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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 171, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/171\hfil Existence of solutions]
{Existence of solutions to boundary-value problems
 governed by general non-autonomous nonlinear differential operators}

\author[C. Marcelli \hfil EJDE-2012/171\hfilneg]
{Cristina Marcelli}  

\address{Cristina Marcelli \newline
Dipartimento di Scienze Matematiche \\ 
Universit\`a Politecnica delle Marche\\
Via Brecce Bianche, 60131  Ancona, Italy}
\email{marcelli@dipmat.univpm.it}

\thanks{Submitted September 3, 2012. Published October 4, 2012.}
\subjclass[2000]{34B40, 34C37, 34B15, 34L30}
\keywords{Boundary value problems; unbounded domains;
 heteroclinic solutions; \hfill\break\indent
 nonlinear differential operators; 
$p$-Laplacian operator;  $\Phi$-Laplacian operator}

\begin{abstract}
 This article concerns the existence and non-existence of solutions to
 the  strongly nonlinear non-autonomous boundary-value problem
 \begin{gather*}
 (a(t,x(t))\Phi(x'(t)))' =  f(t,x(t),x'(t)) \quad \text{a.e. } t\in \mathbb{R}  \\
 x(-\infty)=\nu^- ,\quad  x(+\infty)= \nu^+ 
 \end{gather*}
 with $\nu^-<\nu^+$, where $\Phi:\mathbb{R} \to \mathbb{R}$ is a general
 increasing homeomorphism, with $\Phi(0)=0$, $a$ is a positive,
 continuous function and $f$ is a Carathe\'odory nonlinear function.
 We provide sufficient conditions for the solvability which result 
 to be optimal  for a wide class of problems. In particular, 
 we focus on the role played  by the behaviors of $f(t,x,\cdot)$ 
 and $\Phi(\cdot)$ as $y\to 0$ related  to that of $f(\cdot,x,y)$ 
 and $a(\cdot,x) $ as $|t|\to +\infty$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In the previous decade an increasing interest has been devoted to differential 
equations of the type
$$
(\Phi(x'))'=f(t,x,x'),
$$
governed by nonlinear differential operators such as the classical $p$-Laplacian
 or its generalizations.
Various types of differential operators, even singular or non-surjective,
have been considered due to many applications in different fields.
We quote for the scalar case  Bereanu and Mawhin \cite{bm1}-\cite{bm},
Garcia-Huidobro,  Man\'asevich and  Zanolin
\cite{gmz3},  Dosla, Marini and  Matucci \cite{dmm,mr},
Cabada and Pouso \cite{cp1,cp2}, and
Papageorgiou and  Papalini \cite{pp1}.
Moreover,  Man\'asevich and Mawhin treated systems of
equations in \cite{mm}, where they studied a periodic problem.
Finally, in the more general framework of differential inclusions
we quote \cite{fmp}  and the paper   by   Kyritsi,  Matzakos and  Papageorgiou
\cite{kmp1} for systems of differential inclusions involving maximal monotone 
operators and with various boundary conditions.

Different types of differential operators, depending also on $x$ are involved in
reaction-diffusion equations with non-constant diffusivity
 (see, e.g. \cite{acm,mmmm, mmmm1}), in
porous media equations and other models.   So, it
naturally arises the interest in mixed differential
operators, that is strongly nonlinear equations as
\[ 
\big(a(x)\Phi(x')\big )' = f(t,x,x').
\]
In \cite{pp} a periodic problem for a vectorial  differential inclusion
involving an operator of the type $(a(x) \| x' \|^{p-2}x')'$ is
studied, where $a:\mathbb{R} \to \mathbb{R}$ is a positive,
continuous function. Moreover, in \cite{kmp1} the
differential operator is even more general, having  structure \
$(A(x,x'))'$, and the existence of solutions is proved for a
Dirichlet vector problem. In these last two
papers, the boundary-value problem is studied on a compact
interval.

Recently, boundary-value problems on the whole real line, of the type
\begin{gather*}
 \big(a(x(t))\Phi(x'(t))\big)' = f(t,x(t),x'(t)) \quad \text{for a.e. } 
t\in \mathbb{R} \\
 x(-\infty)=\nu^- ,\quad x(+\infty)= \nu^+ 
\end{gather*} 
have been studied in \cite{cmp1} where existence and non-existence
results have been proved for various types of differential operator $\Phi$, 
including the classical $p$-laplacian.
It was proved that the existence of heteroclinic solutions depends on the
 behavior of $\Phi$ and $f(t,x,\cdot)$ at $0$ and
$f(\cdot,x,y)$ at infinity, while the presence of the function $a$ 
does not influence the existence of solutions.

The aim of this article  is to introduce a dependence also on $t$ on the 
function $a$ appearing in the differential operator; that is, to study 
the solvability of the  boundary-value problem
\begin{equation} \label{P}
\begin{gathered}
\big(a(t,x(t))\Phi(x'(t))\big)' = f(t,x(t),x'(t)) \quad \text{for a.e. }
t\in \mathbb{R} \\
 x(-\infty)=\nu^- ,\quad  x(+\infty)= \nu^+
\end{gathered}
\end{equation}
where $\nu^- < \nu^+$ are given constants,  $\Phi:\mathbb{R}\to \mathbb{R}$
is a general increasing homeomorphism, with $\Phi(0)=0$, and $a$ is a positive,
continuous function. We underline that we allow the function $a$
to have null infimum.

Contrary to the autonomous case $a=a(x)$, where the presence of the function $a$ 
does not influence  the existence and non-existence of solutions
(see \cite{cmp1}), in the present setting the dependence on $t$ of the function
 $a$  is instead very relevant.
In more detail, the asymptotic behavior of $a(\cdot,x)$ as $|t|\to +\infty$, 
related to that of $f(\cdot,x,y)$ and compared with the asymptotic behavior 
of $f(t,x,\cdot)$ and $\Phi$ as $y\to 0$, plays a central role
in the existence and non-existence results.

We provide sufficient conditions guaranteeing the solvability of problem \eqref{P}, 
that cannot be improved, in the sense that in a wide range of cases
they are both necessary and sufficient for the existence of solutions.

For instance, when $a$  and $f$  have the product structure
\[ 
f(t,x,y)= h(t) g(x) c(y) , \quad a(t,x)=\alpha(t)\beta(x)
\]
with $h\in L^{q}_{\rm loc}(\mathbb{R})$, for some $1\le q\le \infty$,
satisfying $t  h(t)g(x)\le 0$ for every $(t,x)$ and  $c(y)$ satisfying $c(0)=0$, 
$0<c(y)< K|y|^{2-\frac1q} $ for $y\ne 0$, we have (see Corollary
\ref{c:operativo}):
\begin{gather*}
 \text{ if } \alpha(t) \sim  \text{const.} |t|^p  \text{ and } 
 |h(t)| \sim \text{const.} |t|^\delta 
\text{ as } |t| \to +\infty,  \\
 |c(y)| \sim \text{const.} |y|^\beta , \quad 
|\Phi(y)| \sim \text{const.}  |y|^\mu   \text{ as } y\to 0
\end{gather*}
with $0<\mu\le 1$,  $ \mu(\delta+1) > p\beta$ and $\mu<\beta\le \mu(2-\frac1q)$,
 then  \eqref{P} admits solutions if and only if  $\mu> \beta +p -\delta-1$.


We underline that in the framework of autonomous functions $a$ treated 
in \cite{cmp1}, only the case when $\delta \ge -1$ can be handled 
(see also \cite{cmp2}).
Here, the dependence on $t$ of the function $a$
 allows us to avoid this limitation, provided that $p<0$; that is, when
$a(t,x)$ vanishes as $|t|\to +\infty$.

To the best of our knowledge, the results  presented here are new even 
if for $\Phi(y)\equiv y$; that is, for differential equations
of the form
\[ 
\big(a(t,x(t)) x'(t)\big)' = f(t,x(t),x'(t)) \quad \text{a.e. } t.
\]
Moreover,  the operators considered here are quite general and extend 
the classical $p$-laplacian. Nevertheless,
when dealing just with the $p$-laplacian the results can be slightly improved,
 by using the positive homogeneity of the operator,
as we will show in a forthcoming paper.

\section{Notation and auxiliary results}

In the whole paper we will consider a general increasing homeomorphism
 $\Phi$ on $\mathbb{R}$, such that $\Phi(0)=0$,
a positive continuous function
$a:\mathbb{R}^2\to \mathbb{R}$ and a Carath\'eodory function 
$f:\mathbb{R}^3\to \mathbb{R}$.
We will deal with the  nonlinear differential equation
\begin{equation}
\big((a(t,x(t)) \Phi(x'(t)\big)' = f(t,x(t),x'(t)) \quad \text{a.e. } t
\label{e:E}\end{equation}
We will adopt the following notation:
\begin{gather*}
 m(t):=\min_{x\in [\nu^-\nu^+]} a(t,x),  \quad 
 M(t):=\max_{x\in [\nu^-\nu^+]} a(t,x), \\
m^*(t):=\min_{(s,x)\in [-t,t]\times [\nu^-\nu^+]} a(s,x) , \quad  
M^*(t):=\max_{(s,x)\in [-t,t]\times [\nu^-\nu^+]} a(s,x).
\end{gather*}
Of course, $M^*(t)\ge M(t)\ge m(t)\ge m^*(t) >0$ for every 
$t\in \mathbb{R}$, with ${\inf_{t\in \mathbb{R}} m(t)}$ possibly null.

