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\begin{document}
\title{
%\begin{flushleft}
%{\small{\it Advanced Nonlinear Studies} {\bf 1} (2001), xxx--xxx}
%\end{flushleft}
\vspace{0.5in}
{\bf\Large  Weak Solutions and Energy Estimates for a Class of Nonlinear
Elliptic Neumann Problems}}
\author{{\bf\large Gabriele Bonanno,}\hspace{2mm}
{\bf\large Giovanni Molica Bisci,}\\{\bf\large Vicen\c{t}iu D. R\u{a}dulescu}\footnote{The author acknowledges the support through  Grant CNCS PCE--47/2011.}\vspace{1mm}\\
{\it\small Department of Science for Engineering and Architecture (Mathematics Section),
}\\ {\it\small Engineering Faculty, University of Messina},
{\it\small 98166  Messina, Italy}\\
{\it\small e-mail: bonanno@unime.it}\vspace{1mm}\\
{\it\small Dipartimento MECMAT, University of Reggio Calabria,
}\\ {\it\small Via Graziella, Feo di Vito},
{\it\small 89124 Reggio Calabria, Italy}\\
{\it\small e-mail: gmolica@unirc.it}\vspace{1mm}\\
{\it\small Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,
}\\ {\it\small 014700 Bucharest, Romania} \\{\small\&}\\
{\it\small Department of Mathematics, University of Craiova,
}\\ {\it\small 200585 Craiova, Romania}
\\
{\it\small e-mail: vicentiu.radulescu@imar.ro}\vspace{1mm}
\date{\small{Received May 29, 2012} \\
{\it\small Communicated by Norman Dancer}}  }

\maketitle
\newpage
\begin{center}
{\bf\small Abstract}

\vspace{3mm}
\hspace{.05in}\parbox{4.5in}
{{\small We establish existence results and energy estimates of solutions for a homogeneous
Neumann problem involving the $p$--Laplace operator. The case of large dimensions corresponding to the lack of compactness is also considered.
In particular, for a precise localization of the parameter, it is established the existence of a non-zero
 solution, without requiring any
asymptotic condition at zero or at infinity of the nonlinear term. In the case of
$(p-1)$--sublinear terms at the origin, we deduce the existence of solutions for small positive values of the parameter and we obtain that the corresponding solutions have smaller and smaller energies as the parameter goes to zero.
Finally, a multiplicity result is obtained
and concrete examples of applications are provided.
A basic ingredient in our arguments is a recent local minimum theorem for differentiable
functionals.}}
\end{center}

\noindent
{\it \footnotesize 2010 Mathematics Subject Classification}. {\scriptsize 35J60, 47J30, 58E05}.\\
{\it \footnotesize Key words}. {\scriptsize Critical point; Variational methods; Neumann problem; sub-critical growth.}




\section{Introduction}

The aim of this paper is to investigate the existence and the qualitative properties of  nontrivial weak solutions of the following non-autonomous elliptic
Neumann problem
\begin{equation} \label{N}
\left\{
\begin{array}{ll}
-\Delta_p u+\alpha(x)|u|^{p-2}u= \lambda f(x,u)& \rm in \quad \Omega
\\ \displaystyle \frac{\partial u}{\partial\nu}=0&\rm on \,\,
\partial \Omega ,\\
\end{array}
\right.
\end{equation}
 where $p>1$ and $\Omega\subset \R^N$ is a bounded open subset with boundary of class $C^1$.\par
 \indent As usual, $\Delta_p$ denotes the $p$-Laplace operator, namely
$\Delta_pu:=\div(|\nabla u|^{p-2}\nabla u)$. We assume that $\alpha\in
L^{\infty}(\Omega)$,  with $\essinf_{x\in {\Omega}}\alpha(x)> 0$,
 $\lambda$ is a positive real parameter and $\nu$ denotes the outward
unit normal to $\partial\Omega$.\par
  Throughout this paper we assume that
 $f:\Omega\times\R\rightarrow\R$ is a Carath\'{e}odory function such that
\begin{equation}\label{S}
|f(x,t)|\leq a_1+a_2|t|^{q-1},\qquad\forall\, (x,t)\in\Omega\times\R,
\end{equation}
for some non-negative constants $a_1,a_2$, where $q\in ]1, p^*[$ and $p^*$ stands for the critical Sobolev exponent, that is
$$
p^*:=\left\{
\begin{array}{ll}
\displaystyle\frac{pN}{N-p} &{\rm if}\quad p<N\\
+\infty &{\rm if} \quad p\geq N.
\end{array}
\right.
$$

Finally, we denote by
${\mathfrak{F}}_q$ the class of all Carath\'{e}odory functions for
which condition (\ref{S}) holds true.

The question of existence of multiple non-zero solutions has been
studied for Dirichlet problems also involving the
$p$-Laplace operator and there are several such papers in the literature,
using different hypotheses and techniques.
 On the other hand, recently there have been established some multiplicity results within the
Neumann setting, see  the papers
\cite{BD,BMBi,BMB4, BMBR} and \cite{BMW,R13,R17}. In these works, the
authors establish the existence of weak solutions for
certain nonlinear elliptic problems, by imposing different kinds of
oscillatory behavior on the nonlinear term.

 In all the above cited contributions
it is imposed the  basic assumption $p>N$, which corresponds to low-dimensional problems.
This dimensionality condition implies that the Sobolev space $
W^{1,p}(\Omega)$ is compactly embedded in $C^0(\bar{\Omega})$ and
this fact is used extensively in the aforementioned
works; see Remark \ref{gh}.
 In particular, in \cite{R4} and
\cite{R9}, the authors consider nonlinear Neumann
eigenvalue problems and prove a ``three solutions theorem" using an
abstract multiplicity result of Ricceri \cite{R16}; see also the related papers \cite{KMR,KR}. Condition $p>N$ is also assumed in a
recent work of Wu and Tan \cite{R18}, in which there are used minimax techniques combined with critical point theory tools.

The difficulties caused by the lack of a compact
embedding of $
W^{1,p}(\Omega)$ in $C^0(\bar{\Omega})$ if $p\le N$ are overcome by a different procedure. Indeed, to apply  variational techniques to Neumann
problems in the case $p\le N$,
  a suitable condition on the nonlinear term $f$, which implies that the problem admits the zero solution, is customary requested. Thus, three non-zero solutions
  can be surely obtained only when the problem is perturbed; see \cite{R9,Ricc4} and \cite{RiccN}.

Very recently, by using variational methods, a precise interval of
values of the parameter $\lambda$, for which problem (\ref{N})
admits at least three non-zero weak solutions has been achieved in
\cite[Theorem 3.1]{DaMo} without explicit perturbations of the
nonlinear term and in higher dimension.
We also recall
that in the last few years, several authors have treated
Neumann problems with $p$-Laplace operators (with $p\leq N$), by using
completely different techniques. We refer to
\cite{GP,MaMO} and the references therein; see also the works \cite{BMBR3,KP,KPV}.

 We point out that problem (\ref{N}) arises in the study of mathematical models in biological formation theory governed by diffusion and cross-diffusion systems \cite{niams}. We refer to the recent monograph by
Krist\'aly, R\u adulescu and Varga \cite{KRV} for several related results and examples.

