\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 239, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/239\hfil Continuous dependence of solutions]
{Continuous dependence of solutions for indefinite semilinear elliptic problems}

\author[E. A. B. Silva, M. L. Silva \hfil EJDE-2013/239\hfilneg]
{Elves A. B. Silva, Maxwell L. Silva}  % in alphabetical order

\address{Elves A. B. Silva \newline
Universidade de Bras\'ilia, Departamento de Matem\'atica,
Cep 70 910 900, Bras\'ilia-DF, Brazil}
\email{elves@unb.br}

\address{Maxwell L. Silva \newline
 Universidade Federal de Goi\'as, Instituto de Matem\'atica e Estat\'istica,
 Cep 74 001 970, Goiania-GO, Brazil}
\email{maxwell@mat.ufg.br}

\thanks{Submitted August 12, 2011. Published October 24, 2013.}
\subjclass[2000]{35J20, 35J60, 35Q55}
\keywords{Positive solution; constrained minimization; eigenvalue problem;
\hfill\break\indent Neumann boundary condition;  unique continuation}

\begin{abstract}
 We consider the superlinear elliptic problem
 $$
 -\Delta u + m(x)u = a(x)u^p
 $$ 
 in a bounded smooth domain under  Neumann boundary conditions,
 where $m \in L^{\sigma}(\Omega)$,  $\sigma\geq N/2$ and
 $a\in C(\overline{\Omega})$ is a sign changing function.
 Assuming that the associated first eigenvalue of the operator
 $-\Delta + m $ is zero, we use constrained minimization methods to
 study the existence of a positive solution when $\widehat{m}$
 is a suitable perturbation of $m$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

In this article, we consider the continuous dependence of positive 
solution for the semilinear elliptic problem 
\begin{equation} \label{ePm}
\begin{gathered}
 -\Delta u+m(x)u  = a(x)u^p \quad\text{in }\Omega,\\
 \frac{\partial u}{\partial\eta} =0 \quad\text{on }
\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^N$, $N\geq 3$, is a bounded smooth domain,
$\eta$ is the outward unit normal, $1<p<\frac{N+2}{N-2}$,
$m\in L^{\sigma}(\Omega)$ for some $\sigma\geq N/2$, and
$a\in C(\overline{\Omega})$ is a sign changing function, more
specifically, $a$ satisfies
\begin{itemize}
\item[(A1)] $\Omega^{+}:=\{x\in\Omega:a(x)>0\}\neq \emptyset$ and
$\Omega^{-}:=\{x\in\Omega:a(x) <0\}\neq \emptyset$.
\end{itemize}

The existence of solutions for indefinite semilinear elliptic problems has 
been intensively studied in the literature;  see for example 
\cite{Afrouzi,AlDel,AlTar1,BCDN,BCDN2,RTT,Tehrani} and references therein. 
As it has been established in several articles 
\cite{Afrouzi,BCDN,Tehrani}, the existence of solution for  \eqref{ePm}
 depends on the interaction between the nonlinear term and the eigenfunctions 
corresponding to the associated eigenvalue problem
\begin{equation} \label{EPml}
\begin{gathered}
 -\Delta u+m(x)u = \lambda u\quad\text{in }\Omega,\\
 \frac{\partial u}{\partial\eta} =0 \quad\text{on }\partial\Omega.
 \end{gathered}
\end{equation}
Noting that the first eigenvalue of \eqref{EPml} is simple,
isolated and that the associated eigenfunction does not change sign
in $\Omega$ (see Section 3), we assume:
\begin{itemize}
\item[(A2)] the first eigenvalue for \eqref{EPml} is zero,
 $\lambda_1(-\Delta+m)=0$,

\item[(A3)] $\int_{\Omega} a(x)\varphi_1^{p+1}<0$,
\end{itemize}
where $\varphi_1$ is the first eigenfunction for \eqref{EPml} which is
positive in $\Omega$ and normalized in
$L^2(\Omega)$.

Our main interest in this article is to establish the existence of a 
positive solution for \eqref{ePm} with $\widehat{m}$ instead of $m$, when  
$\widehat{m}\in L ^{\sigma}(\Omega)$ is a suitable perturbation 
of the function $m$. More specifically, throughout this paper we will 
consider $\{m_j\}_j\subset L^{\sigma}(\Omega)$ satisfying
\begin{itemize}
\item[(M1)]  $ m_j\rightharpoonup m$ weakly in 
$L^{\sigma}(\Omega),\sigma\geq N/2$.
 If $\sigma=N/2$ we suppose in addition that $|m_j|\leq f$  a.e. in 
$\Omega$ for some $f\in L^{N/2}(\Omega)$.
\end{itemize}

It is worthwhile mentioning that the hypothesis (M1) has been motivated 
by the works of Tehrani \cite{Tehrani} and Afrouzi \cite{Afrouzi} 
that studied the existence of positive solutions for \eqref{ePm} with $m_j$ 
instead of $m$,  supposing $m, \{m_j\}_j\subset C(\bar{\Omega})$ and 
$m_j \to m$ uniformly  in $\bar{\Omega}$.
We should emphasize that when the linear operator $-\Delta+m_j$ is 
coercive; i.e., when the associated first eigenvalue,
 $\lambda_1(-\Delta+m_j)$, is  positive, 
it may be proved that Problem \eqref{ePm} with $m_j$ instead
of $m$ always has a positive 
solution (see e.g. \cite{AlTar1,BCDN}). The key point in this paper is that, 
under our hypotheses,  the operator $-\Delta+m_j$ is not necessarily coercive, 
and the eigenvalue $\lambda_1(-\Delta+m_j)$ may even be strictly negative.
 Now we may state our main results:

\begin{theorem}\label{convfraca}
 Suppose {\rm (A1)--(A3), (M1)}.
Then there exists $j_1\in\mathbb{N}$ such that \eqref{ePm} with $m_j$ 
instead of $m$  possesses a positive solution $w_j$ for all $j\geq j_1$.
\end{theorem}

\begin{theorem}\label{convsol} 
Suppose {\rm (A1)--(A3), (M1)}. Let $\{w_j\}_j $ be the sequence of solutions 
given by Theorem \ref{convfraca}. Then $\{w_j\}_j$ has a subsequence
which converges strongly in $H^1(\Omega)$ to a solution of \eqref{ePm}.
\end{theorem}

As a direct consequence of Theorem \ref{convfraca},
the perturbed problem \eqref{ePm} with $\hat{m}$ instead of $m$ 
 has a positive solution if $\hat{m}$
 is close to $m$ in $L^{\sigma}(\Omega)$.

\begin{corollary}\label{melhorateherani}
 Assume {\rm (A1)--(A3)}. Then there exists $R>0$ such that \eqref{ePm} 
with $\widehat{m}$ instead of $m$
 possesses a positive solution whenever 
$\|m-\widehat{m}\|_{L^{\sigma}(\Omega)}<R$.
\end{corollary}

Our next result establishes the existence of a positive solution for 
 \eqref{ePm} with $\widehat{m}$ instead of $m$, when the first 
eigenvalue $\lambda_1(-\Delta +\hat{m})$ is near to $\lambda_1(-\Delta +m)$, 
without supposing that $\hat{m}$ is  close to $m$ in $L^{\sigma}(\Omega)$:

\begin{theorem}\label{theorem2.1}
Assume {\rm (A1)--(A3)}. 
Given $R>0$ (or $f\in L^{N/2}(\Omega)$ when $\sigma = N/2$), 
then  there exists $\mu>0$ such that the Problem \eqref{ePm} with 
$\widehat{m}$ instead of $m$, possesses a positive solution for all 
$\widehat{m}\in L^{\sigma}(\Omega)$ satisfying
\begin{itemize}
\item[(i)] $\|m-\widehat{m}\|_{L^{\sigma}(\Omega)}\leq R$, 
($|\widehat{m}|\leq f$ a.e in $\Omega$ if $\sigma=\frac{N}{2}$),

\item[(ii)] $\int_{\Omega}|\nabla\varphi_1|^2
+\widehat{m}(x)\varphi_1^2\leq \mu$,

\item[(iii)] $-\mu<\lambda_1(-\Delta+\widehat{m})<0$.
\end{itemize}
\end{theorem}

 Afrouzi \cite{Afrouzi} proved Theorem \ref{convfraca}, 
considering the Dirichlet boundary conditions, under the  
assumptions $m\in C(\overline{\Omega})$ and 
$\|m-m_j\|_{C(\overline{\Omega})}\to 0$. In \cite{Tehrani}, also considering 
the norm $\|\cdot\|_{C(\bar\Omega)}$, Tehrani established Corollary 
\ref{melhorateherani} for $m\in C(\overline{\Omega})$ and 
Theorem \ref{theorem2.1} for $m\in C^{0,\alpha}(\overline{\Omega})$
supposing the additional hypothesis 
$\int_{\Omega} a(x)\varphi_{\widehat{m}}^{p+1}<0$, where 
$\varphi_{\widehat{m}}>0$ is the eigenfunction associated with the 
first eigenvalue $\lambda_1(-\Delta+\widehat{m})$. 
It is worthwhile  mentioning that the condition (A3) 
is a necessary condition for the existence of positive solution for 
the Problem \eqref{ePm} in the Theorem \ref{convfraca} (see \cite{Tehrani}).


Finally we observe that Beresticky, Capuzzo-Dolcetta and Nirenberg \cite{BCDN} 
proved the existence of a positive solution for \eqref{ePm} with 
$m-\tau$ instead of $m$, when 
$m\in C(\overline{\Omega})$ satisfies (A2) and $\tau\geq0$ is sufficiently 
small.
We observe that, as a consequence of Theorem \ref{convfraca}, we may 
assert the existence of a positive solution for
\eqref{ePm} with $m_j-\tau$ instead of $m$, 
 under the hypothesis (M1) whenever $\tau\geq 0$ 
is sufficiently small (see Remark \ref{versaoBCDN}).

