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\title{Existence of solutions for an impulsive  boundary value problem with nonlinear derivative dependence on unbounded intervals  via variational methods}
\author{Sihem Boumaraf}
\address{Laboratory of Fixed Point Theory and Applications\\
Ecole Normale Sup�rieure, Kouba, Algiers, Algeria,}
\email{boumaraf.sihem@gmail.com}
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\author{Toufik Moussaoui}
\address{Laboratory of Fixed Point Theory and Applications\\
Ecole Normale Sup�rieure, Kouba, Algiers, Algeria,}
\email{moussaoui@ens-kouba.dz}
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\subjclass{34B37, 34B40, 35A15, 47J25}
\keywords{Impulsive BVPs, unbounded interval, nonlinear derivative dependence, iterative methods, variational methods}
\begin{abstract}
In this paper, we employ the critical point theory and iterative methods to establish  the existence of  solutions for an impulsive boundary value problem with  nonlinear derivative dependence on the half-line.
\end{abstract}
\maketitle

\section{Introduction}
In this paper, we consider the solvability of an impulsive boundary value problem    with nonlinear derivative dependence on the
half-line. More precisely, we consider the problem
\begin{equation}\label{01}
\left\{
\begin{array}{rlll}
-(p(t)u'(t))'&=&f(t,u(t), u'(t)), & \mbox{a.e. }\, t\ge0,\;t\not=t_{j}, \\
u(0)= u(+\infty)&=&0,& \\
\triangle (p(t_{j})u'(t_{j}))&=&g(t_{j})I_{j}(u(t_{j})), & j\in\{1,2,\ldots\},
\end{array}
\right.
\end{equation}
where $f: [0,+\infty)\times\mathbb{R}\times\mathbb{R}\longrightarrow\mathbb{R}$  is measurable in $t\in [0,+\infty) $ for each $(x, \xi) \in \mathbb{R}\times\mathbb{R},$ and  continuous in $(x, \xi) \in \mathbb{R}\times\mathbb{R}$ for a.e. $t \in [0, +\infty).$
We assume that the impulsive functions $I_{j}: \mathbb{R}\longrightarrow\mathbb{R}$ are continuous where $t_{0}=0<t_{1}<t_{2}<\ldots<t_{j}<\ldots<t_{m}\to+\infty$, as $m\to\infty$, are the impulse points.\\
The coefficient $p: [0,+\infty)\longrightarrow(0,+\infty)$ satisfies
$\frac{1}{p}\in L^{1}(0,+\infty),$ and
$$
M=\int_{0}^{+\infty}\left(\int_{t}^{+\infty}\frac{1}{p(s)}ds\right)dt<+\infty.
$$
We  define the jump
$$
\triangle(p(t_{j})u'(t_{j}))=p(t^{+}_{j})u'(t^{+}_{j})-p(t_{j}^{-})u'(t^{-}_{j}),
$$
where $u'(t^{+}_{j})=\lim\limits_{t\to t_{j}^{+}} u'(t)$ and $u'(t^{-}_{j})=\lim\limits_{t\to t_{j}^{-}}u'(t)$
stand for the right and the left limits of $u'$ at $t_j$, respectively.
Finally $g: [0,+\infty)\longrightarrow[0,+\infty)$  is a continuous function that satisfies $\sum_{j=1}^{+\infty}g(t_{j})<+\infty.$\\

Recently, in \cite{briki, chen}, the authors obtained the existence of solutions for BVPs  associated to impulsive  equations  on unbounded domains by using  variational methods. In \cite{fgm}, de Figueiredo, Girardi and Matzeu  proved the existence of solution  for semilinear elliptic equations with dependence on the gradient through an iterative technique. However,  there are few papers that have studied the existence of solutions for impulsive boundary value problems similar to  the problem (\ref{01}) by using variational methods coupled with the iterative methods. \\

In order to use variational methods, we consider a family of  boundary value problems with no dependence on the derivative. Namely, for each $w \in H_{0, p}^{1}(0, +\infty)$, we consider the problem
\begin{equation}\label{02}
\left\{
\begin{array}{rlll}
-(p(t)u'(t))'&=&f(t,u(t), w'(t)), & \mbox{a.e. }\, t\ge0,\;t\not=t_{j}, \\
u(0)= u(+\infty)&=&0,& \\
\triangle (p(t_{j})u'(t_{j}))&=&g(t_{j})I_{j}(u(t_{j})), & j\in\{1,2,\ldots\}.
\end{array}
\right.
\end{equation}
The class of  problems (\ref{02}) is  of variational type and
we can resolve them  by variational methods and
the existence of a solution for the initial  problem
is obtained by iterative methods.\\

Now we need to define the following Banach  space and this before giving the variational formulation of (\ref{02}).
$$
H^{1}_{0,p}(0,+\infty)=\{u\in AC[0,+\infty),\mathbb{R})\mid\, u(0)=u(+\infty)=0,\,\sqrt{p}u'\in L^{2}(0,+\infty)\},
$$
equipped with the norm
$$
\|u\|_{0,p}=\sqrt{\int_{0}^{+\infty}p(t)u'^{2}(t)dt+\int_{0}^{+\infty}u^{2}(t)dt},
$$
or the equivalent norm
$$
\|u\|_{p}= \|u\|_{L^{2}} + \|\sqrt{p}u'\|_{L^{2}}.
$$
Moreover the space $ H^{1}_{0,p}(0,+\infty)$ is reflexive (see \cite{briki}).\\
\begin{lemma}
On $H^{1}_{0,p}(0,+\infty),$  the quantity $\|u \|=\sqrt{\int_{0}^{+\infty}p(t)u'^{2}(t)dt}$ is a norm which is equivalent to the
$H^{1}_{0,p}(0, +\infty)$-norm.
\end{lemma}
%%%%%%%%%%%%%%%%%%%

Now let us recall the following essential embeddings (see \cite{briki}).


\begin{lemma}
$(H^{1}_{0,p}(0,+\infty),\Vert\cdot\Vert)$ embeds in $(C_{0}[0,+\infty),\|u\|_{\infty})$, where
$C_{0}[0,+\infty)=\{u\in C([0,+\infty),\mathbb{R})\mid\;\lim_{t\to+\infty}u(t)=0\}$ and $\|u\|_{\infty}=\sup_{t\in[0,+\infty)}|u(t)|.$
\end{lemma}
%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{lemma2}
$H^{1}_{0,p}(0, +\infty)$ embeds continuously in $C_{0}[0, +\infty)$ and in $L^2(0,+\infty)$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{lemma1}
The embedding $H^{1}_{0,p}(0,+\infty)\hookrightarrow C_{0}[0,+\infty)$
is compact with
\begin{eqnarray*}
\Vert u\Vert_{\infty}\leq M_1\Vert u\Vert,
\end{eqnarray*}
where $M_1= \sqrt{\Vert\frac{1}{p}\Vert_{L^1}}$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
First we recall some basic definitions and  lemmas which are used in this paper.
\begin{lemma}(Minimization Principle\cite{badiale})\label{lemma3}
Let $X$ be a reflexive Banach space and $J$ a functional defined on $X$ such that
\item (1) $\lim\limits_{{\|u\|\to+\infty}}J(u)=+\infty$ (coercivity condition),
\item (2) $J$ is sequentially weakly lower semi-continuous.\\
Then $J$ is lower bounded on $X$ and achieves its lower bound at some point $u_{0}$.
\end{lemma}

\begin{definition}
Let $X$ be a real Banach space, $J\in C^1(X,\mathbb{R})$. If any sequence $(u_{n})\subset X$ for which $(J(u_{n}))$ is bounded in $\mathbb{R}$ and $J'(u_{n})\longrightarrow 0$ as $n\to+\infty$ in $X'$ possesses a convergent subsequence, then we say that $J$ satisfies the Palais-Smale condition (PS condition for brevity).
\end{definition}