The approach we adopt  to handle the nonlinear problem on the whole real line 
is based on a sequential technique, considering boundary-value
problems in compact intervals exhausting $\mathbb{R}$. 
The next lemma is just the key result for the convergence of sequences of
solutions in compact intervals towards a solution of  \eqref{e:E} in $\mathbb{R}$.
It was proved in the case the operator $a$ is autonomous; that is,
$a(t,x)\equiv a(x)$, but the same proof works also with the dependence on $t$.


\begin{lemma}[{\cite[Lemma 2.2]{cmp1}}]\label{l:con1} 
For all $n\in \mathbb{N}$ let $I_n:=[-n,n]$ and let $u_n\in C^1(I_n)$ 
be such that the function
$t\mapsto a(t,u_n(t))\Phi(u_n'(t))$ belongs to
 $ W^{1,1}(I_n)$, the sequences $(u_n(0))_n$ and $(u'_n(0))_n$ are
  bounded and 
  \[ 
(a(t,u_n(t))\Phi(u_n'(t)))' = f(t,u_n(t),u_n'(t)) \quad \text{for a.e. } t\in I_n.
\]
 Assume that there exist two functions $H,\gamma \in L^1(\mathbb{R})$ such
  that
 \[  
|u_n'(t)| \le H(t), \quad |a(t,u_n(t))\Phi(u_n'(t))| \le \gamma (t) \quad 
\text{a.e. on }   I_n, \text{ for all } n \in \mathbb{N}.
\]
Then, the sequence $(x_n)_n$ of continuous functions on $\mathbb{R}$ defined 
by  $ x_n(t):= u_n(t)$ for   $ t\in I_n$ and constant outside $I_n$,
 admits a subsequence uniformly convergent in $\mathbb{R}$
to a function $x\in C^1(\mathbb{R})$, such that the composition
$t\mapsto a(t,x(t))\Phi(x'(t))$ belongs to $W^{1,1}(\mathbb{R})$,
and it is a solution of \eqref{e:E}.
Moreover, if\ $ {\lim_{n\to +\infty} u_n(-n)}=\nu^-$ and
$ {\lim_{n\to +\infty} u_n(n)}=\nu^+$,
then we have that ${\lim_{t\to -\infty} x(t)}=\nu^-$ and 
${\lim_{t\to +\infty} x(t)}=\nu^+$.
\end{lemma}

To achieve the solvability of the boundary-value problem in compact intervals, 
we will use the following existence result proved in \cite{fp}, 
concerning a two-point functional differential problem.

\begin{theorem}\cite[Theorem 1]{fp} \label{t:fix}
Let $I=[a,b]\subset \mathbb{R}$ and let $A:C^1(I)
\to C(I)$, $x \mapsto A_x$, and $F:C^1(I) \to L^1(I)$, $x\mapsto
F_x$, be two continuous functionals.
Suppose that  $A$ maps bounded sets of $C^1(I)$ into uniformly
continuous sets in $C(I)$.
Moreover, assume that
\begin{equation} m \le A_x(t) \le M \quad
  \text{for every } x \in C^1(I), t \in I, \text{ for some } M>m>0;
\label{ip:F1}
\end{equation}
and that there exists $\eta \in L^1(I)$ such that
\begin{equation} 
 |F_x(t)|\le \eta(t), \quad \text{a.e. on }  I, \text{ for every } x \in C^1(I).
\label{ip:F3}\end{equation}
Then, there exists a function
$u \in C^1(I)$ such that $A_u \cdot (\Phi\circ u')\in W^{1,1}(I)$
and
\begin{gather*}
    (A_u(t)\Phi(u'(t)))'=F_u(t) ,\quad \text{a.e. on } I \\
    u(a)=\nu^-,\quad u(b)=\nu^+.
  \end{gather*}
\end{theorem}

For  recent results on  two-point boundary-value problems in different settings
see  \cite{bmt1,bmt2,cpr,cr,cm1}.

\section{Existence and non-existence theorems}\label{sez3}


We begin with an existence result for differential operators
growing at most linearly at infinity.

\begin{theorem}\label{t:ex1} Let $\Phi$ be such that
\begin{equation} 
\limsup_{|y|\to +\infty}\frac{ |\Phi(y)|}{|y|}<+\infty.
\label{e:phidominnew}
\end{equation}
Assume that
\begin{equation} 
f(t,\nu^-,0)\le 0 \le f(t,\nu^+,0) \quad \text{for a.e. }
 t\in\mathbb{R} \label{e:const}
\end{equation}
and that there exist constants $L,H>0$, a continuous function
$\theta:\mathbb{R}^+\to \mathbb{R}^+$ and a function $\lambda\in L^q([-L,L])$,
with $1\le q \le \infty$, such that
\begin{gather} 
|f(t,x,y)| \le \lambda(t) \theta(a(t,x) |\Phi(y)|) \quad
\text{for a.e. } |t| \le L, \text{ every } x \in [\nu^-, \nu^+], \
|y| \ge H, \label{eq:nagumo1} 
\\  \int^{+\infty}
   \frac{\tau^{1-\frac{1}{q}}}{\theta(\tau) }\,{\rm d}\tau =
+\infty \label{eq:nagumo2} 
\end{gather}
(with  $\frac{1}{q}=0$ if\ $q=+\infty$).
Also assume that there exists a constant $\gamma > 1$ such that
for every $C>0$ there exist a function $\eta_C\in L^1(\mathbb{R})$ and a
function $K_C \in W_{\rm loc}^{1,1}([0, +\infty))$, null in $[0,L]$ and
strictly increasing in $[L,+\infty)$, such that
\begin{gather} 
N_C(t):=\Phi^{-1}\Big( \frac{1}{m(t)}\Big\{ (M^*(L)\Phi(C))^{1-\gamma} 
+ (\gamma-1) \big|\int_0^t \frac{K_C'(|s|)}{M(s)^\gamma} \,{\rm d}s\big| \Big\}^\frac{1}{1-\gamma} \Big)
\in L^1(\mathbb{R}),\label{e:newL1} 
\\
\begin{gathered}
 f(t,x,y)\le -K_C'(t) \Phi(|y|)^\gamma, \quad 
f(-t,x,y) \ge K_C'(t) \Phi(|y|)^\gamma \\
\text{ for a.e. } t \ge L, \text{ every } x \in
[\nu^-, \nu^+],\ |y| \le N_C(t),
\end{gathered}  \label{eq:fminoraz}
\\
|f(t,x,y)| \le \eta_C(t) \quad
\text{if } x \in [\nu^-, \nu^+],  |y|\le N_C(t), \text{ for a.e. } t\in \mathbb{R}. 
 \label{eq:fdomin2}
\end{gather}
Then, there exists a function $x \in C^1(\mathbb{R})$, such that 
$t\mapsto a(t,x(t))\Phi(x'(t))$ belongs to $  W^{1,1}(\mathbb{R})$ and 
\begin{gather*}
\big(a(t,x(t))\Phi(x'(t))\big)' = f(t,x(t),x'(t))  \quad
 \text{for a.e. } t\in \mathbb{R}  \\
\nu^-\le x(t)\le \nu^+  \quad\text{for every } t\in \mathbb{R}  \\
x(-\infty)=\nu^- , \quad  x(+\infty)= \nu^+.
\end{gather*} 
\end{theorem}

\begin{proof} 
 By \eqref{e:phidominnew} we have
\begin{equation} 
|\Phi(y)|\le K|y| \quad \text{for every } |y|\ge H\label{e:phidomin}
\end{equation}
 for some constant $K>0$, and $H>\frac{\nu^+-\nu^-}{2L}$.  Moreover,  by
 \eqref{eq:nagumo2} there exists a constant
\[
C>\max\{\Phi^{-1}(\frac{M^*(L)}{m^*(L)}\Phi( H)), 
 -\Phi^{-1}(\frac{M^*(L)}{m^*(L)}\Phi(-H))\} \ge H
\]
such that
\begin{equation}
\int_{M^*(L)\Phi(H)}^{m^*(L)\Phi(C)}
\frac{\tau^{1-\frac{1}{q}}}{\theta(\tau)}\,{\rm d}\tau > \|\lambda\|_q [KM^*(L) (\nu^+-\nu^-)]^{1
-\frac{1}{q}} \label{eq:nagumoint}
\end{equation}
and
\begin{equation}
\int_{-M^*(L)\Phi(-H)}^{-m^*(L)\Phi(-C)}
\frac{\tau^{1-\frac{1}{q}}}{\theta(\tau)}\,{\rm d}\tau > \|\lambda\|_q [KM^*(L) (\nu^+-\nu^-)]^{1
-\frac{1}{q}}. \label{eq:nagumoint2}
\end{equation}
 Fix $n \in \mathbb{N}$, $n>L$, and put $I_n:=[-n,n]$.
Consider the truncation operator $T:W^{1,1}(I_n)\to W^{1,1}(I_n)$ defined by
\begin{equation} 
T(x):= T_x \quad \text{where } T_x(t):=\max\{\nu^-, \min\{\nu^+, x(t)\} \}.
\label{eq:Tdef} 
\end{equation} 
Of course, $T$ is well-defined and
$T_x'(t)=x'(t)$ for a.e. $t\in I_n$ such that
$\nu^-<x(t)<\nu^+$, whereas $T_x'(t)=0 $ for a.e.
$t$ such that \ $x(t)\le \nu^-$ or \ $x(t)\ge \nu^+$.
For every $x\in W^{1,1}_{\rm loc}(\mathbb{R})$, put 
\begin{equation}
 Q_x(t):= \max\{-N_C(t), \min\{ T_x'(t),
N_C(t)\}\}.\label{e:Qdefin}
\end{equation} 
Moreover, for every $x\in \mathbb{R}$, put
$ w(x):=\max\{x-\nu^+, 0\} + \min\{ x-\nu^-,0\}$. 
Of course, $w(x)=0$
if  $\nu^-\le x \le \nu^+$, $w(x)>0$ if $x>\nu^+$ and $w(x)<0$ if $x>\nu^-$.