The method employed in this article is variational, more specifically, 
we use a constrained minimization method. In order to prove our main results,
 we establish continuity results for the first and the second eigenvalues 
and their associated eigenfunctions for   \eqref{EPml} with $m_j$ instead of $m$ under
the condition (M1), generalizing a version for the uniform convergence 
provided by \cite{Afrouzi}.  We also verify an uniform equivalence result 
between the $H^1(\Omega)$-norm and a norm, depending on $m_j$, for the 
orthogonal space to the first eigenfunction of \eqref{EPml} with $m_j$ instead of $m$.

The organization of this paper is as follows: in Section 2 we introduce 
some notation and preliminary results. In Section 3, based on the 
arguments used by Manes-Micheletti \cite{MM} and 
deFigueiredo \cite{dF} for Dirichlet boundary conditions, 
we present the technical properties and a convergence result, under 
the condition (M1), for the eigenvalues and eigenfunctions of 
\eqref{EPml} with $m_j$ instead of $m$. The proofs of Theorems
 \ref{convfraca}, \ref{convsol} and \ref{theorem2.1} are presented in
 Section 4. In this section we also present an example that illustrate 
the application of Theorems \ref{convfraca} and \ref{convsol} on a 
setting where the condition (M1) is satisfied.  
In the Appendix  we establish the regularity results for the solutions 
of \eqref{ePm} and \eqref{EPml} and we prove Theorems 
\ref{teoparalambda1} and \ref{exisparalambda2} from Section 3.

\section{Preliminaries}

In this section we introduce the definitions and the technical results 
that will be used throughout the text. First we recall the  
strong unique continuation property (SUCP for short) due to 
Jerison and Kenig \cite{JK} which  guarantees that the solutions 
and the first eigenfunction that we find are positive.
 Next, in the main result of this section, we establish the convergence  
$\int_{\Omega} m_j(x)v_j^2\to \int_{\Omega} m(x)v_o^2$, under the hypothesis 
(M1), when $v_j\rightharpoonup v_o$ weakly in $H^1(\Omega)$.

We use $\|\cdot \|_s$ to denote the norm in $L^s(\Omega)$. 
If $E\subset\Omega$ is a proper subset, the norm in $L^s(E)$ is 
represented by $\|\cdot \|_{L^s(E)}$. $|A|$ denotes the Lebesgue's measure 
of $A\subset\mathbb{R}^N$. $B_{\rho}(x_o)$ denotes the open ball of radius 
$\rho>0$ centered in $x_o$. The symbol $\rightharpoonup$ denotes the weakly 
convergence.

\begin{definition} \label{def2.1} \rm
(i) We say that $f$ possesses a zero of infinite order if there exists 
$x_o\in\Omega$ such that $\int_{|x-x_o|<\epsilon}f^2(x)dx=O(\epsilon^k)$
for every $k\in\mathbb{N}$.

(ii)  Let $c:\Omega\to\mathbb{R}$ be a function. We say that the differential 
inequality 
\begin{equation}\label{pcu}
|\Delta f(x)|\leq |c(x)f(x)|,\quad x\in\Omega
\end{equation} 
has the SUCP in the Sobolev space $ W^{2,q}_{\rm loc}(\Omega)$, if $f\equiv 0$
whenever $f$ belongs to $ W^{2,q}_{\rm loc}(\Omega)$, satisfies \eqref{pcu}
almost everywhere in $\Omega$, $f\in L^{2}_{\rm loc}(\Omega)$ and has a zero
of infinite order.
\end{definition}

\begin{theorem}[\cite{JK}] \label{KENIG}
Let $\Omega\subset \mathbb{R}^N$ be a domain, $N\geq 3$, and 
$c\in L^{N/2}_{\rm loc}(\Omega)$. Then the differential inequality
\eqref{pcu} has the SUCP in $W^{2,\frac{2N}{N+2}}_{\rm loc}(\Omega)$.
\end{theorem} 

To employ the SUCP we may use  the following result.

\begin{theorem}[\cite{DG}] \label{DG}
 Let $\Omega\subset \mathbb{R}^N$, be a bounded domain, $N\geq 3$ and 
$c\in L^{N/2}_{\rm loc}(\Omega)$. Suppose that
$u\in H^{1}_{\rm loc}(\Omega)$ is such that 
$\int_{\Omega} \nabla u\nabla v+c(x)uv=0$, for all 
$v\in C_o^{\infty}(\Omega)$. If $u=0$ on a set $E$ of positive measure, 
then $u$ possesses a zero of infinite order in almost everywhere point of $E$.
\end{theorem}


\begin{remark}\label{positive} \rm
If $u\geq 0$ is a weak solution for  \eqref{ePm}, after regularization 
(see Appendix A) $u\in W^{2,\frac{2N}{N+2}}(\Omega)\cap L^{t}(\Omega)$ 
for all $t\geq 1$. So, taking $c(x):=a(x)u^{p-1}-m(x)$, 
$$
|\Delta u(x)|=|c(x)u(x)|
$$ 
for almost everywhere $x \in\Omega$, and $c\in L^{N/2}(\Omega)$. 
If  $|\{x\in \Omega: u(x)=0\}|>0$, by Theorems \ref{DG} and \ref{KENIG}, 
$u\equiv 0$ in $\Omega$. Therefore, if $u\geq 0$ is a nontrivial solution 
for \eqref{ePm}, then $u>0$ almost everywhere in $\Omega$. 
The same conclusion is valid if $\varphi\geq 0$ is the first eigenfunction 
for \eqref{EPml}.
\end{remark} 

As a direct consequence of the Sobolev imbedding theorem, we have the following
result.

\begin{lemma}\label{intnegativa}
 Suppose {\rm (A3)} is satisfied. Then there exists $\beta>0$ such that
 \[
\int_{\Omega} a(x)|\varphi_1+w|^{p+1}
< \frac{1}{2}\int_{\Omega} a(x)\varphi_1^{p+1},\quad \forall
 \|w\|_{H^1(\Omega)}<\beta.
\]
\end{lemma}

The next result is standard and it is also based on the Sobolev imbedding theorem.

\begin{lemma}[\cite{W}]\label{EGOROV} 
Assume $m\in L^{N/2}(\Omega)$ and $\{\omega_j\}_j\subset H^1(\Omega)$.
 If $\omega_j\rightharpoonup\omega_o$ weakly in $H^1(\Omega)$, then 
$$
\int_{\Omega}\! |m(x)(\omega_j^2-\omega_o^2)|\to 0.
$$
\end{lemma}

Now we present a version of Lemma \ref{EGOROV} when the sequence $\{m_j\}$ 
satisfies the hypothesis (M1).

\begin{lemma}\label{EGOROV2}
Assume {\rm (M1)} is satisfied. If $\omega_j\!\rightharpoonup \!\omega_o$ 
weakly in $H^1(\Omega)$, then 
$$
\int_{\Omega} m_j(x)\omega_j^2\to\int_{\Omega}m(x)\omega_o^2.
$$
\end{lemma}

\begin{proof} Given  arbitrary subsequences of $\{m_j\}$ and $\{\omega_{j}\}$,
 (still denoted by $\{m_j\}$ and $\{\omega_{j}\})$, we may write 
$$
\int_{\Omega}\;m_j(x)\omega_j^2- \int_{\Omega}\;m(x)\omega_o^2
=\int_{\Omega}\;m_j(x)(\omega_j^2-\omega_o^2)
+\int_{\Omega}\;\omega_o^2(m_j(x)-m(x)).
$$
Since $m_j-m\rightharpoonup 0$ weakly in $L^\sigma(\Omega),\sigma\geq N/2$, and 
$\omega_o \in H^1(\Omega)$, we have 
\[
\int_{\Omega}\;\omega_o^2[m_j(x)-m(x)] \to 0.
\]
Therefore,  to prove Lemma \ref{EGOROV2}, it suffices to verify that
$$
\big|\int_{\Omega}\;m_j(x)(\omega_j^2-\omega_o^2) \big|\to 0.
$$
If $\sigma=N/2$, by (M1) and
Lemma \ref{EGOROV}, we obtain
$$
\int_{\Omega}\;|m_j(x)(\omega_j^2-\omega_o^2)| 
\leq \int_{\Omega}\;|f(x)|\;|\omega_j^2-\omega_o^2|\to 0.
$$
On the other hand, if  $\sigma>N/2$, taking a subsequence if necessary, 
we may suppose that
$\|\omega_j^2-\omega_o^2\|_{\frac{\sigma}{\sigma-1}}\to 0$  since 
$2\sigma/(\sigma-1)<2^{\ast}$. Hence, by H\"older's inequality,
 $$
\int_{\Omega}\;|m_j(x)(\omega_j^2-\omega_o^2) |
\leq \|m_j\|_r \|\omega_j^2-\omega_o^2\|_{\frac{\sigma}{\sigma-1}}\to 0.
$$
The  proof of Lemma \ref{EGOROV2} is complete.
\end{proof}

\begin{remark} \label{rmk2.8} \rm
Applying the same argument used in the proof of the above lemma, we 
have 
$$
\int_{\Omega} m_j(x)\omega_jz\to \int_{\Omega} m(x)w_oz,\quad
\forall z\in H^1(\Omega).
$$
\end{remark}
Furthermore, as a direct consequence of Lemma \ref{EGOROV2}, we have
the following result.

\begin{corollary}\label{compara-t-e-v} 
Assume {\rm (M1)}. Then, for any given $\epsilon>0$, 
there exists $j_{\epsilon}>0$ such that
 $$
|\int_{\Omega} [m_j(x)-m(x)]\omega^2|\leq \epsilon\|\omega\|^2_{H^1(\Omega)}
\quad \forall \omega\in H^1(\Omega),\;j\geq j_{\epsilon}.
$$
\end{corollary}

\section{The eigenvalue problem}

In this section we establish some basic properties for the eigenvalue 
problem \eqref{EPml}. We start by observing that its first eigenvalue is simple, 
isolated and the associated eigenfunction does not change sign. 
After that we establish a continuity result for the first and the second 
eigenvalues and their corresponding eigenfunctions for \eqref{EPml} with $m_j$ 
instead of $m$ under the condition (M1).
 We finalize this section by proving the uniform equivalence  between the 
standard $H^1(\Omega)$ norm and a norm associated with the sequence 
$\{m_j\}$ on the subspace of $H^1(\Omega)$ orthogonal to the first 
eigenfunction of \eqref{EPml} with $m_j$ instead of $m$.