\begin{lemma} \label{lemma4} (\protect{\cite[Theorem 2.2]{rabinowitz2}}, \protect{\cite[Theorem 3.1]{struwe}}) [Mountain Pass Theorem]
Let  $X$ be a real Banach space and $J \in C^{1}(X,\mathbb{R})$
satisfying the  (PS) condition. Suppose that $J(0) = 0$ and
\item (1) there are constants $\rho, \alpha >0$ such that $J(u)\geq \alpha$ for all $u\in X$ with $\Vert u\Vert =\rho$,
\item (2) there exists $u_{0}\in X$ such that $\|u_{0}\|>\rho$ and $J(u_{0})<\alpha.$\\
Then  $J$ possesses a critical value such that $ c\geq \alpha.$ Moreover, $c$ can be characterized as
$$ c = \displaystyle\inf_{\gamma\in \Gamma}\displaystyle\max_{u\in \gamma([0,1])}J(u),$$
where  $$ \Gamma = \big\{\gamma\in C([0,1],X): \gamma(0)= 0,\; \gamma(1) = u_{0}\big \}.$$
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Variational setting}
Take $v\in H^{1}_{0,p}(0,+\infty)$, multiply the equation in problem (\ref{01}) by $v$ and integrate over $(0,+\infty)$, we obtain
$$
-\int_{0}^{+\infty}(p(t)u'(t))'v(t)dt=\int_{0}^{+\infty}f(t,u(t), u'(t))v(t)dt.
$$
The first term is
\begin{equation*}
\begin{array}{rll}
-\int_{0}^{+\infty}(p(t)u'(t))'v(t)dt&=&-\displaystyle\sum_{j=0}^{+\infty}\int_{t_{j}}^{t_{j+1}}(p(t)u'(t))'v(t)dt\\
&=& \displaystyle\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u(t_{j}))v(t_{j})+ \int_{0}^{+\infty} p(t)u'(t)v'(t)dt.
\end{array}
\end{equation*}
Hence
$$
\int_{0}^{+\infty} p(t)u'(t)v'(t)dt=-\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u(t_{j}))v(t_{j})+\int_{0}^{+\infty}f(t,u(t), u'(t))v(t)dt.
$$
\begin{definition}
We say that a function $u\in H^{1}_{0,p}(0,+\infty)$ is a weak solution of Problem (\ref{01}) if
$$
\int_{0}^{+\infty}p(t)u'(t)v'(t)dt+\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u(t_{j}))v(t_{j})-\int_{0}^{+\infty} f(t,u(t), u'(t))v(t)dt=0,
$$
for every $v \in H^{1}_{0,p}(0,+\infty).$¸
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{03}
Suppose that the following conditions hold:\\
$(H_1)$ There exists constant $\sigma >2$ and two positive functions  $\varphi, \psi$ such that $\varphi \in L^1(0, +\infty), \psi \in L^{\infty}(0, +\infty)$   with
$$
|  f(t,x, \xi)|\leq \varphi(t)|x|^{\sigma}\psi(\xi), \;\mbox{ for a.e. } \,t\in[0,+\infty), x \in\mathbb{R}, \xi\in \mathbb{R}.
$$
$(I_0)$ There exist  positive constants $c_0$ and $\nu$  such that
$$\, |I_j(x)|\leq c_0|x|^{\nu}, \quad \forall x\in \mathbb{R}, \;j\in \left\lbrace 1, 2, \ldots  \right\rbrace .$$
Then, for each $w\in H^{1}_{0,p}(0,+\infty)$ fixed, the functional  $J_{w}: H^{1}_{0,p}(0,+\infty)\longrightarrow\mathbb{R}$ defined by
$$
J_{w}(u)=\frac{1}{2}\|u \|^{2}+\sum_{j=1}^{+\infty}g(t_{j})\int_{0}^{u(t_{j})}I_{j}(\tau)d\tau-\int_{0}^{+\infty}F(t,u(t), w'(t))dt,
$$
where $F(t, u, \xi) =\int_{0}^{u}f(t,s, \xi)ds,$ is continuous, differentiable and
\begin{equation}\label{04}
\begin{array}{rll}
(J_{w}'(u),v)&=& \displaystyle\int_{0}^{+\infty}p(t)u'(t)v'(t)dt+\displaystyle\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u(t_{j}))v(t_{j})\\
&&-\displaystyle\int_{0}^{+\infty}f(t,u(t), w'(t))v(t)dt,
\end{array}
\end{equation}
for all $v\in H^{1}_{0,p}(0,+\infty)$.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
\item[\it{Claim 1.}] {\it Let $w\in H^{1}_{0,p}(0,+\infty)$ fixed. Then $J_{w}$ is G\^{a}teaux-differentiable. }\\
Indeed, for all  $v\in H^{1}_{0,p}(0,+\infty)$, we have
\begin{eqnarray*}
J_{w}(u+hv)-J_{w}(u)&=& \frac{1}{2}\int^{+\infty}_{0}p(t)(u'(t)+hv'(t))^{2}dt\\
&+&  \sum_{j=1}^{+\infty}g(t_{j})\int_{0}^{u(t_{j})+hv(t_j)}I_{j}(\tau)d\tau \\
&-& \int_{0}^{+\infty}F(t,u(t)+hv(t),w'(t))dt\\
&-&\frac{1}{2}\int^{+\infty}_{0}p(t)u'^{\, 2}(t)dt- \sum_{j=1}^{+\infty}g(t_{j})\int_{0}^{u(t_{j})}I_{j}(\tau)d\tau \\
&+&\int_{0}^{+\infty}F(t,u(t),w'(t))dt\\
&=& h\int_{0}^{+\infty}p(t)u'(t)v'(t)dt + \frac{h^2}{2}\int_{0}^{+\infty}p(t)v'^{\, 2}(t)dt \\
&+& \sum_{j=1}^{+\infty}g(t_{j}) \left[ \int_{0}^{u(t_{j})+hv(t_j)}I_{j}(\tau)d\tau - \int_{0}^{u(t_{j})}I_{j}(\tau)d\tau \right] \\
&-& \int_{0}^{+\infty}\Big[F(t,u(t)+hv(t),w'(t))-F(t,u(t), w'(t)) \Big]dt
\end{eqnarray*}
\begin{eqnarray*}
J_{w}(u+hv)-J_{w}(u)&=& h\int_{0}^{+\infty}p(t)u'(t)v'(t)dt + \frac{h^2}{2}\int_{0}^{+\infty}p(t)v'^{\, 2}(t)dt\\
&+& h  \sum_{j=1}^{+\infty}g(t_{j})I_{j}(u(t_{j})+c_hv(t_j))v(t_j) \\
&-& h\int_{0}^{+\infty} f(t, u(t)+\theta_h v(t), w'(t))v(t)dt,
\end{eqnarray*}
where  $0 < \theta_h < 1$ and $0 < c_h < 1$ from the Mean Value Theorem. Thus
\begin{eqnarray*}
\frac{J_{w}(u+hv)-J_{w}(u)}{h} &=&\int_{0}^{+\infty}p(t)u'(t)v'(t)dt + \frac{h}{2}\int_{0}^{+\infty}p(t)v'^{\, 2}(t)dt\\
&+&   \sum_{j=1}^{+\infty}g(t_{j})I_{j}(u(t_{j})+c_hv(t_j))v(t_j) \\
&-& \int_{0}^{+\infty} f(t, u(t)+\theta_h v(t), w'(t))v(t)dt.
\end{eqnarray*}
By $(H_1)$, $(I_0)$ and the Lebesgue Dominated Convergence Theorem, we obtain
\begin{eqnarray*}
\lim_{h\rightarrow 0}\frac{J_{w}(u+hv)-J_{w}(u)}{h}&=&\int_{0}^{+\infty}p(t)u'(t)v'(t)dt + \sum_{j=1}^{+\infty}g(t_{j})I_{j}(u(t_{j}))v(t_j) \\
&&- \int_{0}^{+\infty} f(t, u(t), w'(t))v(t)dt,
\end{eqnarray*}
so that, $J_{w}$ is G\^{a}teaux-differentiable and
\begin{eqnarray*}
(J'_{w}(u), v )&=&\int_{0}^{+\infty}p(t)u'(t)v'(t)dt +
\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u(t_{j}))v(t_j)\\
&&- \int_{0}^{+\infty} f(t, u(t), w'(t))v(t)dt ,
\end{eqnarray*}
for all $v\in H^{1}_{0,p}(0,+\infty)$. Therefore a critical point of $J_{w}$ is a weak solution of Problem (\ref{02}).\\
%%%%%%%%%%%%%%%%%%%%%%
\item[\it{Claim 2.}] {\it $J'_{w}$ is continuous.}\\
Indeed, let $(u_n)$ be a sequence in $ H^{1}_{0,p}(0,+\infty)$ such that $u_n\longrightarrow u $ as $n\longrightarrow +\infty$.
From Lemma \ref{lemma1}, we have $(u_{n})$ converges uniformly to $u$ on $[0,+\infty)$ as $n\longrightarrow +\infty$.
Since  $f$  and $I_j$ are continuous, then $f(t, u_{n}(t), w'(t)) \longrightarrow  f(t, u(t), w'(t)), \quad  I_j(u_n(t_j))\longrightarrow I_j(u(t_j))$  as $n\longrightarrow +\infty$ and
 it follows from $(H_1)$ that
\begin{eqnarray*}
   \vert f(t, u_{n}(t), w'(t))\vert &\leq& \varphi(t)\vert u_{n}(t)\vert^{\sigma} \vert \psi(w'(t)) \vert\\
 &\leq& \varphi(t) \Vert u_{n}\Vert_{\infty}^{\sigma} |\psi(w'(t))| \\
   &\leq&  M_1^{\sigma}\varphi(t)\Vert u_{n}\Vert^{\sigma} |\psi(w'(t))|.
\end{eqnarray*}
And  by  $(I_0)$, we have
\begin{eqnarray*}
\vert I_j( u_{n}(t_j))\vert &\leq& c_0\vert u_{n}(t_j)\vert^{\nu} \\
 &\leq &  c_0 \Vert u_{n}\Vert_{\infty}^{\nu} \\
   &\leq &  M_1^{\nu}c_0\Vert u_{n}\Vert^{\nu}.
\end{eqnarray*}
Then from the Lebesgue Dominated Convergence Theorem, we obtain
\begin{equation*}
\lim_{n\to +\infty}\int_{0}^{+\infty} f(t, u_{n}(t),w'(t))dt = \int_{0}^{+\infty} f(t,u(t),w'(t))dt,
\end{equation*}
and
\begin{eqnarray*}
\lim_{n\to +\infty} \sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_n(t_{j}))= \sum_{j=1}^{+\infty}g(t_{j}) I_j(u(t_j)).
\end{eqnarray*}
So
\begin{eqnarray*}
( J'_{w}(u_n)-J'_{w}(u), v ) &=&
 \int_{0}^{+\infty}p(t)(u_n'(t)-u'(t))v'(t)dt\\
&+& \sum_{j=1}^{+\infty}g(t_{j})\Big[I_{j}(u_n(t_{j}))- I_j(u(t_j))\Big]v(t_j)\\
&-& \int_{0}^{+\infty} \Big[f(t,u_n(t),w'(t))-f(t,u(t),w'(t)) \Big] v(t)dt.
\end{eqnarray*}
Passing to the limit in $( J'_{w}(u_n)-J'_{w}(u), v )$ when $n \longrightarrow +\infty$, using assumptions $(H_1)$, $(I_0)$ and the Lebesgue Dominated Convergence Theorem, we obtain that
$ J'_{w}(u_n)\longrightarrow J'_{w}(u)$, as $n \longrightarrow +\infty$.\\
Consequently, $J_{w} \in C^1(H^{1}_{0,p}(0,+\infty) , \mathbb{R})$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main results}
\subsection{ Nontrivial weak solution}
\begin{theorem}\label{05}
Assume that $f$ satisfies $(H_1)$, $I_j$ satisfies $(I_0 )$ and the following  hypotheses:\\
$(H_2) \underset{x\rightarrow 0}{\lim} \frac{f(t, x, \xi)}{x}=0,$  uniformly in $ t \in [0,  +\infty)$ and $\xi \in \mathbb{R}$.\\
$(H_3)$ There exist positive functions $c_{1}, c_{2}\in L^{1}(0, +\infty),$  and $\mu>2$ such that
$$
\begin{array}{c}
\hspace{-1cm}(a)\,F(t,x, \xi )\geq c_{1}(t)|x|^{\mu}-c_{2}(t),\;  \;\mbox{for  a.e. } \,t\ge0,\; \mbox{ and all }\, x\in\mathbb{R}, \,  \xi \in \mathbb{R},\\\\
(b)\,\mu F(t,x, \xi)\leq x  f(t, x, \xi), \qquad  \;\mbox{for  a.e.  }\, t\ge0,\; \mbox{ and all }\, x\in\mathbb{R},\,  \xi \in \mathbb{R}.
\end{array}
$$
$(I_{1})$ There exists $0<\gamma\leq 2$ such that
$$
\gamma\int_{0}^{x}I_{j}(s)ds\geq xI_{j}(x)>0, \;\forall\,x\in\mathbb{R}\setminus\{0\},\; \forall\,j\in\{1,2,\ldots\}.
$$
Then  there exist  positive constants $d_{1}, d_{2}$  such that, for each $w\in H^{1}_{0,p}(0, +\infty)$, Problem (\ref{02}) has at least one nontrivial weak solution $u_{w}$ satisfying
$$d_{1}\leq \Vert u_{w} \Vert \leq d_{2}.$$
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
\item[\it{Claim 1.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. Then $J_{w}$ satisfies the (PS) condition.}\\
Indeed, let $(u_{n}) \subset H^{1}_{0,p}(0,+\infty) $  such that   $(J_w(u_{n}))$ is bounded and \\
$J_w'(u_{n}) \longrightarrow 0$ as $n \longrightarrow +\infty.$ Using $(H_3)(b)$ and $(I_1)$,  there exists some $d>0$ such that

