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\title[Robin (p-q)-equations with singular and superlinear terms]{Robin (p-q)-equations with singular and superlinear terms}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author[N.S. Papageorgiou]{Nikolaos S. Papageorgiou}
\address{National Technical University, Department of Mathematics,
                Zografou Campus, Athens 15780, Greece}
\email{\tt npapg@@math.ntua.gr}

\author[V. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{University of Craiova, Department of Mathematics, Street A.I.Cuza 13,
        200585 Craiova, Romania \\
        and Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,
          014700 Bucharest, Romania}
\email{\tt vicentiu.radulescu@imar.ro}

\author[Dusan Repovs]{Dusan Repovs}
\address{Faculty of Education and Faculty of Mathematics and Physics, University of Ljublijiana, Karadeljeva Ploscad 16, SI-1000 Ljubljana, SLOVENIA}
\email{\tt dusan.repovs@guest.arnes.si}

\keywords{Nonhomogeneous differential operator, nonlinear regularity theory, truncations strong comparison, positive solutions\\
\phantom{aa} 2010 AMS Subject Classification: 35J75, 35J92, 35P30}


\begin{abstract}
We consider a nonlinear Robin problem driven by the sum of a $p$-Laplacian plus a $q$-Laplacian (a (p,q)-ecuation). In the reaction there are the competing effects of a singular term and of a parametric perturbation $\lambda f(z,x)$ which is Caratheodory and variational tools together with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parametre $\lambda>0$ varies.
\end{abstract}

\maketitle

\section{Introduction}

Let $\Omega\subseteq\RR^N$ be a bounded domain with a $C^2$-boundary $\partial\Omega$. In this paper, we study the following nonlinear Robin problem
\begin{equation}
    \left\{
        \begin{array}{ll}
            -\Delta_p u(z)-\Delta_q u(z) + \xi(z) u(z)^{p-1} = u(z)^{-\gamma} + \lambda f(z,u(z))\ \mbox{in}\ \Omega,\\
            \frac{\partial u}{\partial n_{pq}} + \beta(z) u^{p-1}=0\ \mbox{on}\ \partial\Omega, u>0, \lambda>0, 0<\gamma<1, 1<q<p.
        \end{array}
    \right\}\tag{$P_{\lambda}$}\label{eqp}
\end{equation}

For every $r\in (1,\infty)$ by $\Delta_r$ we denote the $r$-Laplace differential operator defined by
$$
\Delta_r u=div(|Du|^{r-2}Du)\ \mbox{for all}\ u\in W^{1,r}(\Omega).
$$

The differential operator of \eqref{eqp} is the sum of a $p$-Laplacian and of a $q$-Laplacian. Such an operator is not homogeneous and appears in the mathematical models of various physical processes. We mention the works of Cherfils-Ilyasov \cite{1} (reaction-diffusion systems) and Zhikov \cite{18} (elasticity theory). The potential function $\xi\in L^\infty(\Omega)$ and $\xi(z)\geq0$ for almost all $z\in\Omega$. In the reaction (right hand side of \eqref{eqp}), we have the combined effects of two nonlinearities of different nature. One nonlinearity, is the singular term $u^{-\gamma}$ and the other nonlinearity is the parametric term $\lambda f(z,x)$ where $f(z,x)$ is a Caratheodory function (that is, for all $x\in\RR\ z\to f(z,x)$ is measurable and for almost all $z\in\Omega\ x\rightarrow f(z,x)$ is continuous), which exhibits $(p-1)$-superlinear growth near $+\infty$ but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition (the AR-condition for short). In the boundary condition, $\frac{\partial u}{\partial n_{pq}}$ denotes the conormal derivative correspondin to the $(p,q)$-Laplace differential operator. Then according to the nonlinear Green's identity (see Gasinski-Papageorgiou \cite{2}, p.210), we have
$$
\frac{\partial u}{\partial n_{pq}} = (|Du|^{p-2}Du + |Du|^{q-2}Du,n)\ \mbox{for all}\ u\in C^1(\overline\Omega),
$$

with $n(\cdot)$ being the outward unit normal on $\partial\Omega$. The boundary coefficient $\beta\in C^{0,\alpha}(\partial\Omega)$ with $0<\alpha<1$ and $\beta(z)\geq0$ for all $z\in\partial\Omega$.

In the past nonlinear singular problems were studied only in the context of Dirichlet equations driven by the $p$-Laplacian (a homogeneous differential operator). We mention the works of Giacomoni-Schnidler-Taka\v c \cite{15}, Papageorgiou-Radulescu-Repovs \cite{10}, \cite{11}, Papageorgiou-Smyrlis \cite{13}, Papageorgiou-Winkert \cite{14}, Perera-Zhang \cite{16}. A comprehensive study of semilinear singular problems, can be found in the book of Gherghu-Radulescu \cite{4}.

Using variational methods based on the critical point theory together with suitable truncation and comparison techniques, we prove a bifurcation type result, describing in a precise way the dependence of the set of positive solutions of \eqref{eqp} on the parameter. So, we produce a critical parameter value $\lambda^*>0$ such that for all $\lambda\in(0,\lambda^*)$ problem \eqref{eqp} has at least two positive solutions, for $\lambda=\lambda^*$ problem \eqref{eqp} has at least one positive solution and for $\lambda>\lambda^*$ there are no positive solutions for problem \eqref{eqp}.

\section{Mathematical Background-Hypotheses}

Let $X$ be a Banach space. By $X^*$ we denote the topological dual of $X$. Given $\varphi\in C^1(X,\RR)$, we say that $\varphi(\cdot)$ satisfies the "C-condition", if the following property holds
$$
\begin{array}{ll}
\mbox{"Every sequence}\ \{u_n\}_{n\geq1}\subseteq X\ \mbox{such that} \\
\{\varphi(u_n)\}_{n\geq1}\subseteq\RR\ \mbox{is bounded} \\
\mbox{and}\ (1+||u_n||)\varphi'(u_n)\rightarrow0\ \mbox{in}\ X^*\ \mbox{as}\ n\rightarrow\infty,\\
\mbox{admits a strongly convergent subsequence"}
\end{array}
$$

This a compactness type condition on the functional $\varphi$ which leads to the minimax theory of the critical values of $\varphi(\cdot)$.

The two main spaces in the analysis of problem \eqref{eqp}, are the Sobolev space $W^{1,p}(\Omega)$ and the Banach space $C'(\overline\Omega)$. By $||\cdot||$ we denote the norm of the Sobolev space $W^{1,p}(\Omega)$. We have
$$
||u||=\left[||u||^p_p + ||Du||^p_p\right]^\frac{1}{p}\ \mbox{for all}\ u\in W^{1,p}(\Omega).
$$

The Banach space $C^1(\overline\Omega)$ is ordered with positive (order) cone given by
$$
C_+=\{u\in C^1(\overline\Omega):u(z)\geq0\ \mbox{for all}\ z\in\overline\Omega\}.
$$

This cone has a nonempty interior which contains the set
$$
D_+ = \{u\in C_+:u(z)>0\ \mbox{for all}\ z\in\overline\Omega\}.
$$

Note that $D_+$ is the interior of $C_+$ when the latter is endowed with the weaker $C(\overline{\Omega})$-norm topology.

To take care of the Robin boundary condition, we will also use the "boundary" Lebesgue spaces $L^q(\partial\Omega) (1\leq q\leq\infty)$. More precisely, on $\partial\Omega$ we consider the $(N-1)$-dimensional Hausdorff (surface) measure $\sigma(\cdot)$. Using this measure on $\partial\Omega$ we can define in the usual way the Lebesgue spaces $L^q(\partial\Omega) (1\leq q\leq\infty)$. We know that there exists a continuous, linear map $\gamma_0 W^{1,p}(\Omega)\rightarrow L^p(\partial\Omega)$, known as the "trace map" such that
$$
\gamma_0(u)=u|_{\partial\Omega}\ \mbox{for all}\ u\in W^{1,p}(\Omega)\cap C(\overline\Omega).
$$

So, the trace map extends the notion of boundary values to all Sobolev functions. We have
$$
im\gamma_0= W^{\frac{1}{p},p}(\partial\Omega)(\frac{1}{p}+\frac{1}{p'}=1)\ \mbox{and}\ ker\gamma_0 = W^{1,p}_0(\Omega).
$$

The trace map $\gamma_0$ is compact into $L^q(\partial\Omega)$ for all $q\in \left[1,\frac{(N-1)p}{N-p}\right)$ if $N>p$ and into $L^q(\partial\Omega)$ for all $q\geq1$ if $p\geq N$. In the sequel for the sake of notational simplicity, we drop the use of the trace map $\gamma_0(\cdot)$. All restrictions of Sobolev functions on $\partial\Omega$ are understood in the sense of traces.

