#  Generalized eigenproblem and nonlinear elliptic system with nonlinear
boundary conditions

###### Abstract.

We will study solvability of nonlinear second-order elliptic system of partial
differential equations with nonlinear boundary conditions. We study the
generalized Steklov–Robin eigensystem (with possibly matrices weights) in
which the spectral parameter is both in the system and on the boundary. We
prove the existence of solutions for nonlinear system when both nonlinearities
in the differential system and on the boundary interact, in some sense, with
the generalized spectrum of Steklov-Robin. The method of proof makes use of
Leray-Schauder degree.

##  1\. Notations definitions

To put our results into the context, we have collected in this shore section
some relevant notations and definitions for our purposes. Throughout this
work, $H^{1}_{0},$ $H^{1}(\Omega)$ denotes the usual real Sobolev space of
functions on $\Omega$.

  * •

$k\in\mathbb{N}$

  * •

$H(\Omega)=H^{1}(\Omega)\times H^{1}(\Omega)\times\cdots\times
H^{1}(\Omega)\times H^{1}(\Omega)=[H^{1}(\Omega)]^{k}$

  * •

$H_{0}(\Omega)=H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)\times\cdots\times
H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)=[H^{1}_{0}(\Omega)]^{k}$

  * •

$L^{p}_{k}(\Omega)=L^{p}(\Omega)\times L^{p}(\Omega)\times\cdots\times
L^{p}(\Omega)\times L^{p}(\Omega)=[L^{p}(\Omega)]^{k}$

  * •

$L^{p}_{k}(\partial\Omega)=L^{p}(\partial\Omega)\times
L^{p}(\partial\Omega)\times\cdots\times L^{p}(\partial\Omega)\times
L^{p}(\partial\Omega)=[L^{p}(\partial\Omega)]^{k}$

  * •

$[W^{2}_{p}(\Omega)]^{k}=W^{2}_{p}(\Omega)\times
W^{2}_{p}(\Omega)\times\cdots\times W^{2}_{p}(\Omega)\times W^{2}_{p}(\Omega)$

######  Definition 1.1.

Cooperative-plus Cooperative-plus matrix means that all the entries of the
matrix are non-negative

##  2\. Introduction

This paper is concerned existence of (weak) solutions to the following
nonlinear elliptic system

$\begin{gathered}-\Delta U+A(x)U=f(x,U)\quad\text{in }\Omega,\\\
\frac{\partial U}{\partial\nu}+\Sigma(x)U=g(x,U)\quad\text{on
}\partial\Omega,\end{gathered}$ (2.1)

where $\Omega\subset\mathbb{R}^{N}$ , $N\geq 2$ is a bounded domain with
boundary $\partial\Omega$ of class $C^{0,1}$,

$U=[u_{1},\dots,u_{k}]^{T}\in H(\Omega)$ and the matrix

$A(x)=\left[\begin{array}[]{cccc}a_{11}(x)&a_{12}(x)&\cdots&a_{1k}(x)\\\
a_{21}(x)&a_{22}(x)&\cdots&a_{2k}(x)\\\ \vdots&\vdots&\ddots&\vdots\\\
a_{k1}(x)&a_{k2}(x)&\cdots&a_{kk}(x)\end{array}\right]$

Verifies the following conditions:

  1. $(A1$)

The functions $a_{ij}:\Omega\to\mathbb{R},$ $a_{ij}(x)\geq
0,~{}~{}\forall~{}i,j=1,\cdots,k,~{}~{}x\in\Omega$ with strict inequality on a
set of positive measure of $\Omega$.

  2. $(A2)$

$A(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$ almost
everywhere $x\in\Omega,$ and $A(x)$ is positive definite on a set of positive
measure of $\Omega$ with $a_{ij}\in
L^{p}(\Omega)~{}~{}\forall~{}i,j=1,\cdots,k,$ for $p>\frac{N}{2}$ when $N\geq
3$, and $p>1$ when $N=2$

The function $F:\bar{\Omega}\times\mathbb{R}^{k}\to\mathbb{R}^{k},$

$f(x,U)=f(x,u_{1},\cdots,u_{k})=\begin{bmatrix}f_{1}(x,U)\\\\[3.00003pt]
\vdots\\\ f_{k}(x,U)\end{bmatrix}$

Satisfies the following conditions:

  1. $(F1)$

$F$ is Carathédory function

  2. $(F2)$

There exist constants $b1>0,$ such that

$|f(x,U)|\leq b_{1}(1+||U||_{L^{p}_{k}(\Omega)}^{q_{1}-1}),~{}~{}{\rm
with}~{}~{}2\leq q_{1}\leq\frac{2N}{N-2},$

where $||U||=\displaystyle\sum_{i=1}^{k}||u_{i}||_{L^{p}(\Omega)}$

$\partial/\partial\nu:=\nu\cdot\nabla$ is the outward (unit) normal derivative
on $\partial\Omega$, The matrix $\Sigma$ is

$\Sigma(x)=\left[\begin{array}[]{cccc}\sigma_{11}(x)&\sigma_{12}(x)&\cdots&\sigma_{1k}(x)\\\
\sigma_{21}(x)&\sigma_{22}(x)&\cdots&\sigma_{2k}(x)\\\
\vdots&\vdots&\ddots&\vdots\\\
\sigma_{k1}(x)&\sigma_{k2}(x)&\cdots&\sigma_{kk}(x)\end{array}\right]$

Verifies the following conditions:

  1. $(S1)$

The functions $\sigma_{ij}:\partial\Omega\to\mathbb{R},$ $\sigma_{ij}(x)\geq
0,~{}~{}\forall~{}i,j=1,\cdots,k~{}~{}x\in\partial\Omega$

  2. $(S2)$

$\Sigma(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$
almost everywhere $x\in\partial\Omega,$ and $\Sigma(x)$ is positive definite
on a set of positive measure of $\partial\Omega$ with $\sigma_{ij}\in
L^{q}(\partial\Omega)~{}~{}\forall~{}i,j=1,\cdots,k,$ for $q\geq N-1$ when
$N\geq 3$, and $q>1$ when $N=2$

######  Remark 2.1.

  3. (1)

Conditions $(A2),(S2)$ are equivalent to

$\int_{\Omega}\langle
A(x)U,U\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)U,U\rangle\,dx>0$

  4. (2)

We can put condition $(S2)$ in $A$ and condition $(A2)$ in $\Sigma$ i.e.;
interchange conditions in $A$ and $\Sigma$

$G:\partial\Omega\times\mathbb{R}^{k}\to\mathbb{R}^{k},$

$g(x,U)=g(x,u_{1},\cdots,u_{k})=\begin{bmatrix}g_{1}(x,U)\\\\[3.00003pt]
\vdots\\\ g_{k}(x,U)\end{bmatrix}$

satisfies the following conditions

  1. $(G1)$

$G$ is Carathédory function

  2. $(G2)$

There exist constants $a_{1}>0$ such that

$|g(x,U)|\leq a_{1}(1+||U||^{q_{2}-1}_{L^{p}_{k}(\partial\Omega)})~{}~{}{\rm
with}~{}~{}2\leq q_{2}\leq\frac{2N}{N-2},$

where

$||U||{L^{p}_{k}(\partial\Omega)}=\sum_{i=1}^{k}||u_{i}||_{L^{p}(\partial\Omega)}.$

  3. $(G3)$

And for every constant $r>0$ there is a constant $K=K(r)$ such that

$|g(x,U)-g(x,V)|\leq K(|x-y|+|U-V|)$

for all $x,y\in\partial\Omega$ and all $U,V\in H(\Omega)$ with $|U|\leq r,$
and $|V|\leq r$

Where the nonlinear reaction-function $f(x,U)$ and nonlinearity on the
boundary $g(x,U)$ interact, in some sense, with the generalized Steklov-Robin
spectrum of the following linear system problem (with possibly matrices
$(M,P)$-weights)

$\begin{gathered}-\Delta U+A(x)U=\mu M(x)U\quad\text{in }\Omega,\\\
\frac{\partial U}{\partial\nu}+\Sigma(x)U=\mu P(x)U\quad\text{on
}\partial\Omega,\end{gathered}$ (2.2)

Where

$M(x)=\left[\begin{array}[]{cccc}m_{11}(x)&m_{12}(x)&\cdots&m_{1k}(x)\\\
m_{21}(x)&m_{22}(x)&\cdots&m_{2k}(x)\\\ \vdots&\vdots&\ddots&\vdots\\\
m_{k1}(x)&m_{k2}(x)&\cdots&m_{kk}(x)\end{array}\right]$

and

$P(x)=\left[\begin{array}[]{cccc}\rho_{11}(x)&\rho_{12}(x)&\cdots&\rho_{1k}(x)\\\
\rho_{21}(x)&\rho_{22}(x)&\cdots&\rho_{2k}(x)\\\
\vdots&\vdots&\ddots&\vdots\\\
\rho_{k1}(x)&\rho_{k2}(x)&\cdots&\rho_{kk}(x)\end{array}\right]$

Note that the eigensystem (2.2) includes as special cases the weighted Steklov
eigenproblem for a class of elliptic system when ( $\Sigma(x)=M(x)\equiv 0$,
$A(x)\in\mathbb{R}^{2\times{2}}$ and $P(x)=I$ where $I$ is identity matrix
that was considered in [GMR2013]. For scalar case generalized Steklov-Robin
spectrum of the scalar case that was considered in [Mav2012] as well as
weighted Robin-Neumann eigenproblem when ($P(x)\equiv 0$ and $M(x)=I$) that
was considered in [Auc2012] and [GMR2013]. Under conditions of
$A(x),\Sigma(x)$ together with the hypothesis on $\Omega$, we have

  * •

The embedding of $H^{1}(\Omega)$ into $L^{p}(\Omega)$ is continuous for $1\leq
p\leq p(N)$ and compact for $1\leq p\leq p(N)$ where $p(N)=\frac{2N}{N-2}$ if
$N\geq 3$ and $p(N)=\infty$ if $N=2$ (see [EG1992] for more details)

  * •

$\langle x,y\rangle=\sum_{i=1}^{k}x_{i}y_{i}~{}~{}\forall~{}x,y\in H(\Omega)$
then

The main difficulty is to lead with trace operator. In the scalar we would
like to cite papers by [GMR2013], [Mav2012], [MN2012], [MN2011] and [MN2010],
while for the system case, to best of our knowledge, we not able to find any
reference for (2.1), which we describe herein for the first time.