Setting $Q_m(u):=\int_{\Omega}|\nabla u|^2+m(x)u^2$, we say that a function 
$\varphi$ is associated with $\lambda$ if $\int_{\Omega} \varphi^2=1$ 
and $Q_m(\varphi)=\lambda$. We define   
\begin{equation}\label{deflambda1}
\lambda_1:=\lambda_1(-\Delta+m)=\inf\big\{Q_m(u): u\in H^1(\Omega),\;
 \|u\|_2=1\big\}.
\end{equation}

\begin{remark}\label{smallestev} \rm
If $\varphi\in H^1(\Omega)$ is a (normalized) solution for \eqref{EPml}, then
\begin{equation}\label{formafraca}
\int_{\Omega} \nabla\varphi\nabla u+m(x)\varphi u
=\lambda\int_{\Omega} \varphi u,\quad \forall u\in H^1(\Omega).
\end{equation} 
Using $\varphi$ as test function in \eqref{formafraca}, we obtain
\begin{equation}\label{menorautovalor}
\lambda=\lambda\int_{\Omega} \varphi^2=Q_m(\varphi)\geq\lambda_1.
\end{equation}
Thus, any eigenvalue for \eqref{EPml} is greater than $\lambda_1$.
 Moreover, if there exists a nontrivial solution $\varphi$ for
 \eqref{EPml} with $\lambda_1$ instead of $\lambda$, then $\lambda_1$ 
is the smallest eigenvalue for \eqref{EPml}. 
\end{remark}

\begin{theorem}\label{teoparalambda1} 
Suppose $m\in L^{N/2}(\Omega)$. Then the Problem \eqref{EPml} has its 
first eigenvalue given by \eqref{deflambda1}. Moreover this first eigenvalue 
is simple and we may suppose that the associated eigenfunction, $\varphi_1$, 
is positive.  
\end{theorem}

To state our second result, we define 
\begin{equation}\label{deflab2}
\lambda_2:=\lambda_2(-\Delta+m)
=\inf\{Q_m(v): v\in V\;,\|v\|_2=1\},
\end{equation} 
where $V:=\{ v\in H^1(\Omega): \int_{\Omega} \varphi_1 v=0\}$. 
The next theorem shows that $\lambda_2$ is the second eigenvalue for \eqref{EPml}.

\begin{theorem}\label{exisparalambda2}
Suppose $m\in L^{N/2}(\Omega)$. Then the second eigenvalue for  \eqref{EPml} 
is given by \eqref{deflab2}. Moreover $\lambda_1<\lambda_2$ and any 
eigenfunction associated with $\lambda_2$ changes sign in $\Omega$.
\end{theorem}

The arguments used in the proofs of the above theorems  are due to Manes 
and Micheletti \cite{MM}(see also \cite{dF}). In order to verify the 
simplicity of the first eigenvalue, including the case $\sigma = N/2$, 
we apply  Theorems \ref{KENIG} and \ref{DG} (see Remark \ref{positive}). 
For the sake of completeness we present the proofs in the appendix.

Now let $\{m_j\}_j\subset L^{N/2}(\Omega)$ be the sequence satisfying (M1). 
Replacing $m$ by $m_j$ in  Definition \eqref{deflambda1},
 we obtain the first eigenvalue $\lambda_1^j:=\lambda_1(-\Delta+m_j)$ 
for \eqref{EPml} with $m_j$ instead of $m$ and its positive first 
eigenfunction $\varphi_{1,j}$. In a similar way, the second eigenvalue 
for \eqref{EPml} with $m_j$ instead of $m$ is given by
 $$
\lambda_2^j:=\lambda_2(-\Delta+m_j)
=\inf\{Q_{m_j}(v): v\in V_j\;,\|v\|_2=1\},
$$ 
where $V_j=\{v\in H^1(\Omega): \int_{\Omega} \varphi_{1,j}v=0\}$. 
We denote by $\varphi_{2,j}$ the  eigenfunctions associated with $\lambda_2^j$. 
Hereafter we will always suppose that the given eigenfunctions are  
eigenfunctions normalized in $L^2(\Omega)$.  
Hence $Q_{m_j}(\varphi_{1,j})=\lambda_1^j$ and 
$Q_{m_j}(\varphi_{2,j})=\lambda_2^j$.
The main result of this section is the following  continuity result.

\begin{theorem}\label{lema21mod} 
Assume {\rm (M1)} is satisfied. Then
\begin{itemize}
  \item[(i)] $\lim_{j\to\infty}\lambda_1^j=\lambda_1$ and 
$\varphi_{1,j}\to \varphi_{1}$ strongly in $H^1(\Omega)$;

\item[(ii)] $\lim_{j\to\infty}\lambda_2^j=\lambda_2$ and 
$\varphi_{2,j}\to \varphi$ strongly in $H^1(\Omega)$, where 
$\varphi$ is an eigenfunction associated with $\lambda_2$.
\end{itemize}
\end{theorem}

 \begin{proof}  First note that, by (M1), 
$\int_{\Omega} m_j(x)\varphi_1^2\to \int_{\Omega} m(x)\varphi_1^2$. 
Therefore, in view of \eqref{deflambda1}, 
\begin{equation}\label{limsup1}
\limsup_{j\to\infty}\;\lambda_1^j\leq \limsup_{j\to\infty}Q_{m_j}(\varphi_1)
=\lambda_1.
\end{equation}
Next we claim that there exists $M>0$ such that 
$\|\varphi_{1,j}\|_{H^1(\Omega)}<M$, for all $j\in\mathbb{N}$. 
Indeed, otherwise we may suppose that  
 $\|\nabla\varphi_{1,j}\|_2\to\infty$. Defining 
$\omega_j:=\frac{\varphi_{1,j}}{\|\nabla\varphi_{1,j}\|_2}$ 
and taking a subsequence if necessary, we have that
 $\omega_j\rightharpoonup 0$ weakly in $H^1(\Omega)$ since 
$\| \varphi_{1,j}\|_2\equiv 1$. Applying  Lemma \ref{EGOROV2},
 $\int_{\Omega} m_j(x)\omega_j^2 \to 0$. 
Thus, using the characterization for the first eigenvalue given 
by \eqref{deflambda1}, we have  
$$
\frac{\lambda_1^j}{\|\nabla \varphi_{1,j}\|_2^2}
=1+\int_{\Omega} m_j(x)\omega_j^2\to 1.
$$ 
Consequently $\lambda_1^j\to\infty$. However this contradicts \eqref{limsup1}. 
The claim is proved. Invoking the above claim, we may suppose  
$\varphi_{1,j}\rightharpoonup z$ weakly in $H^1(\Omega)$ and 
$\varphi_{1,j}\to z$ strongly in $L^{2}(\Omega)$. From Lemma \ref{EGOROV2}, 
\begin{equation}\label{convparamj}
\int_{\Omega} m_j(x)\varphi_{1,j}^2\to\int_{\Omega} m(x) z^2.
\end{equation} 
Therefore, since 
$\liminf_{j\to\infty}\|\nabla \varphi_{1,j}\|_2^2\geq \|\nabla z\|_2^2$, 
\begin{equation}\label{liminf3}
\liminf_{j\to\infty} \lambda_1^j
=\liminf_{j\to\infty} Q_{m_j}(\varphi_{1,j})\geq Q_{m}(z)
\geq \lambda_1.
\end{equation}
 By \eqref{limsup1} and \eqref{liminf3}, 
$Q_{m}(z)=\lim_{j\to\infty} \lambda_1^j=\lambda_1$. Hence, from 
Theorem \ref{teoparalambda1}, $z=\varphi_1$. Moreover, 
by \eqref{convparamj}, we have  
$$
\int_{\Omega} |\nabla\varphi_{1,j}|^2 
=\lambda_1^j-\int_{\Omega} m_j(x)\varphi_{1,j}^2\to 
\lambda_1-\int_{\Omega} m(x)\varphi_1^2
=\int_{\Omega} |\nabla\varphi_1|^2
$$ 
and, consequently,  $\varphi_{1,j}\to \varphi_1$ strongly in $H^1(\Omega)$. 
This concludes the proof of  part (i).

