. J. Elliptic Parabol. Equ., 1, 307–323, 2015.
  * [7] Michel Chipot. On some elliptic problems in unbounded domains. Chinese Ann. Math. Ser. B, 39(3), 597–606, 2018.
  * [8] Michel Chipot, Juan Dávila, and Manuel del Pino. On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains. J. Fixed Point Theory Appl., 19(1), 205–213, 2017.
  * [9] Michel Chipot and Sorin Mardare. Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction. J. Math. Pures Appl. (9), 90(2), 133–159, 2008.
  * [10] Michel Chipot and Sorin Mardare. The Neumann problem in cylinders becoming unbounded in one direction. J. Math. Pures Appl. (9), 104(5), 921–941, 2015.
  * [11] Michel Chipot, Aleksandar Mojsic, and Prosenjit Roy. On some variational problems set on domains tending to infinity. Discrete Contin. Dyn. Syst., 36(7), 3603–3621, 2016.
  * [12] Michel Chipot and Arnaud Rougirel. On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded. Trans. Amer. Math. Soc., 360(7), 3579–3602, 2008.
  * [13] Michel Chipot, Prosenjit Roy, and Itai Shafrir. Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity. Asymptot. Anal., 85(3-4), 199–227, 2013.
  * [14] Michel Chipot and Yitian Xie. On the asymptotic behaviour of the p-Laplace equation in cylinders becoming unbounded. In Nonlinear partial differential equations and their appli- cations, volume 20 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 16–27. Gakkotosho, Tokyo, 2004.
  * [15] Michel Chipot and Yitian Xie. Asymptotic behavior of nonlinear parabolic problems with periodic data. In Elliptic and parabolic problems, volume 63 of Progr. Nonlinear Differential Equations Appl., pages 147–156. Birkhäuser, Basel, 2005.
  * [16] Michel Chipot and Stephanie Zube. On the asymptotic behaviour of the pure Neumann problem in cylinder-like domains and its applications. Asymptot. Anal., 108(3), 163–185, 2018.
  * [17] Indranil Chowdhury, Gyula Csató, Prosenjit Roy, and Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete Contin. Dyn. Syst., 41(6), 2993–3020, 2021.
  * [18] Indranil Chowdhury and Prosenjit Roy. On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity. Commun. Contemp. Math., 19(5), 1650035, 21, 2017.
  * [19] Indranil Chowdhury and Prosenjit Roy. On fractional Poincaré inequality for unbounded domains with finite ball conditions: counter example. Nonlinear Anal., 228, Paper No. 113189, 16, 2023.
  * [20] Pavel Drábek and Stephen B. Robinson. Resonance problems for the p-Laplacian. J. Funct. Anal., 169(1), 189–200, 1999.
  * [21] Luca Esposito, Prosenjit Roy, and Firoj Sk. On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded. Asymptot. Anal., 123(1-2), 79–94, 2021.
  * [22] S. Kesavan. Topics in functional analysis and applications. John Wiley & Sons, Inc., New York, 1989.
  * [23] An Lê. Eigenvalue problems for the p-Laplacian. Nonlinear Anal., 64(5), 1057–1099, 2006.
  * [24] Hervé Le Dret and Amira Mokrane. On problems in the calculus of variations in increasingly elongated domains. Chinese Ann. Math. Ser. B, 39(2), 163–182, 2018.
  * [25] Peter Lindqvist. A nonlinear eigenvalue problem. In Topics in mathematical analysis, volume 3 of Ser. Anal. Appl. Comput., pages 175–203. World Sci. Publ., Hackensack, NJ, 2008.
  * [26] Vladimir Lotoreichik and Jonathan Rohleder. Eigenvalue inequalities for the Laplacian with mixed boundary conditions. J. Differential Equations, 263(1), 491–508, 2017.
  * [27] Michael Struwe. Variational methods. Springer-Verlag, Berlin, 1990. Applications to nonlinear partial differential equations and Hamiltonian systems.
  * [28] Karen Yeressian. Asymptotic behavior of elliptic nonlocal equations set in cylinders. Asymptot. Anal., 89(1-2), 21–35, 2014.