Let us consider the auxiliary boundary-value problem on
the compact interval $I_n$: 
\begin{equation} \label{Pn*} 
\begin{gathered}
\big(a(t,T_x(t))\Phi(x'(t))\big)'= f(t,T_x(t),Q_x(t))+ \arctan(w(x(t))), \quad
\text{a.e. in } I_n \\
x(-n)=\nu^- ,  \quad x(n)=\nu^+.
\end{gathered}
\end{equation}
Let us now prove that this problem admits solutions for every
$n >L$. To this aim, let $A:C^1(I_n)\to C(I_n)$, $x\mapsto A_x$, and
$F:C^1(I_n)\to L^1(I_n)$, $x\mapsto F_x$, be the functionals defined
by
\[
A_x(t):= a(t,T_x(t)) , \quad  F_x(t):= f(t, T_x(t), Q_x(t)) +
\arctan(w(x(t))).
\]
As it is easy to check,  by \eqref{eq:fdomin2} the functionals $A, F$
are well-defined, continuous and they respectively satisfy
assumptions \eqref{ip:F1}, \eqref{ip:F3} of Theorem \ref{t:fix},
taking $m:=m^*(n)$ and $M:=M^*(n)$. Furthermore, by the uniform
continuity of $a(\cdot,\cdot)$ in $[-n,n]\times [\nu^-, \nu^+]$,
for every $\epsilon>0$ there exists $\delta=\delta(\epsilon)>0$ such that
 $|a(t_1,\xi_1) - a(t_2,\xi_2)| < \epsilon$   whenever
$|t_2-t_1|\le \delta$ and $|\xi_1-\xi_2|<\delta$. Let
$D$ be a bounded subset of $C^1(I_n)$; i.e., there exists $S>0$ such
that $\|x\|_{C^1(I)} \le S$ for every $x\in D$. Put
$\rho:=\min\{\delta,\frac{\delta}{S}\}$, if $|t_1-t_2|<\rho$ we have
\[
|T_x (t_1) - T_x(t_2)| \le |x(t_1)-x(t_2)|
 \le \big|\int_{t_1}^{t_2} |x'(\tau)| d\tau \big|
\le S |t_1-t_2| < \delta
\]
for all $x \in D$ and consequently
$ |A_x (t_1) - A_x(t_2)|< \epsilon$ for every $x \in D$, whenever
$|t_1-t_2|<\rho$, that is $A$ maps bounded sets of $C^1(I_n)$ into
uniformly continuous sets of $C(I_n)$. Therefore, we can apply
Theorem \ref{t:fix} and obtain the existence of a function
$u_n \in C^1(I_n)$ such that
$t\mapsto a(t,u_n(t)) \Phi(u'_n(t)) \in W^{1,1}(I_n)$, solution of  \eqref{Pn*}.

 Now we will show that $u_n$ is a solution
of  \eqref{e:E}, in order to apply Lemma \ref{l:con1}. To
this aim, split the proof in steps.


\noindent\textbf{Step 1.}
 We have  $\nu^-\le u_n(t) \le \nu^+$ for all $t \in I_n$.
Indeed, let $t_0$ be such that $u_n(t_0)=\min_{t \in I_n}u_n(t) $. 
If  $u_n(t_0)<\nu^-$, by the boundary conditions in \eqref{Pn*},
$t_0$ belongs to a compact interval $[t_1,t_2]\subset I_n$ satisfying
$u_n(t_1)=u_n(t_2)=\nu^-$ and
$u_n(t)<\nu^-$ for every $t \in (t_1,t_2)$. 
 Hence, by \eqref{eq:Tdef} we have $T_{u_n}(t)\equiv \nu^-$ and $Q_{u_n}(t)\equiv
0$ in $[t_1,t_2]$, and by \eqref{e:const}
for a.e. $t\in (t_1,t_2)$ we have
$$ 
\big(a(t,u_n(t))\Phi(u_n'(t))\big)'=  f(t,\nu^-,0)+ \arctan (u_n(t)-\nu^-) <0.
$$ 
Thus, the function  $ t\mapsto a(t,u_n(t))\Phi(u_n'(t))$ is strictly decreasing
in $(t_1,t_2)$ and being $u_n'(t_0)=0$ we have 
\[
a(t,u_n(t))\Phi(u_n'(t))<a(t_0,u_n(t_0))\Phi(u_n'(t_0))=0\quad \text{for every }
t\in (t_0,t_2).
\] 
Hence, $u_n'(t)<0$ in $(t_0,t_2)$, in contradiction
with the definition of $t_0$. Similarly one can show that
$u_n(t)\le \nu^+$ for every $t\in I_n$.

\noindent\textbf{Step 2.}
 The function $u_n$ is increasing in $[-n,-L]$ and in $[L,n]$. Moreover, 
if $u_n'(t_0)=0$ for some $|t_0|>L$, then $u_n'(t)=0$ whenever $|t|>|t_0|$.
To prove this claim, first observe that the function 
$t\mapsto a(t,u_n(t))\Phi(u'_n(t))$
is increasing in $[-n,-L]$ and decreasing in  $[L,n]$. 
In fact, since  $u_n$ is a solution of \eqref{Pn*} and
$|Q_{u_n}(t)|\le N_C(t)$, using  Step 1 and  assumption \eqref{eq:fminoraz} 
for a.e. $t\ge L$ we have
\begin{equation} 
\big(a(t,u_n(t))\Phi(u_n'(t))\big)'=f(t,u_n(t),Q_{u_n}(t))\le
-K_C'(t)\Phi(|Q_{u_n}(t)|)^{\gamma}\le 0 \label{e:negativ} 
\end{equation} 
and we obtain the monotonicity in  $[L,n]$. Analogously we can proceed for
the interval $[-n,-L]$.

Suppose now, by contradiction, that $u_n'(\bar t)< 0$ for some $\bar t \in [L,n)$.  
Then
\[
a(t,u_n(t))\Phi(u_n'(t)) \le  a(\bar t,u_n(\bar t))\Phi(u_n'(\bar t))< 0 \quad 
\text{for every } t\in [\bar t,n] 
\] 
 and so $u_n'(t) < 0$ for every $t\in [\bar t,n]$.
This contradicts what proved in Step 1, since $u_n(n)=\nu^+$. 
Hence $u_n$ is increasing in $[L,n]$. Similarly we can reason in the 
interval $[-n,-L]$.
Finally, if  $u_n'(t_0)=0$ for some $t_0\in [L,n)$,  for every $t\in (t_0,n)$ 
 we have 
$a(t,u_n(t))\Phi(u_n'(t)) \le a(t_0, u_n(t_0))\Phi(u_n'(t_0))=0$, 
hence $u_n'(t)\le 0$ in $[t_0,n]$. Therefore,  since $u_n$ is increasing 
in the same interval, we deduce that $u_n$ is constant in $[t_0,n]$.


\noindent\textbf{Step 3.} We have   $|u_n'(t)|<C $ for every  $t
\in [-L,L]$.
As it is easy to check, put $g(t):=a(t,u_n(t))\Phi(u_n'(t))$, the claim will 
be proved if we show that
\begin{equation}
m^*(L) \Phi(-C) < g(t) < m^*(L)\Phi(C) \quad \text{for every } t\in [-L,L].
\label{eq:gtdomin}
\end{equation}
To this aim, note that by the Lagrange Theorem there exists a point
$\tau_0\in I_n$ such that  
$$
|u_n'(\tau_0)| = \frac{1}{2L}|u_n(L)-u_n(-L)|\le \frac{\nu^+ -\nu^-}{2L}< H<C, 
 $$
so
\[
m^*(L)\Phi(-C)<  M^*(L) \Phi(-H)< g(\tau_0)< M^*(L)\Phi(H) < m^*(L)\Phi(C). 
\]
Assume, by contradiction, the
existence of an interval $(\tau_1,\tau_2)\subset [-L,L]$ such
that $  M^*(L)\Phi(H) < g(t) < m^*(L)\Phi(C)$ in $(\tau_1,\tau_2)$ and  
$g(\tau_1)=  M^*(L)\Phi(H) $,
$g(\tau_2)=  m^*(L)\Phi(C)$ or viceversa.

Then we have $H < u_n'(t)< C$ in $(\tau_1,\tau_2)$
and since \ $N_C(t)= \Phi^{-1} (\frac{M^*(L)}{m(t)} \Phi(C)) \ge C$ for every
$t\in [-L,L]$, we have $|u_n'(t)|<N_C(t)$ for every
$t\in(\tau_1,\tau_2)$. Then, by Step 1, the definition of \eqref{Pn*}
and  assumption \eqref{eq:nagumo1}, for a.e. $t\in(\tau_1,\tau_2)$
we have
\[
 |g'(t)| = |f(t,u_n(t),u_n'(t))| \le \lambda(t) \theta(g(t)).
\]
Therefore, by the H\"older inequality and \eqref{e:phidomin}, we obtain
\begin{align*}
 \int_{M^*(L)\Phi(H)}^{m^*(L)\Phi(C)} \frac{\tau^{1-\frac{1}{q}}}{\theta(\tau)}
\,{\rm d}\tau
& \le \int_{\tau_1}^{\tau_2}  \frac{g(t)^{1-\frac{1}{q}}}{\theta(g(t))}
   |g'(t)| \,{\rm d}t 
\le\int_{\tau_1}^{\tau_2} \lambda(t) g(t)^{1-\frac{1}{q}}\,{\rm d}t  \\
& \le \|\lambda\|_q \Big( M^*(L) \int_{\tau_1}^{\tau_2} |\Phi(u_n'(t))| \,{\rm d}t 
\Big)^{1-\frac{1}{q}}\\
&\le \|\lambda\|_q \Big(KM^*(L) \int_{\tau_1}^{\tau_2} |u_n'(t)| \,{\rm d}
t\Big)^{1-\frac{1}{q}}  \\ 
& \le \|\lambda\|_q [KM^*(L)(\nu^+-\nu^-)]^{1-\frac{1}{q}}
\end{align*}
in contradiction with \eqref{eq:nagumoint}.