Now we prove (ii): since $Q_{m_j}(\varphi_{2,j})=\lambda_2^j$, 
$\varphi_{1,j}\to \varphi_1$ strongly in $H^1(\Omega)$ and $\{m_j\}_j$ 
satisfies (M1), we have
\begin{gather}\label{l2est}
\int_{\Omega}\varphi_2\varphi_{1,j}\to\int_{\Omega}
\varphi_2\varphi_{1}=0; \\
\label{l2est2} 
Q_{m_j}(\varphi_2)\to Q_{m}(\varphi_2)=\lambda_2; \\
\label{l2est3}
\int_{\Omega} \nabla\varphi_{1,j}\nabla\varphi_2+m_j(x)\varphi_{1,j}\varphi_2
\to \int_{\Omega} \nabla\varphi_{1}\nabla\varphi_2+m(x)\varphi_{1}
\varphi_2;
\end{gather} 
where $\varphi_2$ is an eigenfunction associated with $\lambda_2$.
Defining $v_j:=\varphi_2-t_j\varphi_{1,j}$ with 
$t_j:=\int_{\Omega} \varphi_2\varphi_{1,j}$ and using (\ref{l2est}),   
 \begin{equation}\label{l2est4}
\|v_j\|_2^2=\int_{\Omega} \varphi_2^2\,dx-t_j^2\to 1.
\end{equation} 
Setting $w_j:=\frac{v_j}{\|v_j\|_2}$, we obtain 
$\|w_j\|_{L^2(\Omega)}=1$ and $\int_{\Omega} \varphi_{1,j}w_j=0$,
 for all $j\in\mathbb{N}$.
Hence $w_j\in V_j$ and, by \eqref{l2est4} and the definition of 
$\lambda_2^j$, 
\begin{align*}
Q_{m_j}(\varphi_{2,j})
&\leq Q_{m_j}(w_j)\\
&=\frac{Q_{m_j}(\varphi_2)-2t_j\int_{\Omega}(\nabla \varphi_2\nabla \varphi_{1,j}
 +m_j(x)\varphi_2\varphi_{1,j})+t_j^2\lambda_1^j}{\|v_j\|^2_2}.
\end{align*}
 Consequently, from \eqref{l2est}--\eqref{l2est4} and the item (i),
\begin{equation}\label{limitalambda2}
\limsup_{j\to\infty} \lambda_2^j=\limsup_{j\to\infty} Q_{m_j}(\varphi_{2,j})
\leq \limsup_{j\to\infty} Q_{m_j}(w_j)=\lambda_{2}.
\end{equation}
Arguing as in the item (i), we may suppose that the sequence 
 $\{\varphi_{2,j}\}_j$ is bounded in $H^1(\Omega)$, 
$\varphi_{2,j}\rightharpoonup \varphi$ weakly in $H^1(\Omega)$, with 
$\int_{\Omega} \varphi^2=1$ and $\varphi\in V$. We assert that $\varphi$ 
is an eigenfunction associated with $\lambda_2$.
 Indeed, invoking Lemma \ref{EGOROV2} one more time, 
\begin{equation}\label{convewlsc}
\int_{\Omega} m_j(x)\varphi_{2,j}^2\to \int_{\Omega} m(x)\varphi^2.
\end{equation} 
Therefore 
\begin{equation}\label{limitaliminf2}
\liminf_{j\to\infty} Q_{m_j}(\varphi_{2,j})\geq  Q_{m}(\varphi)
\geq \lambda_2.
\end{equation}
Hence, from \eqref{limitalambda2},
 \begin{equation}\label{convparal2}
\lambda_2^j\to \lambda_2, \quad Q_{m}(\varphi)=\lambda_2.
\end{equation} 
Since $\varphi\in V$ and $\int_{\Omega} \varphi^2=1$, we may apply 
Theorem \ref{exisparalambda2} to  conclude that $\varphi$ is an 
eigenfunction associated with $\lambda_2$ as asserted. 
Finally, by \eqref{convewlsc} and \eqref{convparal2}, 
$$
\int_{\Omega} |\nabla\varphi_{2,j}|^2=\lambda_2^j
-\int_{\Omega} m_j\varphi_{2,j}^2 \to \lambda_2
-\int_{\Omega} m\varphi^2=\int_{\Omega} |\nabla\varphi|^2.
$$ 
Consequently $\varphi_{2,j}\to\varphi$ strongly in $H^1(\Omega)$. 
The proof of Theorem \ref{lema21mod} is complete.
\end{proof}

\begin{remark} \label{rmk3.5} \rm
(i) Afrouzi \cite{Afrouzi} proved Theorem \ref{lema21mod} for the Dirichlet 
boundary conditions supposing $\{m_j\}_j\subset C(\overline{\Omega})$ and 
$m_j\to m$ uniformly in $\overline{\Omega}$.

 (ii) Note that Theorem \ref{lema21mod} holds if $ m_j\to m$ strongly 
in $L^{N/2}(\Omega)$ since in this case the condition (M1) is satisfied.
 \end{remark}

Given $u\in H^1(\Omega)$, we may write $u=t_j\varphi_{1,j}+v_j$,
 where $t_j=\int_{\Omega} u\varphi_{1,j}\in\mathbb{R}$ and 
$v_j=u-t_j\varphi_{1,j}\in V_j$. Defining 
\begin{equation}\label{normavj}
\|v\|^2_{V_j}=\int_{\Omega} |\nabla v|^2+m_j(x)v^2
\end{equation} 
and recalling that $\lambda_{2,j}\to\lambda_2>0$, it is not hard to 
see that $\|\cdot\|_{V_j}$ is an equivalent norm to
 $\|\cdot\|_{H^1(\Omega)}$ in $V_j$ if $j$ is large enough. 
Actually we have the following uniform equivalence result between 
the $H^1(\Omega)$-norm and the norm given by \eqref{normavj} on $V_j$.

\begin{lemma}\label{equivalenciauniforme} 
Assume {\rm (M1), (A2)} are satisfied. Then there exist
 $j_o\in\mathbb{N}$ and constants $A,B>0$ such that
 \begin{equation}\label{equnif}
 A\|v\|^2_{H^1(\Omega)}\leq \|v\|^2_{V_j}
\leq B\|v\|^2_{H^1(\Omega)},\quad \forall v\in V_j,\;\forall j\geq j_o.
\end{equation}
\end{lemma}

\begin{proof} The existence of $B>0$ follows from H\"older's inequality 
and the Sobolev Imbedding Theorem. Thus it suffices to show the existence 
of $A>0$ in \eqref{equnif}. Arguing by contradiction, 
we suppose that  there exists a sequence $\{v_j\}_j$, with $v_j\in V_j$ 
for each $j$, such that $\|v_j\|^2_{H^1(\Omega)}\equiv 1$ and
 $ \|v_j\|^2_{V_j}<1/j$. Hence
$$
0=\lim_{j\to\infty}\int_{\Omega} |\nabla v_j|^2+m_j(x)v_j^2
\geq \lim_{j\to\infty}\lambda_{2,j}\int_{\Omega} v_j^2.
$$ 
By [(A2)] and Theorems \ref{exisparalambda2} and \ref{lema21mod}, 
$\lambda_{2,j}\to \lambda_2>0$. Up to a subsequence, 
$v_j\rightharpoonup 0$ weakly in $H^1(\Omega)$. 
Applying Lemma \ref{EGOROV2}, $\int_{\Omega} m_j(x)v_j^2\to 0$. 
Consequently  $\int_{\Omega} |\nabla v_j|^2\to 0$ and 
$\|v_j\|_{H^1(\Omega)}\to 0$.
 This contradicts $\|v_j\|_{H^1(\Omega)}\equiv 1$. 
The lemma is proved.
\end{proof}

\begin{remark}\label{normaalternativa} \rm
For further reference we remark that, since $\lambda_2>0$, 
$$
\|v\|_{V_o}:=\Big(\int_{\Omega} |\nabla v|^2+m(x)v^2\Big)^{1/2}
$$ 
is a norm equivalent to the norm  $\|\cdot \|_{H^1(\Omega)}$ in $V_o:=V$.
 Moreover,  
$$
\|u\|:=\Big(t^2+\|v\|^2_{V_o}\Big)^{1/2}, \quad
u=t\varphi_1+v,\;t\in\mathbb{R},\;v\in V_o,
$$
is a norm equivalent to the standard norm in $H^1(\Omega)$.
\end{remark}

\section{Proofs of Theorems}

In this section, using a constrained minimization method, we present
 the proof of Theorem \ref{convfraca}. After that we prove  
Theorems \ref{convsol} and \ref{theorem2.1}.
To prove Theorem \ref{convfraca}, we define 
$J(u)=\int_{\Omega} a(x)|u|^{p+1}$ and set 
$$
S_g:=\{u\in H^1(\Omega): Q_{g}(u)=1\},
$$ 
for every $g\in L^{N/2}(\Omega)$, where 
$Q_{g}(u)=\int_{\Omega} |\nabla u|^2+g(x)u^2$. 
Consider the maximization problem: 
 $$
\alpha_g:=\sup_{u\in S_g}\;J(u).
$$

\begin{lemma}\label{lemaauxiliar}
Given $g\in L^{N/2}(\Omega)$, then $S_g\neq \emptyset$. Furthermore, if
{\rm (A1)} holds, then $\alpha_g>0$.
\end{lemma}

\begin{proof} 
Given an arbitrary subdomain $\hat{\Omega}$ of $\Omega$, we let  
$\{\omega_j\}$  be a sequence in $H^1_0(\hat{\Omega})$ such that
$$
\omega_j\rightharpoonup 0\quad  \text{weakly in } H^1_0(\hat{\Omega})
\text{ and } \int_{\Omega} |\nabla \omega_j|^2 = 1  \text{ for every }  j.
$$
Noting that $H^1_0(\hat{\Omega})$ is a closed subspace of $H^1(\Omega)$, 
we also have that $\omega_j\rightharpoonup 0$ weakly in $H^1(\Omega)$. 
Therefore,  invoking Lemma \ref{EGOROV} and the fact that 
$g\in L^{N/2}(\Omega)$, we  obtain
$$
Q_{g}(\omega_j)=\int_{\Omega} |\nabla \omega_j|^2
+ \int_{\Omega}g(x){\omega_j}^2 \to 1.
$$
Hence,  for  $j$ sufficiently large, we may find $t_g = t_g(j)>0$ such that 
$Q_{g}(t_g\omega_j)=t_g^2Q_{g}(\omega_j)  =1$. 
This implies that $S_g\neq  \emptyset$.