 The main result of this paper (Theorem \ref{TT1}) ensures
the existence of precise values of the parameter $\lambda$ for which
(\ref{N}) admits at least one non-zero weak solution, without assuming
any asymptotic condition at zero or at infinity.
A special case is also pointed out in Corollary \ref{CC2}.
A related consequence, where the unique condition requested on
the datum is the $(p-1)$-sublinearity at the origin, is also presented in this paper; see Theorem \ref{ultimo}.\par
We also observe that, when the nonlinear term is
$(p-1)$-sublinear at infinity, then the corresponding energy functional is
coercive, hence  the existence of one solution (possibly zero) comes from the direct methods theorem; see Remark \ref{direct}. It is worth noticing that,
in our cases, the potential may be also not coercive; see, for instance, Example
\ref{Ex1}. Moreover, also in presence of coercivity, our results
ensure the existence of at least one non-zero weak solution.\par
 A basic tool used in the proofs is a recent critical point theorem obtained by Bonanno in \cite[Theorem 5.1]{Bonanno} for functionals of the form $J_\lambda:=\Phi-\lambda\Psi$, where $\lambda$ is a positive parameter; see Theorem \ref{T2} below.

 Consider the Sobolev space $W^{1,p}(\Omega)$
endowed with the norm
$$
\|u\|:=\Big(\int_\Omega |\nabla u(x)|^pdx+\int_\Omega\alpha(x)|u(x)|^pdx\Big)^{1/p},
$$
which is equivalent to the usual one.

We state here, as an example, the following
special case of our results; see also Remark \ref{Introrem}.

\begin{theorem}\label{intro}
Let $p>1$ and $f:\R\to\R$ be a continuous function such that
\begin{equation}  \label{cv}
\lim_{t\rightarrow0^+} \frac{f(t)}{t^{p-1}}=+\infty
,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\lim_{|t|\rightarrow
\infty}\frac{f(t)}{|t|^{s}}=0,
\end{equation} 
\noindent for some $0\leq s<p^*-1$.\par
 \noindent Then, there exists
$\lambda^{\star}>0$ such that, for every $\lambda\in
]0,\lambda^\star[$, the following autonomous Neumann problem
\begin{equation}  \label{N3}
\left\{
\begin{array}{ll}
-\Delta_p u+\alpha(x)|u|^{p-2}u= \lambda f(u)&\rm in
\quad \Omega
\\ \partial u/\partial\nu=0&\rm on \,\,
\partial \Omega ,\\
\end{array}
\right.
\end{equation}
admits at least one non-zero weak solution $u_{0,\lambda}\in W^{1,p}(\Omega)$.
Moreover, we have $\|u_{0,\lambda}\|\rightarrow 0$ as
$\lambda\rightarrow 0^+$ and the mapping
$$
\lambda\mapsto \frac{1}{p}\Big(\int_\Omega |\nabla u_{0,\lambda}(x)|^pdx+\int_\Omega\alpha(x)|u_{0,\lambda}(x)|^pdx\Big)-\lambda\int_\Omega\Big(\int_0^{u_{0,\lambda}(x)}f(x,t)dt\Big)dx,
$$
is negative and strictly decreasing in $]0,\lambda^{\star}[$.
\end{theorem}

 We observe that Theorem \ref{intro} ensures the existence of one non-zero weak solution even if $f(0)=0$ provided that (\ref{cv}) holds, and, if $\alpha$ is not constant, the solution is not constant. Moreover, a concrete expression for $\lambda^{\star}$, that appears in the above result, is given in Remark \ref{maximalinterval}. Furthermore, we just point out that in Corollary \ref{CAR}, adding to
hypotheses of Theorem \ref{ultimo} the classical Ambrosetti and
Rabinowitz condition, namely $(\rm AR)$, a second weak solution
is achieved; see Example \ref{Ex1Final}.
It is worth noticing that in Corollary \ref{CAR} the assumptions are different from those usually requested applying the mountain pass theorem; see Remark \ref{rem10}.

 The paper is organized as follows. In Section 2, we
recall some basic definitions and our main tool, while Section 3 is
devoted to our main results and examples.

\section{Preliminaries}
Let $W^{1,p}(\Omega)$ endowed with the norm $\|\cdot\|$. From standard variational arguments, the weak solutions of (\ref{N}) are
the critical points of the $C^1$-functional given by
$$
J_\lambda(u):=\frac{1}{p}\Big(\int_\Omega |\nabla u(x)|^pdx+\int_\Omega\alpha(x)|u(x)|^pdx\Big)-\lambda\int_\Omega\Big(\int_0^{u(x)}f(x,t)dt\Big)dx,\,
$$
\noindent for every $u\in \H$.

A function $u:\Omega\rightarrow \R$ is said to be a weak
solution of problem (\ref{N}) if $u\in W^{1,p}(\Omega)$ and
$$
\int_\Omega |\nabla u(x)|^{p-2}\nabla u(x)\cdot\nabla v(x) dx+\int_\Omega \alpha(x)|u(x)|^{p-2}u(x)v(x) dx
=\lambda\int_\Omega f(x,u(x))v(x)dx,\quad
$$
\noindent for every $v\in W^{1,p}(\Omega).$

Let $X$ be a real Banach space. We say that a continuously G\^ateaux differentiable functional $J:X\rightarrow \R$ verifies the {\it Palais-Smale condition}
(in short $(\rm PS)$-condition) if any sequence $\left\{u_n \right\}$
such that
\begin{itemize}
\item[$({\textrm{j}}_1)$] $\{J(u_n)\}$ is bounded,
\item[$({\textrm{j}}_2)$] $\displaystyle\lim_{n \to \infty}\|J'(u_n)\|_{X^*}=0$,
\end{itemize}
has a convergent subsequence.

\indent For an exhaustive treatment of these topics we refer to
\cite{Motreanu-Radulescu,Rabinowitz} and the references
therein.

Let $\Phi, \Psi:X \to \R$ be two continuously G\^ateaux
differentiable functions. Set $$J=\Phi-\Psi,$$
and fix $r_1, r_2 \in
[-\infty,+\infty]$, with $r_1<r_2$; we say that function $J$
verifies the {\it Palais-Smale condition cut off lower at $r_1$ and
upper at $r_2$} (in short $^{[r_1]}({\rm PS})^{[r_2]}$-condition) if any
sequence $\left\{u_n \right\}$ such that $({\textrm{j}}_1)$, $({\textrm{j}}_2)$ hold and
\begin{itemize}
\item[$({\textrm{j}}_3)$] $r_1<\Phi(u_n)<r_2$,\,\, $\forall\, n\in \N$,
\end{itemize}
has a convergent subsequence.

\indent Clearly, if $r_1=-\infty$ and $r_2=+\infty$ it coincides
with the classical $({\rm PS})$-condition. Moreover, if $r_1=-\infty$ and
$r_2\in \R$ it is denoted by $({\rm PS})^{[r_2]}$, while if $r_1\in
\R$ and $r_2=+\infty$ it is denoted by $^{[r_1]}({\rm PS})$. Furthermore, if
$J$ satisfies $^{[r_1]}({\rm PS})^{[r_2]}$-condition, then it
satisfies $^{[\rho_1]}({\rm PS})^{[\rho_2]}$-condition for all
$\rho_1,\rho_2 \in [-\infty,+\infty]$ such that $r_1\le
\rho_1<\rho_2\le r_2$.