Similarly, assuming that  $  m^*(L)\Phi(-C) < g(t) < M^*(L)\Phi(-H)$ in
 $(\tau_1,\tau_2)$ and  $g(\tau_1)=  m^*(L)\Phi(-C) $,
$g(\tau_2)=  M^*(L)\Phi(-H)$ or viceversa, reasoning as above we obtain 
a contradiction to \eqref{eq:nagumoint2}.
Thus, \eqref{eq:gtdomin} holds and the claim is proved.



\noindent\textbf{Step 4.}  We have \ $|u_n'(t)|\le N_C(t)$ for a.e. $t\in I_n$.
Observe that by virtue of what we proved in Step 3, for every
$t\in [-L,L]$ we have $|u_n'(t)| < C \le N_C(t)$. Moreover, in force
of Step 2, we have $u_n'(t)\ge 0$ for every  $t\in I_n
\setminus [-L,L]$. Hence, in order to prove the claim, it remains to
show that $u_n'(t)\le N_C(t)$ for every $t\in I_n\setminus [-L,L]$.

To this aim, let $\hat t:=\sup \{t>L: u_n'(\tau) < N_C(\tau) \text{
in } [L,t]\}$. By Step 3, $\hat t$ is well defined. Assume, by
contradiction, $\hat t<n$.  By Step 1 and the definition \eqref{e:Qdefin}  we have
\[
\big(a(t,u_n(t))\Phi(u_n'(t))\big)'=f(t,T_{u_n}(t),Q_{u_n}(t))=
f(t,u_n(t),u_n'(t))\quad \text{a.e. in $[L, \hat t]$}.
\] 
Since $u_n'(t)\ge 0$ in $[L,n)$, by \eqref{e:negativ}  we have
\[
 (a(t,u_n(t))\Phi(u_n'(t)))' \le -K_C'(t) \Phi(u_n'(t))^\gamma \le
- \frac{K_C'(t)}{M(t)^\gamma} (a(t,u_n(t))\Phi(u_n'(t)))^\gamma 
\] 
for a.e. $t\in  [L, \hat t]$.
 Then
\begin{align*}
&\frac{1}{1-\gamma} [  (a(t,u_n(t))\Phi(u_n'(t)))^{1-\gamma} -
(a(L,u_n(L))\Phi(u_n'(L)))^{1-\gamma}]  \\
&=\int_L^t \frac{(a(u_n(s))\Phi(u_n'(s)))'}{(a(u_n(s))\Phi(u_n'(s)))^\gamma}
\,{\rm d}s \le - \int_L^t \frac{K_C'(s)}{M(s)^\gamma} \,{\rm d}s 
=  - \int_0^t \frac{K_C'(s)}{M(s)^\gamma} \,{\rm d}s
\end{align*} 
for every $t\in [L, \bar t]$. Therefore,
\begin{align*}
(a(t,u_n(t))\Phi(u_n'(t)))^{1-\gamma}
&\ge (a(L,u_n(L))\Phi(u_n'(L)))^{1-\gamma}
+ (\gamma-1)\int_0^t \frac{K_C'(s)}{M(s)^\gamma} \,{\rm d}s \\
& > (M^*(L)\Phi(C))^{1-\gamma} + (\gamma-1)\int_0^t \frac{K_C'(s)}{M(s)^\gamma} 
\,{\rm d}s
\end{align*}
implying that
\[
u_n'(t) <
\Phi^{-1}\Big( \frac{1}{m(t)} \big\{(M^*(L)\Phi(C))^{1-\gamma} +
(\gamma-1)\int_0^t \frac{K_C'(s)}{M(s)^\gamma} \,{\rm d}s\big\}
^\frac{1}{1-\gamma} \Big) =N_C(t).
\]
for every $t \in [L, \hat t]$, a contradiction when $\hat t<n$.
So, $\hat t =n$ and the claim is proved. The
same argument works in the interval $[-n,-L]$ too.


Summarizing, taking account of the properties proved in Steps 1-4,
we infer that
$$
(a(t,u_n(t)) \Phi(u'_n(t)))' = f(t,u_n(t),u'_n(t)) \quad
 \text{a.e. } t\in I_n 
$$ 
for every $n\in \mathbb{N}$.
Therefore, by \eqref{eq:fdomin2} the sequence of solutions $(u_n)_n$
satisfies all the assumptions of Lemma \ref{l:con1}, applied with
$H(t)=N_C(t)$ and $\gamma(t)= \eta_C(t)$, for 
$t\in \mathbb{R}$, where $C$ is the constant fixed at the beginning of the
proof. So, we obtain the
existence of a solution $x$ of equation \eqref{e:E},  such that
$t\mapsto a(t,x(t))\Phi(x'(t))$ belongs to $W^{1,1}(\mathbb{R})$, satisfying
$x(-\infty)=\nu^-$, $x(+\infty)=\nu^+$.
\end{proof}

It is also possible to deal with differential operators having superlinear 
growth at infinity, provided that condition \eqref{eq:nagumo2} is  
strengthened requiring that the Nagumo function
has sublinear growth at infinity, as the following result states.


\begin{theorem}\label{t:ex2} 
Suppose that all the assumptions of Theorem \ref{t:ex1} are satisfied, 
with the exception of \eqref{e:phidominnew}, and with \eqref{eq:nagumo2}
 replaced by
\begin{equation} \label{eq:nagumo2ter} 
\lim_{y \to +\infty} \frac{\theta(y)}{y}=0.
\end{equation} Then the assertion of Theorem
\ref{t:ex1} follows.
\end{theorem}


\begin{proof}
The proof is quite similar to that of the previous theorem. 
Few modifications only regard Step 3, the unique part in which we used
assumption \eqref{eq:nagumo2}. Indeed, notice that the new assumption 
\eqref{eq:nagumo2ter} implies that
\begin{gather*} 
\lim_{\xi \to +\infty } \frac{1}{\xi^{1-\frac1q}}
\int_{M^*(L)\Phi(H)}^{m^*(L)\xi} 
\frac{\tau^{1-\frac1q}}{\theta(\tau)}
 \,{\rm d}\tau =+\infty ; 
 \\
\lim_{\xi \to -\infty } \frac{1}{|\xi|^{1-\frac1q}}
\int_{-M^*(L)\Phi(-H)}^{-m^*(L)\xi} 
\frac{\tau^{1-\frac1q}}{\theta(\tau)}
 \,{\rm d}\tau  =+\infty
\end{gather*}
hence we can choose the constant $C$ in such a way that
\begin{gather*}
 \int_{M^*(L)\Phi(H)}^{m^*(L)\Phi(C)} \frac{\tau^{1-\frac1q}}{\theta(\tau)} 
 \,{\rm d}\tau > \|\lambda\|_q (2LM^*(L)\Phi(C))^{1-\frac1q}, 
\\
\int_{-M^*(L)\Phi(-H)}^{-m^*(L)\Phi(-C)} 
\frac{\tau^{1-\frac1q}}{\theta(\tau)} > 
\|\lambda\|_q (-2LM^*(L)\Phi(-C))^{1-\frac1q} , 
\end{gather*}
which respectively replace conditions \eqref{eq:nagumoint} 
and \eqref{eq:nagumoint2}.
From now on the proof proceeds as in the previous result, with the exception
 of the last chain of inequalities of Step 3, which now becomes
\begin{align*}
\int_{M^*(L)\Phi(H)}^{m^*(L)\Phi(C)} \frac{\tau^{1-\frac1q}}{\theta(\tau)} 
 \,{\rm d}\tau 
&\le \| \lambda\|_q \Big(M^*(L) \int_{\tau_1}^{\tau_2} |\Phi(u_n'(t))| 
\,{\rm d}t \Big)^{1-\frac1q} \\
&\le \| \lambda\|_q (2LM^*(L) \Phi(C))^{1-\frac1q}. \qedhere
\end{align*}
\end{proof}


The key tools in the previous existence Theorems is
the summability of function $N_C(t)$ (condition \eqref{e:newL1}) 
joined with assumption \eqref{eq:fminoraz}. Such conditions are not improvable
in the sense that if \eqref{eq:fminoraz} is satisfied with the reversed
inequality and $N_C$ is not summable, then problem \eqref{P} does not admit
 solutions, as the following result states.


\begin{theorem} \label{t:nonex1} 
 Suppose that there exist two constants $\rho>0$, $\gamma>1$ and a 
positive strictly increasing function
 $K\in W^{1,1}_{\rm loc}([0,+\infty))$, 
such that the following pair of conditions hold: 
\begin{gather} 
f(t,x,y) \ge - K'(t) \Phi(y)^\gamma \quad \text{for a.e. } t\ge 0, \text{
every } x\in [\nu^-,\nu^+],\ y\in(0,\rho), \label{eq:nonex1} 
\\
f(t,x,y) \le K'(-t) \Phi(y)^\gamma \quad \text{for a.e. } t\le 0, \text{ every }
 x\in [\nu^-,\nu^+],\ y\in(0,\rho) \label{eq:nonex2}
\end{gather}
and for every constant $C$  the function
\begin{equation} 
N_C(t):=\Phi^{-1}\Big( \frac{1}{M(t)}\Big\{ C + (\gamma-1) 
\big| \int_0^t \frac{K'(|s|)}{m(s)^\gamma} \,{\rm d}s\big|\Big\}^\frac{1}{1-\gamma}
 \Big) \label{eq:defNCnonex}
\end{equation}
does not belong to $L^1(\mathbb{R})$.