##  3\. Generalized Steklov-Robin eigensystem

In this section, we will study the generalized spectrum that will be used for
the comparison with the nonlinearities in the system (2.1). This spectrum
includes the Steklov, Neumann and Robin spectra, We therefore generalize the
results in [Auc2012], and [GMR2013]
. Consider the elliptic system

$\begin{gathered}-\Delta U+A(x)U=\mu M(x)U\quad\text{in }\Omega,\\\
\frac{\partial U}{\partial\nu}+\Sigma(x)U=\mu P(x)U\quad\text{on
}\partial\Omega,\end{gathered}$ (3.1)

Where

$M(x)=\left[\begin{array}[]{cccc}m_{11}(x)&m_{12}(x)&\cdots&m_{1k}(x)\\\
m_{21}(x)&m_{22}(x)&\cdots&m_{2k}(x)\\\ \vdots&\vdots&\ddots&\vdots\\\
m_{k1}(x)&m_{k2}(x)&\cdots&m_{kk}(x)\end{array}\right]$

and

$P(x)=\left[\begin{array}[]{cccc}\rho_{11}(x)&\rho_{12}(x)&\cdots&\rho_{1k}(x)\\\
\rho_{21}(x)&\rho_{22}(x)&\cdots&\rho_{2k}(x)\\\
\vdots&\vdots&\ddots&\vdots\\\
\rho_{k1}(x)&\rho_{k2}(x)&\cdots&\rho_{kk}(x)\end{array}\right]$

  1. (M1)

Where $M(x)$ is positive definite matrix on $\mathbb{R}^{{k}\times{k}}$,
almost everywhere $x\in\Omega,$ The functions $m_{ij}:\Omega\to\mathbb{R},$
$m_{ij}(x)\geq 0,~{}~{}\forall~{}i,j=1,\cdots,k~{}~{}x\in\Omega,$ with strict
inequality on a set of positive measure of $\Omega$, $m_{ij}\in
L^{p}(\Omega)~{}~{}\forall~{}i,j=1,\cdots,k,$ for $p\geq\frac{N}{2}$ when
$N\geq 3$, and $p>1$ when $N=2.$

  2. (P1)

Where $P(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$
almost everywhere $x\in\partial\Omega,$ and positive define on a set of
positive measure of $\partial\Omega$ with $\rho_{ij}\in
L^{q}(\partial\Omega)~{}~{}\forall i,j=1,\cdots,k,$ for $q\geq N-1$ when
$N\geq 3$, and $q>1$ when $N=2.$

  3. (MP)

And $(m_{ij},\rho_{ij})>0$; that is $m_{ij}(x)\geq 0$ a.e in $\Omega$,
$\rho_{ij}(x)\geq 0$ a.e on $\partial\Omega$ for all $i,j=1,\cdots,k$ such
that

$\int_{\Omega}m_{ij}(x)dx+\int_{\partial\Omega}\rho_{ij}(x)dx>0~{}\forall~{}i,j=1,\cdots,k,$

Assume that $A(x),~{}\Sigma(x),~{}M(x),~{}P(x)$ are Verifies the following
assumption :

######  Assumption 1.

  1. $A(x)$ is positive definite on a set of positive measure of $\Omega$ with $A(x)\in L^{p}(\Omega)$ for $p>\frac{N}{2}$ when $N\geq 3$, and $p>1$ when $N=2.$

  2. OR

$\Sigma(x)$ is positive definite on a set of positive measure of
$\partial\Omega$ with $\Sigma(x)\in L^{q}(\partial\Omega)$ for $q\geq N-1$
when $N\geq 3$, and $q>1$ when $N=2$

  3. AND

$M(x)$ is positive definite on a set of positive measure of $\Omega$ with
$M(x)\in L^{p}(\Omega)$ for $p>\frac{N}{2}$ when $N\geq 3$, and $p>1$ when
$N=2.$

  4. OR

$P(x)$ is positive definite on a set of positive measure of $\partial\Omega$
with $P(x)\in L^{q}(\partial\Omega)$ for $q\geq N-1$ when $N\geq 3$, and $q>1$
when $N=2.$

######  Remark 3.1.

Since $A(x),~{}\Sigma(x),~{}M(x),~{}P(x)$ are satisfied
$(A2),~{}(S2),~{}(M1),~{}(P1)$ respectively and they are Cooperative-plus
matrices, then we can write them in following from (i.e.; eigen-decomposition
of a positive semi-definite matrix or diagonalization)

$A(x)=Q_{A}^{T}(x)D_{A}(x)Q_{A}(x).$
$\Sigma(x)=Q_{\Sigma}^{T}(x)D_{\Sigma}(x)Q_{\Sigma}(x).$
$M(x)=Q_{M}^{T}(x)D_{M}(x)Q_{M}(x).$ $P(x)=Q_{P}^{T}(x)D_{P}(x)Q_{P}(x).$

Where $Q_{J}(x)^{T}Q_{J}(x)=I$ ($Q_{J}^{T}(x)=Q_{J}^{-1}(x)$ i.e.; are
orthogonal matrix ) are the normalized eigenvectors, $I$ is identity matrix,
$D_{J}(x)$ is diagonal matrix in the diagonal of $D_{J}(x)$ the eigenvalues of
$J(x)$ (i.e.;
$D(x)_{J}=diag\left(\lambda_{1}^{J}(x),\cdots,\lambda_{k}^{J}(x)\right)$) and
$J(x)=\\{A(x),~{}\Sigma(x),~{}M(x),~{}P(x)\\}$

######  Remark 3.2.

The weight matrices $M(x)$ and $P(x)$ may vanish on subset of positive
measure.

######  Definition 3.1.

The generalized Steklov-Robin eigensystem is to find a pair
$(\mu,\varphi)\in\mathbb{R}\times H(\Omega)$ with $\varphi\not\equiv 0$ such
that

$\begin{gathered}\int_{\Omega}\triangledown\varphi.\triangledown
U\,dx+\int_{\Omega}\langle
A(x)\varphi,U\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)\varphi,U\rangle\,dx=\\\
\mu\left[\int_{\Omega}\langle
M(x)\varphi,U\rangle\,dx+\int_{\partial\Omega}\langle
P(x)\varphi,U\rangle\,dx\right]~{}~{}\forall~{}~{}U\in
H(\Omega)\end{gathered}$ (3.2)

######  Remark 3.3.

Let $U=\varphi$ in (3.2), if there is such an eigenpair, then $\mu>0$ and

$\int_{\Omega}\langle
M(x)\varphi,\varphi\rangle\,dx+\int_{\partial\Omega}\langle
P(x)\varphi,\varphi\rangle\,dx>0$
$\left(i.e;\sum_{j=1}^{k}\sum_{i=1}^{k}\left(\int_{\Omega}m_{ij}(x)\varphi_{j}\varphi_{i}\,dx+\int_{\partial\Omega}\rho_{ij}(x)\varphi_{j}\varphi_{i}\,dx\right)>0\right)$

and

######  Remark 3.4.

If $\int_{\Omega}\langle
M(x)\varphi,\varphi\rangle\,dx+\int_{\partial\Omega}\langle
P(x)\varphi,\varphi\rangle\,dx=0.$ Then

$\int_{\Omega}|\triangledown\varphi|^{2}\,dx+\int_{\Omega}\langle
A(x)\varphi,\varphi\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)\varphi,\varphi\rangle\,dx=0$

We have that $\int_{\Omega}|\triangledown\varphi|^{2}\,dx=0$ this implies that
$\varphi={\rm~{}constant}$ and

$\int_{\Omega}\langle A(x)\varphi,\varphi\rangle\,dx=0$ this implies that
$A(x)=0,~{}a.e.$ in $\Omega$ and
$\int_{\partial\Omega}\langle\Sigma(x)\varphi,\varphi\rangle\,dx=0,$ then
$\Sigma(x)=0~{}a.e$ on $\partial\Omega.$ So we have that, $\varphi$ would be a
constant vector function; which would contradict the assumptions (assumption
1) imposed on $A(x)$ and $\Sigma(x)$

######  Remark 3.5.

If $A(x)\equiv 0$ and $\Sigma\equiv 0$ then $\mu=0$ is an eigenvalue of the
system (3.1) with eigenfunction $\varphi={\rm constant~{}vector~{}function}$
on $\bar{\Omega},$

It is therefore appropriate to consider the closed linear subspace of
$H(\Omega)$ under assumption 1 defined by

$\mathbb{H}_{(M,P)}(\Omega):=\\{U\in H(\Omega):\int_{\Omega}\langle
M(x)U,U\rangle\,dx+\int_{\partial\Omega}\langle P(x)U,U\rangle\,dx=0\\}.$

Now all the eigenfunctions associated with (3.2) belongs to the
$(A,\Sigma)-$orthogonal complement
$H_{(M,P)}(\Omega):=[\mathbb{H}_{(M,P)}(\Omega)]^{\perp}$

$i.e.;H_{(M,P)}(\Omega)=[\mathbb{H}_{(M,P)}(\Omega)]^{\perp}=$ $\\{U\in
H(\Omega):\int_{\Omega}\langle M(x)U,U\rangle\,dx+\int_{\partial\Omega}\langle
P(x)U,U\rangle\,dx>0~{}{\rm and}~{}~{}\langle
U,V\rangle_{(A,\Sigma)}=0,\forall~{}V\in\mathbb{H}_{(M,P)}(\Omega)\\}$

of this subspace in $H(\Omega)$

######  Definition 3.2.

$\Omega(M):=\\{x\in\Omega:M(x)>0\\}.$
$\partial\Omega(P):=\\{x\in\partial\Omega:P(x)>0\\}.$