For every $r\in(1,+\infty)$ let $A_r:W^{1,r}(\Omega)\rightarrow W^{1,r}(\Omega)^*$ be defined by
$$
\langle A_r(u),h\rangle = \int_\Omega|Du|^{r-2}(Du,Dh)_{\RR^N}dz\ \mbox{for all}\ u,h\in W^{1,p}(\Omega).
$$

The following proposition sumarizes the main properties of this map (see Gasinski-Papageorgiou \cite{2}).

\begin{prop}\label{prop1}
    The map $A_r(\cdot)$ is bounded (that is, maps bounded sets to bounded sets) continuous, monotone (hence maximal monotone too) and of type $(S)_+$, that is, if $u_n\xrightarrow{w}u$ in $W^{1,r}(\Omega)$ and $\limsup_{n\rightarrow\infty}\langle A_r(u_n),u_n-u\rangle$, then
    $$
    u_n\rightarrow u\ \mbox{in}\ W^{1,r}(\Omega).
    $$
\end{prop}

Evidently the $(S)_+$-property is useful in verifying the C-condition.

Now we introduce the conditions on the potential function $\xi(\cdot)$ and on the boundary coefficient $\beta(\cdot)$.

$H(\xi)$: $\xi\in L^\infty(\Omega)$ and $\xi(z)\geq0$ for almost all $z\in\Omega$.

$H(\beta)$: $\beta\in C^{0,\alpha}(\partial\Omega)$ with $0<\alpha<1$ and $\beta(z)\geq0$ for all $z\in\partial\Omega$.

$H_0$: $\xi\not\equiv0$ or $\beta\not\equiv0$.

\begin{remark}\label{rem1}
    When $\beta\equiv0$ we have the usual Neumann problem.
\end{remark}

The next two propositions can be found in Papageorgiou-Radulescu \cite{9}.

\begin{prop}\label{prop2}
    If $\xi\in L^\infty(\Omega)$, $\xi(z)\geq0$ for almost all $z\in\Omega$ and $\xi\not\equiv0$, then $c_0||u||^p\leq ||Du||^p_p + \int_\Omega \xi(z)|u|^pdz$ for some $c_0>0$, all $u\in W^{1,p}(\Omega)$.
\end{prop}

\begin{prop}\label{prop3}
    If $\beta\in L^\infty(\partial\Omega), \beta(z)\geq0$ for $\sigma$\mbox{-}almost all $z\in\partial\Omega$ and $\beta\not\equiv0$, then $c_1||u||^p\leq ||Du||^p_p + \int_{\partial\Omega}\beta(z)|u|^pd\sigma$ for some $c_1>0$, all $u\in W^{1,p}(\Omega)$.
\end{prop}

In what follows let $\gamma_p:W^{1,p}(\Omega)\rightarrow\RR$ be defined by
$$
\gamma_p(u) = ||Du||^p_p + \int_\Omega\xi(z)|u|^pdz + \int_{\partial\Omega}\beta(z)|u|^pd\sigma\ \mbox{for all}\ u\in W^{1,p}(\Omega).
$$

In hypotheses $H(\xi), H(\beta), H_0$ hold, then from Propositions \ref{prop2} and \ref{prop3} we infer that
\begin{equation}\label{eq1}
    c_2||u||^p \leq \gamma_p(u)\ \mbox{for some}\ c_2>0,\ \mbox{all}\ u\in W^{1,p}(\Omega).
\end{equation}

As we already mentioned in the Introduction, our approach involves also truncation and comparison techniques. So, the next strong comparison principle, a slight variant of Proposition 4 of Papageorgiou-Smyrlis \cite{13}, will be useful.

\begin{prop}\label{prop4} If $\hat\xi\in L^\infty(\Omega)$ with $\hat\xi(z)\geq0$ for almost all $z\in\Omega, h_1, h_2\in L^\infty(\Omega)$,
$$
0<c_3\leq h_2(z)-h_1(z)\ \mbox{for almost all}\ z\in\Omega,
$$
and the functions $u_1,u_2\in C^1(\overline\Omega)\backslash\{0\}, u_1\leq u_2, u_1^{-\gamma}, u_2^{-\gamma}\in L^\infty(\Omega)$ satisfy
$$
\begin{array}{ll}
-\Delta_p u_1 - \Delta_q u_1 + \hat\xi(z) u_1^{p-1} - u_1^{-\gamma}=h_1\ \mbox{for almost all}\ z\in\Omega,\\
-\Delta_p u_2 - \Delta_q     u_2 + \hat\xi(z) u_2^{p-1} - u_2^{-\gamma}=h_2\ \mbox{for almost all}\ z\in\Omega.
\end{array}
$$
then $u_2-u_1\in intC_+$.
\end{prop}

Consider a Caratheodory function $f_0:\Omega\times\RR\rightarrow\RR$ satisfying
$$
|f_0(z,x)|\leq a_0(z)[1+|x|^{r-1}]\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\in\RR,
$$
with $a_0\in L^\infty(\Omega)$ and $1<r\leq p^*=\left\{\begin{array}{ll}\frac{Np}{N-p}&\mbox{if}\ p<N\\+\infty &\mbox{if}\ N\leq p\end{array}\right.$ (the critical Sobolev exponent corresponding to $p$). We set $F_0(z,x)=\int^x_0 f_0(z,s)ds$ and consider the $C^1$-functional $\varphi_0:W^{1,p}(\Omega)\rightarrow\RR$ defined by
$$
\varphi_0(u)=\frac{1}{p}\gamma_p(u) + \frac{1}{q}||Du||^q_q - \int_\Omega F_0(z,u)dz\ \mbox{for all}\ u\in W^{1,p}(\Omega)\ \mbox{(recall $q<p$)}
$$

The next proposition can be found in Papageorgiou-Radulescu \cite{8} and essentially is an outgrowth of the nonlinear regularity theory of Lieberman \cite{6}.

\begin{prop}\label{prop5}
    If $u_0\in W^{1,p}(\Omega)$ is a local $C^1(\overline\Omega)$-minimizer of $\varphi_0$, that is, there exists $\rho_0>0$ such that
    $$
    \varphi_0(u_0)\leq\varphi_0(u_0+h)\ \mbox{for all}\ ||h||_{C^1(\overline\Omega)}\leq\rho_0,
    $$
    then $u_0\in C^{1,\alpha}(\overline\Omega)$ for some $\alpha\in(0,1)$ and $u_0$ is also a local $W^{1,p}(\Omega)$-minimizer of $\varphi_0$, that is, there exists $\rho_1>0$ such that
    $$
    \varphi_0(u_0)\leq\varphi_0(u+h)\ \mbox{for all}\ ||h||\leq\rho_1.
    $$

    The next fact about ordered Banach spaces, is useful in producing upper bounds for functions and can be found in Gasinski-Papageorgiou \cite{3} (Problem 4.180, p.680).
\end{prop}

\begin{prop}\label{prop6}
    If $X$ is an ordered Banach space with positive (order) cone $K$,
    $$
    int K\neq\varnothing\ \mbox{and}\ e\in int K
    $$
    then for every $u\in K$ we can find $\lambda_u>0$ such that $\lambda_u e-u\in K$.
\end{prop}

Under hypotheses $H(\xi), H(\beta), H_0$, the differential operator $\Delta u\rightarrow-\Delta_p u + \xi(z)|u|^{p-2}u$ with the Robin boundary condition, has a principal eigenvalue $\hat\lambda_1(p)>0$ which is isolated, simple and admits the following variational characterization
$$
\hat\lambda_1(p)=inf\left[\frac{\gamma_p(u)}{||u||^p_p}:u\in W^{1,p}(\Omega),u\neq0\right].
$$

The infimum is realized on the corresponding one-dimensional eigenspace, the elements of which have fixed sign. By $\hat{u}_1(p)$ we denote the positive, $L^p$-normalized (that is, $||\hat{u}_1(p)||_p=1$) eigenfunction corresponding to $\hat\lambda_1(p)>0$. The nonlinear Hopf's theorem (see, for example, Gasinski-Papageorgiou \cite{2}, p.738), we have $\hat{u}_1(p)\in D_+$.