  In particular, we deduce that if $J$ satisfies
the classical $({\rm PS})$-condition, then it satisfies
$^{[\rho_1]}({\rm PS})^{[\rho_2]}$-condition for all $\rho_1,\rho_2 \in
[-\infty,+\infty]$ with $\rho_1<\rho_2$.
Set
\begin{equation} \label{beta}
\beta(r_1,r_2):=\inf_{v \in \Phi^{-1}(]r_1,r_2[)} \frac{\displaystyle\sup_{u \in \Phi^{-1}(]r_1,r_2[)} \Psi(u)-\Psi(v)}{r_2-\Phi(v)},
\end{equation}
\noindent and
\begin{equation} \label{rho2}
\rho_2(r_1,r_2):=\sup_{v \in \Phi^{-1}(]r_1,r_2[)} \frac{\Psi(v)-\displaystyle\sup_{u \in \Phi^{-1}(]-\infty,r_1])} \Psi(u)}{\Phi(v)-r_1},
\end{equation}

\noindent for all $r_1, r_2 \in \R$, with $r_1<r_2$.\par

A crucial role in the existence proof of one non-zero weak solution
for problem (\ref{N}) is played by the following version of an
abstract local minimum theorem obtained in ~\cite[Theorem
5.1]{Bonanno}, which we recall  here for convenience.

\begin{theorem} \label{T2}
Let $X$ be a real Banach space and let $\Phi, \Psi :X \to \R$ be two continuously
G\^ateaux differentiable functions.
Assume that there are $r_1$, $r_2 \in \R$, with $r_1<r_2$, such
that
$$\beta(r_1,r_2)<\rho_2(r_1,r_2),$$
where $\beta$ and $\rho_2$ are given by (\ref{beta}) and
(\ref{rho2}), and for each $$\lambda \in
\Big]\frac{1}{\rho_2(r_1,r_2)}, \frac{1}{\beta(r_1,r_2)} \Big[,$$
the function $J_{\lambda}:=\Phi-\lambda\Psi$ satisfies
$^{[r_1]}({\rm PS})^{[r_2]}$-condition.

 \noindent Then, for all $\lambda \in \Big]\frac{1}{\rho_2(r_1,r_2)},
\frac{1}{\beta(r_1,r_2)} \Big[$ there is $u_{0,\lambda} \in
\Phi^{-1}(]r_1,r_2[)$ such that $J_{\lambda}(u_{0,\lambda})\le
J_{\lambda}(u)$ for all $u \in \Phi^{-1}(]r_1,r_2[)$ and
$J_{\lambda}'(u_{0,\lambda})=0$.
\end{theorem}

\begin{remark}\label{R3}
\rm{
 Theorem \ref{T2} has been inspired from the Ricceri variational principle; see \cite[Theorem 2.5]{Ricceri 1}. With respect to the mentioned principle, the above result furnishes a more precise localization of the minimum $u_{0,\lambda}$, and, in particular, since $\Phi(u_{0,\lambda})>r_1$, in many applications we have $u_{0,\lambda}\neq 0$ as, for instance, in Theorem \ref{TT1}. Moreover, we also emphasize that in Theorem \ref{T2} no weak lower semicontinuity assumption is requested on the contrary of \cite[Theorem 2.5]{Ricceri 1}.
  }

\end{remark}

\section{Main results}

 Recall that $p^*$ stands for the critical exponent of the Sobolev embedding $\H\hookrightarrow L^q(\Omega)$. Thus, if $p<N$ then $p^*=Np/(N-p)$ and for every $q\in [1,p^*]$
 there exists a positive constant $\kappa_q$ such that
\begin{equation}\label{Poincare2}
\|u\|_{L^q(\Omega)}\leq \kappa_q \|u\|\,,
\end{equation}
\noindent for every $u\in \H$. When $p\geq N$ this inequality holds
for any $q\in [1,+\infty[$.\par
\indent Moreover, for every two
nonnegative constants $\gamma,\delta$, with $\gamma\neq\delta$, we
set
\begin{equation}\label{g}
a_\gamma(\delta):= \frac{\displaystyle A(\gamma)-q\int_{\Omega}F(x,\delta)dx}{\|\alpha\|_{L^1(\Omega)}(\gamma^p-\delta^p)q},
\end{equation}
\noindent where
\begin{equation}\label{formula5}
A(\gamma):=q\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma\kappa_1a_1+\|\alpha\|_{L^1(\Omega)}^{q/p}\gamma^{q}{\kappa_q^qa_2},
\end{equation} and
$$F(x,\xi):=\int_0^{\xi}f(x,t)dt,$$
\noindent for every $(x,\xi)\in\Omega\times\R$.\par

\indent The following result establishes a qualitative property of solutions to problem $($\ref{N}$)$.

\begin{theorem}\label{TT1}
Let $f\in{\mathfrak{F}}_q$ and assume that there exist three real
constants $\gamma_1,\gamma_2$ and $\delta$, with $0\leq \gamma_1 <
\delta< \gamma_2$, such that
\begin{equation}\label{EE1}
a_{\gamma_2}(\delta)<a_{\gamma_1}(\delta).
\end{equation}
\noindent Then, for each parameter $\lambda$ belonging to
$$
\left]\displaystyle
\frac{1}{pa_{\gamma_1}(\delta)},\frac{1}{pa_{\gamma_2}(\delta)}\right[,
$$
problem $($\ref{N}$)$ admits at least one non-zero weak solution $u_{0,\lambda}\in \H$, such that $$\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma_{1}<\|u_{0,\lambda}\|<\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma_{2}.$$
\end{theorem}

\proof
Our aim is to apply Theorem \ref{T2}. Hence, let $X:=\H$ and consider the
functionals $\Phi,\Psi:X\to\R$ defined by
$$
\Phi(u):={\frac{\|u\|^p}{p}},\,\,\,\,
\displaystyle \Psi(u):=\int_\Omega F(x,u(x))dx,\qquad\forall\, u\in X.
$$
\noindent Clearly, $\Phi:X\to\R$ is a coercive and continuously G\^{a}teaux differentiable functional. On the other
hand,  $\Psi$ is
well-defined and continuously G\^{a}teaux differentiable. More precisely, we have
$$
\Phi'(u)(v)=\int_\Omega |\nabla u(x)|^{p-2}\nabla u(x)\cdot\nabla v(x) dx+\int_\Omega \alpha(x)|u(x)|^{p-2}u(x)v(x) dx,$$
$$
\Psi'(u)(v)=\int_\Omega f(x,u(x))v(x)dx,
$$
for every $u,v\in X$.

\noindent  Fix $\lambda>0$. A critical point of the functional $J_\lambda:=\Phi-\lambda\Psi$ is a
function $u\in X$ such that
$$
\Phi'(u)(v)-\lambda\Psi'(u)(v)=0,
$$
for every $v\in X$. Hence, the critical points of the functional $J_\lambda$ are weak solutions of problem (\ref{N}). At this point, let us observe that $\Phi (0_X)=\Psi (0_X)=0.$
 \noindent Moreover, since $f\in \mathfrak{F}$, we have
\begin{equation}\label{jj}
F(x,\xi)\leq a_1|\xi|+a_2\frac{|\xi|^{q}}{q},
\end{equation}
for every $(x,\xi)\in \Omega\times\R$.\par
\noindent Now, taking into account relation (\ref{jj}), it follows that
$$
\Psi(u)=\int_\Omega F(x,u(x))dx\leq a_1\|u\|_{L^1(\Omega)}+\frac{a_2}{q}\|u\|_{L^q(\Omega)}^q.
$$
Then, for every $u\in X$ such that $\Phi(u)\leq r$, owing to (\ref{Poincare2}), we obtain
$$
\Psi(u)\leq (pr)^{1/p}\kappa_1a_1+\frac{p^{q/p}\kappa_q^qa_2}{q}r^{q/p}.
$$
Therefore
\begin{eqnarray}\label{hh}
\sup_{u\in\Phi^{-1}(]-\infty,r])}\Psi(u)\leq (pr)^{1/p}\kappa_1a_1+\frac{p^{q/p}\kappa_q^qa_2}{q}r^{q/p}.
\end{eqnarray}
\noindent Next, we set
$$
r_{1}:=\frac{\|\alpha\|_{L^1(\Omega)}}{p}\gamma_{1}^{p}, \quad r_{2}:=\frac{\|\alpha\|_{L^1(\Omega)}}{p}\gamma_{2}^{p}, \quad \mbox{and} \quad u_{\delta}(x):=\delta, \,\,\,\mbox{ for every }\; x\in\Omega.
$$

