\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 $$\label{P} \begin{gathered} \big(a(t,x(t))\Phi(x'(t))\big)' = f(t,x(t),x'(t)) \quad \text{for a.e. } t\in \mathbb{R} \\ x(-\infty)=\nu^- ,\quad x(+\infty)= \nu^+ \end{gathered}$$ 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 $c(0)=0$, $0 p\beta$ 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 $$\big((a(t,x(t)) \Phi(x'(t)\big)' = f(t,x(t),x'(t)) \quad \text{a.e. } t \label{e:E}$$ 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 $$m \le A_x(t) \le M \quad \text{for every } x \in C^1(I), t \in I, \text{ for some } M>m>0; \label{ip:F1}$$ and that there exists $\eta \in L^1(I)$ such that $$|F_x(t)|\le \eta(t), \quad \text{a.e. on } I, \text{ for every } x \in C^1(I). \label{ip:F3}$$ 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 $$\limsup_{|y|\to +\infty}\frac{ |\Phi(y)|}{|y|}<+\infty. \label{e:phidominnew}$$ Assume that $$f(t,\nu^-,0)\le 0 \le f(t,\nu^+,0) \quad \text{for a.e. } t\in\mathbb{R} \label{e:const}$$ 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 $$|\Phi(y)|\le K|y| \quad \text{for every } |y|\ge H\label{e:phidomin}$$ 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 $$\int_{M^*(L)\Phi(H)}^{m^*(L)\Phi(C)} \frac{\tau^{1-\frac{1}{q}}}{\theta(\tau)}\,{\rm d}\tau > \|\lambda\|_q [KM^*(L) (\nu^+-\nu^-)]^{1 -\frac{1}{q}} \label{eq:nagumoint}$$ and $$\int_{-M^*(L)\Phi(-H)}^{-m^*(L)\Phi(-C)} \frac{\tau^{1-\frac{1}{q}}}{\theta(\tau)}\,{\rm d}\tau > \|\lambda\|_q [KM^*(L) (\nu^+-\nu^-)]^{1 -\frac{1}{q}}. \label{eq:nagumoint2}$$ 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 $$T(x):= T_x \quad \text{where } T_x(t):=\max\{\nu^-, \min\{\nu^+, x(t)\} \}. \label{eq:Tdef}$$ Of course, $T$ is well-defined and $T_x'(t)=x'(t)$ for a.e. $t\in I_n$ such that $\nu^-0$ if $x>\nu^+$ and $w(x)<0$ if $x>\nu^-$. Let us consider the auxiliary boundary-value problem on the compact interval $I_n$: $$\label{Pn*} \begin{gathered} \big(a(t,T_x(t))\Phi(x'(t))\big)'= f(t,T_x(t),Q_x(t))+ \arctan(w(x(t))), \quad \text{a.e. in } I_n \\ x(-n)=\nu^- , \quad x(n)=\nu^+. \end{gathered}$$ 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))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 $$\big(a(t,u_n(t))\Phi(u_n'(t))\big)'=f(t,u_n(t),Q_{u_n}(t))\le -K_C'(t)\Phi(|Q_{u_n}(t)|)^{\gamma}\le 0 \label{e:negativ}$$ 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 $|u_n'(t)|L: u_n'(\tau) < N_C(\tau) \text{ in } [L,t]\}$. By Step 3, $\hat t$ is well defined. Assume, by contradiction, \hat t (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 everyt \in [L, \hat t]$, a contradiction when$\hat t \|\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 $$N_C(t):=\Phi^{-1}\Big( \frac{1}{M(t)}\Big\{ C + (\gamma-1) \big| \int_0^t \frac{K'(|s|)}{m(s)^\gamma} \,{\rm d}s\big|\Big\}^\frac{1}{1-\gamma} \Big) \label{eq:defNCnonex}$$ does not belong to $L^1(\mathbb{R})$. Moreover, assume that $$\label{eq:nonexglobal} t f(t,x,y)\le 0 \quad \text{for a.e. } t\in \mathbb{R}, \text{ every } (x,y) \in [\nu^-,\nu^+] \times \mathbb{R}$$ 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 $t0$ whenever $\nu^- 0$. 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 $00$; \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 $$m_\infty:=\inf_{t\in \mathbb{R}} \alpha(t) \ge 0. \label{e:defminf}$$ \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 $$t\cdot b(t,x)< 0 \quad \text{for a.e. t such that } |t|\ge L, \text{ every } x \in [\nu^-, \nu^+]\label{eq:csegno}$$ for some $L> 0$ and there exists a function $\lambda\in L^q_{\rm loc}(\mathbb{R})$, $1\le q \le + \infty$, such that $$|b(t,x)| \le \lambda(t) \quad \text{for a.e. } t\in \mathbb{R}, \text{ every } x \in [\nu^-, \nu^+].\label{eq:adomin}$$ Moreover, assume that there exist real constants $\gamma>1$, $p$, $\delta$, with $p<\delta+1$, satisfying $$\delta+1 >p\gamma \label{ip:Kinf}$$ 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 $$\limsup_{y\to 0^+}\frac{\Phi(y)}{y^\mu} >0 \label{ip:limsup}$$ for some positive constant $\mu$ satisfying $$\label{eq:MUdomin} \mu< {\frac{\delta+1 -p}{\gamma-1}}.$$ 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 $$\Phi(N_C(t)) \le \text{const.}\ t^{\frac{\delta +1-p\gamma}{1-\gamma} -p} \quad \text{for t large enough}. \label{eq:Phin}$$ 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 $$c(x,y)\ge k_1 \Phi(|y|)^\gamma \quad \text{whenever } |y|<\rho. \label{eq:c3weak}$$ 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 p\gamma.\label{e:infinity} Furthermore, suppose that $$\label{eq:power2} \limsup_{y\to 0} \frac{\Phi(y)}{y^\mu} < +\infty$$ for some positive constant $\mu$ satisfying $$\mu \ge \frac{\delta+1 -p}{\gamma-1}. \label{e:MUnonex}$$ 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 \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 $$\delta+1 > \frac{p\beta}{\mu}. \label{e:ultimamu}$$ 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 $$\delta+1 < \frac{p\beta}{\mu}. \label{e:ultimamuno}$$ 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 $s+21$, 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 $$\lim_{|y| \to +\infty} \frac{{\max_{x\in [\nu^-,\nu^+]} c(x,y)}}{|\Phi(y)|}=0 \label{e:blimit}$$ 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} \begin{thebibliography}{99} \bibitem{acm} M. Arias, J. Campos, C. Marcelli; \emph{Fastness and continuous dependence in front propagation in Fisher-KPP equations}, Discr. Cont. Dyn. Syst., Ser. B, \textbf{11} (2009), no. 1, 11-30. \bibitem{bmt1} I. Benedetti, L. Malaguti, V. 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