Let us fix some basic notation which we will use throughout this work. So, if $x\in\RR$, we set $x^\pm=\max\{\pm x,0\}$ and the for $u\in W^{1,p}(\Omega)$ we define $u^\pm(z)=u(z)^\pm$ for all $z\in\Omega$. We know that
$$
u^\pm\in W^{1,p}(\Omega), u=u^+-u^-, |u|=u^++u^-.
$$

If $\varphi\in C^1(W^{1,p}(\Omega),\RR)$, then by $K_\varphi$ we denote the critical set of $\varphi$, that is,
$$
K_\varphi = \{u\in W^{1,p}(\Omega):\varphi'(u)=0\}.
$$

Also, if $u,y\in W^{1,p}(\Omega)$, with $u\leq y$, then we define
$$
\begin{array}{ll}
    [u,y]=\{h\in W^{1,p}(\Omega): u(z)\leq h(z)\leq y(z)\ \mbox{for almost all}\ z\in\Omega\},
    [u) = \{h\in W^{1,p}(\Omega): u(z)\leq h(z)\ \mbox{for almost all}\ z\in\Omega\},
\end{array}
$$
$int_{C^1(\overline\Omega)}[u,y]$ the interior in the $C^1(\overline\Omega)$-norm of $[u,y]\cap C^1(\overline\Omega)$.

Now we introduce our hypotheses on the perturbation $f(z,x)$.

$H(f)$: $f:\Omega\times\RR\rightarrow\RR$ is a Caratheodory function such that $f(z,0)=0$ for almost all $z\in\Omega$ and
\begin{itemize}
    \item [(i)] $f(z,x)\leq a(z)[1+x^{r-1}]$ for almost all $z\in\Omega$, all $x\geq0$ with $a\in L^\infty(\Omega), p<r<p^*$;
    \item [(ii)] if $F(z,x)=\int_0^x f(z,s)ds$, then $\lim_{x\rightarrow+\infty}\frac{F(z,x)}{x^p}=+\infty$ uniformly for almost all $z\in\Omega$;
    \item [(iii)] there exists $\tau\in((r-p)\max\{\frac{N}{p},1\},p^*)$ such that
        $$
        0<\hat\beta_0\leq\liminf_{x\rightarrow+\infty}\frac{f(z,x)x-p F(z,x)}{x^\tau}\ \mbox{uniformly for almost all}\ z\in\Omega;
        $$
    \item [(iv)] for every $\vartheta>0$, there exists $m_\vartheta>0$ such that for almost all
\end{itemize}


\begin{equation}
    0 \leq \lambda f(z,\bar{u}(z)) \leq 1 \mbox{ for a.a } z\in \Omega, \mbox{ all } 0 < \lambda \leq \lambda_0.
    \label{17}
    \end{equation}

    We consider the following truncation of the reaction in problem $(p_\Lambda)$


    \begin{equation}
    \partial_\lambda(z,x) = \left\{
     \begin{array}{lr}
                    v(z)^{-\gamma} + \lambda f(z,v(z)) \hspace{5.5mm} \mbox{ if } x < v(z) \\
                    x^{-\gamma} + \lambda f(z,x) \hspace{15mm}\mbox{ if } v(z) \leq x \leq \bar{u}(z) \\
                    \bar{u}(z)^{-\gamma} + \lambda f(z,\bar{u}(z)) \hspace{5.2mm} \mbox{ if } \bar{u}(z) < x.
     \end{array}
    \right.
    \label{18}
    \end{equation}

    This is a Caratheodory function. We set $\theta_\lambda(z,x) = \int_{0}^x \partial_\lambda (z,s) ds$ and consider the functional $\mu\lambda : W^{1,p}(\Omega) \to R\,\, (\lambda \in (0,\lambda_0])$ defined by
    $$\mu_\lambda(u) = \frac{1}{p} \gamma_p (u) + \frac{1}{q} ||D u||_q^q - \int_{\Omega} \theta_\lambda(z,u)dz \mbox{ for all }u\in W^{1,p}(\Omega)$$

    Since $0\leq\bar{u}^{-\gamma} \leq v^{-\gamma} \in L^{\infty}(\Omega)$, we see that $\mu_\lambda \in C'(W^{1,p}(\Omega))$. Also, it is clear from ~\eqref{18} and ~\eqref{1}, that $\mu_\lambda(\cdot)$ is coercive. In addition, it is sequentially weakly lower semicontinuous. So, we can find $u_\lambda \in W^{1,p}(\Omega)$ such that


     $$\mu_\lambda(u_\lambda) = inf \Big[ \mu_\lambda(u): u\in W^{1,p}(\Omega) \Big],$$
     $$\Rightarrow \mu_\lambda^{'} (u_\lambda) = 0,$$
     $$\Rightarrow \langle A_p(u_\lambda),h\rangle + \langle A_q(u_\lambda),h \rangle + \int_{\Omega}\xi(z) u_\lambda^{' p-2} u_\lambda hdz + \int_{\partial\Omega} \beta(z) |u_\lambda|^{p-2} u_\lambda hdo$$
    \begin{equation}
    = \int_{\Omega} \partial_\lambda (z,u_\lambda) hdz \mbox{ for all } h\in W^{1,p}(\Omega).
    \label{19}
    \end{equation}

    In ~\eqref{19} first we choose $h=(u_\lambda -\bar{u})^+ \in W^{1,p}(\Omega)$.Then

    $$\mbox{ In \eqref{19} first we choose } h = (u_\lambda -\bar{u})^+ \in W^{1,p}(\Omega). \mbox{ Then } $$

    $$ \langle A_n(u_\lambda),(u_\lambda - \bar{u})^+\rangle + \langle A_q(u_\lambda),(u_\lambda-\bar{u})^+\rangle + \int_{\Omega}\xi (z) u_\lambda^{p+} (u_\lambda -\bar{u})^{+} dz +  \int_{\partial\Omega} \beta(z) u_\lambda^{p-1} (u_\lambda - \bar{u}) do  $$

    $$ =\int_{\Omega} [\bar{u}^{-\gamma} + \lambda f(z,\bar{u})](u_\lambda -\bar{u})^+ dz \mbox{ (see ~\eqref{18})) } $$

    $$ \leq \int_{\Omega} [\bar{u}^{-\gamma} +1](u_\lambda -\bar{u})^+ dz \mbox{ (see ~\eqref{17}) } $$

    $$ \leq \int_{\Omega} [v^{-\gamma} + 1](u_\lambda -\bar{u})^+ dz \mbox{ (since } v\leq\bar{u}) $$

    $$ = \langle A_p(\bar{u},(u_\lambda -\bar{u}))^+ \langle + \langle A_q(\bar{u}),(u_\lambda-\bar{u})^+ \rangle + \int_{\Omega} \xi (z) \bar{u}^{p-1} (u_\lambda -\bar{u})^+ dz$$

    $$ + \int_{\partial\Omega} \beta(z) \bar{u}^{p-1} (u_\lambda -\bar{u})^+ do \mbox{ (see Proposition 9),} $$

    $$ \Rightarrow u_\lambda \leq \bar{u}. $$

    Next in ~\eqref{19} we choose $h=(v-u_\lambda)^+ \in W^{1,p}(\Omega).$ Then

    %pag 21 , este () sau || la u
    $$ \langle A_p(u_\lambda),(v-u_\lambda)^+\rangle + \langle A_q(u_\lambda),(v-u_\lambda)^+ \rangle + \int_{\Omega} \xi (z) |u_\lambda|^{p-2} u_\lambda (v-u_\lambda)^+ dz + \int_{\partial\Omega} \beta(z) |u_\lambda|^{p-2} u_\lambda(v-u_\lambda)^+ do $$

    $$ =\int_{\Omega} [v^{-\gamma} + \lambda f(z,v) ] (v-u_\lambda)^+ dz \mbox{ (see ~\eqref{18})} $$

    $$ \geq \int_{\Omega} v^{-\gamma} (v-u_\lambda)^+ dz \mbox{ (since } f\geq 0) $$

    $$ =\,\,\langle A_p(v),(v-u_\lambda)^+\rangle + \langle A_q(v),(v-u_\lambda)^+\rangle + \int_{\lambda} \xi (z) v^{p-1} (v-u_\lambda)^+ dz $$
    $$+ \int_{\partial\Omega}\beta (z) v^{p-1} (v-u_\lambda)^+ do \mbox{ (see Proposition 8),}$$

    $$ \Rightarrow v \leq u_\lambda. $$

    So, we have proved that

    \begin{equation}
    u_\lambda \in [v,\bar{u}].
    \label{20}
    \end{equation}