Moreover, assume that
\begin{equation} \label{eq:nonexglobal} t
f(t,x,y)\le 0 \quad \text{for a.e. } t\in \mathbb{R}, \text{ every }
 (x,y) \in [\nu^-,\nu^+] \times \mathbb{R}
\end{equation}
and there exist two constants $\mu, H>0$ such that
\begin{gather}
a(t,x_1) \le H a(t+\delta, x_2) \quad \text{for every } t\ge 0, \;
  x_1,x_2\in [\nu^-,\nu^+], \;  0<\delta<\mu, \label{ip:auc1}
\\
a(t+\delta,x_1) \le H a(t, x_2) \quad \text{for every } t\le 0, \;
  x_1,x_2\in [\nu^-,\nu^+], \;  0< \delta<\mu.
\label{ip:auc2}
\end{gather}
 Then, \eqref{P} does not admit solutions such that $\nu^-\le x(t) \le \nu^+$;
that is,  no function $x\in C^1(\mathbb{R})$, with $t\mapsto a(t,x(t))\Phi(x'(t))$ 
almost everywhere differentiable and $\nu^-\le x(t) \le \nu^+$, exists solving
 problem \eqref{P}.
\end{theorem}

\begin{proof}
Let $x\in C^1(\mathbb{R})$, with $a(t,x(t))\Phi(x'(t))$ almost everywhere 
differentiable and $\nu^-\le x(t)\le \nu^+$
(not necessarily belonging to $W^{1,1}(\mathbb{R})$), be a solution of \eqref{P}.

First of all let us prove that the function $x$ is monotone increasing. Indeed,
notice that by assumption \eqref{eq:nonexglobal} we deduce that
the function $t\mapsto a(t,x(t))\Phi(x'(t))$ is decreasing in $[0, +\infty)$ 
and increasing in $(-\infty,0]$.
Then, if $x'(t_0)=0$ for some $t_0\ge 0$, we have 
 $ a(t,x(t)) \Phi(x'(t)) \le a(t_0,x(t_0))\Phi(x'(t_0))=0$ for every
$t>t_0$; hence, $x'(t)\le  0$ for every $t\ge t_0$. Since $\nu^-\le x(t)\le \nu^+$ 
and $x(+\infty)=\nu^+$, this implies that $x(t) \equiv \nu^+$
in $[t_0,+\infty)$.
Therefore, for every $t\ge 0 $ we have $x'(t)\ge 0$ and $x'(t)>0$ whenever 
$x(t)<\nu^+$. Similarly, if
$x'(t_0)=0$ for some $t_0\le 0$, we have 
 $ a(t,x(t)) \Phi(x'(t)) \le a(t_0,x(t_0))\Phi(x'(t_0))=0$ for every
$t<t_0$; hence, $x'(t)\le  0$ for every $t\le t_0$, implying $x'(t)=0$ in 
$(-\infty,t_0]$.
 Therefore, we have $x'(t)\ge 0$ for every $t\in \mathbb{R}$ and $x'(t)>0$ 
 whenever $\nu^-<x(t) < \nu^+$.

Let us now prove that\ ${\lim_{t\to \pm\infty}}\ x'(t)= 0$. Since $x$ is increasing, 
it suffices to prove that
$\ell:{\limsup_{t\to \pm\infty}}\ x'(t)= 0$.
Assume, by contradiction, that $\ell > 0$. Then there exists
an interval $[t_1,t_2] \subset [0,+\infty)$ such that $|t_1-t_2|<\mu$,
 $0 < x'(t)< \rho$ in $[t_1,t_2]$ and $\Phi(x'(t_2)) > H \Phi(x'(t_1))$, where
 $\mu $ and $H$ are the constants appearing in assumption \eqref{ip:auc1}.
Hence, 
\[ \Phi(x'(t_2)) > H \Phi(x'(t_1)) \ge \frac{a(t_1, x(t_1))}{a(t_2,x(t_2))} 
\Phi(x'(t_1)
\]
so
\[
 a(t_2,x(t_2))\Phi(x'(t_2)) > a(t_1,x(t_1))\Phi(x'(t_1)) 
\]
a contradiction, since the function
$t\mapsto a(t,x(t))\Phi(x'(t))$ is decreasing in $[0, +\infty) $.
Similarly, by using \eqref{ip:auc2} we obtain 
 ${\limsup_{t\to -\infty}}\ x'(t)= 0$.
Then, ${\lim_{t\to \pm\infty}}\ x'(t)= 0$.

Let us now define
$t^*:=\inf\{ t \ge 0 : x'(t)<\rho \text{ in } [t,+\infty) \}$ and assume by 
contradiction that $x'(t^*)>0$.
Put $T:=\sup\{t: x(t)<\nu^+\}$, so that $0<x'(t)<\rho$ in $(t^*,T)$. By
\eqref{eq:nonex1} for every $t\in (t^*, T)$ we obtain 
\begin{align*}
&\frac{1}{1-\gamma}[(a(t,x(t))\Phi(x'(t)))^{1-\gamma} 
- (a(t^*, x(t^*))\Phi(x'(t^*)))^{1-\gamma}]\\
&= \int_{t^*}^t \frac{(a(s,x(s))\Phi(x'(s)))'}{(a(s,x(s)) 
\Phi(x'(s)))^{\gamma}}\,{\rm d}s \ge \int_{t^*}^t - \frac{K'(s)}{m(s)^\gamma}
 \,{\rm d}s
\end{align*}
therefore,
\[ 
a(t,x(t))\Phi(x'(t)) \ge \Big( (a(t^*,x(t^*))\Phi(x'(t^*)))^{1-\gamma}
 + (\gamma-1) \int_{t^*}^t
\frac{K'(s)}{m(s)^\gamma} \,{\rm d}s \Big)^\frac{1}{1-\gamma}
\]
and finally
\[ 
x'(t) \ge \Phi^{-1}\Big( \frac{1}{M(t)}\Big\{ (a(t^*,x(t^*))\Phi(x'(t^*)))^{1-\gamma} 
+ (\gamma-1) \int_{t^*}^t  \frac{K'(s)}{m(s)^\gamma} \,{\rm d}s
\Big\}^\frac{1}{1-\gamma} \Big).
\]
Then, if $T<+\infty$, necessarily we have $x'(T)=0$, in contradiction with the 
above inequality. Therefore, $T=+\infty$ and again by the above inequality
we deduce $x(+\infty)=+\infty$ since the function on the right side in not summable 
by assumption \eqref{eq:defNCnonex}.
Therefore, $x'(t^*)=0$, implying $t^*=0$ and $x(0)=\nu^+$. Similarly one can show
 that $x(0)=\nu^-$, a contradiction,
by using \eqref{eq:nonex2}.
\end{proof}

\begin{remark}\label{r:nonexcond} \rm
Assumptions \eqref{ip:auc1}, \eqref{ip:auc2} in the previous non-existence 
theorem have been introduced just to deal with non-autonomous differential operators.
Notice that when dealing with autonomous operators, that is for $a(t,x)=a(x)$, 
 they are trivially satisfied.
However, also in the non-autonomous case they hold in many relevant situations. 
For instance, when $a(t,x)$  has the product structure
$a(t,x)=\alpha(t) \beta(x)$, then it is easy to check that 
assumptions  \eqref{ip:auc1} and \eqref{ip:auc2}
 hold  if one the following conditions is satisfied:
\begin{itemize}
\item $\alpha(t)$ is decreasing in $(-\infty,0)$  and increasing in  $(0,+\infty)$;
\item  $\alpha$ is uniformly continuous in $\mathbb{R}$ and ${\inf_{t\in \mathbb{R}}} \alpha(t)>0 $;
\item  $\alpha(t)\sim |t|^{-p}$ as $|t|\to +\infty$ for some $p>0$.
\end{itemize} \label{r:auc}
\end{remark}

\section{Some asymptotic criteria}

In this section we present some operative criteria applicable for operators and
right-hand side having the product structure
$$
a(t,x)= \alpha(t)\beta(x) \quad \text{and } \quad f(t,x,y)=b(t,x)c(x,y).
$$
We will focus on the link between the local behaviors
of $c(x,\cdot)$ at $y=0$ and of $b(\cdot, x)$, $\alpha(\cdot)$ at infinity, which
play a key role for the existence or non-existence of solutions.