\noindent Then $u_\delta\in X$ and
\begin{eqnarray}\label{norm}
\Phi(u_\delta)=\frac{1}{p}\Big(\int_\Omega|\nabla u_\delta(x)|^pdx+\int_\Omega\alpha(x)|u_\delta(x)|^pdx\Big)=\frac{\delta^p}{p}\|\alpha\|_{L^1(\Omega)}.
\end{eqnarray}
Taking into account that $\gamma_1<\delta<\gamma_2$, by a direct computation,
one has
$r_1<\Phi(u_\delta)<r_2$.
\noindent Moreover,
\begin{eqnarray}\label{l}
\Psi(u_\delta)=\int_\Omega F(x,u_\delta(x))\;dx=\int_{\Omega}F(x,\delta)dx.
\end{eqnarray}
>From (\ref{hh}) it follows that
\begin{equation}\label{EE5}
\sup_{u\in\Phi^{-1}(]-\infty, r_2[)}\Psi(u) \leq (pr_2)^{1/p}\kappa_1a_1+\frac{p^{q/p}\kappa_q^qa_2}{q}r_2^{q/p}.
\end{equation}
and
\begin{equation}\label{EE6}
\sup_{u\in\Phi^{-1}(]-\infty, r_1])}\Psi(u) \leq (pr_1)^{1/p}\kappa_1a_1+\frac{p^{q/p}\kappa_q^qa_2}{q}r_1^{q/p}.
\end{equation}
We have $r_1<\Phi(u_\delta)<r_2$. On the other hand,
$$\beta(r_1,r_2):=\inf_{v \in \Phi^{-1}(]r_1,r_2[)} \frac{\displaystyle\sup_{u \in \Phi^{-1}(]r_1,r_2[)} \Psi(u)-\Psi(v)}{r_2-\Phi(v)}\leq \frac{\displaystyle\sup_{u \in \Phi^{-1}(]-\infty,r_2[)} \Psi(u)-\Psi(u_\delta)}{r_2-\Phi(u_\delta)}$$
and
$$\rho_2(r_1,r_2):=\sup_{v \in \Phi^{-1}(]r_1,r_2[)} \frac{\Psi(v)-\displaystyle\sup_{u \in \Phi^{-1}(]-\infty,r_1])} \Psi(u)}{\Phi(v)-r_1}\geq \frac{\Psi(u_\delta)-\displaystyle\sup_{u \in \Phi^{-1}(]-\infty,r_1])} \Psi(u)}{\Phi(u_\delta)-r_1}.$$
\noindent Hence, by using the notation (\ref{g}), from (\ref{l}) and (\ref{EE5}) together with (\ref{EE6}), it follows that
$$
\beta(r_1,r_2)\leq pa_{\gamma_2}(\delta),\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\rho_2(r_1,r_2)\geq pa_{\gamma_1}(\delta).
$$
Finally, hypothesis (\ref{EE1}) yields
$$\beta(r_1,r_2)<\rho_2(r_1,r_2).$$
\noindent Now, from \cite[Proposition 2.1]{Bonanno}, the functional $J_\lambda$ satisfies $^{[r_1]}({\rm PS})^{[r_2]}$-condition for all
$r_1$ and $r_2$ with $r_1<r_2<+\infty$. Therefore,
owing to Theorem \ref{T2}, for each $\lambda \in \left]\displaystyle
\frac{1}{pa_{\gamma_1}(\delta)},\frac{1}{pa_{\gamma_2}(\delta)}\right[$,
the functional $J_\lambda$ admits at least one critical point
$u_{0,\lambda}$ such that
$$r_1 < \Phi (u_{0,\lambda}) < r_2,$$ that is
$$\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma_{1}<\|u_{0,\lambda}\|<\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma_{2}.$$
This completes the proof.
%\qed


\begin{remark}\label{costanti}
\rm{
  If $p<N$, a concrete upper bound for the constants $\kappa_q$ in Theorem \ref{TT1} (hence for the values of constant $A$ defined as in (\ref{formula5}) can be obtained considering an open and convex set $\Omega\subset {\R}^{N}$ of diameter $d$ and Lebesgue measure $``\meas(\Omega)"$. In this case, if $q\in[1,p^{*}[$, we have
\begin{equation}\label{inequality2}
\displaystyle \kappa_q\leq
S_q\frac{(1+N)\meas(\Omega)^{\frac{p-q}{pq}}}{\min\left\{1,\displaystyle\essinf_{x\in\Omega}\alpha(x)\right\}^{1/p}},
\end{equation}
 where
\begin{equation}\label{K}
\displaystyle S_q:=\left\{
\begin{array}{l}\displaystyle
\max\left\{1,\frac{d}{N}\right\}\quad\, {\rm{if}}\quad
q\in [1,p[, \\\\
{\displaystyle\max\left\{1,\frac{d}{N}\left(\frac{pq-N(q-p)}{pq+(p-q)}\right)^{\frac{q-p(1+q)}{pq}}\right\}\quad {\rm{if}}\quad q\in [p,p^{*}[,}\\
\end{array}
\right.
\end{equation}
\noindent see, for instance, Remark 3.2 in \cite{DaMo} and, more generally, the exhaustive book \cite{Burenkov}. Moreover, we point out that the following alternative holds: either $q\in [p,p^*[$ and the
diameter of $\Omega$ is sufficiently small, that is,
$$
d<N\displaystyle
\left(\frac{pq-N(q-p)}{pq+(p-q)}\right)^{\frac{p(1+q)-q}{pq}},
$$
\noindent or $q\in [1,p[$ and $d<N$. Inequality (\ref{inequality2})
can be rewritten as
\begin{equation}\label{u}
\displaystyle \kappa_q\leq
\frac{(1+N)\meas(\Omega)^{\frac{p-q}{pq}}}{\min\left\{1,\essinf_{x\in\Omega}\alpha(x)\right\}^{1/p}},
\end{equation}
\noindent as observed in \cite[Remark 3.3]{DaMo}.\par \noindent On
the other hand, in the low-dimensional case that corresponds to $p>N$, assuming $\Omega$
to be convex and by using \cite[Remark 2.1]{BMBi}, an easy
computation ensures that for every $q\geq 1$, we have
$$
\kappa_q\leq
2^{\frac{p-1}{p}}\max\left\{\frac{1}{\displaystyle\|\alpha\|_{L^1(\Omega)}^{\frac{1}{p}}},\frac{d}{N^{\frac{1}{p}}}\left(\frac{p-1}{p-N}\meas(\Omega)\right)^{\frac{p-1}{p}}\frac{\|\alpha\|_{\infty}}{\|\alpha\|_{L^1(\Omega)}}\right\}\meas(\Omega)^{1/q}.
$$}
\end{remark}

\begin{remark} \label{C0}
{\rm
If $p>N$ and $\Omega$ is convex, by using again \cite[Remark 2.1]{BMBi}, a concrete estimation of the solution $u_{0,\lambda}$ in $C^0(\bar \Omega)$ can be achieved. Precisely, we have
$$
\|u_{0,\lambda}\|_{\infty}\leq
2^{\frac{p-1}{p}}\max\left\{\frac{1}{\displaystyle\|\alpha\|_{L^1(\Omega)}^{\frac{1}{p}}},\frac{d}{N^{\frac{1}{p}}}\left(\frac{p-1}{p-N}\meas(\Omega)\right)^{\frac{p-1}{p}}\frac{\|\alpha\|_{\infty}}{\|\alpha\|_{L^1(\Omega)}}\right\}\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma_{2}.
$$
}
\end{remark}

\indent Now, we point out a particular case of Theorem \ref{TT1}.