We will show that indeed the $H_{(M,P)}(\Omega)$ is subspace of $H(\Omega).$
Let $U,~{}V\in H_{(M.P)}(\Omega)$ and $\alpha\in\mathbb{R}$ we show that
$\alpha U\in H_{(M.P)}(\Omega)$ and $U+V\in H_{(M.P)}(\Omega)$

$\left(\int_{\Omega}\langle M(x)(\alpha U),\alpha
U\rangle\,dx+\int_{\partial\Omega}\langle P(x)(\alpha U),\alpha
U\rangle\,dx\right)=$ $\alpha^{2}\left(\int_{\Omega}\langle
M(x)U,U\rangle\,dx+\int_{\partial\Omega}\langle
P(x)U,U\rangle\,dx\right)=^{U\in H_{(M.P)}}0$

Therefore $\alpha U\in H_{(M.P)}(\Omega)$. Now we show that $U+V\in
H_{(M.P)}(\Omega)$

$\int_{\Omega}\langle M(x)(U+V),(U+V)\rangle\,dx+\int_{\partial\Omega}\langle
P(x)(U+V),(U+V)\rangle\,dx=$ $\int_{\Omega}\langle
M(x)U,U\rangle\,dx+\int_{\partial\Omega}\langle
P(x)U,U\rangle\,dx+\int_{\Omega}\langle
M(x)V,V\rangle\,dx+\int_{\partial\Omega}\langle P(x)V,V\rangle\,dx$
$+2\int_{\Omega}\langle M(x)U,V\rangle\,dx+2\int_{\partial\Omega}\langle
P(x)U,V)\rangle\,dx$

Case 1
If $x\in\partial\Omega(P),$ this implies that $U=0$ on $\partial\Omega(P)$

  * •

$M(x)=0$ on $\Omega$ (Steklov problem)

  * •

$M(x)>0$ on set of positive measure of $\Omega$ this implies that $U=0$ on
$\Omega$ Now we use and 1, 3.1 we have that

$0=\int_{\Omega}\langle M(x)U,U\rangle\,dx=\int_{\Omega}\langle
Q^{T}_{M}(x)D_{M}(x)Q_{M}(x)U,U\rangle\,dx=\int_{\Omega}\langle
D_{M}(x)Q_{M}(x)U,Q_{M}(x)U\rangle\,dx$

Since $D_{M}(x)>0$ this implies that $Q_{M}(x)U=0$ since $Q_{M}(x)$ is
invertible this implies that $U=0$ on $\Omega(M)$

Case 2
If $x\in\Omega(M),$ this implies that $U=0$ on $\Omega$

  * •

$P(x)=0$ on $\partial\Omega$ (usual spectrum problem )

  * •

$P(x)>0$ on set of positive measure of $\partial\Omega$ this implies that
$U=0$ on $\partial\Omega$ Now we use and 1, 3.1 we have that

$0=\int_{\partial\Omega}\langle
P(x)U,U\rangle\,dx=\int_{\partial\Omega}\langle
Q^{T}_{P}(x)D_{P}(x)Q_{P}(x)U,U\rangle\,dx=\int_{\partial\Omega}\langle
D_{P}(x)Q_{P}(x)U,Q_{P}(x)U\rangle\,dx$

Since $D_{P}(x)>0$ this implies that $Q_{P}(x)U=0$ since $Q_{P}(x)$ is
invertible this implies that $U=0$ on $\Omega(P)$

Therefor $U+V\in\mathbb{H}_{(M,P)}(\Omega),$ so we have that
$\mathbb{H}_{(M,P)}(\Omega)$ subspace of $H(\Omega).$ So that

$\mathbb{H}_{(M,P)}(\Omega):=\\{U\in H(\Omega):U=0~{}a.e.;~{}{\rm
in}~{}\Omega(M)~{}{\rm and}~{}\Gamma U=0~{}a.e.;{\rm
on}~{}~{}\partial\Omega(P)\\}$

######  Remark 3.6.

.

  1. (1)

If $M(x)\equiv 0$ in $\Omega$ and $x\in\partial\Omega(P)$, then the subspace
$\mathbb{H}_{(M,P)}(\Omega)=H_{0}(\Omega)$

  2. (2)

If $P(x)\equiv 0$ in $\partial\Omega$ and $x\in\Omega(M)$, then the subspace
$\mathbb{H}_{(M,P)}(\Omega)=\\{0\\}$

Thus, one can split the Hilbert space $H(\Omega)$ as a driect
$(A,\Sigma)-$orthogonal sum in the following way

$H(\Omega)=\mathbb{H}_{(M,P)}(\Omega)\oplus_{(A,\Sigma)}[\mathbb{H}_{(M,P)}(\Omega)]^{\perp}$

######  Remark 3.7.

If $(\mu,\varphi)\in\mathbb{R}\times H(\Omega)$ is an eigenpair of 3.2, then
it follows from the definition of $H_{0}(\Omega)$ that

$<\varphi,V>_{(A,\Sigma)}=\int_{\Omega}[\triangledown\varphi.\triangledown
V+\langle
A(x)\varphi,V\rangle]\,dx+\int_{\partial\Omega}\langle\Sigma(x)\varphi,V\rangle\,dx=0,~{}~{}~{}$

$\forall~{}V\in\mathbb{H}_{(M,P)}(\Omega)$ that is,
$\varphi\in[\mathbb{H}_{(M,P)}(\Omega)]^{\bot}$

  * •

We shall make use in what follows the real Lebesgue space
$L^{q}_{k}(\partial\Omega)$ for $1\leq q\leq\infty$, and of the continuity and
compactness of the trace operator.

$\Gamma:H(\Omega)\to L^{q}_{k}(\partial\Omega)~{}~{}for~{}~{}1\leq
q<\frac{2(n-1)}{n-2}$

is well-defined it is a Lebesgue integrable function with respect to Hausdorff
$N-1$ dimensional measure, sometime we will just use $U$ in place of $\Gamma
U$ when considering the trace of function on $\partial\Omega$ Throughout this
work we denote the $L^{2}_{N}(\partial\Omega)-$ inner product by

$\langle U,V\rangle_{\partial}:=\int_{\partial\Omega}U.V\,dx$

and the associated norm by

$||U||^{2}_{\partial}:=\int_{\partial\Omega}U.U~{}\forall U,V\in H(\Omega)$

(see [KJF1997], [Nec̆1967] and the references therein for more details )

  * •

The trace mapping $\Gamma:H(\Omega)\to L^{2}_{N}(\partial\Omega)$ is compact
(see [Gri1985])

  * •

$\begin{gathered}\langle U,V\rangle_{(M,P)}=\int_{\Omega}\langle
M(x)U,V\rangle\,dx+\int_{\partial\Omega}\langle
P(x)U,V\rangle\,dx\end{gathered}$ (3.3)

defines an inner product for $H(\Omega)$, with associated norm

$\begin{gathered}||U||_{(M,P)}^{2}:=\int_{\Omega}\langle
M(x)U,U\rangle\,dx+\int_{\partial\Omega}\langle
P(x)U,U\rangle\,dx\end{gathered}$ (3.4)

######  Theorem 3.8.

Assume that $A2,S2,C1-C3$ as above. Then we have the following.

i:

The eigensystem (3.1) has a sequence of real eigenvalues

$0<\mu_{1}\leq\mu_{2}\leq\mu_{3}\leq\cdots\leq\mu_{j}\leq\cdots\to\infty~{}as~{}j\to\infty$

each eigenvalue has a finite-dimensional eigenspace.

ii:

The eigenfunctions $\varphi_{j}$ corresponding to the eigenvalues $\mu_{j}$
from an $(A,\Sigma)-$orthogonal and $(M,P)-$orthonormal family in
$[\mathbb{H}_{M,P}(\Omega)]^{\bot}$ ( a closed subspace of $H(\Omega)$

iii:

The normalized eigenfunctions provide a complete $(A,\Sigma)-$orthonormal
basis of $[\mathbb{H}_{M,P}(\Omega)]^{\bot}$. Moreover, each function in
$U\in[\mathbb{H}_{M,P}(\Omega)]^{\bot}$ has a unique representation of the
from

$\begin{gathered}U=\displaystyle\sum^{\infty}_{j=1}c_{j}\varphi_{j}~{}with~{}c_{j}:=\frac{1}{\mu_{j}}\langle
U,\varphi_{j}\rangle_{(A,\Sigma)}=<U,\varphi_{j}>_{(M.P)}\\\
||U||^{2}_{(A,\Sigma)}=\displaystyle\sum^{\infty}_{j=1}\mu_{j}|c_{j}|^{2}\end{gathered}$
(3.5)

In addition,

$||U||^{2}_{(M,P)}=\displaystyle\sum^{\infty}_{j=1}|c_{j}|^{2}$

The following proposition shows the principality of the first eigenvalue
$\mu_{1}.$

######  Proposition 3.9.

The first eigenvalue $\mu_{1}$ is simple if and only if the associated
eigenfunction $\varphi_{1}$ does not changes sign (i.e.; $\varphi_{1}$ is
strictly positive or strictly negative in $\Omega$ .

######  Remark 3.10.

Note that if we have smooth data and $\partial\Omega$ in proposition 3.9, then
the eigenfunction $\varphi_{1}(x)$ on $\partial\Omega$ as well, by the
boundary point lemma (see for example [Eva99]). or Hopf’s Boundary Point
Lemma.