%Then there exists a constant $d > 0$ such that
%$$
%|J_w(u_{n})|\leq d,\,\,\,\,\,  \|J_w'(u_{n})\| \leq d \mu, \,\,\,\,\,\forall n \in \mathbb{N}.
%$$

\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
  d &\geq & \mu J_{w}(u_{n}) -(J_{w}'(u_{n}), u_{n})\\
&\geq &(\frac{\mu}{2}-1)\|u_{n}\|^{2}\\
&&-\int_{0}^{+\infty}\Big(\mu F(t,u_{n}(t), w'(t))-f(t,u_{n}(t), w'(t))u_{n}(t)\Big)dt\\
&&+\sum_{j=1}^{+\infty}g(t_{j})\left(\mu\int_{0}^{u_{n}(t_{j})}I_{j}(\tau)d\tau -I_{j}(u_{n}(t_{j}))u_{n}(t_{j})\right)\\
&\geq&(\frac{\mu}{2}-1)\|u_{n} \|^{2}.
\end{eqnarray*}
Since $\mu >2$, it follows that $(u_{n})$ is bounded in $H^{1}_{0,p}(0,+\infty).$\\
Then there exists a subsequence of $(u_{n})$ still denoted $(u_{n})$ such that $(u_{n})$ converges weakly to some $u$ in $H^{1}_{0,p}(0,+\infty)$ because $(u_{n})$ is bounded in the reflexive Banach space $H^{1}_{0,p}(0,+\infty)$.  Lemma \ref{lemma1} implies that  $(u_{n})$ converges uniformly to $u$ on $[0,+\infty)$ . Thus
\begin{equation*}
\lim_{n\to+\infty}\sum_{j=1}^{+\infty}g(t_{j})\Big(I_{j}(u_{n}(t_{j}))-I_{j}(u(t_{j}))\Big)(u_{n}(t_{j})-u(t_{j}))=0
\end{equation*}
and
\begin{equation*}
\lim_{n\to+\infty}\int_{0}^{+\infty}\Big(f(t, u_{n}(t), w'(t))-f(t,u(t),w'(t))\Big)\left(u_{n}(t)-u(t)\right)dt=0.
\end{equation*}
Since $\lim_{n\to+\infty} J'(u_{n})=0$ and $(u_{n})$ converges weakly to some $u,$ we get
\begin{equation*}
\lim_{n\to+\infty}( J_w'(u_{n})-J_w'(u), u_{n}-u) =0.
\end{equation*}
From (\ref{04}), we have
\begin{eqnarray*}
( J_w'(u_{n})- J_w'(u),u_{n}-u)&=&\|u_{n}-u\|^{2}\\
&+&\sum_{j=1}^{+\infty}g(t_{j})\big( I_{j}(u_{n}(t_{j}))
-I_{j}(u(t_{j}))\big)(u_{n}(t_{j})- u(t_{j}))\\
&-&\int_{0}^{+\infty}\big(f(t,u_{n}(t),w'(t))-f(t,u(t), w'(t))\big)\left(u_{n}(t)-u(t)\right)dt.
\end{eqnarray*}
Hence $\lim_{n\to+\infty}\|u_{n}-u\|=0.$
Thus $(u_{n})$ converges strongly to $u$ in $H^{1}_{0,p}(0, +\infty).$
Consequently $J_{w}$ satisfies the $(PS)$ condition.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
\item[\it{Claim 2.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. Then there exist $\rho >0$ and $\alpha >0$, independent of $w$, such that $J_{w}(u)\geq \alpha , \quad \forall u \in H^{1}_{0,p}(0, +\infty), \Vert u\Vert =\rho$.}\\
Indeed, let $0<\varepsilon<\frac{1}{M}$. By $(H_{2})$, there exists $\delta>0$ such that
$$
|x|\leq\delta\, \Longrightarrow |f(t,x, \xi)|\leq\varepsilon |x|, \quad \forall t\in [0, +\infty), \xi \in \mathbb{R}.
$$
We have  $\Vert u\Vert^2_{L^2}\leq M\Vert u\Vert^2$ (see \cite{briki}) , so we deduce that
$$
\int_{0}^{+\infty}|F(t,u(t),w'(t))dt|\leq\frac{\varepsilon}{2}\Vert u\Vert^{2}_{L^2}\le\frac{\varepsilon}{2}M\Vert u\Vert^{2}, \;\mbox{ for a.e. }\, t\geq 0,
$$
whenever $\Vert u\Vert_\infty\le\delta$.\\
By choosing $0<\rho\le\frac{\delta}{M_1}$ and $\alpha=\frac{1}{2}(1-\varepsilon M)\rho^{2},$ hence for $\|u\|=\rho$ (note $\Vert u\Vert_\infty\le\delta$ ), we get
\begin{eqnarray*}
J_w(u)&=&\frac{1}{2}\|u\|^{2}+\sum_{j=1}^{+\infty}g(t_{j})\int_{0}^{u(t_{j})}I_{j}(\tau)d\tau-\int_{0}^{+\infty} F(t,u(t),w'(t))dt\\
&\geq&\frac{1}{2}\|u\|^{2}-\int_{0}^{+\infty} F(t,u(t),w'(t))dt\\
&\geq&\frac{1}{2}(1-\varepsilon M)\Vert u\Vert^{2}=\alpha.
\end{eqnarray*}
So there are $\rho > 0$ and $\alpha > 0$ such that $J_{w}(u) \geq \alpha, \ \ \forall u \in H^{1}_{0,p}(0, +\infty)$ with $\Vert u\Vert= \rho$.
%%%%%%%%%%%%%%%%%%%%%%
\item[\it{Claim 3.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. Then there exists $T_{0} >0$, independent of $w$, such that $J_{w}(\vartheta u^{\ast})\leq 0 , \quad \forall \vartheta\geq T_{0},$ where $u^{\ast} \in H^{1}_{0,p}(0, +\infty)$ with $\|u^{\ast}\| = 1$.}\\
Indeed, from $(I_{1}),$ there exists $c_3>0$ such that
$$
\int_{0}^{x}I_{j}(s)ds\leq c_3 |x|^{\gamma}, \;\mbox{ for every }\, x \in\mathbb{R}.
$$
Take an arbitrary $u^{\ast} \in H^{1}_{0,p}(0, +\infty)$ with $\|u^{\ast}\| = 1$ and using Lemma \ref{lemma1}, $(H_3)(a)$, we obtain
\begin{eqnarray*}
J_w(\vartheta u^{\ast})&=&\frac{1}{2}\vartheta^{2}\|u^{\ast}\|^{2}+\sum_{j=1}^{+\infty}g(t_{j})\int_{0}^{\vartheta u^{\ast}(t_{j})}I_{j}(\tau)d\tau-\int_{0}^{+\infty}F(t,\vartheta u^{\ast}(t), w'(t))dt\\
&\leq &\frac{1}{2}\vartheta^{2}+c_3 |\vartheta |^{\,\gamma} \Vert u^{\ast}\Vert^{\vartheta}_{\infty}\sum_{j=1}^{+\infty} g(t_{j})\\
&&-|\vartheta|^{\mu}\int_{0}^{+\infty}c_{1}(t)|u^{\ast}(t)|^{\mu}dt+\int_{0}^{+\infty}c_{2}(t)dt\\
&\leq &\frac{1}{2}\vartheta^{2}+c_3 |\vartheta |^{\,\gamma}M_1^{\,\gamma}\sum_{j=1}^{+\infty} g(t_{j})\\
&&-|\vartheta |^{\mu}\int_{0}^{+\infty}c_{1}(t)|u^{\ast}(t)|^{\mu}dt+\int_{0}^{+\infty}c_{2}(t)dt \leq 0,
\end{eqnarray*}
when $\vartheta \geq T_{0}$ for some  $T_{0}$ large, since $\mu>2\geq\gamma.$ \\