When (A1) holds,  taking $\hat{\Omega}\subset \Omega^+$,    
the above argument may be used to conclude that   $\alpha_g>0$. 
Indeed by construction  $\omega_j \in H^1_0(\hat{\Omega})\setminus \{0\}$ and, 
consequently,  $J(t_gw_j)=t_g^{p+1}\int_{\Omega^+} a(x)|w_j|^{p+1}>0$.
 The proof of Lemma \ref{lemaauxiliar} is complete. 
\end{proof}

Considering the sequence $\{m_j\}_j$, given by (M1), we set
 $S_j:= S_{m_j}\neq \emptyset$ and $\alpha_j:= \alpha_{m_j}>0$. 
Moreover, as in Section 3, we set $m_o:=m$.
In the next results we verify the existence of a nonnegative 
function $u_j\in S_j$ such that $J(u_j)=\alpha_j$. 
After that we prove Theorem \ref{convfraca} by rescaling $u_j$ and using 
Lagrange's Theorem and Theorems \ref{KENIG} and \ref{DG}.

Applying Lemmas \ref{intnegativa} and \ref{equivalenciauniforme}, 
we may find $\delta>0$ such that, for every $j\geq j_o$, $\|\cdot\|_{V_j}$ 
satisfies \eqref{equnif} and  
 \begin{equation}\label{intneg2}
\int_{\Omega} a(x)|\varphi_{1,j}+v|^{p+1}
<\frac{1}{2}\int_{\Omega} a(x)\varphi_1^{p+1}<0,\quad \forall
v\in V_j,\;\|v\|^2_{V_j}<\delta.
\end{equation}


\begin{lemma}\label{lema4.2} 
Assume {\rm (A1)--(A3),  (M1)}. Then there exists $j_1\in \mathbb{N}$ such that, 
for every $j\geq j_1$, we may find $u=u_j\in S_j$, $u\geq 0$ and
 $J(u)=\alpha_j>0$.
\end{lemma}

\begin{proof} 
Considering $j\geq j_o$, where $j_o$ is given by \eqref{intneg2}, 
from (A2), (M1) and Theorem \ref{lema21mod}, we may find $j_1\geq j_o$ 
such that $|\lambda_{1,j}|<\delta/2$ for every $j\geq j_1$. 
For the rest of this article, we fix $j\geq j_1$. Let $\{u_{k}\}_k\subset S_j$ 
be sequence such that 
\begin{equation}\label{positivo}
J(u_{k})\to  \alpha_j>0\quad \text{as }k\to\infty.
\end{equation}
Using the decomposition $H^1(\Omega)=\mathbb{R}\varphi_{1,j}\oplus V_j$, 
we may write $u_{k}=t_{k}\varphi_{1,j}+v_{k}$, with 
$t_{k}=\int_{\Omega} u_{k}\varphi_{1,j}\in\mathbb{R}$ and 
$v_{k}=u_{k}-t_{k}\varphi_{1,j}\in V_j$. By Theorem \ref{lema21mod}
 and Lemma \ref{equivalenciauniforme}, there is $C>0$ such that,   
\begin{equation}\label{estimaparanorma}
\|u_{ k}\|^2_{H^1(\Omega)}\leq 2[t_{k}^2\|\varphi_{1,j}\|^2_{H^1(\Omega)}
+\|v_{k}\|^2_{H^1(\Omega)}]\leq C[t_{k}^2+\|v_{k}\|^2_{V_j}].
\end{equation}
Furthermore, 
\begin{equation}\label{1}
1=Q_{m_j}(u_{k})=\|v_{k}\|^2_{V_j}+\lambda_{1,j}\;t_{k}^2.
\end{equation}
We assert that $\{u_{k}\}_k$ is a bounded sequence in $H^1(\Omega)$. 
Since $|\lambda_{1,j}|<\delta/2$, in view of \eqref{1}
 and \eqref{estimaparanorma}, it suffices to show that 
$\{t_{k}\}_k\subset\mathbb{R}$ is bounded. Arguing by contradiction 
and taking a subsequence if necessary, we suppose that $|t_{k}|\to\infty$. 
Consequently, from \eqref{1},
  $$
\limsup_{k\to\infty}\|\frac{v_{k}}{t_{k}} \|_{V_j}^2
\leq  \limsup_{k\to\infty}\{\frac{1}{t_k^2}+|\lambda_{1,j}|\} 
=|\lambda_{1,j}|\leq \frac{\delta}{2}.
$$
Hence, using \eqref{intneg2} one more time, 
$$
\lim_{k\to\infty}J(u_{k})=\lim_{k\to\infty}|t_{k}|^{p+1}
\int_{\Omega} a(x)|\varphi_{1,j}+\frac{v_{k}}{t_{k}}|^{p+1}=-\infty.
$$ 
This contradicts \eqref{positivo}. The sequence $\{u_{k}\}_k\subset H^1(\Omega)$ 
is bounded as asserted. Using this assertion, we may suppose that  
$t_{k}\to t\in\mathbb{R}$, $v_{k}\rightharpoonup  v$ weakly in $V_j$. 
Consequently $u_{k}\rightharpoonup  u=t\varphi_{1,j}+v$ weakly in 
$H^1(\Omega)$ and 
\begin{equation}\label{alphajpositivo}
0<\alpha_j=\lim_{k\to\infty} \int_{\Omega} a(x)|u_{k}|^{p+1}
=\int_{\Omega}\;a(x)|u|^{p+1}.
\end{equation} 
It remains to show that $u\in S_j$. Applying Lemma \ref{EGOROV}, we have
$$
1=\liminf_{k\to\infty} Q_{m_j}(u_{k})\geq \int|\nabla u|^2+m_j(x)u^2=Q_{m_j}(u).
$$
Actually $Q_{m_j}(u)=1$. Indeed, if  
$Q_{m_j}(u)=\|v\|^2_{V_j}+\lambda_{1,j} t^2\leq 0$, 
then $\|v\|^2_{V_j}=0$ for $t=0$, or 
$\|v/t\|_{V_j}^2\leq |\lambda_{1,j}|<\delta/2$ if $t\neq 0$. 
In both cases, by \eqref{intneg2}, $J(u)\leq 0$. This contradicts 
\eqref{alphajpositivo}. Therefore we may suppose that 
$0<Q_{m_j}(u)\leq1$ and that there exists $\rho\geq1$ such that
 $Q_{m_j}(\rho u)=1$. Since $J(\rho u)=\rho^{p+1} J (u)=\rho^{p+1}\alpha_j$, 
we obtain $\rho=1$, $Q_{m_j}(u)=1$ and $u\in S_j$. 
Finally, observing that $J$ and $Q_{m_j}$ are even, we may suppose 
$u\geq 0$. The proof of Lemma \ref{lema4.2} is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{convfraca}]
 By Lemmas \ref{lemaauxiliar}, \ref{lema4.2} and  Lagrange's Theorem, 
there exist $j_1\in\mathbb{N}$ and $u_j\in S_j,u_j\geq 0$, for every 
$j\geq j_1$, such that  
\begin{equation}\label{formafracauj}
\int_{\Omega} \nabla u_j\nabla w +m_j(x)u_jw
=\frac{1}{\alpha_j}\int_{\Omega} a(x)|u_j|^{p-1}u_jw,\quad \forall 
w\in H^1(\Omega).
\end{equation}
That is, $u_j$ is a nonnegative weak solution for the problem
 $$
-\Delta u+m_j(x)u=\frac{1}{\alpha_j} a(x)u^p,\quad x\in\Omega,\;
u\in H^1(\Omega),
$$
 under the Neumann boundary conditions. Applying Theorems \ref{KENIG} 
and \ref{DG}, we obtain  $u_j>0$. Setting 
$\omega_j:=(1/\alpha_j)^{1/(p-1)}u_j$, we have that $\omega_j>0$ 
is a weak solution for \eqref{ePm} with $m_j$ instead of $m$  
for all $j\geq j_1$. The proof of Theorem \ref{convfraca} is complete.
\end{proof}

 Before proving Theorem \ref{convsol}, we  consider the following lemma.

\begin{lemma}\label{comportamento-alpha-j}
Assume {\rm (A1)--(A3), (M1)}. Then there exists $M>0$ and 
$j_2\in\mathbb{N}$, such that $\|u_j\|_{H^1(\Omega)}\leq M$ for each 
$u_j\in S_j$ satisfying $J(u_j)=\alpha_j$, for all $j\geq j_2$.
\end{lemma}

\begin{proof} 
Writing $u_j=s\varphi_{1}+v\in \mathbb{R}\varphi_{1}\oplus V$, if 
$s\neq  0$, 
$$
0<\alpha_j=J(u_j)=|s|^{p+1}\int_{\Omega} a(x)|\varphi_1+\frac{v}{s}|^{p+1}.
$$ 
By Lemma \ref{intnegativa}, $\|v\|_{H^1(\Omega)}>\beta|s|$. 
From Remark \ref{normaalternativa}, there exists $C>1$  such that 
\begin{equation}\label{limitacaopara uj}
\|u_j\|_{H^1(\Omega)}
\leq\big(\frac{1}{\beta}\|\varphi_1\|_{H^1(\Omega)}+1\big)\|v\|_{H^1(\Omega)}
\leq C\|v\|_{V}.
\end{equation} 
Note that \eqref{limitacaopara uj} is trivially true when $s=0$. 
Hence, by Corollary \ref{compara-t-e-v}, given $0<\epsilon<1/C$, 
there exists $j_{2}\in\mathbb{N}$ such that
\begin{align*}
1&=Q_{m_j}(u_j)=\|v\|^2_{V}+\int_{\Omega} [m_j(x)-m(x)]u_j^2\\
&\geq\|v\|^2_{V}-\epsilon\|u_j\|_{H^1(\Omega)}^2\\
&\geq\big(\frac{1}{C}-\epsilon\big)\|u_j\|_{H^1(\Omega)}^2,\quad
 \forall j\geq j_{2}.
\end{align*}
The proof of Lemma \ref{comportamento-alpha-j} is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{convsol}]
 Defining  $u_j:=(\alpha_j)^{1/(p-1)}w_j$, where $\{w_j\}_j$ is the sequence 
of solutions from Theorem \ref{convfraca}, we obtain $u_j>0$, $u_j\in S_j$ 
and $J(u_j)=\alpha_j$. Moreover $u_j$ satisfies \eqref{formafracauj}. 
From (M1), Lemma \ref{comportamento-alpha-j}, Lemma \ref{EGOROV}, 
Corollary \ref{compara-t-e-v} and taking a subsequence if necessary, 
we may suppose that
\begin{gather}\label{convdeujlfa}
u_j\rightharpoonup u\quad \text{weakly in } H^1(\Omega),\\
\label{convdeujlp}
u_j\to u\quad \text{strongly in }L^{\tau}(\Omega),\;1\leq\tau<2^{\ast},\\
\label{convdeuj2}
\int_{\Omega} m_j(x)u_j^2\to\int_{\Omega} m(x)u^2,\\
\label{fracamjuj}
\int_{\Omega} m_j(x)u_j\phi\to\int_{\Omega} m(x)u\phi,\quad \forall 
 \phi\in H^1(\Omega),\\
\label{convdeuj7}
\int_{\Omega} [m_j(x)-m(x)]u_j^2\to 0.
\end{gather}