######  Definition 3.3.

Reflexive Space Let $E$ be a Banach space and let $J:E\to E^{**}$ ( Where The
Bidual $E^{**}$ Orthogonality )be the canonical injection from $E$ into
$E^{**}.$ The space $E$ is said to be reflexive if $J$ is surjective, i.e;
$J(E)=E^{**}.$ (See [Bz2011])

######  Theorem 3.11.

Assume that $E$ is a reflexive Banach space and let
$\\{x_{n}\\}_{n=1}^{\infty}\subset E$ be bounded sequences in $E.$ Then there
exists a subsequence $\\{x_{n_{l}}\\}_{l=1}^{\infty}\subset E$ that converges
in the weak topology $\sigma(E,E^{*})$ ( for proof see [Bz2011])

##  4\. Main Results

We will consider special case from equation 2.1 when the $f(x,U)=0$ and when
$M(x)=0$ in equation 3.1 then we have that

$\begin{gathered}-\Delta U+A(x)U=0\quad\text{in }\Omega,\\\ \frac{\partial
U}{\partial\nu}+\Sigma(x)U=g(x,U)\quad\text{on }\partial\Omega,\end{gathered}$
(4.1)

$\begin{gathered}-\Delta U+A(x)U=0\quad\text{in }\Omega,\\\ \frac{\partial
U}{\partial\nu}+\Sigma(x)U=\mu P(x)U\quad\text{on
}\partial\Omega,\end{gathered}$ (4.2)

We obtain solution for the problem (4.1) which the nonlinearities at the
boundary are compared with higher eigenvalues of equation(4.2), we obtain form
TheoremLABEL:th1 problem (4.2) has sequence of real eigenvalues

$0<\mu_{1}\leq\mu_{2}\leq\mu_{3}\leq\cdots\leq\mu_{j}\leq\cdots\to\infty~{}as~{}j\to\infty$

each eigenvalue has a finite-dimensional eigenspace. In this section we impose
conditions on the asymptotic behavior of the ’slopes’ of the boundary
nonlinearities $g(x,U)$ i.e.; on $\frac{\langle g(x,U),U\rangle}{|U|^{2}}$ as
$|U|\to\infty.$ These conditions are of nonuniform type since the asymptotic
ration $\frac{\langle g(x,U),U\rangle}{|U|^{2}}$ need not be (uniformly)
bounded away from consecutive Steklov-Robin eigenvalues. We mention that
results below, the boundary nonlinearity $g(x,U)$ may be replaced by
$g(x,U)+H(x)$ where
$H\in[W^{1-1/p}_{P}(\partial\Omega)]^{k}\subset[C(\partial\Omega)]^{k}.$ We
approach which based in on topological degree theory on suitable boundary-
trace spaces, here in this work we generalization of [MN2011] from Scalar case
to system case. If for the scalar case this is in contrast recent approaches
where variational methods were used for problems with nonlinear conditions,
but in the case of system the variational methods doesn’t work because of the
potentials doesn’t exists as integrations of $f(x,U)$ and $g(x,U)$, since that
potential function the hart of the variational methods therefore doesn’t work.
In case $f(x,U)\neq 0$ the system will be very hard to study even in in the
topological degree methods and comparison with eigenvalues of Steklov-Robin
that the reason we will study the case when $(Fx,U)=0$ and we will leave the
case when $f(x,U)\neq 0$ as the future work.

######  Remark 4.1.

In the scalar when $f(x,U)\neq$ comparison with eigenvalues of Steklov-Robin
used the variational method was done by Mavinga see [Mav2012] of strong
Solution for equation (2.1) in scalar case . For Multiple solution (weak) for
equation ( 2.1) in scalar case and $\Sigma(x)=0$ used variational Methods was
done by Yao see [Yao2014].

######  Remark 4.2.

Let $U\neq 0$, and assume that $(P1)$ hold, the eigenvalue of $P(x)$ are
nonnegative Let the $\lambda_{p}(x)\geq 0$ be the eigenvalue of $P(x)$ from
Rayleigh quotient we have that $\min\lambda_{P}(x)\leq\frac{\langle
P(x)U,U\rangle}{\langle U,U\rangle}$ and $\max\lambda_{P}(x)\geq\frac{\langle
P(x)U,U\rangle}{\langle U,U\rangle}$

###  4.1. Main Theorem

The main result of this chapter is given in the following existence theorem.

######  Theorem 4.3.

Nonuniform non-resonance between consecutive Steklov-Robin eigenvalues Assume
that $(A1),~{}(A2),~{}(S1),~{}(S2)~{},(G1)-(G3)$ and assumption 1 are holds,
assume there exist constants $a,b>0$, and Let
$\alpha(x):=a(\min\lambda_{P}(x))$ and ,$\beta(x)=b(\max\lambda_{P}(x))$ such
that

$\mu_{j}\leq\alpha(x)\leq\liminf_{|U|\to\infty}\frac{\langle
g(x,U),U\rangle}{|U|^{2}}\leq\limsup_{|U|\to\infty}\frac{\langle
g(x,U),U\rangle}{|U|^{2}}\leq\beta(x)\leq\mu_{j+1}$

uniformly for a.e. $x\in\partial\Omega$ with

$\int_{\partial\Omega}(\alpha(x)-\mu_{j})\langle\varphi(x),\varphi(x)\rangle\,dx>0,~{}~{}~{}\forall~{}~{}\varphi\in
E_{j+1}\setminus\\{0\\}$

and

$~{}~{}\int_{\partial\Omega}(\mu_{j+1}-\beta(x))\langle\varphi(x),\varphi(x)\rangle\,dx>0,~{}~{}\forall~{}~{}\varphi\in
E_{j+1}\setminus\\{0\\},$

Where $i\in\mathbb{N},$ $E_{i}$ denotes the Steklov-Robin nullspace (finite-
dimensional) with the Steklov-Robin eigenvalue $\mu_{i}$. Then the nonlinear
system (4.2) has at least one (strong) solution $U\in[W^{2}_{p}(\Omega)]^{k}$

In contrast to some recent approaches in the literature for system which
nonlinear boundary conditions , we first the system in terms of nonlinear
compact perturbations of the identity on appropriate boundary-trace as follows

##  5\. THE HOMOTOPY

Let $\delta$ between and consecutive Steklov-Robin eigenvalues
($i.e.;\mu_{j}<\delta<\mu_{j+1})$ and $\delta$ isn’t eigenvalue of Steklov-
Robin system. system 4.1 equivalently

$\begin{gathered}-\Delta U+A(x)U=0\quad\text{in }\Omega,\\\ \frac{\partial
U}{\partial\nu}+\Sigma(x)U-\delta U=g(x,U)-\delta U\quad\text{on
}\partial\Omega,\end{gathered}$ (5.1)

Let us associate to equation (5.1) the homotopy, with $\lambda\in[0,1]$

$\begin{gathered}-\Delta U+A(x)U=0\quad\text{in }\Omega,\\\ \frac{\partial
U}{\partial\nu}+\Sigma(x)U-\delta P(x)U=\lambda[g(x,U)-\delta
P(x)U]\quad\text{on }\partial\Omega,\end{gathered}$ (5.2)

or equivalent,

$\begin{gathered}-\Delta U+A(x)U=0\quad\text{in }\Omega,\\\ \frac{\partial
U}{\partial\nu}+\Sigma(x)U=(1-\lambda)\delta P(x)U+\lambda g(x,U)\quad\text{on
}\partial\Omega,\end{gathered}$ (5.3)

Note that for $\lambda=0$ we have that we have a linear system which admits
only the trivial solutions since $\delta$ is in the resolvent of the linear
Steklov-Robin problem ( for scalar case see [Mav2012] and [Auc2012] and the
references therein ). Whereas for $\lambda=1$ we have system (4.1) We define
the linear (Steklov-Robin) boundary operator

$\mathbb{L}:Dom(\mathbb{L})\subset[W^{2}_{p}(\Omega)]^{k}\Subset[W^{1-1/p}_{p}(\partial\Omega)]^{k}\to[W^{1-1/p}_{p}(\partial\Omega)]^{k}$

(’$\Subset$’ compact embedding. Here, the compact ‘containment’
$[W^{2}_{p}(\Omega)]^{k}\Subset[W^{1-1/p}_{p}(\partial\Omega)]^{k}$ must be
understood in the sense of the trace (i.e.;
$[W^{2}_{p}(\Omega)]^{k}\hookrightarrow[W^{1-1/p}_{p}(\partial\Omega)]^{k}$ is
a compact linear operator ( see [Eva99] and [GT83] ) ) by

$\begin{gathered}\mathbb{L}U:=\frac{\partial U}{\partial\nu}+\Sigma(x)U-\delta
P(x)U\end{gathered}$ (5.4)

where

$Dom(\mathbb{L}):=\\{U\in[W_{p}^{2}]^{k}:-\Delta U+A(x)U=0~{}a.e.~{}{\rm
in}~{}\Omega\\}$

###  5.1. Nemytskǐi Operator

$\v{N}:[W^{1-1/p}_{p}(\partial\Omega)]^{k}\subset[C(\partial\Omega)]^{k}\to[W^{1-1/p}_{p}(\partial\Omega)]^{k}$

by

$\begin{gathered}\v{N}U:=G(.,U)-\delta P(.)U\end{gathered}$ (5.5)

System 4.1 is then equivalent to finding $U\in Dom(\mathbb{L})$ such that

$\begin{gathered}\mathbb{L}U=\v{N}U\end{gathered}$ (5.6)

Whereas the homotopy system7.2 is equivalent to

$\begin{gathered}\mathbb{L}U=\lambda\v{N}U\,\,\,\,\,U\in
Dom(\mathbb{L})\end{gathered}$ (5.7)

##  6\. Properties of Operators $\mathbb{L},~{}$Ň$~{},\mathbb{L}^{-1}$

I:

$Dom(\mathbb{L}):=\\{U\in[W_{p}^{2}]^{k}:-\Delta U+A(x)U=0~{}a.e.~{}{\rm
in}~{}\Omega\\}$ is a closed linear subspace of $[W_{p}^{2}]^{k}$