\begin{corollary}\label{CC2}
Let $f\in{\mathfrak{F}}_q$ and assume that there exist two positive constants $\gamma$ and $\delta$, with $\gamma>\delta$, such that
\begin{equation} \label{F1}
    \frac{\displaystyle\int_\Omega F(x,\delta)dx}{\delta^p}>\frac{A(\gamma)}{q\gamma^p}.
\end{equation}
Then, for each parameter $\lambda$ belonging to
$$
\left]\frac{\|\alpha\|_{L^1(\Omega)}\delta^p}{\displaystyle
p\int_\Omega
F(x,\delta)dx},\frac{\|\alpha\|_{L^1(\Omega)}\gamma^pq}{\displaystyle
pA(\gamma)} \right[,
$$
problem $($\ref{N}$)$ admits at least one non-zero weak solution $u_{0,\lambda}\in \H$, such that $$\|u_{0,\lambda}\|<\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma.$$
\end{corollary}

\proof
Our aim is to apply Theorem \ref{TT1}. To this end we pick $\gamma_1=0$ and $\gamma_2:=\gamma$.
 Bearing in mind (\ref{g}), we obtain
 $$
 a_{\gamma}(\delta)=\frac{\displaystyle A(\gamma)-q\int_{\Omega}F(x,\delta)dx}{\|\alpha\|_{L^1(\Omega)}(\gamma^p-\delta^p)q},
 $$
 as well as
 $$
 a_{0}(\delta)=\frac{\displaystyle\int_\Omega F(x,\delta)dx}{\delta^p\|\alpha\|_{L^1(\Omega)}}.
 $$
 Now, inequality (\ref{F1}) immediately yields
 $$
 a_{\gamma}(\delta)<a_{0}(\delta).
 $$
 Hence, Theorem \ref{TT1} ensures the conclusion.
%\qed

A direct consequence of Corollary \ref{CC2} is the following property.

\begin{theorem}\label{ultimo}
Let $f\in{\mathfrak{F}}_q$
 and assume that
\begin{equation}\label{EC3}
\lim_{\xi\rightarrow0^+} \frac{\displaystyle\int_\Omega F(x,\xi)dx}{\xi^p}=+\infty.
\end{equation}
\noindent Furthermore, let $\gamma>0$ and set
$$
\lambda^\star_\gamma:=\frac{q\|\alpha\|_{L^1(\Omega)}}{p}\frac{\gamma^p}{\displaystyle A(\gamma)}.
$$
Then, for every $\lambda\in ]0,\lambda^\star_\gamma[$,
the problem $($\ref{N}$)$ admits at least one non-zero weak solution $u_{0,\lambda}\in \H$ such that $\|u_{0,\lambda}\|<\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma$ and
$$
\lim_{\lambda\rightarrow 0^+}\|u_{0,\lambda}\|=0.
$$
\end{theorem}
\proof
Fix $\lambda\in ]0,\lambda^{\star}_\gamma[$. From
(\ref{EC3}) there exists a positive constant $\delta$ with $\delta<\gamma$ such that
$$
\frac{\|\alpha\|_{L^1(\Omega)}\delta^p}{\displaystyle p\int_\Omega F(x,\delta)dx}<\lambda<\frac{\|\alpha\|_{L^1(\Omega)}\gamma^pq}{\displaystyle A(\gamma)p}.
$$
Hence, owing to
Corollary \ref{CC2},
problem $($\ref{N}$)$ admits at least one non-zero weak solution $u_{0,\lambda}$, such that $\|u_{0,\lambda}\|<\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma.$ Then, for every $\lambda\in ]0,\lambda^{\star}_{\gamma}[$, there exists at least one non-zero weak solution $u_{0,\lambda}\in \Phi^{-1}(]0,r_2[)$ of problem $($\ref{N}$)$  and one has
\begin{equation}\label{bounded0}
\|u_{0,\lambda}\|<\|\alpha\|_{L^1(\Omega)}^{1/p}\gamma.
\end{equation}
\noindent for every $\lambda\in ]0,\lambda^{\star}_{\gamma}[$.\par
\noindent Therefore, from (\ref{S}), taking into account (\ref{Poincare2}) and (\ref{bounded0}), it follows that
\begin{equation}\label{bounded}
\left|\int_\Omega f(x, u_{0,\lambda}(x))u_{0,\lambda}(x)dx\right|\leq \|\alpha\|_{L^1(\Omega)}^{1/p}\gamma\kappa_1a_1+\|\alpha\|_{L^1(\Omega)}^{q/p}\gamma^{q}{\kappa_q^qa_2},
\end{equation}
\noindent for every $\lambda\in ]0,\lambda^{\star}_\gamma[$.\par
\noindent Now, $J'_\lambda(u_{0,\lambda})=0$, for every $\lambda\in ]0,\lambda^{\star}_\gamma[$ and in particular $J'_\lambda(u_{0,\lambda})(u_{0,\lambda})=0$, that is,
$$
\|u_{0,\lambda}\|^p=\lambda\int_\Omega f(x, u_{0,\lambda}(x))u_{0,\lambda}(x)dx,
$$
\noindent for every $\lambda\in ]0,\lambda^{\star}_\gamma[$.\par
\noindent Then, from (\ref{bounded}), it follows that
$$
\lim_{\lambda\rightarrow 0^+}\|u_{0,\lambda}\|^p=\lim_{\lambda\rightarrow 0^+}\lambda\Psi'(u_{0,\lambda})(u_{0,\lambda})=0,
$$
\noindent that implies
$
\lim_{\lambda\rightarrow 0^+}{\|u_{0,\lambda}\|}=0.$
The proof is complete.
%\qed

\begin{remark}\label{nn}
\rm{
\noindent We claim that under the above assumptions, the mapping $\lambda\mapsto J_{\lambda}(u_{0,\lambda})$ is negative and strictly decreasing in $]0,\lambda^{\star}_{\gamma}[$. Indeed,  the restriction
of the functional $J_{\lambda}$ to
$\Phi^{-1}(]0,r_2[)$, where $r_{2}:=({\|\alpha\|_{L^1(\Omega)}}/{p})\gamma_{2}^{p}$, admits a global minimum, which is a
critical point $($local minimum$)$ of $J_{\lambda}$ in $X$. Moreover, since $u_\delta:=\delta\in\Phi^{-1}(]0,r_2[)$ and
$$
\frac{\Phi(u_\delta)}{\Psi(u_\delta)}=\frac{\|\alpha\|_{L^1(\Omega)}\delta^p}{\displaystyle p\int_\Omega F(x,\delta)dx}<\lambda,
$$
we have
$$
J_\lambda(u_{0,\lambda})\leq J_\lambda(u_{\delta})=\Phi(u_\delta)-\lambda\Psi(u_\delta)<0.
$$
 Next, we observe that
$$
J_{\lambda}(u)=\lambda\left(\frac{\Phi(u)}{\lambda}-\Psi(u)\right),
$$
for every $u\in X$ and fix $0<\lambda_1<\lambda_2<\lambda^{\star}_{\gamma}$. Set
$$
m_{\lambda_1}:=\left(\frac{\Phi(u_{0,\lambda_1})}{\lambda_1}-\Psi(u_{0,\lambda_1})\right)=\inf_{u\in \Phi^{-1}(]0,r_2[)}\left(\frac{\Phi(u)}{\lambda_1}-\Psi(u)\right),
$$
and
$$
m_{\lambda_2}:=\left(\frac{\Phi(u_{0,\lambda_2})}{\lambda_2}-\Psi(u_{0,\lambda_2})\right)=\inf_{u\in \Phi^{-1}(]0,r_2[)}\left(\frac{\Phi(u)}{\lambda_2}-\Psi(u)\right).
$$
Clearly, as claimed before, $m_{\lambda_i}<0$ (for $i=1,2$), and $m_{\lambda_2}\leq m_{\lambda_1}$ thanks to $\lambda_1<\lambda_2$.
Then the mapping $\lambda\mapsto J_{\lambda}(u_{0,\lambda})$ is strictly decreasing in $]0,\lambda^{\star}_{\gamma}[$ owing to
$$
J_{\lambda_2}(u_{0,\lambda_2})=\lambda_2m_{\lambda_2}\leq \lambda_2m_{\lambda_1}<\lambda_1m_{\lambda_1}=J_{\lambda_1}(u_{0,\lambda_1}).
$$ This concludes the proof of our claim.}
\end{remark}