Applying the same arguments used in the proof of Lemma \ref{lema4.2}, 
we find a  positive solution $u_o\in H^1(\Omega)$ for the  maximization 
problem  
$$
0<\alpha_o:=\sup_{Q_m(u)=1} J(u)=\int_{\Omega} a(x)|u_o|^{p+1}<\infty.
$$
 We claim that $\alpha_j\to \alpha_o$. Indeed, by (M1), $Q_{m_j}(u_o)\to 1$. 
Thus there exist $j_o$ and $\{\theta_j\}_j\subset (0,\infty)$ such that 
$Q_{m_j}(\theta_ju_o)\equiv 1$ for all $j\geq j_o$. Since 
$\theta_j^2Q_{m_j}(u_o)\equiv 1$, 
$$
\lim_{j\to\infty}\theta_j=1.
$$ 
Using this and the fact that  $\theta_ju_o\in S_j$, 
\begin{equation}\label{infalphaj}
\liminf_{j\to\infty} \alpha_j\geq \liminf_{j\to\infty} 
\int_{\Omega} a(x)|\theta_j u_o|^{p+1}=\liminf_{j\to\infty}
 |\theta_j |^{p+1}\alpha_o=\alpha_o>0 .
\end{equation}
On the other hand, from \eqref{convdeuj7},
$$
Q_{m}(u_j)=Q_{m_j}(u_j)+\int_{\Omega} [m(x)-m_j(x)]u_j^2
= 1+\int_{\Omega} [m(x)-m_j(x)]u_j^2\to 1.
$$
 Thus, taking $\beta_j>0$ such that $Q_m(\beta_ju_j)\equiv 1$, 
we have that $\beta_j\to 1$. Hence, from the definition of $\alpha_o$, 
\begin{gather}
\alpha_o\geq J(\beta_ju_j)=\beta_j^{p+1}\alpha_j, \nonumber \\
\limsup_{j\to\infty} \alpha_j\leq \limsup_{j\to\infty} 
\frac{\alpha_o}{\beta_j^{p+1}}=\alpha_o. \label{supalphaj}
\end{gather} 
We conclude from \eqref{infalphaj} and \eqref{supalphaj} that 
$\alpha_j\to\alpha_o>0$. The claim is proved.

By the above claim and \eqref{convdeujlp},
\begin{equation}\label{convdeuj3}
\alpha_o=\lim_{j\to\infty}\alpha_j
=\lim_{j\to\infty}\int_{\Omega} a(x)|u_j|^{p+1}
=\int_{\Omega} a(x)|u|^{p+1}.
\end{equation}
Hence $u\neq 0$. Since $u_j>0$ satisfies \eqref{formafracauj} for each 
$j\geq j_1$, using \eqref{convdeujlfa}, \eqref{convdeujlp},\eqref{convdeuj2}
 and \eqref{fracamjuj}, we obtain 
$$
\int_{\Omega} \nabla u\nabla\varphi+m(x)u\varphi
=\frac{1}{\alpha_o}\int_{\Omega} a(x)|u|^{p-1}u\varphi\quad
 \forall\;\varphi\in H^1(\Omega).
$$ 
In particular $w_o:=\alpha_o^{-1/(p-1)}u$ is a solution for \eqref{ePm}. 
A similar argument shows that
\begin{align*}
\lim_{j\to\infty}\int_{\Omega} |\nabla u_j|^2
&=\lim_{j\to\infty}\big\{ \frac{1}{\alpha_j}\int_{\Omega} a(x)|u_j|^{p+1}
-\int_{\Omega} m_j(x)u_j^2\big\}\\
&=\frac{1}{\alpha_o}\int_{\Omega} a(x)|u|^{p+1} 
 - \int_{\Omega} m(x)u^2=\int_{\Omega} |\nabla u|^2.
\end{align*}
 Consequently, we have that $u_j\to u$ strongly in $H^1(\Omega)$ and $Q_m(u)=1$. 
Thus $u\in S_o=S_m$ and, by \eqref{convdeuj3}, $J(u)=\alpha_o$. 
As before  by Theorems \ref{KENIG} and \ref{DG}, $u>0$. 
Moreover 
$$
w_j:=\big(\frac{1}{\alpha_j}\big)^{\frac{1}{p-1}}u_j\to w_o\quad
\text{in }H^1(\Omega).
$$
 The proof of Theorem \ref{convsol} is complete. 
\end{proof}

\begin{remark}\label{versaoBCDN} \rm
  Note that under the hypotheses (A1)--(A3)  and (M1), there exist $r_0>0$ 
and $j=j_0$ such that \eqref{ePm}, with $m_{j_n}-\tau_n$ instead of $m$,
possesses a positive solution 
whenever $\tau\in[0,r_0)$ and $j\geq j_0$. Indeed, arguing by contradiction,
 we find sequences $\{\tau_n\}\in [0,\infty)$ and $\{j_n\}$ such that 
\eqref{ePm}, with $m_{j_n}-\tau_n$ instead of $m$, has no positive solution 
and  $\tau_n\to 0$, $j_n\to\infty$ as $n\to\infty$. However, by 
Theorem \ref{convfraca}, for every $n$ sufficiently large, 
\eqref{ePm}, with $m_{j_n}-\tau_n$ instead of $m$, 
 possesses a positive solution  since $m_{j_n}-\tau_n\rightharpoonup m$ 
weakly in $L^{\sigma}(\Omega)$ as $n\to\infty$. 
This contradiction implies that the result holds.

It is worthwhile mentioning that, by Theorem \ref{lema21mod},  
$\lambda_1(-\Delta+m_j-\tau)\to \lambda_1(-\Delta+m-\tau)=-\tau<0$ 
for every $\tau>0$. We also note that the existence of positive 
solutions for \eqref{ePm}, with $m-\tau$ instead of $m$,  was considered 
in \cite{BCDN}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{theorem2.1}]
Arguing by contradiction, we suppose that there exist $R>0$ 
(or $f\in L^{N/2}(\Omega)$ if $\sigma=N/2$) and a sequence 
$\{m_j\}_j\subset L^r(\Omega)$ such that 
(i) $\|m- m_j\|_{L^{\sigma}(\Omega)}\leq R$ 
 (or $|m_j|\leq f$ a.e. in $\Omega$, if $\sigma=N/2$),
(ii) $Q_{m_j}(\varphi_1)\leq 1/j$, 
(iii) $-1/j<\lambda_1(-\Delta+m_j)<0$, and \eqref{ePm} with $m_j$ instead of $m$ 
 has no solution for all $j\in\mathbb{N}$. 
Taking a subsequence if necessary, $m_j\rightharpoonup \widehat{m}$ 
weakly in $L^{\sigma}(\Omega)$. Then, from 
(A2), Theorem \ref{lema21mod} and (ii) and (iii) above,
\begin{gather*} 
\lambda_1(-\Delta+\widehat{m})=\lim_{j\to\infty}\lambda_1(-\Delta+m_j)=0, \\
0\leq\int_{\Omega}|\nabla\varphi_1|^2 +\widehat{m}(x)\varphi_1^2
=\lim_{j\to\infty}\int_{\Omega}|\nabla\varphi_1|^2 +m_j(x)\varphi_1^2\leq 0.
\end{gather*}
Hence $\varphi_1$ is an eigenfunction for \eqref{EPml}, with 
$\widehat{m}$ instead of $m$,
 associated with $\lambda_1(-\Delta+\widehat{m})=0$. Consequently 
$$
\int_{\Omega}\nabla \varphi_1\nabla \omega+\widehat{m}(x)\varphi_1 \omega=0
=\int_{\Omega}\nabla \varphi_1\nabla \omega+m(x)\varphi_1 \omega,\quad
\forall \omega\in H^1(\Omega),
$$ 
and 
$$
\int_{\Omega}(\widehat{m}(x)-m(x))\varphi_1 \omega=0\quad
\forall \omega\in H^1(\Omega).
$$ 
This implies that $(\widehat{m}-m)\varphi_1= 0$ a.e. in  $\Omega$.
Since $\varphi_1>0$, $\widehat{m}\equiv m$. 
Therefore, $m_j\rightharpoonup m$ weakly in $L^{\sigma}(\Omega)$ and 
$\{m_j\}_j$ satisfies the condition (M1). From (A1), (A2), (A3)
 and  Theorem  \ref{convfraca}, \eqref{ePm} with $m_j$ instead of $m$  
possesses a solution for $j$ sufficiently large. 
This contradiction concludes the proof of Theorem
  \ref{theorem2.1}.
\end{proof}

 We conclude this article by presenting an application of 
Theorems \ref{convfraca} and \ref{convsol} to the problem
\begin{gather*}
-\Delta u+\varphi(x)\cos{(jx_N)}u=a(x)u^{p},\quad x\in\Omega,\\
\frac{\partial u}{\partial\eta} = 0 \quad x\in \partial\Omega,
\end{gather*}
where $\varphi\in C_o^{\infty}(\Omega)$ is a nontrivial function,  
$1<p <(N+2)/(N-2)$  and $x_N\in\mathbb{R}$ is the last component of 
$x=(x_1,\dots,x_N)\in\Omega$.