    From ~\eqref{18},~\eqref{19},~\eqref{20} it follows that

    \begin{equation}
    \left\{
     \begin{array}{lr}
                -\Delta_p u_\lambda(z) -\Delta_q u_\lambda(z) + \xi(z) u_\lambda (z)^{p-1} = u_\lambda (z)^{-\gamma} + \lambda f (z,u_\lambda(z)) \mbox{ for a.a } z\in \Omega, \\
                    \frac{\partial u_\lambda}{\partial n_{pq}} + \beta(z) u_\lambda^{p-1} = 0 \mbox{ on } \partial\Omega,
     \end{array}
    \right\}
    \label{21}
    \end{equation}

    $$ \mbox{ (see [7])}. $$

    From ~\eqref{21} and Proposition 7 of Papageorgiou-Radulescu [8], we have that $u_\lambda\in L^{\infty}(\Omega).$ So, the nonlinear regularity theory of Lieberman [6] implies that $u_\lambda \in D_+$ (see ~\eqref{20}). Therefore we have proved that

    $$ (0,\lambda_0] \leq L \neq \varnothing \mbox{ and } S_\lambda \subseteq D_+.$$

    \begin{flushright}
    \underline{\underline{QED}}
    \end{flushright}

    Next we establish a lower bound for the elements of $S_\lambda$

    \underline{Proposition 11}: \underline{If} hypotheses $H(\xi),H(\beta),H_0,H(f)$ hold, $\lambda\in L$ and $u\in S_\lambda$, \underline{then} $v\leq u$.
    %small sau smau sau sman

    \underline{Proof}: From Proposition 10 we know that $u\in D_+$. Then Proposition 7 implies that for $\eta >0$ small we have $\tilde{u}_\eta \leq u.$ So, we can define the following Caratheodory function

    \begin{equation}
    e(z,x) = \left\{
     \begin{array}{lr}
                \tilde{u}_\eta (z)^{-\gamma} \hspace{3.3mm} \mbox{ if } x<\tilde{u}_\eta (z)\\
                x^{-\gamma} \hspace{10mm}\mbox{ if } \tilde{u}_\eta(z) \leq x \leq u(z) \\
                u(z)^{-\gamma} \hspace{5mm}\mbox { if } u(z) <x
     \end{array}
    \right.
    \label{22}
    \end{equation}

    We set $E(z,x)= \int_{0}^x e(z,s) ds$ and consider the functional $d:W^{1,p}(\Omega) \to R$ defined by

    $$ d(u) = \frac{1}{p} \gamma_p(u) + \frac{1}{q} ||D u||_q^q - \int_\lambda E(z,u) dz \mbox{ for all } u\in W^{1,p}(\Omega). $$

    As before we have $d\in C'(W^{1,p}(\Omega))$. Also, $d(\cdot)$ is coercive (see ~\eqref{22}) and weakly lower semicontinuous. Hence we can find $\hat{v} \in W^{1,p}(\Omega) $ such that

    $$ d(\hat{u})= inf [d(u): u\in W^{1,p}(\Omega)], $$

    $$ \Rightarrow d'(\hat{v}) = 0,$$

    \begin{equation}
    \begin{aligned}
    \Rightarrow \langle A_p(\hat{v}),h\rangle + \langle A_q(\hat{v}),h\rangle + \int_{\Omega} \xi (z) |\hat{v}|^{p-2} \hat{v} hdz \\
    + \int_{\partial\Omega} \beta(z) |\hat{v}|^{p-2} \hat{v} hdo = \int_{\Omega} e(z,\hat{v}) hdz \mbox { for all } h\in W_{1,p}(\Omega).
    \end{aligned}
    \label{23}
    \end{equation}

    In ~\eqref{23} first we choose $h=(\hat{v}-u)^+ \in W^{1,p}(\Omega).$ Exploiting the fact that $u\in S_\lambda$ and recalling that $f\geq 0$, we obtain $\hat{v} \leq u$. Next in ~\eqref{23} we test with $h=(\tilde{u}_\eta -v)^+\in W^{1,p}(\Omega)$

    Using ~\eqref{22},~\eqref{9} and Proposition 7, we otain $\tilde{u}_\eta \leq \hat{v}$. Therefore

    \begin{equation}
    \hat{v} \in [\tilde{u}_\eta,u].
    \label{24}
    \end{equation}

    From ~\eqref{22},~\eqref{23},~\eqref{24} and Proposition 8, we conclude that a

    $$ \hat{v} = v, $$

    $$ \Rightarrow v \leq u \mbox{ for all } u\in S_\lambda. $$

    \begin{flushright}
    \underline{\underline{QED}}
    \end{flushright}

    Now we can have a structural property of $L$

    \underline{Proposition 12}: \underline{If} hypotheses $H(\xi),H(\beta),H_0,H(f)$ hold, $\lambda \in L$, $0<\mu < \lambda$ and $u_\lambda \in S_\lambda \subseteq D_+,$ \underline{then} $\mu \in L$ and we can find $u_\mu \in S_\mu \subseteq D_+$ such that $u_\lambda -u_\mu \in \mbox{ int }C_+$.

    \underline{Proof}: From Proposition 11 we know that $v \leq u_\lambda.$ Then we can define the following Caratheodory function

    \begin{equation}
    \hat{k}_\mu(z,x) = \left\{
     \begin{array}{lr}
                x(z)^{-\gamma} + \mu f(z,v(z)) \hspace{9.8mm} \mbox{ if } x< v(z)\\
                x^{-\gamma} + \mu f (z,x) \hspace{19.5mm} \mbox{ if } v(z) \leq x \leq u_\lambda (z) \\
                u_\lambda(z)^{-\gamma} + \mu f(z,u_\lambda(z)) \hspace{6mm} \mbox{ if } u_\lambda(z) < x.
     \end{array}
    \right.
    \label{25}
    \end{equation}

    We set $\hat{k}_\mu(z,x) = \int_{0}^{x} \hat{k}_\mu (z,s) ds$ and consider the C'-functional $\hat{\psi}_\mu:W^{1,p}(\Omega) \to R$ defined by

    $$ \hat{\psi}_\mu (u) = \frac{1}{p} \gamma_p(u) + \frac{1}{q} || D u ||_q^q - \int_{\Omega} \hat{k}_\mu (z,u) dz \mbox{ for all } u \in W^{1,p}(\Omega). $$

    Evidently $\hat{\psi}_\mu(\cdot)$ is coercive (see ~\eqref{25}) and sequentially weakly lower semicontinuous. So, we can find $u_\mu \in W^{1,p}(\Omega)$ such that

    $$ \hat{\psi}_\mu(u_\mu) = \mbox{ inf } \Big[ \hat{\psi}_\mu(u): u\in W^{1,p}(\Omega) \Big] ,$$

    $$ \Rightarrow \hat{\psi}_\mu (u_\mu) = 0, $$
    $$ \Rightarrow \langle A_p(u_\mu),h> + <A_q(u_\mu),h\rangle + \int_{\Omega} \xi(z) |u_\mu|^{p-2} u_\mu hdz + \int_{\partial\Omega} B(z) |u_\mu|^{p-2} u_\mu hdo $$

    \begin{equation}
    = \int_{\Omega} \hat{k}_\mu (z,u\mu) hdz \mbox{ for all } n \in W^{1,p}(\Omega).
    \label{26}
    \end{equation}

    In ~\eqref{26} first we choose $h = (u\mu - u_\lambda)^+ \in W^{1,p}(\Omega).$ Using ~\eqref{25}, the fact that $\mu < \lambda$ and that $f \geq 0$ and recalling that $u_\lambda \in S_\lambda$, we conclude that $u_\mu \leq u_\lambda$. Next in ~\eqref{26} we choose $h=(v-u\mu)^+ \in W^{1,p}(\Omega).$ From ~\eqref{25}, the fact that $f \geq 0$ and Proposition 8, we infer that $v \leq u_\mu$. Therefore we have proved that

    \begin{equation}
    u_\mu \in [v,u_\lambda]
    \label{27}
    \end{equation}

    From ~\eqref{25},~\eqref{26},~\eqref{27} it follows that

    $$ u_\mu \in S_\mu \subseteq D_+ \mbox{ (see Proposition 10).}$$

    Let $p= ||u_\lambda||_{\infty} $ and let $\hat{\xi}_{p}^\lambda > 0$ be as postulated by hypothesis $H(f)(\underline{v})$. We have

    \begin{equation}
    \begin{aligned}
    &-\Delta_p u_\lambda (z) - \Delta_q u_\mu (z) + \Big[\xi(z) + \hat{\xi}_p^{\lambda}\Big] u_\mu(z)^{p-1} - u_\mu(z)^{-\gamma}\\
    &= \mu f(z,u_\mu(z)) + \hat{\xi}_p^{\lambda} u_\mu(z)^{p-1}\\
    &=\lambda f(z,u_\mu(z)) + \hat{\xi}_p^{\lambda} u_\mu (z)^{p-1} - (\lambda - \mu) f(z,u_\mu (z)) \\
    &< \lambda f (z,u_\mu(z)) + \hat{\xi}_p^{\lambda} u_\lambda (z)^{p-1} \mbox{ (recall } \lambda > \mu ) \\
    &\leq \lambda f (z,u_\mu(z)) + \hat{\xi}_p^{\lambda} u_\mu(z)^{p-1} \mbox{ (see ~\eqref{27} and hypothesis } H(f)(v)) \\
    &= -\Delta_p u_\lambda (z) - \Delta_q u_\lambda (z) + \Big[ \xi(z) + \hat{\xi}_p^{\lambda} \Big] u_\lambda(z)^{p-1} - u_\lambda(z)^{-\lambda} \mbox{ for a.a } z \in \lambda \\
    &\mbox{ (recall }u_\lambda \in S_\lambda)
    \end{aligned}
    \label{28}
    \end{equation}