By  Proposition \ref{03}, the functional $j_w$ is in $C^1(H^{1}_{0,p}(0, +\infty), \mathbb{R})$.  Lemma \ref{lemma4} guarantees that $J_w$ possesses a critical point which is a weak solution of Problem (\ref{02}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[\it{Claim 4.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. Then there is a constant $d_{1}>0$, independent of $w$, such that $\Vert u_{w}\Vert \geq d_{1},$ for all solution $u_{w}$ obtained above.}\\
Indeed, let $u_{w}$ be a solution of Problem (\ref{02}). Then
\begin{eqnarray*}
\| u_w\|^2+\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_w(t_{j}))u_w(t_{j})=\int_{0}^{+\infty}f(t,u_w(t), w'(t))u_w(t)dt.
\end{eqnarray*}
It follows from $(H_1)$ and $(H_2)$ that,
\begin{eqnarray*}
|f(t, x, \xi)|\leq  \varepsilon |x|+  \varphi(t) |x|^{\sigma} \psi(\xi),\quad\mbox{for}\; t\in [0, +\infty), x\in \mathbb{R}, \xi\in \mathbb{R}.
\end{eqnarray*}
Then
\begin{eqnarray*}
\| u_w\|^2 &\leq& \| u_w\|^2+\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_w(t_{j}))u_w(t_{j})\\
&=& \int_{0}^{+\infty}f(t,u_w(t), w'(t))u_w(t)dt\\
&\leq& \varepsilon \int_{0}^{+\infty} |u_w(t)|^2 dt+  \int_{0}^{+\infty}\varphi(t)|u_w(t)|^{\sigma +1} \psi(w'(t))dt\\
&\leq & \varepsilon M \| u_w\|^2+  \Vert\varphi\Vert_{L^1}\; \Vert \psi\Vert_{L^{\infty}} \| u_w\|^{\sigma +1}_{\infty}\\
&\leq & \varepsilon M \| u_w\|^2+ M_1^{\sigma +1} \Vert\varphi\Vert_{L^1}\, \Vert \psi\Vert_{L^{\infty}} \| u_w\|^{\sigma +1},
\end{eqnarray*}
which implies that
\begin{eqnarray*}
(1- \varepsilon M)\| u_w\|^2 \leq M_1^{\sigma +1} \Vert\varphi\Vert_{L^1}\, \Vert \psi\Vert_{L^{\infty}} \| u_w\|^{\sigma +1}.
\end{eqnarray*}
Hence
\begin{eqnarray*}
\Vert u_{w}\Vert \geq d_{1}, \quad \mbox{for}\, \,  \mbox{some}\,\, d_{1}>0.
\end{eqnarray*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[\it{Claim 5.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. Then there is a constant $d_{2}>0$, independent of $w$, such that $\Vert u_{w}\Vert \leq d_{2},$ for all solution $u_{w}$ obtained above.}\\
Indeed, by the characterization of the critical point and $(H_3)$, it follows that \begin{eqnarray*}
\vert J_{w}(u_{w})\vert \leq \max_{\vartheta \in[0, +\infty)} J_{w}(\vartheta u^{\ast}),
\end{eqnarray*}
where $u^{\ast}$ is given in Claim $3$.\\
From $(H_3)(a)$, we get
\begin{eqnarray*}
\vert J_{w}(u_{w})\vert &\leq & \max_{\vartheta \in[0, +\infty)}\left\lbrace \frac{1}{2}\vartheta^{2}+c_3 |\vartheta |^{\,\gamma}M_1^{\,\gamma}\sum_{j=1}^{+\infty} g(t_{j})-|\vartheta |^{\mu}\int_{0}^{+\infty}c_{1}(t)|u^{\ast}(t)|^{\mu}dt \right.\\
& & \left. +\int_{0}^{+\infty}c_{2}(t)dt \right\rbrace.
\end{eqnarray*}
We define $K$ on $[0, +\infty)$ such that
\begin{eqnarray*}
K(\vartheta)=\frac{1}{2}\vartheta^{2}+c_3 |\vartheta |^{\,\gamma}M_1^{\,\gamma}\sum_{j=1}^{+\infty} g(t_{j})-|\vartheta |^{\mu}\int_{0}^{+\infty}c_{1}(t)|u^{\ast}(t)|^{\mu}dt+\int_{0}^{+\infty}c_{2}(t)dt ,
\end{eqnarray*}
and since $\mu >2$, $K(\vartheta)$ can achieve its maximum at some $\vartheta_{0}$.\\
Hence
\begin{eqnarray*}
\vert J_{w}(u_{w})\vert \leq K(\vartheta_{0}).
\end{eqnarray*}
On the other hand, we have
\begin{eqnarray*}
(1-\frac{2}{\mu}) \Vert u_{w}\Vert^{2}&=& 2J_{w}(u_{w})- \frac{2}{\mu}(J'_{w}(u_{w}), u_{w})\\
&+&2\int_{0}^{+\infty} \Big[F(t, u_{w}(t), w'(t))- \frac{u_{w}(t)}{\mu}f(t, u_{w}(t), w'(t))\Big]dt\\
&+& 2\sum_{j=1}^{+\infty} g(t_{j})\Big[ \frac{u_{w}(t_{j})}{\mu}I_{j}(u_{w}(t_{j}))-\int_{0}^{u_{w}(t_{j})}I_{j}(\tau)d\tau \Big] .
\end{eqnarray*}
Using $(H_3)(b), (I_1)$ and $(J'_{w}(u_{w}), u_{w})=0$, we obtain
\begin{eqnarray*}
(1-\frac{2}{\mu}) \Vert u_{w}\Vert^{2}\leq K(\vartheta_{0}).
\end{eqnarray*}
Hence
\begin{eqnarray}\label{12}
\nonumber \Vert u_{w}\Vert &\leq &\left( \dfrac{K(\vartheta_{0})}{1-\frac{2}{\mu}}\right) ^{\frac{1}{2}}\\
&\leq & d_{2},
\end{eqnarray}
we can choose $d_{2}= \left( \dfrac{K(\vartheta_{0})}{1-\frac{2}{\mu}}\right) ^{\frac{1}{2}},$ which is  independent of $w.$
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{13}
Assume hypotheses $(H_1)-(H_3), (I_0), (I_1)$ hold and \\
$(H_4)$ there exist positive constants $L_1$ and $L_2$ such that
\begin{eqnarray*}
\vert f(t, x, \xi)-f(t, y, \xi)\vert\leq L_1 \vert x-y\vert, \quad \forall t\in [0, +\infty), \, x,y\in [0; M_1d_2], \, \xi \in \mathbb{R},\\
\vert f(t, x, \xi)-f(t, x, \xi')\vert\leq L_2 \vert \xi-\xi'\vert, \quad \forall t\in [0, +\infty), \, x\in [0; M_1d_2], \, \xi, \xi' \in \mathbb{R},
\end{eqnarray*}
$(I_2)$ there exist   positive constants $\alpha_j$ such that
\begin{eqnarray*}
\vert I_j(x)-I_j(y) \vert \leq \alpha_j \vert x-y \vert, \quad \forall x,y \in  [0; M_1d_2],\,  j\in \{1,2,\ldots\}.
\end{eqnarray*}
Then Problem (\ref{01}) has at least one nontrivial weak solution provided  that $0< \frac{ L_2  M}{1-L_1 M-M^2_1 \sum_{j=1}^{+\infty}g(t_{j}) \alpha_j}<1$.
\end{theorem}
%%%%%%%ù%%%%%%%%ù%%%%%%%%%%%%%%
\begin{proof}
We construct a sequence $(u_n)\subset H^{1}_{0,p}(0, +\infty) $ as solutions of the problem
\begin{equation*}
(P_n)\left\{