###### Proof.

Clearly $Dom(\mathbb{L})\subset[W_{p}^{2}]^{k}$ is not empty (if $U=0$ then
$0\in Dom(\mathbb{L})$) , let $U,W\in Dom(\mathbb{L})$ then we have that
$-\Delta(U+W)+A(x)(U+W)=-\Delta U+A(x)U-\Delta W+A(x)W=0,$ therefore $U+W\in
Dom(\mathbb{L}),$ now let $\alpha\in\mathbb{R}$ such that $-\Delta\alpha
U+\alpha A(x)U=\alpha(-\Delta U+A(x)U)=0$ so that $\alpha U\in
Dom(\mathbb{L}).$ Therefore $Dom(\mathbb{L})$ is linear subspace of
$[W_{p}^{2}]^{k}.$ For the closed let $\\{U_{n}\\}_{n=1}^{\infty}\in
Dom(\mathbb{L}),$ and $U_{n}\to U$ in $[W_{p}^{2}]^{k}.$ then $-\Delta
U_{n}\to-\Delta U$ and $A(x)U_{n}\to A(x)U$ as $n\to\infty$ then $-\Delta
U+A(x)U=0$ so that $U\in Dom(\mathbb{L})$ Therefore $Dom(\mathbb{L})$ is
closed linear subspace of $[W_{p}^{2}]^{k}.$ ∎

II:

$\ker{\mathbb{L}}=0$, onto, and continuous

###### Proof.

Clearly is continuous for the definition Let $U\in\ker{\mathbb{L}}\subset
Dom(\mathbb{L})$ this implies that $-\Delta U+A(x)U)=0$ and by the definition
of $\mathbb{L}$ (5.4) we have that

$\begin{gathered}\mathbb{L}U=0=\frac{\partial
U}{\partial\nu}+\Sigma(x)U-\delta P(x)U\,x\in\partial\Omega\end{gathered}$
(6.1)

Then you have that

$\begin{gathered}-\Delta U+A(x)U=0~{}a.e.~{}{\rm in}~{}\Omega\\\
\frac{\partial U}{\partial\nu}+\Sigma(x)U=\delta
P(x)U\,x\in\partial\Omega\end{gathered}$ (6.2)

system 6.2 has trivial solution. Therefore the $\ker{\mathbb{L}}=0$ i.e.;
$\mathbb{L}$ one-to-one, Since the domain of $\mathbb{L}$ closed linear and
$\mathbb{L}:Dom(\mathbb{L})\to[W^{1-1/p}_{p}(\partial\Omega)]^{k}$ is
continuous, clearly $\mathbb{L}$ is onto. ∎

III:

$\mathbb{L}$ It is a Fredholm operator of index zero

###### Proof.

Since

1:

$\ker{\mathbb{L}}=0$ has a finite dimension; $dim\ker{\mathbb{L}}=1$

2:

$Im{\mathbb{L}}$ closed and has a finite co-dimension (i.e.;
$R({\mathbb{L}})=[W^{1-1/p}_{p}(\partial\Omega)]^{k}$ )
$dim([W^{1-1/p}_{p}(\partial\Omega)]^{k}/R({\mathbb{L}}))=1$ Indeed it is
Fredholm operator of index zero by the definition of Fredholm operator of
index zero. ∎

IV:

The inverse of $\mathbb{K}=\mathbb{L}^{-1}$ exists

###### Proof.

From

II:

the inverse $\mathbb{K}$ exists ∎

V:

$\mathbb{K}$ is a compact

###### Proof.

Owing to the Compactness of the trace operator
$Dom({\mathbb{L}})\hookrightarrow[W^{1-1/p}_{p}(\partial\Omega)]^{k},$ we
deduce that

$\mathbb{K}=\mathbb{L}^{-1}:[W^{1-1/p}_{p}(\partial\Omega)]^{k}\to^{\rm~{}continuous~{}}Dom({\mathbb{L}})\hookrightarrow^{\rm
compact}[W^{1-1/p}_{p}(\partial\Omega)]^{k}$

Since we that left compositions and $T\circ S$ of compact $T$ with continuous
$S$ produce compact operators for the proof see [GT83]. Therefor $\mathbb{K}$
is a compact linear operator from $[W^{1-1/p}_{p}(\partial\Omega)]^{k}$ into
$[W^{1-1/p}_{p}(\partial\Omega)]^{k}.$ ∎

VI:

$\v{N}$ is continuous operator

###### Proof.

Since the function $G$ satisfied $(G3)$ and
$[W^{1-1/p}_{p}(\partial\Omega)]^{k}\subset[C(\partial\Omega)]^{k}$ Through
the surjectivity of the trace operator
$[W^{1-1/p}_{p}(\partial\Omega)]^{k}\to[W^{1-1/p}_{p}(\partial\Omega)]^{k}$
and the embedding
$[W^{1-1/p}_{p}(\partial\Omega)]^{k}\Subset[C(\partial\Omega)]^{k},$ for $p>N$
it follows that nonlinear operator $\v{N}$ is continuous operator. ∎

VII:

The operator $\mathbb{K}\v{N}$ is continuous and compact (i.e.; completely
continuous)

###### Proof.

$\mathbb{K}\v{N}:[W^{1-1/p}_{p}(\partial\Omega)]^{k}\to[W^{1-1/p}_{p}(\partial\Omega)]^{k}$
Since $\mathbb{K}$ is compact from $V:$, and $\v{N}$ is continuous from $VI:$,
then the compositions of $\mathbb{K}\v{N}$ is completely continuous (see
[Eva99]). ∎

Now we will discuss system (5.7). Thus system 5.7 is equivalent to

$\begin{gathered}U=\lambda\mathbb{L}^{-1}\v{N}U\,{\rm
with}\,\lambda\in[0,1]\,\,\,U\in[W^{1-1/p}_{p}(\partial\Omega)]^{k}\,{\rm
or}\\\ U=\lambda\mathbb{K}\v{N}U\,{\rm
with}\,\lambda\in[0,1]\,\,\,U\in[W^{1-1/p}_{p}(\partial\Omega)]^{k}\,{\rm
or}\\\ (I-\lambda\mathbb{K}\v{N})U=0\,{\rm
with}\,\lambda\in[0,1]\,\,\,U\in[W^{1-1/p}_{p}(\partial\Omega)]^{k}\end{gathered}$
(6.3)

Which shows that, for each $\lambda\in[0,1]$, the operator
$\lambda\mathbb{K}\v{N}$ is a nonlinear compact perturbations of the identity
on $[W^{1-1/p}_{p}(\partial\Omega)]^{k}$. It suffices to show that
$\mathbb{K}\v{N}$ has fixed point $U\in[W^{1-1/p}_{p}(\partial\Omega)]^{k}.$
By properties of $\mathbb{K}$, It follows that a fixed point $U\in
Dom(\mathbb{L})$. Hence, $U\in[W^{2}_{p}(\Omega)]^{k},$ and is a (strong)
solution of the nonlinear system (4.1). We show that all possible solutions to
the homotopy (5.2) (equivalently to (7.2) and (5.7)) are uniformly bounded in
$[W^{1-1/p}_{p}(\partial\Omega)]^{k}$ independent of $\lambda\in[0,1]$ (
actually we show that they are bounded in $[W^{2}_{p}(\Omega)]^{k},$) and then
use the topological degree theory to show the existence of strong solution. We
first prove following lemma which provides intermediate a priori estimates.

##  7\. The A Priori Estimates

######  Lemma 7.1.

Assume that $(A1),~{}(A2),~{}(S1),~{}(S2)~{},(G1)-(G3)$ and assumption 1 are
holds, assume there exist constants $a,b>0$, and Let
$\alpha(x):=a(\min\lambda_{P}(x))$ and ,$\beta(x)=b(\max\lambda_{P}(x))$ such
that

$\mu_{j}\leq\alpha(x)\leq\liminf_{|U|\to\infty}\frac{\langle
g(x,U),U\rangle}{|U|^{2}}\leq\limsup_{|U|\to\infty}\frac{\langle
g(x,U),U\rangle}{|U|^{2}}\leq\beta(x)\leq\mu_{j+1}$

uniformly for a.e. $x\in\partial\Omega$ with

$\int_{\partial\Omega}(\alpha(x)-\mu_{j})\langle\varphi(x),\varphi(x)\rangle\,dx>0,~{}~{}~{}\forall~{}~{}\varphi\in
E_{j+1}\setminus\\{0\\}$

and

$~{}~{}\int_{\partial\Omega}(\mu_{j+1}-\beta(x))\langle\varphi(x),\varphi(x)\rangle\,dx>0,~{}~{}\forall~{}~{}\varphi\in
E_{j+1}\setminus\\{0\\},$

Where $i\in\mathbb{N},$ $E_{i}$ denotes the Steklov-Robin nullspace (finite-
dimensional) with the Steklov-Robin eigenvalue $\mu_{i}$. Then all possible
solutions to

$\begin{gathered}-\Delta U+A(x)U=0\quad\text{in }\Omega,\\\ \frac{\partial
U}{\partial\nu}+\Sigma(x)U=(1-\lambda)\delta P(x)U+\lambda g(x,U)\quad\text{on
}\partial\Omega,\end{gathered}$ (7.1)

are (uniformly) bounded in $H(\Omega)$ independently of $\lambda\in[0,1]$

###### Proof.

By the way of contradiction. Assume the lemma doesn’t hold. Then there are
sequences $\\{\lambda_{n}\\}_{n=1}^{\infty}\subset[0,1]$ and
$\\{U_{n}\\}_{n=1}^{\infty}\subset H(\Omega)$, such that
$||U_{n}||_{(A,\Sigma)}\to\infty$ and put $U=U_{n}\,,$ $\lambda=\lambda_{n}$
in 5.2 we have that

$\begin{gathered}-\Delta U_{n}+A(x)U_{n}=0\quad\text{in }\Omega,\\\
\frac{\partial U_{n}}{\partial\nu}+\Sigma(x)U_{n}=(1-\lambda_{n})\delta
P(x)U_{n}+\lambda_{n}g(x,U_{n})\quad\text{on }\partial\Omega,\end{gathered}$
(7.2)

Now multiple the first equation in 7.2 by $V$ and integration by parts we have
that