 In what follows we assume that $\alpha,\beta$ are continuous positive functions,
$b$ is a Carath\'eodory function and  $c$ is a continuous function satisfying
$$
c(x,y)>0 \quad   \text{for every }  y \ne 0 \textrm{ and }   x\in [\nu^-,\nu^+]; \quad
c(\nu^-,0)=c(\nu^+,0)=0.
$$ 
In this framework, put ${\tilde m:=\min_{x\in [\nu^-,\nu^+]} \beta(x)}$
and ${\tilde M:=\max_{x\in [\nu^-,\nu^+]} \beta(x)}$, we have
\[
m(t)= \tilde m \alpha(t), \quad M(t)=\tilde M \alpha(t), \quad \text{for every } 
 t \in \mathbb{R} 
\]
where recall that ${m(t):=\min_{x\in [\nu^-,\nu^+]} a(t,x)}$ and
 ${M(t):=\max_{x\in [\nu^-,\nu^+]} a(t,x)}$.
We put
\begin{equation}
 m_\infty:=\inf_{t\in \mathbb{R}} \alpha(t) \ge 0. \label{e:defminf}
\end{equation}


\subsection{Case of $\Phi$ growing at most linearly}

In this subsection we deal with differential operators $\Phi$ satisfying 
condition  \eqref{e:phidominnew}; that is,
such that $|\Phi(y)|\le \Lambda |y|$ whenever every $|y|>H$, 
for some $H,\Lambda>0$. With this class of operators we cover
differential equations of the type
\[ 
(a(t,x(t)) x'(t))' = f(t,x(t),x'(t)).
\]
The first two existence theorems are applications of Theorem \ref{t:ex1}.

\begin{proposition} \label{t:pro} 
Suppose that 
\begin{equation} t\cdot b(t,x)< 0 \quad \text{for a.e. $t$ such that } 
|t|\ge L, \text{ every } x \in
[\nu^-, \nu^+]\label{eq:csegno}
\end{equation}
for some $L> 0$ and  there exists a function 
$\lambda\in L^q_{\rm loc}(\mathbb{R})$, $1\le  q \le + \infty$, such that 
\begin{equation} 
|b(t,x)| \le \lambda(t) \quad \text{for a.e. } t\in \mathbb{R}, \text{ every } x
\in [\nu^-, \nu^+].\label{eq:adomin}
\end{equation}  
Moreover, assume that there exist real constants
 $\gamma>1$, $p$, $\delta$, with $p<\delta+1$,  satisfying
 \begin{equation}
\delta+1 >p\gamma \label{ip:Kinf}
\end{equation}
 such that  for  every $x\in [\nu^-, \nu^+]$ we have
\begin{gather} 
h_1 |t|^p\le \alpha(t) \le  h_2 |t|^p,\quad \text{a.e. } |t| > L, \label{eq:a1} 
\\ 
h_1 |t|^\delta\le |b(t,x)| \le  h_2 |t|^\delta, \quad \text{a.e. } |t| > L. 
\label{eq:c1} 
\\
c(x,y)\ge k_1\Phi(|y|)^\gamma \quad \text{for every } y\in \mathbb{R},
\label{eq:c3}
\\
 c(x,y) \le  k_2 \Phi(|y|)^\gamma, \quad \text{whenever  }  |y| < \rho, \label{eq:b1}
\\
 c(x,y) \le k_2 |\Phi(y)|^{2-\frac{1}{q}} \quad \text{whenever } |y|
> H \label{eq:b2}
\end{gather}
for certain positive constants $h_1,h_2,k_1,k_2, \rho, H$.
Let \eqref{e:phidominnew} be satisfied and assume that
\begin{equation} 
\limsup_{y\to 0^+}\frac{\Phi(y)}{y^\mu} >0 \label{ip:limsup}
\end{equation}
for some positive constant $\mu$ satisfying
\begin{equation} \label{eq:MUdomin} 
\mu< {\frac{\delta+1 -p}{\gamma-1}}.
\end{equation}
Then, problem \eqref{P} admits  solutions.
\end{proposition}

\begin{proof}
Without loss of generality we can assume 
$H>\max\{L,\frac{\nu^+ - \nu^-}{2L}\}$. Put 
 $\theta(r):=k_2 (\frac{r}{m^*(L)})^{2-\frac{1}{q}}$  for $r>0$, from
\eqref{eq:adomin} and \eqref{eq:b2} it is immediate to verify the
validity of conditions \eqref{eq:nagumo1} and \eqref{eq:nagumo2}.
Put 
\[
K(t):=\begin{cases} 
k_1  \int_L^t \min\{\min_{x \in [\nu^-, \nu^+]} b(-\tau,x),
 -\max_{x \in [\nu^-, \nu^+]} b(\tau,x)\} \,{\rm d}\tau,   & t\ge L \\
 0, &  0\le t\le L. 
\end{cases}
 \]
By condition \eqref{eq:adomin} we have 
$K\in W_{\rm loc}^{1,1}([0, +\infty))$ and by
\eqref{eq:csegno} we have that $K$  is strictly increasing
for $t \ge L$.
Observe that by \eqref{eq:c3} it follows that
$$ 
f(t,x,y) = b(t,x) c(x,y) \le k_1   b(t,x) \Phi(|y|)^{\gamma} \le
- K'(t)\Phi(|y|)^{\gamma} 
$$ 
and
$$
f(-t,x,y)= b(-t,x)c(x,y)\ge k_1  \,  b(-t,x) \Phi(|y|)^{\gamma}\ge
K'(t)\Phi(|y|)^{\gamma}
$$ 
for a.e. $t \ge L$, every $x \in [\nu^-,\nu^+]$ and every $y\in \mathbb{R}$. 
Then, condition \eqref{eq:fminoraz} of Theorem \ref{t:ex1} holds.

Now, from  \eqref{eq:c1} it follows that  
$  h_1 k_1 t^{\delta} \le K'(t)$   for a.e. $ t \ge L$ and by \eqref{eq:a1} 
we deduce that
 \[ 
\int_L^t \frac{K'(\tau)}{\alpha(\tau)^{\gamma}}\,{\rm d}\tau 
\ge \frac{h_1k_1}{h_2^{\gamma}} \int_L^t \tau^{\delta -p\gamma} \,{\rm d}\tau 
\quad \text{for every  } t>L. 
\]
Hence, by \eqref{ip:Kinf} we obtain 
${\int_L^t \frac{K'(\tau)}{\alpha(\tau)^{\gamma}} \,{\rm d}\tau \to +\infty}$ as 
$t\to +\infty$  and so by condition \eqref{eq:a1} we
deduce that for every fixed $C\in \mathbb{R}$ the function $N_C(t) $ 
defined in \eqref{e:newL1} satisfies
\begin{equation}
\Phi(N_C(t)) \le  \text{const.}\ t^{\frac{\delta +1-p\gamma}{1-\gamma} -p}
\quad \text{for $t$ large enough}. \label{eq:Phin}
\end{equation}
Since $p<\delta+1$, we obtain $\frac{\delta +1-p\gamma}{1-\gamma} -p<0$, 
so $N_C(t)\to 0$ as $t\to +\infty$. Therefore, by \eqref{ip:limsup} and 
\eqref{eq:Phin} we deduce
\[ 
N_C(t) \le \text{const.}\ t^{\frac{\delta +1-p\gamma}{\mu(1-\gamma)} 
-\frac{p}{\mu}} \quad \text{for $t$ large enough}.
\]
implying that $N_C(t)\in L^1(\mathbb{R})$ by  \eqref{eq:MUdomin}. 
Then also \eqref{e:newL1} holds.

Since ${\lim_{|t| \to +\infty}} N_C(t)=0$  a constant
$L_C^* > L$ exists such that $N_C(t)\le \rho$ for every $|t| \ge
L_C^*$. Let us define $\hat C:={\max_{|t|\le L_C^*}} N_C(t)$ and 
\[ 
 \eta_C (t):=  \begin{cases}
{\max_{x \in  [\nu^-, \nu^+]}|b(t,x)| \cdot \max_{(x,y)\in
 [\nu^-, \nu^+]\times [-\hat C,\hat C]}}\ c(x,y) &  \text{if } |t| \le L_C^* \\
 h_2  k_2 |t|^{\delta} \Phi(N_C(t))^{\gamma}  &  \text{if } |t| > L_C^*.
\end{cases} 
\]
By \eqref{eq:c1} and \eqref{eq:b1}, for a.e. $t\in \mathbb{R}$, for
 every  $x \in [\nu^-, \nu^+]$ and every $y \in \mathbb{R}$
 such that  $|y|\le N_C(t) $  we have
$$ 
|f(t,x,y)|= |b(t,x)|  c(x,y)\le \eta_C(t),
$$  so
it remains to prove that $\eta_C\in L^1(\mathbb{R})$.