    We know that

    $$ 0 \leq u_\lambda^{-\gamma},u_\lambda^{-\gamma} \leq v^{-\gamma} \in L^{\infty}(\Omega) $$

    Also, from hypothesis $H(f)(\underline{iv})$ and since $u_\mu \in D_+,$ we have

    $$ 0 < c_8 \leq (\lambda - \mu)f(z,u_\mu(z)) \mbox{ for a.a } z\in \Omega $$


    Invoking Proposition 4, from ~\eqref{28} we conclude that

    $$ u_\lambda - u_\mu \in \mbox{int} C_+. $$

    \begin{flushright}
    \underline{\underline{QED}}
    \end{flushright}

    \underline{Proposition 13}: \underline{If} hypotheses $H(E),H(\beta),H_0,H(f)$ hold, \underline{then} $\lambda^* < + \infty$

    \underline{Proof}: On account of hypotheses $H(f)(\underline{i}) \to (\underline{iv})$, we can find $\lambda_0 >0$ big such that

    \begin{equation}
    x^{-\gamma} + \lambda_0 f(z,x) \geq x^{p-1} \mbox{ for a.a } z\in \Omega , \mbox{ all } x \geq 0.
    \label{29}
    \end{equation}

    Let $\lambda > \lambda_0$ and suppose that $\lambda + L$. Then we can find $u_\lambda \in S_\lambda \subseteq D_+$ (see Proposition 10). Then $m_\lambda = \min_{\bar{\Omega}} u_\lambda >0$. For $\delta\in (0,1)$ we set $m_\lambda^{\delta} = m_\lambda + \delta$ and for $p = ||u_\lambda||_{\infty}$ let $\hat{\xi}_p^{\lambda} > 0$ be as postulated by hypothesis $H(f)(\underline{v})$. We have

    \begin{equation}
    \begin{aligned}
    & -\Delta_p m_\lambda^\delta - \Delta_q m_\lambda^{\delta} + [\xi(z) + \hat{\xi}_p](m_\lambda^{\delta})^{p-1} - (m_\lambda^{\delta})^{-\gamma} \\
    & =[\xi(z) + \hat{\xi}_p^{\lambda}]m_{\lambda}^{p-1} - m_\lambda^{-\gamma} + \chi(\delta) \mbox{ with } \chi(\delta) \to 0^{+} \mbox{ as } \delta \to 0^+ \\
    & < \xi(z) m_\lambda^{p-1} + (1+ \hat{\xi}_p^{\lambda}) m_\lambda^{p-1} - m_\lambda^{-\gamma} + \chi(\delta)\\
    & \leq \lambda_0 f(z,m_\lambda) + [\xi(z) + \hat{\xi}_p^{\lambda}] m_\lambda^{p-1} + \chi(\delta) \mbox{ (see ~\eqref{29})} \\
    & \leq \lambda_0 f(z,u_\lambda) + [\xi(z) + \hat{\xi}_p^{\lambda}] u_\lambda^{p-1} + \chi(\delta) \mbox { (see hypothesis } H(f)(\underline{v})) \\
    % small sau smau iara ????
    & = \lambda f(z,u_\lambda) + [\xi(z) + \hat{\xi}_p^{\lambda}] u_\lambda^{p-1} -(\lambda -\lambda_0) f(z,u_\lambda) + \chi(\delta)\\
    & = \lambda f(z,u_\lambda) + [\xi(z) + \hat{\xi}_p^{\lambda}] u_\lambda^{p-1} \mbox { for } \delta \in (0,1) \mbox { small } \\
    &\quad \mbox{ (recall } u_\lambda \in D_+ \mbox{ and see } H(f)(\underline{iv})) \\
    & = -\Delta_p u_\lambda - \Delta_q u_\lambda + [\xi(z) + \hat{\xi}_p^{\lambda}] u_\lambda^{p-1} - u_\lambda^{-\gamma}
    \end{aligned}
    \label{30}
    \end{equation}
    %? iara pagina 26 wtf small ? sau ce
    Since $(\lambda - \lambda_0)f(z,u_\lambda) - \chi(\delta) \geq c_9 > 0$ for a.a $z \in \Omega$ and for $d\in(0,1)$ small (just recall that $u_\lambda \in D_+$ and use hypothesis H(f)(\underline{iv}), invoking Proposition 4, from ~\eqref{30} we infer that

    $$ u_\lambda - m_\lambda^{\delta} \in \mbox{ int } C_+ \mbox{ for all }\delta \in (0,1) \mbox{ small}.$$

    But this contradicts the definition of $m_\lambda$.

    It follows that $\lambda \notin L$ and so $\lambda^* \leq \lambda_0 < +\infty$

    \begin{flushright}
    \underline{\underline{QED}}
    \end{flushright}

    Therefore we have

    $$ (0,\lambda^*) \leq L \leq (0,\lambda*] $$

    \underline{Proposition 14}: \underline{If} hypotheses $H(\xi),H(\beta),H_0,H(f)$ hold and $\lambda \in (0,\lambda^*)$,
    \underline{then} problem $(p_\lambda)$ has at least two positive solutions

    $$ u_0, \hat{u} \in D_+, u_0 \neq \hat{u}. $$

    \underline{Proof}: Let $0<\mu<\lambda<\eta <\lambda^*$. According to Proposition 12, we can find $u_\eta \in S_\eta \subseteq D_+, u_0\in S_\lambda \subseteq D_+$ and $u_\mu \in S_\mu \subseteq D_+ $ such that

    \begin{equation}
    \begin{aligned}
    & u_\eta - u_0 \in \mbox { int } C_+  \mbox{ and } u_0 -u_\mu \in \mbox{ int } C_+,\\
    & \Rightarrow u_0 \in \mbox{ int }_{C' (\hat{\Omega})} [u_\mu,u_\eta].
    \end{aligned}
    \label{31}
    \end{equation}

    We introduce the following Caratheodory function

    \begin{equation}
    \tilde{T}_\lambda (z,x) = \left\{
     \begin{array}{lr}
                u_\mu(z)^{-\gamma} + \lambda f (z,u_\mu(z)) \hspace{5.9mm} \mbox{ if } x < u_\mu(z) \\
                x^{-\gamma} + \lambda f (z,x) \hspace{20mm} \mbox{ if } u_\mu(z) \leq x \leq u_\eta(z) \\
                u_\eta(z)^{-\gamma} + \lambda f (z,u_\eta(z)) \hspace{6.7mm} \mbox{ if } u_\eta(z) <x.
     \end{array}
    \right.
    \label{32}
    \end{equation}

    Set $\tilde{T}_\lambda(z,x) = \int_{0}^{x} \tilde{T}_\lambda (z,s)ds$ and consider the C'-functional $\tilde{\psi}_\lambda: W^{1,p}(\Omega) \to R$ defined by

    $$ \tilde{\psi}_\lambda(u) = \frac{1}{p}\gamma_p(u) + \frac{1}{q} ||D u||_q^q - \int_{\lambda}\tilde{T}_\lambda (z,u)dz \mbox{ for all } u\in W^{1,p}(\Omega) $$

    Using ~\eqref{32} and the nonlinear regularity theory, we can easily check that

    \begin{equation}
    K_{\tilde{\psi}_\lambda} \leq [u_\mu,u_\eta] \cap D_+.
    \label{33}
    \end{equation}