$\begin{gathered}\int_{\Omega}\triangledown U_{n}.\triangledown
V\,dx+\int_{\Omega}\langle
A(x)U_{n},V\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)U_{n},V\rangle\,dx=\\\
(1-\lambda_{n})\delta\int_{\partial\Omega}\langle
P(x)U_{n},V\rangle\,dx+\lambda_{n}\int_{\partial\Omega}\langle
g(x,U_{n}),V\rangle\,dx~{}~{}\forall~{}~{}V\in H(\Omega)\end{gathered}$ (7.3)

Let $V_{n}=\frac{U_{u}}{||U_{n}||_{(A,\Sigma)}}$ Clearly $V_{n}$ is bounded in
$H(\Omega).$ There exist a subsequence (relabeled) $V_{n}$ which convergent
weakly to $V_{0}$ in $H(\Omega),$ and $V_{n}$ convergent strongly to $V_{0}$
in $L^{2}(\partial\Omega)$. Without loss of generality
$\lambda_{n}\to\lambda_{0}\in[0,1]$ Due to the at most linear growth condition
on the boundary nonlinearity $G$ (See conditions $(G1-G3$), it follows that
$\frac{\langle g(x,U_{n}),V\rangle}{||U_{n}||_{(A,\Sigma)}}$ is bounded in
$L^{2}(\partial\Omega)$ Using the fact that $L^{2}(\partial\Omega)$ is a
reflexive Banach space, by theorem 3.11 we get that $\frac{\langle
g(x,U_{n}),V\rangle}{||U_{n}||_{(A,\Sigma)}}$ converges weakly to $g_{0}$ in
$L^{2}(\partial\Omega)$. Dividing equation (7.3) by $||U_{n}||_{(A,\Sigma)}$

$\begin{gathered}\int_{\Omega}\frac{\triangledown U_{n}.\triangledown
V}{||U_{n}||_{(A,\Sigma)}}\,dx+\int_{\Omega}\frac{\langle
A(x)U_{n},V\rangle}{||U_{n}||_{(A,\Sigma)}}\,dx+\int_{\partial\Omega}\frac{\langle\Sigma(x)U_{n},V\rangle}{||U_{n}||_{(A,\Sigma)}}\,dx=\\\
(1-\lambda_{n})\delta\int_{\partial\Omega}\frac{\langle
P(x)U_{n},V\rangle}{||U_{n}||_{(A,\Sigma)}}\,dx+\lambda_{n}\int_{\partial\Omega}\frac{\langle
g(x,U_{n}),V\rangle}{||U_{n}||_{(A,\Sigma)}}\,dx~{}~{}\forall~{}~{}V\in
H(\Omega)\end{gathered}$ (7.4)

we get that

$\begin{gathered}\int_{\Omega}\triangledown V_{n}.\triangledown
V\,dx+\int_{\Omega}\langle
A(x)V_{n},V\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)V_{n},V\rangle\,dx=\\\
(1-\lambda_{n})\delta\int_{\partial\Omega}\langle
P(x)V_{n},V\rangle\,dx+\lambda_{n}\int_{\partial\Omega}\frac{\langle
g(x,U_{n}),V\rangle}{||U_{n}||_{(A,\Sigma)}}\,dx~{}~{}\forall~{}~{}V\in
H(\Omega)\end{gathered}$ (7.5)

take the limit as $n\to\infty$ we get that

$\begin{gathered}\int_{\Omega}\triangledown V_{0}.\triangledown
V\,dx+\int_{\Omega}\langle
A(x)V_{0},V\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)V_{0},V\rangle\,dx=\\\
(1-\lambda_{0})\delta\int_{\partial\Omega}\langle
P(x)V_{0},V\rangle\,dx+\lambda_{0}\int_{\partial\Omega}\langle
g_{0},V\rangle\,dx~{}~{}\forall~{}~{}V\in H(\Omega)\end{gathered}$ (7.6)

Let $V=V_{0}$ we have that

$\begin{gathered}||V_{0}||_{(A,\Sigma)}^{2}=\int_{\Omega}\triangledown
V_{0}.\triangledown V_{0}\,dx+\int_{\Omega}\langle
A(x)V_{0},V_{0}\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)V_{0},V_{0}\rangle\,dx=\\\
(1-\lambda_{0})\delta\int_{\partial\Omega}\langle
P(x)V_{0},V_{0}\rangle\,dx+\lambda_{0}\int_{\partial\Omega}\langle
g_{0},V_{0}\rangle\,dx~{}~{}\forall~{}~{}V\in H(\Omega)\end{gathered}$ (7.7)

Therefore

$\begin{gathered}||V_{0}||_{(A,\Sigma)}^{2}=(1-\lambda_{0})\delta\int_{\partial\Omega}\langle
P(x)V_{0},V_{0}\rangle\,dx+\lambda_{0}\int_{\partial\Omega}\langle
g_{0},V_{0}\rangle\,dx~{}~{}\forall~{}~{}V\in H(\Omega)\end{gathered}$ (7.8)

Now let $V=V_{n}$ in equation (7.5) we get that

$1=||V_{n}||_{(A,\Sigma)}=(1-\lambda_{n})\delta\int_{\partial\Omega}\langle
P(x)V_{n},V_{n}\rangle\,dx+\lambda_{n}\int_{\partial\Omega}\frac{\langle
g(x,U_{n}),V_{n}\rangle}{||U_{n}||_{(A,\Sigma)}}\,dx$

Taking the limit as $n\to\infty$ and using equation (7.8) and the fact that
$\frac{\langle g(x,U_{n}),V_{n}\rangle}{||U_{n}||_{(A,\Sigma)}}$ converges
weakly to $g_{0}$ in $L^{2}(\partial\Omega)$ we have that

$\begin{gathered}||V_{0}||_{(A,\Sigma)}^{2}=(1-\lambda_{0})\delta\int_{\partial\Omega}\langle
P(x)V_{0},V_{0}\rangle\,dx+\lambda_{0}\int_{\partial\Omega}\langle
g_{0},V_{0}\rangle\,dx=1~{}~{}\forall~{}~{}V\in H(\Omega)\end{gathered}$ (7.9)

######  Claim 7.1.

$V_{0}=0$ which will lead to a contradiction

###### Proof.

From equation (7.9), notice that $V_{0}$ is a weak solution of the linear
equation

$\begin{gathered}-\Delta U+A(x)U=0~{}~{}a.e.~{}~{}{\rm in}~{}\Omega\\\
\frac{\partial U}{\partial\nu}+\Sigma(x)U=(1-\lambda_{0})\delta
P(x)U+\lambda_{0}g_{0}~{}~{}~{}~{}{\rm on}~{}\partial\Omega\end{gathered}$
(7.10)

Let us mention here that equation (7.10) implies that $\lambda_{0}\neq 0.$
Otherwise, Since $\delta$ is in the Steklov-Robin resolvent, we deduce that
$V_{0}=0;$ which contradicts the fact that $||V_{0}||_{(A,\Sigma)}^{2}=1$. In
order to bring out all the properties of the function $V_{0}$, we need to
analyze a little bit more carefully the function $(1-\lambda_{0})\delta
P(x)V_{0}(x)+\lambda_{0}g_{0}(x).$ Let us denote by $k(x)$ the function
defined by

$\begin{gathered}k(x)=\left\\{\begin{array}[]{lr}(1-\lambda_{0})\delta
P(x)V_{0}+\lambda_{0}\frac{g_{0}(x)}{||V_{0}||_{(A,\Sigma)}}&{\rm
if}~{}~{}V_{0}(x)\neq 0\\\ 0&{\rm
if}~{}~{}V_{0}(x)=0\end{array}\right.\end{gathered}$ (7.11)

From the definition of $\delta$ and the condition in theorem 4.3, it turns out
that

$\begin{gathered}\mu_{j}\leq\alpha(x)\leq\langle
k(x)V_{0},V_{0}(x)\rangle\leq\beta(x)\leq\mu_{j+1}~{}~{}{\rm
for}~{}~{}V_{0}(x)\neq 0\end{gathered}$ (7.12)

Therefore $V_{0}$ is a weak solution to the linear equation

$\begin{gathered}\left\\{\begin{array}[]{lr}-\Delta U+A(x)U=0&a.e.~{}~{}{\rm
in}~{}~{}\Omega\\\ \frac{\partial U}{\partial\nu}+\Sigma(x)U=k(x)U&{\rm
on}~{}~{}\partial\Omega\end{array}\right.\end{gathered}$ (7.13)

that is

$\begin{gathered}\int_{\Omega}\triangledown V_{0}.\triangledown
V\,dx+\int_{\Omega}\langle
A(x)V_{0},V\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)V_{0},V\rangle\,dx=\int_{\partial\Omega}\langle
k(x)V_{0},V\rangle\,dx~{}~{}\forall~{}~{}V\in H(\Omega)\end{gathered}$ (7.14)

we claim that this implies that either $V_{0}\in E_{j}$ or $V_{0}\in E_{j+1}$
only ( see Lemma 7.2 below ). Let us assume that for the time being that this
holds and finish the proof.
If $V_{0}\in E_{j},$ then taking $V=V_{0}$ in equation (7.14) we have that

$\mu_{j}\int_{\partial\Omega}\langle
V_{0},V_{0}\rangle\,dx=||V_{0}||^{2}_{(A,\Sigma)}=\int_{\partial\Omega}\langle
k(x)V_{0},V_{0}\rangle\,dx.$

Using equation (7.12) we get that
$\int_{\partial\Omega}(\alpha(x)-\mu_{j})\langle V_{0},V_{0}\rangle\,dx\leq
0$. Since
$\int_{\partial\Omega}(\alpha(x)-\mu_{j})\langle\varphi,\varphi\rangle>0\,dx\leq
0$. for all $\varphi\in E_{j}\setminus\\{0\\},$ we conclude that $V_{0}=0$
which contradicts the fact $||V_{0}||^{2}_{(A,\Sigma)}=1.$
Similarly, if $V_{0}\in E_{j+1}$, then taking $V=V_{0}$ in equation (7.14) we
get that