Assuming that $a$ changes sign and satisfies $\int_{\Omega} a(x)<0$,
 we may assert that this problem has  a positive solution $u_j$ 
for every $j\in\mathbb{N}$ sufficiently large. Moreover, up to a subsequence, 
$u_j\to u$ in $H^1(\Omega)$ where $u>0$ is a solution of 
\eqref{ePm} with $0$ instead of $m$.
Indeed, to derive such result, it suffices to observe that 
$m_j:=\varphi(x)\cos{(jx_N)} \in L^{\infty}(\Omega)$ and 
$m_j\rightharpoonup 0$ weakly in $L^s(\Omega)$, for all $s>1$. 
We  note that $m_j$ does not converge strongly in any $L^s(\Omega)$. 
Hence, in particular, the results from \cite{Afrouzi} and \cite{Tehrani} 
may not be applied to this problem  since $\{m_j\}_j$ does not converge 
uniformly to zero.


\section{Appendix: Regularity of solutions}

For the convenience of the reader,  we present the argument necessary 
to establish  the regularity of the solutions for \eqref{EPml} and \eqref{ePm}.
First we recall the following results.

\begin{lemma}[\cite{WANG}]\label{Wang} 
Suppose $\partial\Omega\in C^1$, $b\in L^{N/2}(\Omega)$ and 
$\alpha\in L^{\infty}(\Omega)$. If $u\in H^1(\Omega)$ is a weak solution of 
\begin{gather*}
-\Delta u=b(x)u\quad\text{in } \Omega, \\
 \frac{\partial u}{\partial\eta}=\alpha(x)u\quad\text{on }
 \partial\Omega,
\end{gather*}
then $u\in L^t(\Omega)$ for all $1\leq t<\infty$.
 \end{lemma}

  \begin{lemma}[\cite{ADN}]\label{ADN} 
Suppose $\partial\Omega\in C^2$, $h\in L^s(\Omega)$ and $g\in H^{1,s}(\Omega)$ 
for some $s\in(1,\infty)$. If $u\in  H^1(\Omega)$ is a weak solution of 
\begin{gather*}
-\Delta u = h(x)\quad\text{in } \Omega\\ 
\frac{\partial u}{\partial\eta}=g(x)\quad\text{on } \partial\Omega,
\end{gather*}
then $u\in W^{2,s}(\Omega)$ and 
$\|u\|_{W^{2,s}(\Omega)}\leq C(\|h\|_{L^{s}(\Omega)}+\|g\|_{W^{1,s}
(\partial\Omega)}+\|u\|_{L^{s}(\Omega)})$ for some $C>0$.
\end{lemma}

  The above lemma due to Agmon, Douglis and Nirenberg is also cited in 
\cite{WANG}. Now, as an application of the above mentioned results, we sate the
following result.

\begin{lemma}
Suppose $m\in L^{\sigma}(\Omega)$, $\sigma\geq N/2$, $\varphi$ 
is an eigenfunction for \eqref{EPml} and $u$ is a solution for \eqref{ePm}. 
Then $\varphi,u\in L^t(\Omega)$, for $1\leq t<\infty$. Moreover 
$\varphi,u\in C^{0,\gamma}(\overline{\Omega})$ provided $\sigma>\frac{N}{2}$. 
If $\sigma=\frac{N}{2}$ then $\varphi,u\in W^{2,s}(\Omega)$ 
for all $s\in[2N/(N+2),N/2)$.
 \end{lemma}

\begin{proof} 
Defining the functions  $b_1:=\lambda-m$ and $b_2:=au^{p-1}-m$, 
from Lemma \ref{Wang} we obtain $\varphi,u\in L^t(\Omega)$ for all 
$1\leq t <\infty$.
Next, considering $\sigma>N/2$ and setting $h_1:=b_1\varphi$ and 
$h_2:=b_2u$, it is easy to see that $h_i\in L^{s}(\Omega)$ for all  
$s\in\left(N/2,\sigma\right),i=1,2$. Indeed, for each $s\in(N/2,\sigma)$, 
choose $\alpha>\sigma/(\sigma-s)$ and take  $\theta>1$ such that 
$1/\theta+s/\sigma+1/\alpha=1$. Then, by H\"older inequality, 
$$
\|m u\|_s^s\leq|\Omega|^{1/\theta} \|m\|_{\sigma}^s \|u\|_{s\alpha}^s<\infty.
$$ 
Since $au^p \in L^{s}(\Omega)$ and $s>N/2$, we may apply the Lemma \ref{ADN} 
and the Sobolev Imbedding Theorem to conclude that 
$$
\varphi,u\in W^{2,s}(\Omega)\hookrightarrow C^{0,\gamma}(\overline{\Omega}) 
\quad\text{for some } 0<\gamma<1.
$$
In the case $\sigma=N/2$, given $s\in \left[2N/(N+2),N/2\right)$,
 we choose $\alpha>N/(N-2s)$ and take $\theta>1$ such that 
$1/\theta+2s/N+1/\alpha=1$. By H\"older inequality $mu\in L^{s}(\Omega)$. 
Applying Lemma \ref{ADN}, we conclude that $\varphi,u\in W^{2,s}(\Omega)$ 
for all $s\in [2N/(N+2),N/2)$.
\end{proof}

\begin{remark} \rm
If $\sigma>N$, then $h_i\in L^s(\Omega)$ for all $s\in(N,\sigma)$. 
Repeating the above steps, 
$\varphi,u\in C^{1,\gamma}(\overline{\Omega})$, $i=1,2$.
\end{remark}

\section*{Proofs of Theorems \ref{teoparalambda1} and \ref{exisparalambda2}}

As observed in Section 3, the arguments employed in the proofs 
of Theorems \ref{teoparalambda1} and \ref{exisparalambda2} are due 
to Manes and Micheletti \cite{MM}.
In order to verify that the first eigenvalue is simple and that the 
first eigenfunction, $\varphi_1$, is positive (almost everywhere) 
in $\Omega$, we apply Theorems \ref{KENIG} and \ref{DG} 
(see Remark \ref{positive}).


\begin{proof}[Proof of Theorem \ref{teoparalambda1}]
By H\"older's inequality and Sobolev Imbedding Theorem, there exists $C>0$ 
such that 
\begin{equation}\label{Qu} 
|Q_m(u)|\leq (1+C\|m\|_{N/2}) \|u\|_{H^1(\Omega)}^2.
\end{equation} 
As a direct consequence of \eqref{Qu}, $\lambda_1<\infty$. 
Now let $\{u_k\}_k\subset H^1(\Omega)$ be a sequence such that $\int u_k^2=1$
and  $Q_m(u_k)\to\lambda_1$. We claim that $\{\|u_k\|_{H^1(\Omega)}\}_k$ 
is bounded. Otherwise, from \eqref{Qu}, up to a subsequence, 
$\|\nabla u_k\|_2\to\infty$. Defining $w_k:=\frac{u_k}{\|\nabla u_k\|_2}$ 
and applying Sobolev Imbedding Theorem one more time, we may suppose
 $w_k\rightharpoonup 0$ weakly in $H^1(\Omega)$ since $\|u_k\|_2\equiv 1$. 
By Lemma \ref{EGOROV}, $\int m(x)w_k^2\to 0$. Thus, since  
\begin{equation}\label{Qinf}
\frac{Q_m(u_k)}{\|\nabla u_k\|_2^2}=1+\int m(x)w_k^2,
\end{equation} 
$Q_m(u_k)\to\infty$. This contradicts $\lambda_1<\infty$. Thus 
$\{u_k\}_k\subset H^1(\Omega)$ is a bounded sequence as claimed. 
Passing to a subsequence if necessary, $u_k\rightharpoonup\varphi_1$ 
weakly in $H^1(\Omega),\|\varphi_1\|_2=1$ and, by Lemma \ref{EGOROV}, 
$$
\int m(x)u_k^2\to\int m(x)\varphi_1^2.
$$ 
Then, since   $\liminf_{k\to\infty}\|\nabla u_k\|_2^2\geq \|\nabla \varphi\|_2^2$,
$$
\lambda_1=\liminf_{k\to\infty}Q_m(u_k)\geq  Q_m(\varphi_1).
$$  
This implies that $Q_m(\varphi_1)=\lambda_1$. Now we show that 
$\varphi_1$ solves  \eqref{EPml} with $\lambda_1$ instead of $\lambda$. 
Given $u\in H^1(\Omega)$, for every $t\in\mathbb{R}\backslash\{0\}$ 
sufficiently small, 
$$
\frac{Q_m(\varphi_1+tu)}{\int (\varphi_1+tu)^2}
=\frac{\int|\nabla(\varphi_1+tu)|^2+m(x)(\varphi_1+tu)^2}{\int (\varphi_1+tu)^2}
\geq \lambda_1.
$$
Consequently, 
\begin{equation}\label{Raylegh1}
2t\int \nabla u\nabla\varphi_1+m(x)u\varphi_1+t^2Q_m(u)
\geq 2t\lambda_1\int u\varphi_1+t^2\lambda_1\int u^2.
\end{equation} 
Dividing \eqref{Raylegh1}
 by $2t$ and taking $t\to 0$ ($t\to 0^+$ and $t\to 0^-$), we obtain 
\begin{equation}\label{Raylegh2}
\int \nabla u\nabla\varphi_1+m(x)u\varphi_1
= \lambda_1\int u\varphi_1,\quad \forall u\in H^1(\Omega).
\end{equation} 
This shows that $\varphi_1$ is a solution for \eqref{EPml} with $\lambda_1$ 
instead of $\lambda$. As observed in Remark \ref{smallestev}, 
this also implies that $\lambda_1$ is the first eigenvalue for \eqref{EPml}. 
Notice that we may suppose $\varphi_1\geq 0$ since $Q_m(|u|)=Q_m(u)$.
 By Remark \ref{positive}, $\varphi_1$
is positive in $\Omega$.