\begin{array}{rlll}
-(p(t)u_n'(t))'&=&f(t,u_n(t), u_{n-1}'(t)), & \mbox{a.e. }\, t\ge0,\;t\not=t_{j}, \\
u_n(0)= u_n(+\infty)&=&0,& \\
\triangle (p(t_{j})u_n'(t_{j}))&=&g(t_{j})I_{j}(u_n(t_{j})), & j\in\{1,2,\ldots\},
\end{array}
\right.
\end{equation*}
given in Theorem \ref{05}, starting with an arbitrary $u_{0}\in H^{1}_{0,p}(0, +\infty).$\\
It follows from (\ref{12}) and Lemma \ref{lemma1} that
\begin{eqnarray*}
\Vert u_{n}\Vert_{\infty}\leq M_1d_2.
\end{eqnarray*}
Using $(P_{n+1})$ and $(P_n)$, we obtain
\begin{align*}
\int_{0}^{+\infty} p(t)u_{n+1}'(t)(u'_{n+1}(t)-u'_n(t))dt=-\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_{n+1}(t_{j}))(u_{n+1}(t_{j})-u_{n}(t_{j}))\\
+\int_{0}^{+\infty}f(t,u_{n+1}(t), u_{n}'(t))(u_{n+1}(t)-u_n(t))dt,	
\end{align*}
and
\begin{align*}
\int_{0}^{+\infty} p(t)u_n'(t)(u'_{n+1}(t)-u'_n(t))dt=-\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_n(t_{j}))(u_{n+1}(t_{j})-u_{n}(t_{j}))\\
+\int_{0}^{+\infty}f(t,u_n(t), u_{n-1}'(t))(u_{n+1}(t)-u_n(t))dt.
\end{align*}
By subtracting, we obtain
\begin{align*}
\Vert u_{n+1}-u_{n}\Vert^{2}= -\sum_{j=1}^{+\infty}g(t_{j})\Big[I_{j}(u_{n+1}(t_{j}))- I_{j}(u_n(t_{j}))\Big] (u_{n+1}(t_{j})-u_{n}(t_{j}))\\
+ \int_{0}^{+\infty}\Big[f(t, u_{n+1}(t),  u'_{n}(t))- f(t, u_{n}(t),  u'_{n-1}(t))\Big](u_{n+1}(t)-u_{n}(t))dt ,
\end{align*}
then
\begin{eqnarray*}
\Vert u_{n+1}-u_{n}\Vert^{2}&=& -\sum_{j=1}^{+\infty}g(t_{j})\Big[I_{j}(u_{n+1}(t_{j}))- I_{j}(u_n(t_{j}))\Big] (u_{n+1}(t_{j})-u_{n}(t_{j}))\\
&+& \int_{0}^{+\infty}\Big[f(t, u_{n+1}(t),  u'_{n}(t))-
f(t, u_{n}(t),  u'_{n}(t))\Big](u_{n+1}(t)-u_{n}(t))dt \\
&+& \int_{0}^{+\infty}\Big[f(t, u_{n}(t),  u'_{n}(t))-
f(t, u_{n}(t),  u'_{n-1}(t))\Big](u_{n+1}(t)-u_{n}(t))dt.
\end{eqnarray*}
By $(H_4)$ and $(I_2)$, we get
\begin{eqnarray*}
\Vert u_{n+1}-u_{n}\Vert^{2}&\leq & \sum_{j=1}^{+\infty}g(t_{j}) \alpha_j |u_{n+1}(t_j)-u_{n}(t_j)|^2\\
&&+L_1\int_{0}^{+\infty} \vert u_{n+1}(t)-u_{n}(t)\vert^{2}dt \\
&&+ L_2 \int_{0}^{+\infty} \vert u'_{n}(t)-u'_{n-1}(t) \vert \, \vert u_{n+1}(t)-u_{n}(t)\vert dt.
\end{eqnarray*}
Using the Cauchy-Schwarz inequality, we have
\begin{eqnarray*}
\Vert u_{n+1}-u_{n}\Vert^{2}&\leq & \Vert u_{n+1}-u_{n}\Vert^{2}_{\infty}\sum_{j=1}^{+\infty}g(t_{j}) \alpha_j +
L_1 \Vert u_{n+1}-u_{n}\Vert^{2}_{L^2}\\
&&+ L_2  \Vert u'_{n}-u'_{n-1}\Vert_{L^2} \, \Vert u_{n+1}-u_{n}\Vert_{L^2} \\
&\leq & M^2_1\Vert u_{n+1}-u_{n}\Vert^{2}\sum_{j=1}^{+\infty}g(t_{j}) \alpha_j  +L_1 M\Vert u_{n+1}-u_{n}\Vert^{2} \\
&&+ L_2  M\Vert u_{n}-u_{n-1}\Vert \, \Vert u_{n+1}-u_{n}\Vert ,
\end{eqnarray*}
which implies that
\begin{eqnarray*}
\Vert u_{n+1}-u_{n}\Vert &\leq & \frac{ L_2  M}{1-L_1 M-M^2_1 \sum_{j=1}^{+\infty}g(t_{j}) \alpha_j} \Vert u_{n}-u_{n-1}\Vert.
\end{eqnarray*}
Since $0<\frac{ L_2  M}{1-L_1 M-M^2_1 \sum_{j=1}^{+\infty}g(t_{j}) \alpha_j}<1$, it follows that $(u_{n})$ is a Cauchy sequence in the reflexive Banach space $ H^{1}_{0,p}(0, +\infty).$ Then  the sequence $(u_{n})$ strongly converges in $H^{1}_{0,p}(0, +\infty)$ to some $u\in H^{1}_{0,p}(0, +\infty)$. Since  $\Vert u_{n}\Vert \geq d_{1}, \,\, \forall n\in \mathbb{N}$, it follows  that $u\neq 0.$\\
Consequently, we obtain a nontrivial solution for Problem (\ref{01}).
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we prove  the existence of a solution  for the problem (\ref{01}) by using the Minimization principle.
\subsection{ The sublinear case}
\begin{theorem}\label{14}
Suppose that the following conditions hold:\\
$(H_{5})$ There exist a constant \,\,$\alpha\in[0,1)$ and  positive functions $a_{1}, b_{1}\in L^{1}(0,+\infty)$ such that
$$
|f(t,x, \xi)|\leq a_{1}(t)|x|^{\alpha}+b_{1}(t),\,\mbox{ for a.e. } \,t\in [0,+\infty)\mbox{ and all } \,x\in\mathbb{R}, \xi\in \mathbb{R} .
$$
$(I_{3})$ There exist constants $c_4>0$ and $\beta\in[0,1)$ such that
$$
|I_{j}(s)|\leq c_4|s|^{\beta}, \;\forall\, s\in\mathbb{R,}\; j\in\{1,2,\ldots\}.
$$
Then  there exists positive constant $d_{3}$  such that, for each $w\in H^{1}_{0, p}(0, +\infty)$, Problem (\ref{02}) has at least one  weak solution $u_{w}$ satisfying
$$\Vert u_{w} \Vert \leq d_{3}.$$
%Then problem (\ref{02}) has at least one nontrivial weak solution.
\end{theorem}
%%%%%%%%%%%%%%%
\begin{proof}
\item[\it{Claim 1.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. The functional $J_w$ is well defined.}\\
Indeed, take $u$ in $H^{1}_{0,p}(0, +\infty)$. From $(H_{5})$, we deduce that
$$
\left\vert F(t,u(t), w'(t))\right\vert\leq\frac{a_{1}(t)}{\alpha +1}|u(t)|^{\alpha +1}+b_{1}(t)|u(t)|.
$$
Thus, by using Lemma  \ref{lemma1}
\begin{eqnarray*}
\left\vert\int_{0}^{+\infty} F(t,u(t), w'(t))dt\right\vert
&\leq &\|u\|^{\alpha +1}_{\infty}\int_{0}^{+\infty}a_{1}(t)dt+\|u\|_{\infty}\int_{0}^{+\infty} b_{1}(t)dt\\
&\leq& \frac{M_1^{\alpha +1}}{\alpha +1}\|u\|^{\alpha +1}\int_{0}^{+\infty}a_{1}(t)dt+M_1\|u\|\int_{0}^{+\infty} b_{1}(t)dt\\