By \eqref{eq:adomin} and the continuity of the function $c$  we have
$\eta_C\in L^1([-L_C^*,L_C^*])$. Moreover, when $|t|>L_C^*$, by
\eqref{eq:Phin} we have
\[ 
\eta_c(t) \le \text{const.} \ |t|^{\delta + \gamma
\frac{\delta +1-p\gamma}{1-\gamma} -p\gamma}
= \text{const. } |t|^\frac{\delta+\gamma-p\gamma}{1-\gamma}
\]
implying that $\eta_c(t)\in L^1(\mathbb{R}\setminus [-L_C^*,L_C^*])$ 
by condition \eqref{ip:Kinf}.
Therefore, Theorem \ref{t:ex1} applies and guarantees the assertion 
of the present result.
\end{proof}


\begin{remark} \label{r:4.8} \rm
 Notice that $\gamma\le 2-\frac1q\le 2$ is a necessary compatibility 
condition in order to have both \eqref{eq:c3} and \eqref{eq:b2} for large $|y|$.
But when $m_\infty >0$ (see \eqref{e:defminf}), then condition \eqref{eq:c3} 
can be weakened, requiring that it holds only for $|y|$ small enough, 
as the following result states.
\end{remark}

\begin{proposition} \label{t:pro48}
 Let all the assumptions of Proposition \ref{t:pro} be satisfied, with the 
exception of \eqref{eq:c3}, replaced by
\begin{equation} 
c(x,y)\ge k_1 \Phi(|y|)^\gamma \quad \text{whenever } |y|<\rho. \label{eq:c3weak}
\end{equation}
Moreover, assume that $m_\infty >0$. Then, problem \eqref{P} admits solutions.
\end{proposition}

\begin{proof}
For every fixed $C>0$ let
\begin{gather*} 
\Gamma_C:=\max \big\{ \rho, \Phi^{-1}\big( \frac{M^*(L)}{m_\infty} \Phi(C)\big)\big\} ,
\quad 
\hat m_C:=\min_{(x,y)\in [\nu^-\nu^+]\times[\rho,\Gamma_C]}c(x,y),
\\
h_C:= \min\{ k_1, \frac{\hat m_C}{\Phi(\Gamma_C)^{\gamma}}\}.
\end{gather*}
Note that
\[ 
c(x,y)\ge h_C \Phi(|y|)^\gamma \quad \text{for every } x\in [\nu^-,\nu^+], 
\text{ whenever }  |y|\le \Gamma_C. 
\]
So, put 
\[
K_C(t):= h_C  \int_L^t \min\{{\min_{x \in [\nu^-, \nu^+]}} b(-\tau,x),
 -{\max_{x \in [\nu^-, \nu^+]}} b(\tau,x)\} \,{\rm d}\tau
\]  
for $ t\ge L $ (and $K_C(t)=0$ for $t\in [0,L]$),
we deduce that   \eqref{eq:fminoraz} holds since $N_C(t)\le \Gamma_C$ 
for every $t\ge L$. From now on,
the proof proceeds as that of Proposition \ref{t:pro}.
\end{proof}

In view of the proof of Proposition \ref{t:pro}, condition \eqref{eq:MUdomin} 
guarantees the summability of the function $N_C(t)$, in the case when 
\eqref{ip:Kinf} holds. The following results cover cases when the reversed
inequality holds.

\begin{proposition}\label{t:pro2}
 Let all the assumptions of Proposition \ref{t:pro} hold, with the exception 
of \eqref{ip:Kinf} and \eqref{eq:MUdomin}, replaced by
\begin{gather} \label{ip:Kinfno}
\delta+1< p\gamma;\\
p>\mu\label{ip:rel1}.
\end{gather}
Then problem \eqref{P} admits  solutions.
\end{proposition}

\begin{proof}
 If $K$ is the function defined in the proof of Proposition \ref{t:pro}, 
by \eqref{ip:Kinfno} we have
\[
\int_L^t \frac{K'(\tau)}{\alpha(\tau)^{\gamma}} \,{\rm d}\tau 
\le \text{const.} \int_L^t \tau^{\delta -p\gamma} \,{\rm d}\tau  \le \text{const.}
\]
Therefore, $\Phi(N_C(T)) \sim \frac{\text{const.}}{\alpha(t)} $ as 
$t\to +\infty$ (see \eqref{e:newL1}),
hence $N_C(t)\le \text{const.} t^{-p/\mu} $ implying that
$N_C$ is summable by virtue of \eqref{ip:rel1}.

Moreover, if $\eta_C$ is defined as in the proof of Proposition  \ref{t:pro}, then
\[
 \eta_C(t)\le \text{const. } |t|^{\delta} \frac{1}{\alpha(t)^{\gamma}}
\le \text{const. } t^{\delta- p\gamma} \quad \text{for $t$ large enough}
\] 
and we conclude that $\eta_C$ is summable by condition \eqref{ip:Kinfno}.
Then, the proof proceeds as that of Theorem \ref{t:pro}
\end{proof}

By the same proof of the previous Proposition one can prove also the following 
result, applicable when $m_\infty>0$.

\begin{proposition}\label{t:pro248}
 Let all the assumptions of Proposition \ref{t:pro48} hold, with the exception 
of \eqref{ip:Kinf} and \eqref{eq:MUdomin}, replaced by
\eqref{ip:Kinfno} and \eqref{ip:rel1}.
Then problem \eqref{P} admits  solutions.
\end{proposition}

We state two non-existence results, obtained by applying Theorem \ref{t:nonex1}.

\begin{proposition} \label{t:npro} 
Suppose that
\[ 
t\cdot b(t,x)\le 0 \quad \text{for a.e. } t\in \mathbb{R} \text{ and every } 
x \in [\nu^-, \nu^+]
\]
and let there exist real constants $\delta$, $\gamma >1$, $\Lambda> 0$
and a positive function $\ell(t)\in L^1([0,\Lambda])$
such that
\begin{gather} 
|b(t,x) |\le \lambda_1 |t|^{\delta}, \quad \text{for every } x
\in [\nu^-, \nu^+], \text{ a.e. } |t| > \Lambda, \label{eq:cc1}  
\\
|b(t,x)|\le \ell(|t|) \quad \text{for a.e. } |t|\le \Lambda,
x\in [\nu^-, \nu^+], \label{eq:cint}
\\
 c(x,y)\le \lambda_2 \Phi(y)^{\gamma}, \quad \text{for every }  x \in [\nu^-, \nu^+],\
0<y<\rho  \label{eq:cc} 
\end{gather} 
for some positive constants $\lambda_1, \lambda_2, \rho$. Moreover, assume that
\eqref{eq:a1} holds for some constants $h_1,h_2, p$  such that
\begin{equation} 
\delta+1> p\gamma.\label{e:infinity}
\end{equation}
Furthermore, suppose that
\begin{equation} \label{eq:power2}
\limsup_{y\to 0} \frac{\Phi(y)}{y^\mu} < +\infty
\end{equation}
for some positive constant $\mu$ satisfying
\begin{equation} 
\mu \ge \frac{\delta+1 -p}{\gamma-1}. \label{e:MUnonex}
\end{equation}
Also suppose that there exist two constants $\epsilon, H>0$ such that
\begin{gather}
\alpha(t) \le H \alpha(t+r) \quad \text{for every } t>0  
\text{ and } 0<r<\epsilon, \label{ip:auc1bis}
\\
\alpha(t+r) \le H \alpha(t) \quad \text{for every } t<0  \text{ and } 0<r<\epsilon.
\label{ip:auc2bis}
\end{gather}
Then, problem \eqref{P} does not admit solutions.
\end{proposition}

\begin{proof} 
Put
\[ 
K(t):=  \begin{cases} \lambda_2\int_0^t \ell(\tau) \,{\rm d}\tau 
& \text{for } t\in [0, \Lambda] \\
 \lambda_2\int_0^\Lambda \ell(\tau) \,{\rm d}\tau  
+ \lambda_1\lambda_2 \int_\Lambda^t \tau^\delta \,{\rm d}\tau 
& \text{for } t> \Lambda \end{cases}
\]
we have that $K$ is a strictly increasing function belonging to 
$W^{1,1}_{\rm loc}([0,+\infty)$ and  one can easily verify that
conditions \eqref{eq:cc1}, \eqref{eq:cint} and \eqref{eq:cc} guarantee 
the validity of \eqref{eq:nonex1} and \eqref{eq:nonex2}.
Moreover, by \eqref{eq:a1} we obtain
\[ 
\int_L^t \frac{K'(\tau)}{\alpha(\tau)^\gamma} \,{\rm d}\tau \ge \text{const. } 
t^{\delta-p\gamma+1} \quad \text{for $t$ large enough,} 
\]
hence by \eqref{e:infinity} we have 
${\int_L^t \frac{K'(\tau)}{\alpha(\tau)^\gamma} \,{\rm d}\tau \to +\infty } $ as 
$t\to +\infty$.
Therefore, by \eqref{eq:a1} and \eqref{eq:cc1},  if $N_C(t)$ is the function
defined in \eqref{eq:defNCnonex} we have
\[ 
\Phi(N_C(t)) \ge \text{const. } t^{\frac{\delta-p\gamma+1}{1-\gamma} - p} \quad 
\text{for $t$ large enough}
\]
implying that
\[ 
N_C(t) \ge  \text{const. } t^{\frac{\delta-p\gamma+1}{\mu(1-\gamma)} 
- \frac{p}{\mu}}\quad \text{for $t$ large enough}
\]
by virtue of \eqref{eq:power2}. Finally, assumption \eqref{e:MUnonex} implies 
that $N_C(t)$ is not summable in $[L,+\infty)$
and the assertion follows as an immediate application of Theorem \ref{t:nonex1}.
\end{proof}

When condition \eqref{e:infinity} does not hold, we can use the following 
non-existence result.

\begin{proposition}\label{t:npro2}
 Let all the assumption of Proposition \ref{t:npro} be satisfied with the
 exception of \eqref{e:infinity} and \eqref{e:MUnonex}, which are replaced by
\begin{gather}
\delta +1 \le p\gamma ,\label{e:noninfinity}\\
p\le \mu. \label{ip:alphamu}
\end{gather}
Then, problem \eqref{P} does not admit solutions.
\end{proposition}

\begin{proof}
 With the same notation of the proof of Proposition \ref{t:npro}, 
notice that under condition \eqref{e:noninfinity} we have
${\limsup_{t\to +\infty}\int_L^t \frac{K'(\tau)}{\alpha(\tau)^\gamma} \,{\rm d}\tau 
< +\infty } $, hence
$\Phi(N_C(t)) \ge \text{const. }  t^{-p} $, implying that
 $N_C(t) \ge \text{const. }  t^{-p/\mu}$. Therefore $N_C$ is not summable
at infinity owing to assumption \eqref{ip:alphamu} and the assertion follows
 from Theorem \ref{t:nonex1}.
\end{proof}