\begin{remark}\label{maximalinterval}
\rm{Roughly speaking, Theorem \ref{ultimo} ensures that if $f$ has the global growth given by (\ref{S}) and the asymptotic condition at zero (\ref{EC3}) is verified then, for every parameter $\lambda$ belonging to the real interval $\Lambda_{\Omega}:=]0,\lambda^{\star}[$, where
$$
\lambda^{\star}:=\frac{q\|\alpha\|_{L^1(\Omega)}}{p}\sup_{\gamma>0}\frac{\gamma^p}{\displaystyle A(\gamma)},
$$
\noindent the problem (\ref{N}) admits at least one non-zero solution $u_{0,\lambda}\in\H$. Moreover $
{\|u_{0,\lambda}\|}\rightarrow 0,$ as $\lambda\rightarrow 0^+$.
 Furthermore, from a direct computation, it follows that
 \begin{equation}\label{j1}
 \lambda^{\star}=
 \left\{
\begin{array}{l}
+\infty \quad\, {\rm if} \quad 1<q<p \\
\displaystyle\frac{1}{\kappa_p^pa_2} \quad {\rm if} \quad q=p \\
\displaystyle\frac{q\|\alpha\|_{L^1(\Omega)}^{(p-1)/p}\overline{\gamma}_{\max}^{p-1}}
{p(q\kappa_1a_1+\|\alpha\|_{L^1(\Omega)}^{(q-1)/p}{\kappa_q^qa_2}\overline{\gamma}_{\max}^{q-1})} \quad {\rm if} \quad q\in \left]p,p^*\right[, \\
\end{array}
\right.
 \end{equation}
\noindent where
$$
\overline{\gamma}_{\max}:=\frac{1}{\|\alpha\|_{L^1(\Omega)}^{1/p}}\left(q\frac{\kappa_1a_1}{\kappa_q^qa_2}\left(\frac{p-1}{q-p}\right)\right)^{1/(q-1)}.
$$
We also note that, in the case $q\in \left]p,p^*\right[$, one has $\|u_{0,\lambda}\|<\|\alpha\|_{L^1(\Omega)}^{1/p}\overline{\gamma}_{\max}$, uniformly for every $\lambda \in \Lambda_{\Omega}$.
}
\end{remark}

\begin{remark} \label{direct}
{\rm We emphasize that, in particular, if $f$ is $(p-1)$-sublinear at infinity, Theorem \ref{ultimo} ensures that,
 problem (\ref{N}) admits at least one non-zero weak solution for each positive parameter $\lambda$. It is worth noticing that, in our case, the attained solution, as affirmed, is non-zero, while the classical direct method approach, that can be adopted in this setting, ensures the existence of at least one solution that may be zero.}
\end{remark}

\begin{remark} \label{positivity}
{\rm
  For completeness we observe that if $f$ is a non-negative function our results guarantee that the attained weak solution is non-negative. From our goal, let $u_0$ be a weak solution of problem $($\ref{N}$)$. Arguing by contradiction, assume that the set $A:=\big\{x \in \Omega : u_0(x)<0\big\}$ has positive Lebesgue measure. Put $\overline v(x)=\min\{0, u_0(x)\}$ for all $x \in \Omega$. Clearly, $\overline v \in W^{1,p}(\Omega)$ and
$$
\int_\Omega |\nabla u_0(x)|^{p-2}\nabla u_0(x)\cdot\nabla \overline
v(x) dx+ \int_\Omega \alpha(x)|u_0(x)|^{p-2}u_0(x)\overline v(x) dx
$$
$$
-\lambda\int_\Omega f(x,u_0(x))\overline v(x)dx=0,\quad
$$
that is,
$$
\int_A |\nabla u_0(x)|^{p}dx+\int_A \alpha(x)|u_0(x)|^{p}dx
=\lambda\int_A f(x,u_0(x))u_0 (x)dx\leq 0.
$$
Hence
$$\int_A |\nabla u_0(x)|^{p}dx+\int_A \alpha(x)|u_0(x)|^{p}dx =0.$$
Then, $u_0=0$ almost everywhere in $A$. This is not possible by the definition of $A$, so it follows that $u_0$ is non-negative.
}
\end{remark}


\begin{remark}\label{moregeneral}
\rm{A careful analysis of the proof of Theorem \ref{ultimo} ensures that the result still remains true after replacing condition (\ref{EC3})
with the more general assumption
\begin{equation}\label{G2}
\limsup_{\xi\rightarrow0^+} \frac{\displaystyle\int_\Omega F(x,\xi)dx}{\xi^p}=+\infty.
\end{equation}
Moreover, in the autonomous case, the previous asymptotic condition at zero assume the form
\begin{equation}\label{G4}
\limsup_{\xi\rightarrow0^+} \frac{\displaystyle F(\xi)}{\xi^p}=+\infty.
\end{equation}}
\end{remark}

\begin{remark}\label{Introrem}
\rm{We just observe that Theorem \ref{intro} in Introduction is a simple consequence of Theorem \ref{ultimo} and Remark \ref{nn}. Indeed, if
\begin{equation}\label{G5}
\lim_{t\rightarrow0^+} \frac{\displaystyle f(t)}{t^{p-1}}=+\infty
\end{equation}
holds, then hypothesis (\ref{G4}) is automatically verified. Moreover, hypothesis
$$
\lim_{|t|\rightarrow
\infty}\frac{f(t)}{|t|^{s}}=0,
$$
where $0\leq s<p^*-1$, ensures that $f$ has a subcritical growth.}
\end{remark}

In the sequel, there is an example of application of our %main
results.

\begin{example} \label{Ex1}
{\rm  Let $\Omega$ be a bounded open subset of $\R^N$ with $1<p<N$
and consider the following problem
\begin{equation} \label{N2}
\left\{
\begin{array}{lll}
-\Delta_p u+|u|^{p-2}u= \lambda
[\eta(x)|u|^{r-2}u+\beta(x)|u|^{s-2}u]& \qquad \rm in\ \Omega
\\ \displaystyle\frac{\partial u}{\partial\nu}=0&\qquad \rm on \
\partial \Omega ,\\
\end{array}
\right.
\end{equation}