&\leq&\frac{M_1^{\alpha+1}}{\alpha +1}\|u\|^{\alpha +1}\Vert a_{1}\Vert_{L^{1}}+M_1\|u\|\Vert b_{1}\Vert_{L^{1}}.
\end{eqnarray*}
It follows from $(I_3)$ that
\begin{eqnarray*}
\left\vert \sum_{j=1}^{+\infty}g(t_{j})\int_{0}^{u(t_{j})}I_{j}(\tau)d\tau\right\vert
&\leq & \frac{c_4 }{\beta +1}\|u\|^{\,\beta +1}_{\infty}\sum_{j=1}^{+\infty}g(t_{j})\\
&\leq &\frac{c_4 M_1^{\,\beta +1}}{\beta +1}\|u\|^{\,\beta +1}\sum_{j=1}^{+\infty}g(t_{j}).
\end{eqnarray*}
Hence
\begin{eqnarray*}
|J_w(u)|&\leq&\frac{1}{2}\|u \|^{2}\,+ \,\frac{c_4 M_1^{\,\beta +1}}{\beta +1}\|u\|^{\,\beta+1}\sum_{j=1}^{+\infty}g(t_{j})\\
&&+\frac{M_1^{\,\alpha +1}}{\alpha +1}\|u\|^{\,\alpha +1}\Vert a_{1}\Vert_{L^{1}}\,+\, M_1\|u\|\Vert b_{1}\Vert_{L^{1}}\\
&<&\infty.
\end{eqnarray*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[\it{Claim 2.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. $J_w$ is sequentially weakly lower semicontinuous.}\\
Indeed, let $(u_{n})$ be a sequence in $H^{1}_{0,p}(0,+\infty)$  such that $u_{n}\rightharpoonup u$ in $H^{1}_{0,p}(0,+\infty)$, as $n\to\infty$. Lemma \ref{lemma1} implies that $(u_n)$ converges uniformly to $u$ on $[0,+\infty)$ and by the fact that the norm is weakly lower semicontinuous, we have
$$
\liminf\limits_{n\to+\infty}\|u_{n}\|\geq\|u\|.
$$
Using the Lebesgue Dominated Convergence Theorem  and the continuity of  the functions $f$ and $I_{j}, j\in\{1,2,\ldots\}$, we obtain
\begin{eqnarray*}
\liminf\limits_{n\to+\infty}J_w(u_{n})&=&\liminf\limits_{n\to+\infty}
\left(\frac{1}{2}\|u_{n}\|^{2}+\sum_{j=1}^{+\infty}g(t_{j})\int_{0}^{u_{n}(t_{j})}I_{j}(\tau)d\tau \right.\\
& & \left.-\int_{0}^{+\infty} F(t,u_{n}(t), w'(t))dt\right)\\
&\geq&\frac{1}{2}\|u\|^{2}+\sum_{j=1}^{+\infty}g(t_{j})\int_{0}^{u(t_{j})}I_{j}(\tau)d\tau-\int_{0}^{+\infty} F(t,u(t), w'(t))dt\\
&=&J(u).
\end{eqnarray*}
Consequently, $J_w$ is sequentially weakly lower semicontinuous.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item[\it{Claim 3.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. $J_w$ is coercive.}\\
Indeed, From $(H_5), (I_3)$ and Lemma \ref{lemma1}, we have
\begin{eqnarray}\label{15}
\nonumber J_w(u)&\geq &\frac{1}{2}\|u \|^{2}\,-\,\frac{c_4 M_1^{\,\beta +1}}{\beta +1}\|u\|^{\,\beta +1}\sum_{j=1}^{+\infty}g(t_{j})\\
& &-\frac{M_1^{\alpha +1}}{\alpha+1}\|u\|^{\alpha+1}\Vert a_{1}\Vert_{L^{1}}\,-\, M_1\|u\|\Vert b_{1}\Vert_{L^{1}}.
\end{eqnarray}
Since $\alpha<1$ and $\beta<1,$ then (\ref{15}) implies that
$$
\lim_{\|u\|\longrightarrow +\infty}J_w(u)=+\infty.
$$
So, by Lemma \ref{lemma3}, $J_w$ has a  minimum  point $u_w$. Under
hypothesis $(H_5)$ and using the same ideas as in Proposition \ref{03},
we get,  $J_w$ is G\^{a}teaux differentiable. Thus $u_w$ is a critical point of $J_w$. \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
\item[\it{Claim 4.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. Then $\Vert u_w\Vert \leq d_3$, for some $d_3 > 0$, for all solutions $u_w$ obtained above.}\\
Indeed, let $u_w$ be a solution of Problem (\ref{02}). Then
\begin{eqnarray*}
\Vert u_w\Vert^2 =\int_{0}^{+\infty} f(t,u_w(t),w'(t))u_w(t)dt-\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_w(t_{j}))u_w(t_{j}).
\end{eqnarray*}
By $(H_5)$ and $(I_3)$, we get
\begin{eqnarray*}
\Vert u_w\Vert^2 &\leq & \int_{0}^{+\infty} a_1(t)|u_w(t)|^{\alpha +1}dt + \int_{0}^{+\infty} b_1(t)|u_w(t)|dt\\
& & +c_4 \sum_{j=1}^{+\infty}g(t_j)|u_w(t_j)|^{\beta +1}\\
&\leq & \Vert u_w\Vert_{\infty}^{\alpha +1}\int_{0}^{+\infty} a_1(t)dt +\Vert u_w\Vert_{\infty} \int_{0}^{+\infty} b_1(t)dt+ c_4 \Vert u_w\Vert_{\infty}^{\beta +1}\sum_{j=1}^{+\infty}g(t_j)\\
&\leq & M_1^{\alpha +1}\Vert u_w\Vert^{\alpha +1} \Vert a_1\Vert_{L^1}+M_1\Vert u_w\Vert \Vert b_1\Vert_{L^1 }+ c_4 M_1^{\beta +1} \Vert u_w\Vert^{\beta +1}\sum_{j=1}^{+\infty}g(t_j).
\end{eqnarray*}
Hence
\begin{eqnarray*}
\Vert u_{w}\Vert \leq d_{3}, \quad \mbox{for}\, \,  \mbox{some}\,\, d_{3}>0.
\end{eqnarray*}
Therefor $u_w$ is a weak solution of Problem (\ref{02}).
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ùù
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
\begin{remark}
In addition, if $u_w\in H^{2}_{p}(t_j,t_{j+1})$, for all $j\in\{1,2,\ldots\}$, where
$$
H^{2}_{p}(t_j,t_{j+1})=\{u\in AC[0,+\infty),\mathbb{R}):\,\,\sqrt{p}u'\in L^{2}(t_j,t_{j+1}),\,(pu')'\in L^{2}(t_j,t_{j+1})\},
$$
then $u_w$ will be called a strong solution of Problem (\ref{02}).
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
In $(H_{5})$, assume that $a_{1}, b_{1}\in L^{2}(0,+\infty)$. Then every weak solution is a strong solution of Problem (\ref{02}).
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
We know that $u_w\in H^{1}_{0,p}(0, +\infty)$ is a critical point of $J_w.$ Then , for any $v\in H^{1}_{0,p}(0,+\infty)$, we have
\begin{eqnarray}\label{16}