Finally we verify that $\lambda_1$ is a simple eigenvalue and that any 
associated eigenfunction $v\in H^1(\Omega)\backslash\{0\}$ does not 
change sign. A simple arithmetic shows that: for every $A,B,X,Y\in\mathbb{R}$ 
with $X,Y >0$, we have either
$$
\frac{A+B}{X+Y}=\frac{A}{X}=\frac{B}{Y}
$$
or
$$
\min\big\{\frac{A}{X},\frac{B}{Y}\big\}
<\frac{A+B}{X+Y}
<\max\big\{\frac{A}{X},\frac{B}{Y}\big\}.
$$
Since 
$$
\lambda_1=\frac{Q_m(v)}{\int v^2}=\frac{Q_m(v^+)+Q_m(v^-)}{\int (v^+)^2
+\int (v^-)^2},
$$
 if $v^+:=\max\{v,0\}\geq 0$ and $v^-:=\max\{-v,0\}\geq 0$ are nontrivial 
functions, we have  
$$
(I)\quad \lambda_1=\frac{Q_m(v^+)}{\int (v^+)^2}
=\frac{Q_m(v^-)}{\int (v^-)^2}
$$
or
$$
(II)\quad \lambda_1>\min\big\{\frac{Q_m(v^+)}{\int (v^+)^2},
\frac{Q_m(v^-)}{\int (v^-)^2}\}.
$$ 
In case I, we also have that $v^+$ and $v^-$ are (after normalization) 
eigenfunctions associated with $\lambda_1$. Applying Theorems \ref{KENIG} 
and \ref{DG}, with $c(x):=m(x)-\lambda_1$, we have simultaneously 
$v^+>0$ and $v^->0$ in $\Omega$, which it is not possible. 
On the other hand, case II contradicts the definition of $\lambda_1$. 
This shows that $v$ does not change sign in $\Omega$.


Now, given any $\alpha\in\mathbb{R}$, by the above claim the function 
$\varphi_1-\alpha v$ does not change sign in $\Omega$. Defining 
$$
A:=\{\alpha\in\mathbb{R}: \varphi_1\geq \alpha v\text{ a.e.}\},\quad 
B:=\{\alpha\in\mathbb{R}: \varphi_1\leq \alpha v\text{ a.e.}\},
$$
it is not difficult to verify that $A$ and $B$ are closed subsets of 
$\mathbb{R}$. Since $A$ and $B$ are nonempty and $A\cup B=\mathbb{R}$, 
there is  $\alpha_o\in A\cap B$ and, consequently, 
$\varphi_1=\alpha_o v$ a.e. in $\Omega$. Therefore, $\lambda_1$ 
is a simple eigenvalue. The verification that $\lambda_1$ is 
isolated will be made in the proof of Theorem \ref{exisparalambda2}. 
The proof of Theorem \ref{teoparalambda1} is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{exisparalambda2}]
By definition \ref{deflab2}, $\lambda_2\geq\lambda_1$. 
Similar argument to the one used in the proof of Theorem 
\ref{teoparalambda1} implies that there exists $\varphi_2\in V$ 
associated with $\lambda_2$. In order to show that $\lambda_1<\lambda_2$ 
we suppose by contradiction  that $\lambda_1=\lambda_2$. 
Then, by Theorem \ref{teoparalambda1}, there exists 
$\alpha\in\mathbb{R}\backslash\{0\}$ such that $\varphi_1=\alpha\varphi_2$. 
But this contradicts  $\int\varphi_1\varphi_2=0$. Moreover it is clear, 
from the definition of $V$, that $\varphi_2$ changes sign in $\Omega$.

Now we show that $\varphi_2$ is a solution  for \eqref{EPml}: 
given $u\in H^1(\Omega)$, we write $u=t\varphi_1+v$, with
 $t=\int \varphi_1u\;\in\mathbb{R}$ and $v=u-t\varphi_1\;\in V$. 
Arguing as in the verification of \eqref{Raylegh2}, 

\begin{equation}\label{fracv}
\int \nabla v\nabla\varphi_2+m(x)v\varphi_2
= \lambda_2\int v\varphi_2,\quad \forall v\in V.
\end{equation} 
Noting that $\int \varphi_1(\varphi_2+sv)=0$ for each $s\in\mathbb{R}$ 
and $v\in V$ and, in view of \eqref{Raylegh2} and \eqref{fracv}), we obtain
\begin{align*}
\int \nabla u\nabla\varphi_2+m(x)u\varphi_2
&=t\int \nabla \varphi_1\nabla\varphi_2+m(x)\varphi_1\varphi_2
 +\int \nabla v\nabla\varphi_2+m(x)v\varphi_2\\
&=\lambda_2\int u\varphi_2, \quad\forall u\in H^1(\Omega).
\end{align*}
Thus $\varphi_2$ is an eigenfunction for \eqref{EPml}. 
Our final task is to verify that \eqref{EPml} has no solution 
if $\lambda\in(\lambda_1,\lambda_2)$. That is, $\lambda_1$ is isolated.
 Otherwise, take $\varphi$ a normalized eigenfunction for \eqref{EPml} 
with $\lambda\in(\lambda_1,\lambda_2)$. Hence 
$$
\lambda\int\varphi\varphi_1=\int \nabla \varphi_1\nabla\varphi
+m(x)\varphi_1\varphi=\lambda_1\int\varphi\varphi_1.
$$ 
Since $(\lambda_1-\lambda)\neq 0$, $\int\varphi\varphi_1=0$. 
Thus $\varphi\in V$. However this contradicts the definition of 
$\lambda_2$, since  $\lambda=\lambda\int\varphi^2=Q_m(\varphi)<\lambda_2$. 
The proof of Theorem \ref{exisparalambda2} is complete. 
\end{proof}


\begin{thebibliography}{00} 

\bibitem{Afrouzi}{G. A. Afrouzi};
\emph{Existence of positive solutions on indefinite superlinear
 elliptic equations}, Applied Math. Computation \textbf{157} (2004) 841--848.

\bibitem{ADN} {S. Agmon}, {A. Douglis}, {L. Nirenberg};
\emph{Estimates near the boundary for solutions of elliptic partial 
differential equations satisfying general boundary conditions I}, 
Comm. Pure Applied Math. \textbf{XII}, 623--727, (1959).

\bibitem{AlDel}{S. Alama}, {M. Del Pino};
\emph{Solutions of elliptic equations with indefinite nonlinearities 
via Morse theory and linking}, Ann. Inst. H. Poincar\'e Anal. 
Non Lineaire \textbf{13} (1996) 95-115.

\bibitem{AlTar1}{S. Alama}, {G. Tarantello};
\emph{On semilinear elliptic equations with indefintite nonlinearities}, 
Calc. Var. \textbf{1} (1993)  439-475.


\bibitem{BCDN} {H. Berestycki}, {I. Capuzzo-Dolcetta,} {L. Nirenberg};
\emph{ Variational methods for indefinite superlinear homogeneous elliptic 
problems}, NoDEA 2 No 4 (1995) 553--572.

\bibitem{BCDN2} {H. Berestycki}, {I. Capuzzo-Dolcetta,} {L. Nirenberg};
\emph{Superlinear indefinite elliptic problems and nonlinear Liouville theorems},
 Topol. Methods Nonlinear Anal. \textbf{4} (1994) 59-78.

\bibitem{dF} {D. deFigueiredo};
\emph{Differential equations},  Lecture Notes in Mathematics 957. 
Springer-Verlag, Berlin-New York, 34-87, 1982.

\bibitem{DG}{D. deFigueiredo}, {J. P. Gossez};
\emph{Strict  monotonicity of the eigenvalues and unique continuation}, 
Comm. Partial Differential Equations  17 no. 1-2 (1992) 339--346 .

\bibitem{JK}{D. Jerison}, {C. E. Kenig};
\emph{Unique continuation and absence of positive eigenvalues for 
Schr\"odinger operators}, Annals of Mathematics \textbf{121} (1985) 463-494.

\bibitem{MM} {M. Manes}, {A. M. Micheletti};
\emph{Un'estensione della teoria variazionale classica degli autovalori 
per operatori ellittici del secondo ordine},
Bolletino U.M.I  7 (1973) 285-301.

\bibitem{RTT}{M. Ramos}, {S. Terracini}, {C. Troestler};
\emph{Superlinear indefinite elliptic problems and Pohozaev type identities}, 
J. Funct. Anal. \textbf{159} (1998) 596-628.

\bibitem{Tehrani} {H. Tehrani};
\emph{On indefinite superlinear elliptic equations}, 
Cal. Var. \textbf{4} (1996) 139-153.

\bibitem{WANG} {Xu-Jia Wang};
\emph{Neumann Problems of Semilinear Elliptic Equations Involving Critical 
Sobolev Exponent}, J. Diff. Eq., \textbf{93}, 283-310, (1991).

\bibitem{W} {M. Willem};
\emph{Minimax Theorems}, Birkh\"auser, Basel, 1996.

\end{thebibliography}

\end{document}