    Also, consider the Caratheodory function

    \begin{equation}
    \tau_\lambda^* (z,x) =  \left\{
     \begin{array}{lr}
                u_\mu(z)^{-\gamma} + \lambda f (z,u_\mu(z)) \hspace{5.9mm} \mbox{ if } x \leq u_\mu (z)\\
                x^{-\gamma} + \lambda f(z,x) \hspace{20mm} \mbox{ if } u_\mu (z) < x.
     \end{array}
    \right.
    \label{34}
    \end{equation}

    We set $T_\lambda^* (z,x) = \int_{0}^{x} \tau_\lambda^* (z,s)ds$ and consider the C'-functional $\psi_\lambda^*: W^{1,p}(\Omega) \to R $ defined by

    $$ \psi_\lambda^*(u) = \frac{1}{p} \gamma_p(u) + \frac{1}{q}||D u||_q^q - \int_{\Omega} T_\lambda^* (z,u) dz \mbox{ for all } u\in W^{1,p}(\Omega). $$

    For this functional using ~\eqref{34}, we show that

    \begin{equation}
    K_{\psi_\lambda^*} \leq [u_\mu) \cap D_+.
    \label{35}
    \end{equation}

    From ~\eqref{32} and ~\eqref{34} we see that

    \begin{equation}
    \tilde{\psi}_\lambda \Big|_{[u_\mu,u_\eta]} = \psi_\lambda^{*} \Big|_{[u_\mu,u_\eta]} \,\,\mbox{ and }\,\, \tilde{\psi}_\lambda^{'}\Big|_{[u_\mu,u_\eta]} = (\psi_\lambda^*)' \Big|_{[u_\mu,u_\lambda]}.
    \label{36}
    \end{equation}

    From ~\eqref{33},~\eqref{35},~\eqref{36}, it follows that without any loss of generality, we may assume that

    \begin{equation}
    K_{\psi_\lambda^*} \cap [u_\mu,u_\eta] = \{u_0\}.
    \label{37}
    \end{equation}

    Otherwise it is clear from ~\eqref{34} and ~\eqref{35} that we already have a second positive smooth solution for problem $(p_\lambda)$ and so we are done.

    Note that $\tilde{\psi_\lambda}(\cdot)$ is coercive (see ~\eqref{32}). Also, it is sequentially weakly lower semicontinuous. So, we can find $\hat{u}_0 \in W^{1,p}(\Omega)$ such that

    \begin{equation}
    \begin{aligned}
    &\tilde{\psi}_\lambda (\hat{u}_0) = \mbox{ inf } \Big[ \tilde{\psi}_\lambda (u): u\in W^{1,p}(\Omega)\Big],\\
    &\Rightarrow \hat{u}_0 \in K_{\tilde{\psi}_\lambda}, \\
    &\Rightarrow \hat{u}_0 \in K_{\psi_\lambda^*} \cap [u_\mu,u_\eta] \mbox{ (see ~\eqref{33},~\eqref{36}) }, \\
    &\Rightarrow \hat{u}_0 = u_0 \in D_+ \mbox{ (see ~\eqref{37}}), \\
    &\Rightarrow u_0 \mbox{ is a local C'} (\bar{\Omega})\mbox{-minimizer of } \psi_\lambda^* \mbox{ (see ~\eqref{31})},\\
    &\Rightarrow u_0 \mbox{ is a local } W^{1,p}(\Omega)\mbox{-minimizer of }\psi_\lambda^* \mbox{ (see proposition 5).}
    \end{aligned}
    \label{38}
    \end{equation}

    We assume that $K_{\psi_\lambda^*}$ is finite. Otherwise on account of ~\eqref{34} and ~\eqref{35} we see that we already have an infinity of positive smooth solutions for problem $(p_\lambda)$ and so we are done. Then ~\eqref{38} implies that we can find $\rho\in(0,1)$ small. such that

    \begin{equation}
    \begin{aligned}
    &\psi_\lambda^*(u_0)< \mbox{ inf } \Big[ \psi_\lambda^*(u): ||u-u_0|| = \rho \Big] = m_\lambda^* \\
    &\mbox{(see Papageorgiou-Radulescu-Repovs [12], Theorem 5.7.6,p.367).}
    \end{aligned}
    \label{39}
    \end{equation}

    On account of hypothesis $H(f)(\underline{ii})$ we have

    \begin{equation}
    \psi_\lambda^* (t\hat{u}_1 (p)) \to -\infty \mbox{ as } t\to +\infty.
    \label{40}
    \end{equation}
    \\
    %% DE AICI am scris primele 10 pagini , si s-a verificat cam vreo 2-4 pagini , trebuie revizuit unde ma oprisem
    \underline{Claim:} $\psi_\lambda^*(\cdot)$ satisfies the C - condition.
    $$ \mbox{Let } \{ u_n \}_{n \geq 1} \,\,\leq \mbox{W}^{1,p}(\Omega) \mbox{ be a sequence such that}  $$
    \begin{equation} |\psi_\lambda^* (u_n) |\leq c_{10} \mbox{ for some }  c_{10} > 0 , \mbox{ all } n \in N,
    \label{41}
    \end{equation}

    \begin{equation} (1 + ||u_n|| ) (\psi_\lambda^*)' (u_n) \to 0 \mbox{ in W }^{1,p}(\Omega)^*.
    \label{42}
    \end{equation}

    From ~\eqref{42} we have
    % pe aici
    \begin{equation}
    \begin{aligned}
    &| \langle A_p(u_n),h\rangle + \langle A_q(u_n),h\rangle + \int_{\Omega} \xi(z) |u_n|^{p-2}u_n h \, dz + \int_{\partial\Omega} \beta(z) |u_n|^{p-2} u_n h do\\
    & - \int_{\Omega} \tau_\lambda^*(z, u_n) h \,dz ) \leq \frac{\epsilon_n ||h||}{1 + ||u_n||} \mbox{ for all } h \in W^{1,p}, \mbox{ with } \epsilon_n \rightarrow 0^+.
    \end{aligned}
    \label{43}
    \end{equation}

    Choosing  $h= -u_n^{-} \in W^{1,p}(\Omega)$, we obtain
     % si aici
    \begin{equation*}
    \begin{split}
     \gamma_p(u_n^{-}) + ||D u_n^{-} ||_q^q \leq c_{11} ||u_n^{-} || \mbox{ for some } c_{11} > 0 , \mbox{ all } n \in N \mbox{ (see ~\eqref{34})}
    \end{split}
    \end{equation*}




    \begin{equation}
    \begin{split}
    \Rightarrow \{u_n^{-} \}_{n \geq 1} \subseteq W^{1,p}(\Omega) \mbox{ is bounded } \mbox{ (see ~\eqref{1} and recall }1<p)
    \end{split}
    \label{44}
    \end{equation}

    Next in ~\eqref{43} we choose $ h = u_n^+ \in W^{1,p}(\Omega)$. Then

    %de la mbox unde trebuie pus
    \begin{equation}
    \begin{aligned}
    &-\gamma_p (u_n^{+} - || Du_n^+ ||_q^q + \int_{\Omega} \tau_\lambda^* (z,u_n) u_n^+ dz \leq \epsilon_n \mbox{ for all } n \in N,\\
    &\Rightarrow -\gamma_p(u_n^+) - ||Du_n^+||_q^q + \int_{\{ u_n \leq u_{\mu} \}} [u_\mu^{-\gamma} + \lambda f(z,u_\mu)] u_n^{+}dz \\
    &+ \int_{\{ u_\mu < u_n \}} [u_n^{-\gamma}+\lambda f(z,u_n)]u_n^+ dz  \, \leq \, \epsilon_n \mbox{ for all } n\in N \mbox{ (see ~\eqref{34})}
    \end{aligned}
    \label{45}
    \end{equation}

    On the other hand from ~\eqref{41} and ~\eqref{44}, we have

    $$ \gamma_p(u_n^+) + \frac{p}{q} || D_u^+ ||_q^q - \int_{ \{u_n\leq u_\mu\}} p[u_\mu - \gamma + \lambda f(z,u_p) ] u_n^+ \, dz $$

    \begin{equation*}
    \begin{aligned}
     &- \int_{\{u_\mu < u_n \}} \bigg[\frac{p}{1-\gamma} (u_n^{1-\gamma} - u_\mu^{1-\gamma}) + p(\lambda F (z,u_n) - \lambda F(z,u_\mu) \bigg] dz \leq \epsilon n \\
    &\quad \mbox{for all } n \in N (see ~\eqref{34})
    \end{aligned}
    \end{equation*}

    \begin{equation}
    \begin{aligned}
     &\Rightarrow \gamma_p(u_n^+) + \frac{p}{q} ||D u_n^+ ||_p^p - \int_{ \{ u_n \leq u_\mu \}} p [u_\mu^{-\gamma} + \lambda f(z,u_\mu)] u_n^+ dz \\
    &- \int_{ \{ u_p < u_n \}} \bigg[ \frac{p}{1-\gamma} u_n^{1-\gamma} + \lambda p F(z,u_n)] dz \leq c_{12}
     \mbox{ for some } c_{12} > 0, \mbox{ all } n \in N.
    \end{aligned}
    \label{46}
    \end{equation}