\nonumber \displaystyle\int_{0}^{+\infty}p(t)u_w'(t)v'(t)dt&+&\displaystyle\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_w(t_{j}))v(t_{j})
\\
&-&\displaystyle\int_{0}^{+\infty}f(t,u_w(t), w'(t))v(t)dt=0.
\end{eqnarray}
For $j\in\{1,2,\ldots\},$ if $v\in H^{1}_{0,p}(t_j,t_{j+1})\;(v=v_j)$, then
$$
\int_{t_{j}}^{t_{j+1}} p(t)u_w'(t)v'(t)dt=\int_{t_{j}}^{t_{j+1}}f(t,u_w(t),w'(t))v(t)dt.
$$
So $u_{w,j}\in H^{1}_{0,p}(t_{j}, t_{j+1})$ is  a solution of the equation:
\begin{equation}\label{17}
-(p(t)u_w')'=f(t,u_w(t), w'(t)), \; t\in\,({t_{j},t_{j+1}}),
\end{equation}
Since, $u_w\in C_0[0, +\infty)$, and by $(H_{5})$, we get
$$\vert f(t,u_w(t), w'(t))\vert^2\le 2\left(a_1(t)^2\Vert u_w\Vert_\infty^{2\alpha}+b_1(t)^2\right),$$
thus $u_{w,j}\in H^{2}_{p}\;(t_j,t_{j+1})$.
Then (\ref{17}), implies that the limits \\
$u'(t^{+}_{j}), u'(t^{-}_{j}),\quad j\in\{1,2,\ldots\}$  exist.\\
Using the integration by parts in (\ref{16}), we obtain
\begin{eqnarray*}
0&=&-\sum_{j=0}^{j=+\infty}\int_{t_j}^{t_{j+1}}(p(t)u_w'(t))'v(t)dt-\sum_{j=1}^{+\infty}\triangle(p(t_{j})u_w'(t_{j}))v(t_{j})\\
& &+\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_w(t_{j}))v(t_{j})-\int_{0}^{+\infty}f(t,u_w(t), w'(t))v(t)dt.
\end{eqnarray*}
Since $u_w$ satisfies the equation in problem (\ref{02}) a.e. on $[0,+\infty),$ we deduce that
$$
\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_w(t_{j}))v(t_{j})=\sum_{j=1}^{+\infty}\triangle(p(t_{j})u_w'(t_{j}))v(t_{j}),\;\mbox{ for all}\, v\in H^1_{0,p}(0,+\infty).
$$
Thus
$$
\triangle(p(t_{j})u_w'(t_{j}))=g(t_{j})I_{j}(u_w(t_{j})), \,\mbox{ for every }\, j\in\{1,2,\ldots\}.
$$
Actually, $u_w$ is even a classical solution, i.e., $u\in C^{2}(t_j,t_{j+1})$, for all \\
$j\in\{1,2,\ldots\}$, when
$f: [0,+\infty)\times\mathbb{R}\times\mathbb{R}\longrightarrow\mathbb{R}$ is  continuous.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{18}
Assume that $(H_4), (H_5), (I_2)$ and $(I_3)$ hold.\\
Then Problem (\ref{01}) has at least one  classical solution provided  that $0< \frac{ L_2  M}{1-L_1 M-M^2_1 \sum_{j=1}^{+\infty}g(t_{j}) \alpha_j}<1$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
The proof is similar to the proof of Theorem \ref{13}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example}
Consider the  impulsive boundary value problem
\begin{equation}\label{Exp1}
\left\{
\begin{array}{rlll}
-(e^tu'(t))'&=&\frac{\sqrt{\vert u\vert}}{(1+t)^{2}}\cos u'+\frac{1}{(1+t)^{3}},&\mbox{a.e. }\, t\ge0,\;t\not=t_{j},\\
u(0)=u(+\infty)&=&0,&\\
\triangle(e^{j}u'(j))&=&\frac{\sqrt[3]{u(j)}}{1+j^{2}},& j\in\{1,2,\ldots\}.
\end{array}
\right.
\end{equation}
We know that all hypotheses of Theorem \ref{14} are satisfied with $f(t,x, \xi)=\frac{\sqrt{\vert x\vert}}{(1+t)^2}\cos \xi+\frac{1}{(1+t)^3}$, $\alpha=1/2$,
$a_1(t)=\frac{1}{(1+t)^2}$, $b_1(t)=\frac{1}{(1+t)^3}$, $I_j(s)=s^{1/3}$ $\beta=1/3$; $c_4=1$, $g(t)=\frac{1}{1+t^2}$, and $\Sigma_{j=1}^\infty g(j)=\frac{\pi}{4}$. Consequently, problem (\ref{Exp1}) has at least one solution.
\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
\subsection{ The limit case $\alpha=1$}
\begin{theorem}\label{19}
Suppose that $(I_{3})$ holds  and \\
$(H_{6})$ there exist positive functions $ a_{2}, b_{2}\in L^{1}(0,+\infty)$ with $||a_{2}||_{L^{1}}<\frac{1}{M_1^{\,2}}$
and
$$
|f(t,x, \xi)|\leq a_{2}(t)|x|+b_{2}(t), \;\mbox{ for a.e. } \,t\in[0,+\infty)\; \mbox{ and  }\,\forall x\in\mathbb{R}, \xi\in\mathbb{R}.
$$
Then  there exists positive constant $d_{4}$  such that, for each $w\in H^{1}_{0, p}(0, +\infty)$, Problem (\ref{02}) has at least one  weak solution $u_{w}$ satisfying
$$\Vert u_{w} \Vert \leq d_{4}.$$
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
\item[\it{Claim 1.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. $J_w$ is sequentially weakly lower semicontinuous.}\\
Indeed, we use the same technique as in the proof of Theorem \ref{14}.
%%%%%%%%%%%%%%%%%%%%%%
\item[\it{Claim 2.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. $J_w$ is coercive.}\\
Indeed, by $(H_{6})$, we obtain
$$
|F(t,u(t), w'(t))|\leq\frac{a_{2}(t)}{2}|u(t)|^{2}+b_{2}(t)|u(t)|,
$$
hence
$$
\begin{array}{rll}
\left\vert\int_{0}^{+\infty} F(t,u(t), w'(t))dt\right\vert&\leq&\int_{0}^{+\infty}\displaystyle\left(\frac{a_{2}(t)}{2}|u(t)|^{2}+b_{\,2}(t)|u(t)|\right)dt\\
&\leq&\displaystyle\frac{M_1^{\,2}}{2}\|u\|^{\,2}\|a_{\,2}\|_{L^{1}}+M_1\|u\|\|b_{\,2}\|_{L^{1}}.
\end{array}
$$
Thus
\begin{eqnarray}\label{20}
\nonumber J_w(u)&\geq &\frac{1}{2}\left(1-M_1^{\,2}\|a_{\,2}\|_{L^{1}}\right)\|u\|^{2}-M_1\|u\|\|b_{2}\|_{L^{1}}\\
&&-\frac{c_4 M_1^{\,\beta +1}}{\beta+1}\|u\|^{\,\beta +1}\sum_{j=1}^{+\infty}g(t_{j}).
\end{eqnarray}
Since $\|a_{2}\|_{L^{1}}<\frac{1}{M_1^{2}}$  and $\beta<1,$ we pass to the limit in (\ref{20}) when $n\rightarrow +\infty$, we get
$$
\lim_{\|u\|\longrightarrow +\infty}J_w(u)=+\infty.
$$
Therefore, $J_w$ is coercive.\\