\noindent where $\eta,\beta:\Omega\to\R$ are two continuous positive
and bounded functions, $1<r<p$ and $p<s< p^*$. Then, for every
$$\lambda\in\left]0,\frac{s\meas(\Omega)^{(p-1)/p}\overline{\gamma}_{\max}^{p-1}}
{2p\max\{\|\eta\|_\infty,\|\beta\|_\infty\}(s\kappa_1+\meas(\Omega)^{(s-1)/p}{\kappa_s^s}\overline{\gamma}_{\max}^{s-1})}\right[,$$
\noindent where
$$
\overline{\gamma}_{\max}:=\frac{1}{\meas(\Omega)^{1/p}}\left(s\frac{\kappa_1}{\kappa_s^s}\left(\frac{p-1}{s-p}\right)\right)^{1/(s-1)}
$$
\noindent the problem $($\ref{N2}$)$ admits at least one non-zero
weak solution $u_{0,\lambda}\in\H$ such that
$$
\|u_{0,\lambda}\|<\left(s\frac{\kappa_1}{\kappa_s^s}\left(\frac{p-1}{s-p}\right)\right)^{1/(s-1)},
$$
and $ \lim_{\lambda\rightarrow 0^+}\|u_{0,\lambda}\|=0. $\par
\noindent To prove this, we can apply Theorem \ref{ultimo} with
$$
f(x,t):=\eta(x)|t|^{r-2}t+\beta(x)|t|^{s-2}t,
$$
\noindent for every $(x,t)\in\Omega\times \R$. Indeed, it is easy to
verify that $f\in{\mathfrak{F}}_s$ since
$$
|f(x,t)|\leq
2\max\{\|\eta\|_\infty,\|\beta\|_\infty\}(1+|t|^{s-1}),\,\,\,\,\forall\,
(x,t)\in \Omega\times\R.
$$
Moreover, a direct computation shows that
$$
\lim_{\xi\rightarrow0^+} \frac{\displaystyle\int_\Omega
F(x,\xi)dx}{\xi^p}\geq 
\frac{\meas(\Omega)\displaystyle\inf_{x\in\Omega}\eta(x)}{r}\lim_{\xi\rightarrow0^+}
\frac{1}{\xi^{p-r}}=+\infty.
$$
\noindent Hence, all the assumptions of Theorem \ref{ultimo} are
verified and the conclusion follows. }
\end{example}


\begin{remark} \label{gh}
{\rm A similar version of Theorem \ref{ultimo} in the low-dimensional case has been studied
in \cite{BOPI} and
\cite{BOSci}. In their setting, taking advantage of the compact
embedding of $W^{1,p}(\Omega)$ into $C^{0}(\bar \Omega)$, condition
(\ref{S}) can be removed and the authors are able to treat also
cases with exponential growth; see, for instance, \cite[Example
3.1]{BOPI}.
Furthermore, a direct comparison with Theorems 1 and 2 of \cite{Zhilong}
can be find in \cite[Remark 3.6]{BOPI}. In conclusion, taking into
account Remarks \ref{positivity} and \ref{moregeneral}, we emphasize
that, Theorem \ref{ultimo} represents an extension of \cite[Theorem
3.3]{BOPI} and \cite[Theorem 4.1]{BOSci} to the case $p\leq N$.
Moreover, in the low-dimensional case, the additional presence of
the growth assumption (\ref{S}) may be determine, in some
case, an improvement of the intervals of parameters for which our
approach guarantee the existence of at least one non-zero weak
solution. Indeed, let $p>N$ and assume
$f:\Omega\times\R\rightarrow \R$ to be $L^1$-Carath\'{e}odory.
%arguing as in
As \cite[Theorem 3.4]{BOSci} shows,
the existence of at least one non-zero weak solution to Neumann
problem is obtained for every $\lambda\in ]0,\lambda_f[$, where
$$
\lambda_f:=\frac{1}{pc^p}\sup_{\gamma>0}\frac{\gamma^p}{\displaystyle\int_\Omega \max_{|\xi|\leq \gamma} F(x,\xi)dx},
$$
and
$$
c:={\displaystyle\sup_{u\in W^{1,p}(\Omega)\setminus\{0\}}\frac{\sup_{x\in \bar{\Omega}}|u(x)|}{\displaystyle\left(\int_{\Omega}|\nabla u(x)|^p dx+\int_{\Omega}\alpha(x)|u(x)|^p dx\right)^{{1}/{p}}}}<+\infty.
$$
If the function $f$ satisfies also the growth condition
(\ref{S}) the necessity of to compare the intervals
$]0,\lambda^{\star}[$ and $]0,\lambda_f[$ naturally arises. We
exhibit a concrete example in which $]0,\lambda_f[\subset
]0,\lambda^\star[$. For our goal, let us take a positive constant
$d$ and set $\Omega=]0,2d[$. At this point, consider the ordinary
non-autonomous Neumann problem
\begin{equation}  \label{N13}
\left\{
\begin{array}{l}
\displaystyle -u''+\frac{ku}{2}= \lambda \beta(x)(1+|u|^{q-1}) \qquad {\rm in}
\quad ]0,2d[
\\u'(0)=u'(2d)=0,\\
\end{array}
\right.
\end{equation}
\noindent where $\beta:[0,2d]\rightarrow \R$ is a positive and continuous function, $q>2$ and
\begin{eqnarray}\label{kkk}
0<k<\frac{\|\beta\|_{L^1(0,2d)}}{2\|\beta\|_{\infty}(cd)^2}.
\end{eqnarray}
\noindent Set
$$
{\mathcal{N}}(c,q):=\frac{1}{2c^2}\left(\frac{q-2}{q-1}\right)\left(\frac{q}{q-2}\right)^{1/(q-1)},
$$
\noindent and
$$
G(\beta,q):=\left(\frac{d}{2\|\beta\|_{L^1(0,2d)}}\right)\left(\frac{q-2}{q-1}\right)\left(\frac{q}{q-2}\right)^{1/(q-1)}.
$$
 Hence, bearing in mind the above results, problem (\ref{N13}) admits a non-zero classical solution for every $\lambda\in ]0,\lambda_f[$ as well as for every $\lambda\in ]0,\lambda^{\star}[$. Now, taking into account that $1/c^2\leq kd$, simple computations yield
$$
\lambda_f=\frac{q}{2c^2\|\beta\|_{L^1(0,2d)}}\sup_{\gamma>0}\left(\frac{\gamma}{q+\gamma^{q-1}}\right)\leq k\left(\frac{qd}{2\|\beta\|_{L^1(0,2d)}}\right)\sup_{\gamma>0}\left(\frac{\gamma}{q+\gamma^{q-1}}\right).
$$
Then, since
$$
\sup_{\gamma>0}\left(\frac{\gamma}{q+\gamma^{q-1}}\right)=\frac{1}{q}\left(\frac{q-2}{q-1}\right)\left(\frac{q}{q-2}\right)^{1/(q-1)},
$$
one has
\begin{eqnarray}\label{Gio1}
\lambda_f\leq k\left(\frac{d}{2\|\beta\|_{L^1}}\right)\left(\frac{q-2}{q-1}\right)\left(\frac{q}{q-2}\right)^{1/(q-1)}=kG(\beta,q).
\end{eqnarray}
\noindent On the other hand, thanks to $\kappa_s\leq (2d)^{1/s}c,$ for every $s\geq 1$, it follows that
$$
\lambda^\star=\frac{q}{\|\beta\|_\infty}\sup_{\gamma>0}\left(\frac{\gamma}{q\sqrt{2}\kappa_1+2^{q/2}\kappa^q_q\gamma^{q-1}}\right)\geq \frac{q}{2d\|\beta\|_\infty} \sup_{\gamma>0}\left(\frac{\gamma}{q\sqrt{2}c+2^{q/2}c^q\gamma^{q-1}}\right).
$$
Hence, since
$$
\sup_{\gamma>0}\left(\frac{\gamma}{q\sqrt{2}c+2^{q/2}c^q\gamma^{q-1}}\right)=\frac{1}{2c^2q}\left(\frac{q-2}{q-1}\right)\left(\frac{q}{q-2}\right)^{1/(q-1)},
$$
\noindent one has
\begin{eqnarray}\label{Gio2}
\lambda^{\star}\geq \left(\frac{1}{2d\|\beta\|_\infty}\right){\mathcal{N}}(c,q).
\end{eqnarray}
\noindent Then, inequalities (\ref{Gio1}) and (\ref{Gio2}) together with condition (\ref{kkk}) yield
$$
\lambda_f\leq kG(\beta,q)<\left(\frac{1}{2d\|\beta\|_\infty}\right){\mathcal{N}}(c,q)\leq \lambda^{\star}.
$$
\noindent In conclusion, the expected strict inclusion $]0,\lambda_f[\subset ]0,\lambda^{\star}[$ holds.
}
\end{remark}