    We add ~\eqref{45} and ~\eqref{46}. Since $p > q$ , we obtain

    \begin{equation*}
    \begin{aligned}
     \lambda \int_{ \{ u_\mu < u_n \} } [f(z,u_n)u_n^+ - pF(z,u_n) ] dz \leq \,\, (p-1) \int_{ \{ u_n \leq u_\mu \}} [u_\mu^{-\gamma} + \lambda f(z,u_\mu] u_n^+ dz \\
    +\bigg(\frac{p}{1-\gamma} - 1 \bigg) \int_{\{ u_\mu < u_n \}} u_n^{1-\gamma} dz
    \end{aligned}
    \end{equation*}
    \begin{equation}
    \begin{split}
    \Rightarrow \lambda \int_{\Omega} [f(z,u_n^+)u_n^+ - p F (z,u_n^+)] dz \,\, \leq \,\, c_{13} \,\big[|| u_n^+ ||_{1} + 1\big] \mbox{ for some } c_{13}>0, \mbox{ all }  n \in N.
    \end{split}
    \label{47}
    \end{equation}

    % such that?
    On account of hypotheses $ H(f)(\underline{i}),(\underline{iii})$ we can find $\hat{\beta}_1 \in (0,\hat{\beta}_0) $ and $c_{14} > 0$ such that

    \begin{equation}
    \hat{\beta}_1 x^\tau - c_{14} \leq f(z,x) - p F(z,x) \mbox{ for a.a } z \in \Omega, \mbox{ all } x \geq 0.
    \label{48}
    \end{equation}

    Using ~\eqref{48} in ~\eqref{47}, we obtain

    %pe aici
    $$ ||u_n^+||_\tau^\tau \leq c_{15} \big[ ||u_n^+||_\tau + 1 \big] \mbox{ for some } c_{15} > 0, \mbox{ all } n\in N, $$

    \begin{equation}
    \Rightarrow \{u_n^+\}_{n \geq 1} \leq L^\tau (\Omega) \mbox{ is bounded}.
    \label{49}
    \end{equation}

    First assume N $\neq $ p . From hypothesis $H(f) (\underline{iii})$ it is clear that we may assume without any loss of generality that $ \tau < r < p^*. $ Let $t\in(0,1)$ be such that

    $$ \frac{1}{r} = \frac{1-t}{\tau} + \frac{t}{p*} $$

    Then from the interpolation inequality (see Papageorgiou - Winkert[15], Proposition 2.3.17,p.116), we have

    $$ || u_n^+ ||_r \leq || u_n^+ ||_\tau^{1-t} ||u_n^+||_{p^*}^{t}, $$

    \begin{equation}
    ||u_n^+||_r^r \leq c_{16} ||u_n^+||^{tr} \mbox{ for some } c_{16} > 0, \mbox{ all } n \in N \mbox{ (see ~\eqref{49})}.
    \label{50}
    \end{equation}

    From hypothesis $H(f)(\underline{i})$ we have
    % all sau a.a?
    \begin{equation}
    f(z,x) x \leq c_{17} [1+ x^r] \mbox{ for all }z \in \Omega , \mbox{ all } x \geq 0, \mbox{ some } c_{17} > 0.
    \label{51}
    \end{equation}

    From ~\eqref{43} with $h=u_n^+ \in W^{1,p} (\Omega)$, we obtain

    \begin{equation}
    \begin{aligned}
    & \gamma_p (u_n^+) + || D u_n^+ ||_q^q - \int_{\Omega} \tau_\lambda^* (z,u_n) u_n^+ dz \leq \epsilon_n \mbox{ for all   } n\in N,\\
    &\Rightarrow \gamma_p (u_n^+) + || D u_n^+ ||_q^q \leq \int_{\Omega} [(u_n^+)^{1-\gamma} + f(z,u_n^+) u_n^+] dz + c_{18} \\
    &\quad\quad\quad\quad\quad\quad\quad\quad\mbox{ for some } c_{18} > 0, \mbox{ all } n \in N \mbox{ (see ~\eqref{34}) } \\
    &\leq c_{19} \big[ 1 + || u_n^+||_r^r \big] \mbox{ for some } c_{19} > 0 , \mbox{ all } n\in N \mbox{ (see ~\eqref{51})}\\
    &\leq c_{20} [1 + ||u_n^+||^{tr}] \mbox{ for some } c_{20} > 0 , \mbox{ all } n\in N \mbox { (see ~\eqref{50})}
    \end{aligned}
    \label{52}
    \end{equation}

    The hypothesis on $\tau$ (see $H(f)(\underline{iii})) $ implies that $tr < p.$ So, from ~\eqref{52} we infer that

    $$ \{ u_n^+ \}_{n \geq 1} \subseteq W^{1,p}(\Omega) \mbox{ is bounded}, $$

    \begin{equation}
    \Rightarrow \{ u_n \}_{n \geq 1} \subseteq W^{1,p}(\Omega)\mbox{ is bounded (see ~\eqref{44})}.
    \label{53}
    \end{equation}
    \setlength{\parindent}{10ex}

    If $N = p,$ then $p^* = + \infty $ and from the Sobolev embedding theorem, we know that $W^{1,p}(\Omega) \hookrightarrow L^s(\Omega)$ for all $1\leq s < \infty$. Then in order for the previous argument to work, we replace $p^* = + \infty$ by $s > r > \tau$ and let $t\in (0,1)$ as before such that

    $$ \frac{1}{r} = \frac{1-t}{\tau} + \frac{t}{s}, $$

    $$ \Rightarrow tr = \frac{s(r-\tau)}{s-\tau}. $$

    Note that $ \frac{s(r-\tau)}{s-\tau} \rightarrow r -\tau $ as $s \to + \infty$. But $r-\tau <p$ (see hypothesis H(f)(iii)). We choose $s>r$ big so that $tr<p$. Then again we have ~\eqref{53}.

    Because of ~\eqref{53} and by passing to a subsequence if neccesary,we may assume that
    \begin{equation}
    u_n \stackrel{w}{\rightarrow} u \mbox{ in } W^{1,p}(\Omega) \mbox{ and } u_n \rightarrow u \mbox{ in } L^r (\Omega) \mbox{ and in }L^p(\partial\Omega)
    \label{54}
    \end{equation}

    In ~\eqref{43} we choose $h = u_n - u \in W^{1,p}(\Omega)$, pass to the limit as $n \rightarrow \infty$ and use ~\eqref{54}.Then

    $$ \lim_{n\to\infty} \big[\langle A_p (u_n),u_n -u\rangle + \langle A_q(u_n),u_n -u\rangle\big] = 0, $$

    \begin{equation*}
    \begin{aligned}
     &\Rightarrow \limsup_{n\to\infty} \big[\langle A_p(u_n),u_n -u\rangle + \langle A_q(u),u_n-u\rangle\big] \leq 0 \\
     &\mbox{( since } A_q(\cdot) \mbox{ is monotone}) \\
     &\Rightarrow\limsup_{n\to\infty} \langle A_p(u_n),u_n -u\rangle \,\, \leq 0, \\
     &\Rightarrow u_n \to u \mbox{ in } W^{1,p}(\Omega) \mbox{ (see Proposition 1).}
    \end{aligned}
    \end{equation*}

    Therefore $\psi_\lambda^*(\cdot)$ satisfies the C-condition. This proves the Claim.

    Then ~\eqref{39},~\eqref{40} and the Claim permit the use of the mountain pass theorem
    So, we can find $\hat{u}\in W^{1,p}(\Omega)$ such that

    $$ \hat{u} \in K_{\psi_\lambda^*} \leq [u_\mu) \cap D_+ \mbox{ (see ~\eqref{35}) } , m_\lambda^* \leq \psi_\lambda^* (\hat{u}) \mbox{ (see ~\eqref{39})  }$$

    Therefore $ \hat{u} \in D_+ $ is a second positive solution of $P_\lambda$  $(\lambda \in (0,\lambda^*))$ distinct from $u_0 \in D_+$.

    \begin{flushright}
    \underline{\underline{QED}}
    \end{flushright}

    Next we examine what can be said in the critical parameter $\lambda^*$.\\

    \underline{Proposition 15}: \underline{If} hypotheses $H(\xi),H(\beta),H_0,H(f)$ hold, \underline{then} $\lambda^* \in L$.