$\int_{\partial\Omega}(\mu_{j+1}-\beta(x))\langle V_{0},V_{0}\rangle\leq 0$

. Since

$\int_{\partial\Omega}(\mu_{j+1}-\beta(x))\langle\varphi,\varphi\rangle>0\,dx\leq
0$. for all $\varphi\in E_{j+1}\setminus\\{0\\},$ we conclude that $V_{0}=0$
which contradicts the fact $||V_{0}||^{2}_{(A,\Sigma)}=1$ again.
The proof of the claim is complete. ∎

Thus all possible solutions of the homotopy (5.2) are uniformly bounded in
$H(\Omega)$ independently of $\lambda\in[0,1]$. The proof is complete. ∎

The following lemma provide some useful information about that function
$V_{0}$ that was used in the proof of preceding lemma

######  Lemma 7.2.

If $U$ is a ( nontrivial) weak solution of equation (7.13) with
$\mu_{j}\leq\alpha(x)\leq\langle
k(x)V(x),V(x)\rangle\leq\beta(x)\leq\mu_{j+1}$, for all $V\in H(\Omega)$ then
either $U\in E_{j}$ or $U\in E_{j+1}.$

###### Proof.

Since $U$ is (also) a weak solution, it satisfies

$\begin{gathered}\int_{\Omega}\triangledown U.\triangledown
V\,dx+\int_{\Omega}\langle
A(x)U,V\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)U,V\rangle\,dx=\int_{\partial\Omega}\langle
k(x)U,V\rangle\,dx~{}~{}\forall~{}~{}V\in H(\Omega)~{}~{}~{}\forall~{}~{}V\in
H(\Omega)\end{gathered}$ (7.15)

Observe that $U\in[H_{(M.P)}(\Omega)]^{\bot}.$ Hence $U=U_{1}+U_{2}$ where
$U_{1}\in\oplus_{l\leq j}E_{l}$ and $U_{2}\in\oplus_{l\geq j+1}E_{l}$ know
that from the properties of the Steklov-Robin eigenfunction

$\begin{gathered}||U_{1}||^{2}_{(A,\Sigma)}\leq\mu_{j}\int_{\partial\Omega}\langle
P(x)U_{1},U_{1}\rangle,\,\,{\rm~{}for~{}all}\,\,U_{1}\in\oplus_{l\leq
j}E_{l}\,\,~{}~{}{\rm and}\\\
||U_{2}||^{2}_{(A,\Sigma)}\geq\mu_{j+1}\int_{\partial\Omega}\langle
P(x)U_{2},U_{2}\rangle,\,\,{\rm~{}for~{}all}\,\,U_{2}\oplus_{l\geq
j+1}E_{l}\end{gathered}$ (7.16)

Tanking $V=U_{1}-U{2}$, in equation (7.15 we get that

$\begin{gathered}\int_{\Omega}|\triangledown
U_{1}|^{2}\,dx+\int_{\Omega}\langle
A(x)U_{1},U_{1}\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)U_{1},U_{1}\rangle\,dx\\\
-\left(\int_{\Omega}|\triangledown U_{2}|^{2}\,dx+\int_{\Omega}\langle
A(x)U_{2},U_{2}\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)U_{2},U_{2}\rangle\,dx\right)=\\\
\int_{\partial\Omega}\langle
k(x)U_{1},U_{1}\rangle\,dx-\int_{\partial\Omega}\langle
k(x)U_{2},U_{2}\rangle\,dx~{}~{}\forall~{}~{}V\in
H(\Omega)~{}~{}~{}\forall~{}~{}V\in H(\Omega)\end{gathered}$ (7.17)

Using equation (7.15), we obtain that

$\int_{\partial\Omega}(\langle k(x)U_{1},U_{1}\rangle-\mu_{j}\langle
P(x)U_{1},U_{1}\rangle)\langle
U_{1},U_{1}\rangle\,dx+\int_{\partial\Omega}(\langle
k(x)U_{2},U_{2}\rangle-\mu_{j+1}\langle P(x)U_{2},U_{2}\rangle)\langle
U_{2},U_{2}\rangle\,dx\leq 0$

Therefore

$\int_{\partial\Omega}(\langle k(x)U_{1},U_{1}\rangle-\mu_{j}\langle
P(x)U_{1},U_{1}\rangle)\langle U_{1},U_{1}\rangle\,dx=0$

and

$\int_{\partial\Omega}(\langle k(x)U_{2},U_{2}\rangle-\mu_{j+1}\langle
P(x)U_{2},U_{2}\rangle)\langle U_{2},U_{2}\rangle\,dx=0$

Let

$S_{1}:=\\{x\in\partial\Omega:U_{1}(x)\neq 0\\}$

and

$S_{2}:=\\{x\in\partial\Omega:U_{2}(x)\neq 0\\}.$

It follows that $\langle k(x)U_{1},U_{1}\rangle=\mu_{j}\langle
P(x)U_{1},U_{1}\rangle~{}a.e.~{}~{}{\rm on}~{}~{}~{}S_{1}$ and $\langle
k(x)U_{2},U_{2}\rangle=\mu_{j+1}\langle
P(x)U_{2},U_{2}\rangle~{}a.e.~{}~{}{\rm on}~{}~{}~{}S_{2}.$ If $meas(S_{1}\cap
S_{2})>0$ we have that $\langle k(x)U_{1},U_{1}\rangle=\mu_{j}\langle
P(x)U_{1},U_{1}\rangle=\langle k(x)U_{2},U_{2}\rangle=\mu_{j+1}\langle
P(x)U_{2},U_{2}\rangle$ for a.e.; $x\in S_{1}\cap S_{2}$ which cannot happen
since $\mu_{j}\neq\mu_{j+1}.$ Now assume that $meas(S_{1}\cap S_{2})=0$ that
is $U_{2}(x)=0~{}~{}a.e.;{\rm on}~{}~{}~{}S_{1}$ and
$U_{1}(x)=0~{}~{}a.e.;{\rm on}~{}~{}~{}S_{2}.$ If $U_{1}\neq 0$, then taking
$V=U_{1}$ in equation (7.15) and using $(A,\Sigma)-$orthogonality, we get that

$\int_{\Omega}|\triangledown U_{1}|^{2}\,dx+\int_{\Omega}\langle
A(x)U_{1},U_{1}\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)U_{1},U_{1}\rangle\,dx=\int_{\partial\Omega}\langle
k(x)U_{1},U_{1}\rangle\,dx.$

Since $\mu_{j}\langle P(x)U_{1},U_{1}\rangle\leq\langle
k(x)U_{1},U_{1}\rangle,$ we have that
$||U_{1}||^{2}_{(A,\Sigma)}\geq\mu_{j}\int_{\partial\Omega}\langle
P(x)U_{1},U_{1}\rangle\,dx.$ It follows from equation (7.16)
$||U_{1}||^{2}_{(A,\Sigma)}=\mu_{j}\int_{\partial\Omega}\langle
P(x)U_{1},U_{1}\rangle\,dx.$ which implies that $U_{1}\in E_{j}.$
Similarly, If $U_{2}\neq 0$ then taking $V=U_{2}$ in equations (7.15) and
using the $(A,\Sigma)-$orthogonality, we get that

$\int_{\Omega}|\triangledown U_{2}|^{2}\,dx+\int_{\Omega}\langle
A(x)U_{2},U_{2}\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)U_{2},U_{2}\rangle\,dx=\int_{\partial\Omega}\langle
k(x)U_{2},U_{2}\rangle\,dx.$

Since $\mu_{j+1}\langle P(x)U_{1},U_{1}\rangle\geq\langle
k(x)U_{1},U_{1}\rangle,$ we have that
$||U_{2}||^{2}_{(A,\Sigma)}\leq\mu_{j+1}\int_{\partial\Omega}\langle
P(x)U_{2},U_{2}\rangle\,dx.$ It follows from equation (7.16) that

$||U_{2}||^{2}_{(A,\Sigma)}=\mu_{j+1}\int_{\partial\Omega}\langle
P(x)U_{2},U_{2}\rangle\,dx$

which implies that $U_{2}\in E_{j+1}.$
Thus $U=U_{1}+U_{2}$ with $U_{1}\in E_{j},$ and $U_{2}\in E_{j+1}.$
Finally, we claim that the function $U$ cannot be written in the
$U=U_{1}-U_{2}$ where $U_{1}\in E_{j}\setminus\\{0\\}$ and $U_{2}\in
E_{j+1}\setminus\\{0\\}.$ Indeed, suppose that this does not hold; that is,
$U=U_{1}+U_{2}$ with $U_{1}\in E_{j}\setminus\\{0\\}$ and $U_{2}\in
E_{j+1}\setminus\\{0\\}.$ Then, by taking $V=U_{1}-U_{2}$ in equation (7.15),
we get equation (7.17) Since $U_{1}\in E_{j}$ and $U_{2}\in E_{j+1}$ and

$\alpha(x)\leq\langle k(x)U,U\rangle\leq\beta(x)~{}a.e.;~{}{\rm
on}~{}~{}\partial\Omega,$

we deduce

$\int_{\partial\Omega}(\alpha(x)-\mu_{j})\langle U_{1},U_{1}\rangle\,dx\leq
0~{}~{}{\rm and}~{}~{}\int_{\partial\Omega}(\mu_{j+1}-\beta(x))\langle
U_{2},U_{2}\rangle\,dx\leq 0$

Which contradicts the fact that
$\int_{\partial\Omega}(\alpha(x)-\mu_{j})\langle\varphi,\varphi\rangle\,dx>0$
for all $\varphi\in E_{j}\setminus\\{0\\}$ and
$\int_{\partial\Omega}(\mu_{j+1}-\beta(x))\langle\varphi,\varphi\rangle\,dx>0$
for all $\varphi\in E_{j+1}\setminus\\{0\\}$ Thus, either $U\in E_{j}$ or
$U\in E_{j+1}$. The proof is complete ∎