By applying  Lemma \ref{lemma3}, we find that  $J_w$ has a  minimum  point $u_w$. Under hypothesis $(H_6)$ and using the same ideas as in Proposition \ref{03}, we get,  $J_w$ is G\^{a}teaux differentiable. Then $u_w$ is a critical point of $J_w$ which is a weak solution of Problem (\ref{02}). \\
%%%%%%%%%%%%%%%%%%%%%%%%
\item[\it{Claim 3.}] {\it Let $w\in H^{1}_{0,p}(0, +\infty)$ fixed. Then $\Vert u_w\Vert \leq d_4$, for some $d_4 > 0$, for all solutions $u_w$ obtained above.}\\
Indeed, let $u_w$ be a solution of Problem (\ref{02}). Then
\begin{eqnarray*}
\Vert u_w\Vert^2 =\int_{0}^{+\infty} f(t,u_w(t),w'(t))u_w(t)dt-\sum_{j=1}^{+\infty}g(t_{j})I_{j}(u_w(t_{j}))u_w(t_{j}).
\end{eqnarray*}
It follows from $(H_6)$ and $(I_3)$ that
\begin{eqnarray*}
\Vert u_w\Vert^2 &\leq & \int_{0}^{+\infty} a_2(t)|u_w(t)|^{2}dt + \int_{0}^{+\infty} b_2(t)|u_w(t)|dt\\
& & +c_4 \sum_{j=1}^{+\infty}g(t_j)|u_w(t_j)|^{\beta +1}\\
&\leq & \Vert u_w\Vert_{\infty}^{2}\int_{0}^{+\infty} a_2(t)dt +\Vert u_w\Vert_{\infty} \int_{0}^{+\infty} b_2(t)dt+ c_4 \Vert u_w\Vert_{\infty}^{\beta +1}\sum_{j=1}^{+\infty}g(t_j)\\
&\leq & M_1^{2}\Vert a_2\Vert_{L^1}\Vert u_w\Vert^{2} +M_1\Vert u_w\Vert \Vert b_2\Vert_{L^1 }+ c_4 M_1^{\beta +1} \Vert u_w\Vert^{\beta +1}\sum_{j=1}^{+\infty}g(t_j).
\end{eqnarray*}
Thus
\begin{eqnarray*}
(1-M_1^{2}\Vert a_2\Vert_{L^1})\Vert u_w\Vert^2 &\leq & M_1\Vert u_w\Vert \Vert b_2\Vert_{L^1 }+ c_4 M_1^{\beta +1} \Vert u_w\Vert^{\beta +1}\sum_{j=1}^{+\infty}g(t_j).
\end{eqnarray*}
Hence
\begin{eqnarray*}
\Vert u_{w}\Vert \leq d_{4}, \quad \mbox{for}\, \,  \mbox{some}\,\, d_{4}>0.
\end{eqnarray*}
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{21}
Assume that $(H_4), (H_6), (I_2)$ and $(I_3)$ hold.\\
Then Problem (\ref{01}) has at least one weak  solution provided  that\\ $0< \frac{ L_2  M}{1-L_1 M-M^2_1 \sum_{j=1}^{+\infty}g(t_{j}) \alpha_j}<1$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Reasoning like in the proof of Theorem \ref{13}, we can prove that Problem (\ref{01}) has at least one weak solution.
%The proof is similar to the proof of Theorem \ref{13} .
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addcontentsline{toc}{section}{References}

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\bibitem{chen} Chen H., Sun J., \emph{An application of variational method to second-order impulsive differential equation on the half-line},  Appl. Math. Comput, \textbf{217}(2010), 1863-1869.

\bibitem{fgm} De Figueiredo D., Girardi M., Matzeu M., \emph{Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques}, Differential Integral Equations. \textbf{17}(2004), 119-126.

\bibitem{rabinowitz2} Rabinowitz P.H., \emph{Minimax Methods in Critical Point Theory with Applications to Differential Equations}, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington DC; by the American Mathematical Society, Providence, RI, 1986.

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\end{document}