For sufficient conditions ensuring the validity of assumptions 
\eqref{ip:auc1bis} and \eqref{ip:auc2bis}, see Remark \ref{r:auc}.
As an immediate application of the previous results, the following 
operative criteria hold.

\begin{corollary}\label{c:operativo}
 Let \eqref{e:phidominnew} be satisfied.
Let $f(t,x,y)=h(t)g(x) c(y)$, where $h\in L^q_{\rm loc}(\mathbb{R})$, for
some $1\le q\le +\infty$, $c$ is continuous in $\mathbb{R}$ and
$g$ is continuous and  positive in $[\nu^-,\nu^+]$.
Assume that $c(y)>0$ for $y\ne 0$; $t\cdot h(t)\le 0$ for
every  $t$ and suppose that there exist constants $C_1,\dots,C_4>0$ such that
\begin{gather} 
\alpha(t) \sim C_1|t|^p \text{ as } |t|\to +\infty, \quad \text{for some }
 p\in \mathbb{R}, \label{eq:ordera} 
\\
 |h(t)| \sim C_2|t|^\delta \text{ as } |t|\to +\infty, \quad \text{for some } 
\delta\in \mathbb{R}, \label{eq:orderc} 
\\
 \Phi(y) \sim C_3 |y|^\mu  \text{ as } y\to 0 , \quad \text{for some }  \mu>0,  
\label{eq:orderPhi} 
\\
 c(y) \sim C_4  |y|^\beta   \text{ as } y\to 0 , \quad \text{for some }  \beta>\mu,  
\label{eq:orderb0} 
\end{gather}
with
\begin{equation} 
\delta+1 > \frac{p\beta}{\mu}. \label{e:ultimamu}
\end{equation}
Then, if conditions \eqref{ip:auc1bis}, \eqref{ip:auc2bis} hold and
 $\mu\le \beta + p -\delta -1 $,  Problem \eqref{P} has no solution.

Viceversa, if $p<\delta+1$,  $ \mu>\beta + p -\delta -1 $ and we further 
assume that
 \begin{gather} 
\limsup_{|y|\to +\infty\ }   c(y)|\Phi(y)|^{\frac{1}{q} -2}   
< +\infty, \label{eq:orderbinf} 
\\
 c(y)\ge k_1 \Phi(|y|)^\frac{\beta}{\mu} \quad \text{for every }
 y\in \mathbb{R} \label{e:nuova} 
\end{gather}
for some $k_1>0$, then \eqref{P} admits solutions.
\end{corollary}

The assertion of the above corollary is an immediate consequence of
Propositions \ref{t:pro} and \ref{t:npro} taking $\gamma=\beta/\mu$.

As observed in Remark \ref{r:4.8}, $\frac{\beta}{\mu} \le 2-\frac1q \le 2$ 
is a necessary compatibility condition to have both
\eqref{eq:orderbinf} and \eqref{e:nuova},
but when $m_\infty>0$ it can be removed, as we state in the following result, 
application of Proposition \ref{t:pro48}.

\begin{corollary}\label{c:operativo48}
 Let all the assumption of Corollary \ref{c:operativo} hold, with the 
exception of \eqref{e:nuova}. Then if $m_\infty>0$, problem \eqref{P} admits
solutions.
\end{corollary}

When assumption \eqref{e:ultimamu} is not satisfied, we can use the following 
result, consequence of Propositions \ref{t:pro2} and \ref{t:npro2}.

\begin{corollary}\label{c:operativo2}
Let all the assumptions of Corollary \ref{c:operativo} be satisfied, with 
the exception of  \eqref{e:ultimamu}, which is replaced by
\begin{equation} 
\delta+1 < \frac{p\beta}{\mu}. \label{e:ultimamuno}
\end{equation}
Then, if conditions \eqref{ip:auc1bis}, \eqref{ip:auc2bis} hold and
 $p\le \mu$,  Problem  \eqref{P} has no solution.

Viceversa, if  $\mu< p $  and we further assume \eqref{eq:orderbinf} and 
\eqref{e:nuova}, then \eqref{P} admits solutions.
\end{corollary}

Finally, a result analogous to Corollary \ref{c:operativo2} holds when 
condition \eqref{e:nuova} is removed, provided that $m_\infty>0$,
as in Corollary \ref{c:operativo48}.

We provide now an  application of the previous results.

\begin{example} \rm
Let $\Phi(y):=y$, $\alpha(t):= |t|^p $, $f(t,x,y)=-t|t|^s |y|^\beta$, 
for some constants $p,s,\beta$
(we avoid to introduce a dependence on $x$ since
we have showed that it does not influence the existence or non-existence of solutions).
In this case we have $\mu=1$. Assume  $s+1\ge 0$ (so we can take $q=+\infty$)
 and  $1<\beta\le 2$ (so that \eqref{eq:orderbinf} holds).


Then, if $s+2>p\beta$, $s+2>p$, problem \eqref{P} admits solutions 
(whatever $\nu^-, \nu^+$ may be),
if and only if $p< s+3-\beta$, as a consequence of Corollary \ref{c:operativo}.
Otherwise, if $s+2<p\beta$, problem \eqref{P} admits solutions if and only 
if $p>1$, as a consequence of Corollary \ref{c:operativo2}.
\end{example}



\subsection{Case of $\Phi$ having superlinear growth}

We now deal with operators $\Phi$ having possibly superlinear growth at infinity, 
that is we now remove condition \eqref{e:phidominnew}.
The non-existence Propositions  \ref{t:npro} and \ref{t:npro2} hold also in this 
case, since they do not require condition \eqref{e:phidominnew}.
As for the existence results, we now use Theorem  \ref{t:ex2} instead of 
Theorem  \ref{t:ex1}, by assuming \eqref{eq:nagumo2ter}.

As it will be clear later, condition \eqref{eq:nagumo2ter} is not compatible 
with \eqref{eq:c3} so from now on we will assume $m_\infty>0$.
However, in the special case of the $p-$laplacian, this condition can be removed, 
as we will show in a forthcoming paper.

 \begin{proposition} \label{t:proTh3}  
Let  all the assumptions of Proposition \ref{t:pro48} 
(or Proposition \ref{t:pro248}) hold, with the exception of \eqref{eq:b2}
which is replaced by 
\begin{equation}\lim_{|y| \to +\infty}
\frac{{\max_{x\in [\nu^-,\nu^+]} c(x,y)}}{|\Phi(y)|}=0
\label{e:blimit}
\end{equation} 
Then, problem \eqref{P} admits solutions.
\end{proposition}

\begin{proof}
Put
\[
\theta(r):=  \max_{(t,x)\in [-L,L]\times [\nu^-,\nu^+]} 
\Big( \max \Big\{c(x, \Phi^{-1}(\frac{r}{a(t,x)})) ,\ 
c(x, \Phi^{-1}(-\frac{r}{a(t,x)}) \Big\} \Big),
\]
it is immediate to check that  $\theta$ is a continuous function on $[0,+\infty)$,
 such that
\[ 
\theta(a(t,x) |\Phi(y)|) \ge  c(x,y) \quad \text{for every } t\in [-L,L],
 x\in [\nu^-,\nu^+], y\in \mathbb{R},
\]
 hence \eqref{eq:nagumo1} holds. Moreover, by
\eqref{e:blimit}, for every $\epsilon>0$ there exists a real
$c_\epsilon$ such that
\[
 c(x,y)\le \epsilon |\Phi(y)| \quad \text{for every } x\in [ \nu^-, \nu^+], 
|y|\ge c_\epsilon. 
\]
Hence, for every $s\ge M^*(L) \max\{\Phi(c_\epsilon), -\Phi(-c_\epsilon)\}$ we have 
$\theta(s)\le \frac{\epsilon}{m^*(L)} s$; that is,
\[ 
\lim_{s\to +\infty} \frac{\theta(s)}{s} = 0.  
\]
Then \eqref{eq:nagumo2ter} holds and the proof proceeds as that of Proposition 
\ref{t:pro48} [or Proposition \ref{t:pro248}], applying
Theorem \ref{t:ex2} instead of Theorem \ref{t:ex1}.
\end{proof}

Note that condition \eqref{e:blimit} is not compatible with \eqref{eq:c3}, 
since $\gamma>1$.
As applications of the previous result, the following operative criteria hold.


\begin{corollary}\label{c:operativoperTh3}
Let $f(t,x,y)=h(t)g(x) c(y)$, where $h\in L^q_{\rm loc}(\mathbb{R})$, for
some $1\le q\le +\infty$, $c$ is continuous in $\mathbb{R}$ and
$g$ is continuous and  positive in $[\nu^-,\nu^+]$.
Assume that $c(y)>0$ for $y\ne 0$; $t\cdot h(t)\le 0$ for
every  $t$ and suppose that there exist constants $C_1,\dots,C_4>0$ such that
\eqref{eq:ordera}, \eqref{eq:orderc}, \eqref{eq:orderPhi},  \eqref{eq:orderb0}, 
\eqref{e:ultimamu} hold with $p<\delta+1$.
Moreover, assume that  $\mu>\beta +p -\delta-1$, $m_\infty>0$ and
\[  
\lim_{|y|\to +\infty} \frac{c(y)}{|\Phi(y)|}=0.
 \]
Then problem  \eqref{P} admits solutions.
\end{corollary}


\begin{corollary}\label{c:operativo2perTh3}
Let all the assumptions of Corollary \ref{c:operativoperTh3} be satisfied,
with the exception of  \eqref{e:ultimamu} which is replaced by \eqref{e:ultimamuno}.
 Then if $p\ge \mu$, problem \eqref{P} admits solutions.
\end{corollary}


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