    \underline{Proof}: Let $\{\lambda_n\}_{n \geq 1} \subseteq (0,\lambda^*) $ be such that $\lambda_n < \lambda^*.$ We can find $u_n\in S_{\lambda_n} \subseteq D_+$ for all $n \in N$.

    We consider the following Caratheodory function

    \begin{equation}
    \mu_n(z,x) = \left\{
     \begin{array}{lr}
                    v(z)^{-\gamma} + \lambda_n f(z,v(z)) \hspace{5.5mm} \mbox{ if } x \leq v(z) \\
                    x^{-\gamma} + \lambda_n f(z,x) \hspace{15mm}\mbox{ if } v(z) < x.
     \end{array}
    \right.
    \label{55}
    \end{equation}

    % pag 34
    We set $M_n (z,x) = \int_{0}^x \mu_n (z,x) ds$ and consider the C'-functional $j_n : W^{1,p}(\Omega) \rightarrow R$ defined by

    $$ j_n(u) = \frac{1}{p} \gamma_p(u) + \frac{1}{q} || D u ||_p^p  - \int_{\Omega}M_n(z,u) dz \mbox{ for all } u\in W^{1,p}(\Omega)$$

    % bullet sau * ?
    Also, we consider the following truncation of $\mu_n(z,*)$

    \begin{equation}
    \mu_n(z,x) = \left\{
     \begin{array}{lr}
                    \mu_n(z,x) \hspace{15mm} \mbox{ if } x \leq u_{n + 1} (z) \\
                    \mu_n(z,u_{n+1}(z)) \hspace{4.3mm} \mbox{ if } u_{n+1} (z) < x
     \end{array}
    \right.
    \label{56}
    \end{equation}
    (recall that $v\leq u_{n+1}$ for all $n\in N$, se Proposition 11). This is a Caratheodory function. We set $\hat{M}_n(z,x) = \int_{0}^x \hat{\mu}(z,x) ds$ and consider the C'-functional $\hat{J}_n : W^{1,p}(\Omega) \rightarrow R$ defined by

    $$ \hat{J}_n (u) = \frac{1}{p} \gamma_p (u) + \frac{1}{q} || D u ||_q^q - \int_{\Omega} \hat{M}_n (z,u) dz \mbox{ for all } u \in W^{1,p}(\Omega). $$

    From ~\eqref{55}, ~\eqref{56} and ~\eqref{1} , it is clear that $\hat{J}_n(\cdot)$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\hat{u}_n \in W^{1,p}(\Omega) $ such that

    \begin{equation}
    \hat{J}_n(\hat{u}_n) = \mbox{inf}\bigg[ \hat{J}_n(u): u \in W^{1,p}(\Omega) \bigg].
    \label{57}
    \end{equation}

    Then we have

    \begin{equation}
    \begin{split}
    \hat{J}_n(\hat{u}_n) \leq \hat{J}_n(v) \\
     &\quad \leq \frac{1}{p} \gamma_p(v) + \frac{1}{q} ||D v||_q^q - \frac{1}{1-\gamma} \int_{\Omega} v^{1-\gamma} dz \\
     &\quad \mbox{ (see ~\eqref{55}, ~\eqref{56} and recall that } f \geq 0 ) \\
     &\quad \leq \langle A_p(v),v\rangle + \langle A_q(v),v\rangle - \int_{\Omega} v^{1-\gamma} dz = 0 \\
     &\quad \mbox{ (see Proposition 8)} .
    \end{split}
    \label{58}
    \end{equation}

    From ~\eqref{57} we have

    \begin{equation}
    \hat{u}_n \in K_{\hat{J}_n} \subseteq [v,u_{n+1}] \cap D_+ \mbox{ for all } n\in N \mbox{ (see ~\eqref{56})}
    \label{59}
    \end{equation}

    Similarly using ~\eqref{55} we obtain

    \begin{equation}
    K_{j_n} \leq [v) \cap D_+
    \label{60}
    \end{equation}
    Note that

    $$ J_n \big|_{[v,u_{n+1}]} = \hat{J}_n \big|_{[v,u_{n+1}]} \mbox{ and } J'\big|_{[v,u_{n+1}]} = \hat{J}'\big|_{[v,u_{n+1}]} \mbox{ (see ~\eqref{55},~\eqref{56}}).$$

    Then from ~\eqref{58},~\eqref{59},~\eqref{60}, we have
    \begin{equation}
    J_n(\hat{u}_n) \leq 0 \mbox{ for all } n\in N \\
    \label{61}
    \end{equation}

    %\aici e xi sau altceva? pag 36 si acolo este hdz sau altceva gen sigma
    \begin{equation}
    \begin{aligned}
    &\langle A_p(\hat{u}_n),h\rangle + \langle A_q(\hat{u}_n),h\rangle + \int_{\Omega} \xi(z)\hat{u}_n^{p-1} h dz + \int_{\partial\Omega} \beta(z)\hat{u}_n^{p-1} hd\sigma = \int_{\Omega} \mu_n(z,\hat{u}_n) h dz \\
    &\quad \mbox{ for all } h \in W^{1,p}(\Omega), \mbox{ all } n\in N.
    \end{aligned}
    \label{62}
    \end{equation}

    Using ~\eqref{61}, ~\eqref{62} and reasoning as in the Claim in the proof of Proposition 14, we show that


    $$ \{\hat{u}_n\}_{n\geq 1} \subseteq W^{1,p}(\Omega) \mbox{ is bounded.} $$

    So, we may assume that
    \begin{equation}
    \hat{u}_n \stackrel{w}{\rightarrow} \hat{u}_* \mbox{ in } W^{1,p}(\Omega) \mbox{ and } \hat{u}_n \rightarrow \hat{u}_* \mbox{ in } L^r(\Omega) \mbox{ and in } L^p(\partial\Omega).
    \label{63}
    \end{equation}

    In ~\eqref{62} we choose $h=\hat{u}_n - \hat{u}_* \in W^{1,p}(\Omega), $ pass to the limit as $n\to\infty$ and use ~\eqref{63}. Then as before (see the proof of Proposition 14), we obtain

    \begin{equation}
    \hat{u}_n \rightarrow \hat{u}_* \mbox{ in } W^{1,p}(\Omega)
    \label{64}
    \end{equation}

    In ~\eqref{62} we pass to the limit as $n\to\infty$ and use ~\eqref{64}. Then

    $$ \langle A_p(\hat{u}_x),h\rangle + \langle A_q(\hat{u}_x),h\rangle + \int_{\Omega} \xi(z) \hat{u}_x^{p-1} hdz + \int_{\partial\Omega}\beta(z) \hat{u}_x^{p-1} hdz $$

    $$ = \int_{\Omega}[\hat{u}_x^{-\gamma} + \lambda_x f(z,\hat{u}_x)] hdz \mbox{ for all } h \in W^{1,p}(\Omega) \mbox{ (see ~\eqref{55},~\eqref{60})}, $$

    $$ \Rightarrow \hat{u}_x \in S_{\lambda^*} \subseteq D_+ \mbox{ and so } \lambda* \in L. $$


    \begin{flushright}
    \underline{\underline{QED}}
    \end{flushright}

    From this proposition it follows that

    $$ L = (0,\lambda*]. $$

    The next bifurcation-type theorem summarizes our findings and provides a complete description of the dependence of the set of positive solutions of problem $(p_\lambda)$ on the parameter $\lambda >0.$

    %aici cu underline a ,b,c trebuie lasate asa sau arenjate ca acolo
    \underline{Theorem 16}: \underline{If} hypotheses $H(\xi),H(\beta),H_0,H(f)$ hold, \underline{then} there exists $\lambda^* >0$ such that

    (\underline{a}) for all $\lambda \in (0,\lambda^*)$ problem $(p_\lambda)$ has at least two positive solutions

    $$ u_0, \hat{u} \in D_+ , u_0 \neq \hat{u};$$

    (\underline{b}) for $\lambda = \lambda^*$ problem $(p_\lambda)$ has at least one positive solution $\hat{u}_*\in D_{+}$;

    (\underline{c}) for all $\lambda > \lambda^*$ problem $ (p_\lambda)$ does not have any positive solutions

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\end{document}