In the sequel we prove how the previous results can be employed in order to pass from the existence of at least one nontrivial solution to the existence of at lest two nontrivial solutions. This goal will be achieved making use of the particular nature of the first solution found, namely it is a local minimum. This information will be used to assure the existence of a second solution as a critical point of mountain pass type. In this direction, we begin with the following theorem, where the celebrated Ambrosetti-Rabinowitz condition is required. As usual, this assumption plays a crucial role in proving that every Palais-Smale sequence is bounded, as well as that the so called `mountain pass geometry' is satisfied.

\begin{corollary}\label{CAR}
Let $f:\R\to\R$ be a continuous function such that
\begin{equation}\label{Su}
|f(t)|\leq a_1+a_2|t|^{q-1},\,\,\,\,\,\forall\, t\in \R,
\end{equation}
for some non-negative constants $a_1,a_2$, where $q\in ]p, p^*[$.
 Furthermore, assume that condition (\ref{G4}) holds in addition to
\begin{itemize}
  \item [$(\rm AR)$]
  there are constants $\mu>p$ and $r>0$ such that, for all $|\xi|\geq
  r$, one has
    $$
    0<\mu F(\xi)\leq \xi f(\xi).
    $$
\end{itemize}
Then, for each $\lambda\in\Lambda_{\Omega}$, problem (\ref{N})
admits at least two weak solutions. If, in addition to the above assumptions, one has $f(0)\neq 0$, the attained solutions are non-zero.
\end{corollary}
\proof
Fix $\lambda\in \Lambda_{{\Omega}}$. Owing to (\ref{Su}) and (\ref{G4}), Theorem
\ref{ultimo} ensures that the problem
(\ref{N}) admits at least one weak non-zero solution $u_1$
which is a local minimum of the functional $J_\lambda$ as defined in
the proof of Theorem \ref{TT1}. Now, we can assume that $u_1$ is a strict
local minimum for $J_\lambda$ in $X$. Therefore, there is $\rho>0$
such that $$\inf_{\|u-u_1\|=\rho}J_{\lambda}(u)>J_{\lambda}(u_1).$$

\noindent Furthermore, from $(\rm AR)$-condition, by standard computations, one has that $J_{\lambda}$ is
unbounded from below. So, there is $u_2$ such that
$J_{\lambda}(u_2)< J_{\lambda}(u_1)$, for which $J_{\lambda}$
satisfies mountain pass geometry. At this point, again exploiting $(\rm AR)$, it follows that the functional $J_{\lambda}$ satisfies the $(\rm PS)$-condition.
Hence, the celebrated Ambrosetti-Rabinowitz theorem ensures the existence of a
critical point $\tilde u$ of $J_{\lambda}$
such that $J_{\lambda}(\tilde u)> J_{\lambda}(u_1)$. So, $u_1$ and
$\tilde u$ are two distinct weak solutions of (\ref{N})
and the proof is complete.
%\qed

\begin{example} \label{Ex1Final}
{\rm  Let $\Omega$ be a bounded open subset of $\R^N$ with $1<p<N$ and consider the following problem
\begin{equation}  \label{N22}
\left\{
\begin{array}{lll}
&\displaystyle -\Delta_p u+\alpha(x)|u|^{p-2}u= \lambda (1+u^{2m-1})&\qquad\rm in\ \Omega
\\ &\displaystyle \frac{\partial u}{\partial\nu}=0&\qquad\rm on \
\partial \Omega ,\\
\end{array}
\right.
\end{equation}

\noindent where $m$ is a positive integer such that $1<p<2m<p^{*}$.\par
\noindent Then, for every
$$\lambda\in\left]0, \frac{2m-p}{p\kappa_1(2m-1)}\left(\frac{2m\kappa_1(p-1)}{\kappa_{2m}^{2m}(2m-p)}\right)^{\frac{p-1}{2m-1}}\right[,$$
\noindent problem $($\ref{N22}$)$ admits at least two non-zero ($f(0)\neq 0$) weak solutions.\par
\noindent To prove this, we can apply Corollary \ref{CAR} with
$$
f(t):=1+t^{2m-1},
$$
\noindent for every $t\in\R$.
Indeed, clearly the function $f$ satisfies (\ref{Su}) and, since
$$
\lim_{t\rightarrow0^+} \frac{f(t)}{t^{p-1}}=+\infty,
$$
also condition (\ref{G4}) holds true. Moreover, taking into account that
$$
\lim_{|\xi|\rightarrow \infty}\frac{\xi f(\xi)}{F(\xi)}=2m\lim_{|\xi|\rightarrow \infty}\frac{\xi^{2m}+\xi}{\xi^{2m}+2m\xi}=2m>p,
$$
there exist $\mu>p$ and $r>0$ such that
$$
    0<\mu F(\xi)\leq \xi f(\xi),
    $$
    for every $|\xi|>r$.\par
\noindent Hence, all the assumptions of Corollary \ref{CAR} are verified and the conclusion follows.
}
\end{example}

\begin{remark} \label{rem10}
{\rm Applying the mountain pass theorem to nonlinear differential problems, involving the Laplace operator $-\Delta u$, it is usual to require that
$\displaystyle\lim_{t\to 0^+}\frac{f(t)}{t}=0$; see, for instance, condition $(p_3)$ in \cite[Theorem 2.15]{Rabinowitz}. On the contrary, in Example  \ref{Ex1Final}, the above assumption at zero is not verified. We also note that, very recently, Neumann problems have been studied without $(\rm AR)$-condition by using a Cerami-type condition; see \cite{GP2} and references therein, as well as Remark \ref{sci} below. Even in this case, hypothesis $\displaystyle\lim_{t\to 0^+}\frac{f(t)}{t}<+\infty$ is requested. Hence, \cite[Theorem 3.6]{GP2} cannot be applied to problem (\ref{N22}) in Example \ref{Ex1Final}.
}
\end{remark}

\begin{remark} \label{sci}
{\rm
The $(\rm AR)$-condition, adopted in Corollary \ref{CAR}, implies
that the energy functional $J_\lambda$ is unbounded
from below and satisfies the classical $(\rm PS)$-condition, so that
the classical mountain pass theorem can be applied.
Therefore, instead of assumption $(\rm AR)$, a Cerami-type condition
could be exploited in Corollary 3.2 in order to obtain the second solution.
We refer, for instance, to \cite{BL,GP2,PR} for more details on related topics.
}
\end{remark}


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\enddocument


\section{\bf Introduction}
\def\theequation{1.\arabic{equation}}\makeatother
\setcounter{equation}{0}
\begin{definition}
{\rm The texts of Definition, Remark, and Examples must be in roman.}
\end{definition}

\begin{proposition}
This is the proposition.
\end{proposition}

\noindent
\proof

\noindent
This is the end of the proof.

\begin{lemma}
This is the lemma.
\end{lemma}

\begin{corollary}
This is the corollary.
\end{corollary}

\section{\bf Title of this section}
\def\theequation{2.\arabic{equation}}\makeatother
\setcounter{equation}{0}
\begin{eqnarray}
A+B=c+d
\end{eqnarray}
$$
\alpha+\beta=\gamma+\delta
$$
The references are cited as \cite{BO89} and \cite{RA83}. {\bf Note that the text and the
mathematical expressions should be adjusted or split so that they do not exceed the right margin}.


\section{\bf Title of this section}
\def\theequation{3.\arabic{equation}}\makeatother
\setcounter{equation}{0}
\begin{definition}
{\rm The texts of Definition, Remark, and Examples must be in roman.}
\end{definition}

\begin{theorem}
This is the theorem.
\end{theorem}


\begin{theorem}
This is another theorem in the same section.
\end{theorem} 