##  8\. Proof of the main Theorem 4.3

Let $(\lambda,U)\in[0,1]\times[W^{1-1/p}_{p}(\partial\Omega)]^{k}$ be a
solution to the homotopy (6.3) (equivalently (5.2)). Since
$\int_{\Omega}\triangledown U.\triangledown V\,dx+\int_{\Omega}\langle
A(x)U,V\rangle\,dx+\int_{\partial\Omega}\langle\Sigma(x)U,V\rangle\,dx=0$ for
all $U\in[C_{0}^{1}(\Omega)]^{k},$ and the trace of
$U\in[W^{1-1/p}_{p}(\partial\Omega)]^{k}\subset[C_{0}^{1}(\partial\Omega)]^{k}$
It follows from Theorem 13.1 in [[LU1968], pp.1999-200] also see [[Eva99],
[Bz2011]]

that there is a constant $C_{0}>0$ (independent of $U$) such that
$\sup_{\Omega}|U(x)|\leq C_{0}||U||_{H(\Omega)},$ and
$\max_{\bar{\Omega}}|U(x)|\leq C_{0}||U||_{H(\Omega)},$ by the continuity of
$U$ on $\bar{\Omega}$ from Lemma 7.1 above and the ( local Lipschitz)
continuity of $g$ we deduce that $\max_{\partial\Omega}|\frac{\partial
U}{\partial\nu}+\Sigma U|=\max_{\partial\Omega}|(1-\lambda)\delta
P(.)U+\lambda g(.,U)$ is bounded independently of $U$ and $\lambda$. Actually,
we deduce from Theorem 2 in ([Lie1988] p 1204) that
$|U|_{[C^{1}(\bar{\Omega})]^{k}}$ is bounded (independently of $U$ and
$\lambda$). Therefore, the continuity of the trace operator
$[C^{1}(\bar{\Omega})]^{k}\subset[W^{1}_{p}(\Omega)]^{k}\to[W^{1-1/p}_{p}(\partial\Omega)]^{k}$
and Lemma 7.1 herein imply that there is a constant $C_{1}>0$ ( independent of
$U$ and $\lambda$) such that

$\begin{gathered}||U||_{[W^{1-1/p}_{p}(\partial\Omega)]^{k}}<C_{1}\end{gathered}$
(8.1)

for all possible solutions to the homotopy 5.2. Now by the homotopy invariance
property of the topological degree (see [Llo1987], [Maw1979]) it follows that

$1=deg(I,B_{C_{1}}(0),0)=deg(I-\mathbb{K}\v{N},B_{C_{1}}(0),0)\neq 0$

where $B_{C_{1}}\subset[W^{1-1/p}_{p}(\partial\Omega)]^{k}$ is ball of radius
$C_{1}>0$ centered at the origin. Thus, by the existence property of the
topological degree (see [Llo1987], [Maw1979]), the nonlinear operator
$\mathbb{K}\v{N}$ has a fixed point in $[W^{1-1/p}_{p}(\partial\Omega)]^{k}$
which is also in $[W^{2}_{p}(\Omega)]^{k}$ as aforementioned. Thus Then the
nonlinear system (4.2) has at least one (strong) solution
$U\in[W^{2}_{p}(\Omega)]^{k}$. The proof of the theorem is complete.

###  8.1. Remarks

######  Remark 8.1.

Notice that, since $g$ is locally Lipschitz, it follows from 8.1 and the
boundary condition in the homotopy 5.2 that
$||U||_{[W^{2-1/p}_{p}(\partial\Omega)]^{k}}<C_{2}$ for some constant
$C_{2}>0$ independent of $U$ and $\lambda$. Therefore
$||U||_{[W^{2}_{p}(\partial\Omega)]^{k}}<C_{3}$ for some constant $C_{3}>0$

######  Remark 8.2.

The case $\mu_{j}=\mu_{1}$ more clearly illustrates the fact that the non-
resonance conditions in Theorem 4.3 are genuinely of nonuniform type. Indeed,
in this case $E_{1}\backslash\\{0\\}$ contains only (continuous) functions
which are either positive or negative on $\bar{\Omega}.$ The condition that
$\alpha(x)\geq\mu_{1}~{}~{}a.e.~{}~{}{\rm on}~{}~{}\partial\Omega$ with
$\int_{\partial\Omega}(\alpha(x)-\mu_{1})\langle\varphi,\varphi\rangle$ for
all $\varphi\in E_{1}\backslash\\{0\\}$ is equivalent to saying that
$\alpha(x)\geq\mu_{1}~{}a.e.~{}~{}{\rm on}~{}~{}\partial\Omega$ with strict
inequality on a subset of $\partial\Omega$ of positive measure. Thus
$\alpha(x)$ need not be (uniformly) bounded away from $\mu_{1}.$

######  Remark 8.3.

Our main result, Theorem 4.3 herein, Still holds true when $A(x)\equiv 0$ and
$\Sigma(x)\equiv 0$ ( The Laplace’s system in the original linear system with
was considered by Auchmuty for the system [Auc2012], [MN2011], and in scalar
case was considered by Steklov on a disk in [Ste1902].) Indeed, a modification
is needed in the proof of Lemma 7.1 as follows. We proceed as in the proof
with $||.||_{(A,\Sigma)}$ ( here $||.||_{H(\Omega)}$ denotes the standard
$H(\Omega)-$norm, and $V_{n}=\frac{U_{n}}{||U||_{H(\Omega)}}$ up to the
equation (7.6) Taking $V=V_{0}$ in 7.6 we now get

$\begin{gathered}\int_{\Omega}|\triangledown
V_{0}|^{2}\,dx=(1-\lambda_{0})\delta\int_{\partial\Omega}\langle
P(x)V_{0},V_{0}\rangle\,dx+\lambda_{0}\int_{\partial\Omega}\langle
g_{0},V_{0}\rangle\,dx\end{gathered}$ (8.2)

Now, taking $V=\frac{U_{n}}{||U_{n}||_{H(\Omega)}}$ in 7.7 where is replaced
by $||U_{n}||_{H(\Omega)}$ we get that

$\begin{gathered}\int_{\Omega}|\triangledown
V_{n}|^{2}\,dx=(1-\lambda_{n})\delta\int_{\partial\Omega}\langle
P(x)V_{n},V_{n}\rangle\,dx+\lambda_{n}\int_{\partial\Omega}\frac{\langle
g(x,U_{n}),V_{n}\rangle}{||U_{n}||_{H(\Omega)}}\,dx\end{gathered}$ (8.3)

Tanking the limit as $n\to\infty$ and using 8.2 and the fact that
$\frac{\langle g(x,U_{n}),V_{n}\rangle}{||U_{n}||_{H(\Omega)}}$ converges
weakly $g_{0}$ in $[L^{2}(\partial\Omega)]^{k}$ $V_{n}$ converges strongly to
$V_{0}$ in $[L^{2}(\partial\Omega)]^{k}$, we have that

$\begin{gathered}\lim_{n\to\infty}\int_{\Omega}|\triangledown
V_{n}|^{2}\,dx=(1-\lambda_{0})\delta\int_{\partial\Omega}\langle
P(x)V_{0},V_{0}\rangle\,dx+\lambda_{0}\int_{\partial\Omega}\langle
g_{0},V_{0}\rangle\,dx=\int_{\Omega}|\triangledown
V_{0}|^{2}\,dx\end{gathered}$ (8.4)

This implies that

$\begin{gathered}||V_{0}||_{H(\Omega)}^{2}=\int_{\Omega}|\triangledown
V_{0}|^{2}\,dx+\int_{\Omega}|V_{0}|^{2}\,dx=\lim_{n\to\infty}\left(\int_{\Omega}|\triangledown
V_{n}|^{2}\,dx+\int_{\Omega}|V_{n}|^{2}\,dx\right)=\lim_{n\to\infty}||V_{n}||_{H(\Omega)}^{2}=1\end{gathered}$
(8.5)

We now proceed as in the proof of Lemma 7.1 after equation (7.9) to Show that
$V_{0}=0$; which is a contradiction with equation 8.5.
The proof of Lemma 7.2 also needs to be modified as follows. The norm
$||.||_{(A,\Sigma)}$ is now replaced by the $H(\Omega)-$ equivalent norm
$||.||$ by

$||U||^{2}:=\int_{\Omega}|\triangledown
U|^{2}\,dx+\int_{\Omega}|U|^{2}\,dx~{}~{}~{}{\rm for}~{}~{}U\in H(\Omega)$

(See [Auc2004]) By the using the decomposition of $H(\Omega)$ given in
[[Auc2004] Theorem 7.3, p337 ] we now proceed with the argument used in the
proof Lemma 7.2 herein to reach its conclusion.

######  Remark 8.4.

Note that the case $A(x)\equiv 0$ and $\Sigma(x)\equiv 0$ even more clearly
illustrates the fact that nonresonance conditions in Theorem 4.3 are genuinely
of nonuniform type. Indeed, in this case $\mu_{1}=0$ and
$E_{1}\backslash\\{0\\}$ contains only constant functions. The condition that
$\alpha(x)\geq\mu_{1}~{}~{}{\rm on}~{}~{}\partial\Omega$ with strict
inequality on a subset of $\partial\Omega$ of positive measure. Thus
$\alpha(x)$ need not be (uniformly) bounded away from $\mu_{1}=0.$ Actually, a
careful analysis of the proofs

of Lemmas 7.1 and 7.2 shows that, in this case one can drop the requirement
that $\alpha(x)\geq 0~{}~{}~{}{\rm on}~{}~{}\partial\Omega$ and require only
that $\int_{\partial\Omega}\alpha(x)\,dx>0.$ Thus a ’crossing’ of the zero
eigenvalue on subset of $\partial\Omega$ of positive measure is allowed that
is $\alpha(x)$ could be negative on a subset of $\partial\Omega$ of positive
measure (i.e $P(x)\leq 0$)

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