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\title[Nonlinear singular problems with indefinite potential terms]{Nonlinear singular problems with indefinite potential 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, indefinite potential, singular term, concave and convex nonlinearities, truncation, comparison principles, nonlinear regularity, nonlinear maximum principle.\\
\phantom{aa} 2010 AMS Subject Classification: 35J75, 35J92, 35P30}

% ce scrie acolo unde e semnu intrebare dupa term fix la abstract
\begin{abstract}
We consider a nonlinear Dirichilet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearity. The concave term ? parametric. We prove a bifurcation-type theory describing the changes in the set of positive solutions as the parameter $\lambda>0$ varies.
\end{abstract}

\maketitle

\section{Introduction}

Let $\lambda \geq \RR^{\NN} $ be a bounded domain with a $C^2$-boundary $\partial\Omega$. In this paper we study the following nonlinear nonhomogeneous parametric singular problem

\begin{equation}
    \left\{
        \begin{array}{ll}
            \mbox{-diva}(Du(z)) + \xi(z)u(z)^{p-1} = \vartheta(u(z)) + \lambda u(z)^{q-1} + f(z,u(z)) \ \mbox{in}\ \Omega, \\
            u|_{\partial\Omega} = 0, u>0, \lambda > 0, 1<q<p<\infty.
        \end{array}
    \right\}\tag{$P_{\lambda}$}\label{eqp}
\end{equation}
%ce scrie acolo dupa diva? acel Du trebuie sa fie scris matematic sau normal? pagina 2

The map $a:\RR^{\NN} \rightarrow \RR^{\NN}$ involved in the differential operator of \eqref{eqp} is strictly monotone, continuous (hence maximal monotone too) and satisfies certain other regularity and growth conditions which are listed in hypotheses $H(a)$ below (see Section 2). These conditions are general enough to incorporate in our framework many differential operators of interest such as the p-Laplacian and the $(p,q)$ Laplacian (that is, the sum of a $p$-Laplacian and of a $q$-Laplacian). The operator $u\rightarrow$diva(Du) is not homogeneous and this is a source of difficulties in the analysis of problem \eqref{eqp}. THe potential function $\xi \in L^{\infty}(\Omega)$ is indefinite (that is, sign changing). So the operator $u \rightarrow $-diva(Du) $+ \xi (z) |u|^{p-2} u $ is not coercive and this is one more difficulty in the analysis of problem \eqref{eqp}. In the reaction (right hand side of \eqref{eqp}), the term $\vartheta(\cdot)$ is singular at $x=0$, while the perturbation contains the combined effects of a parametric concave term $x\rightarrow \lambda x^{q-1} x\geq 0 $ (recall q<p), with $\lambda >0$ being the parameter and of a Caratheodory function $f(z,x)$ (that is, for all $x\in \RR z\rightarrow f(z,x))$ is measurable and for almost all $z\in \Omega x\rightarrow f(z,x)$ is continuous), which is assumed to exhibit $(p-1)$-superlinear growth near $+\infty$, but without satisfying the usual for superlinear problems Ambrosetti-Rabinowitz condition (the AR-condition for short). So in problem \eqref{eqp} we have the competing effects of singular, concave and convex terms. Using variational methods related to the critical point theory, combined with suitable truncation, perturbation and comparison techniques, we produce a critical parameter value $\lambda^x >0$ such that

%arata oribil itemize asta ._. asa e normal? pagina 2

\begin{itemize}
    \item [(i)] for all $\lambda\in (0,\lambda^*)$ problem \eqref{eqp} has at least two positive solutions;
    \item [(ii)] for $\lambda = \lambda^*$ problem \eqref{eqp} has at least one positive solution;
    \item [(iii)] for all $\lambda > \lambda^*$ problem \eqref{eqp} has no positive solutions.
\end{itemize}
%se pune $p$ -Laplacian sau doar p-Laplacian

Our work here continues the recent one by Papageorgiou-Radulescu-Repovs \cite{16}, where $\xi \equiv 0 $ and in the reaction the parametric term is the singular one. Also it is related to the works of Papageorgiou-Smyrlis \cite{17} and Papageorgiou-Winkert \cite{18} where the differential operator is the $p$-Laplacian equations with no potential term and reactions of special form were considered  by Giacomoni-Schindler-Taka\v c \cite{5}, Mohammed \cite{12}, Perera-Zhang \cite{20}, Li-Gao \cite{11}, Chu-Goo-Sun \cite{2}. We also mention the recent work of Papageorgou-Radulescu-Repovs \cite{14}, where the differential operator is the $p$-Laplacian, there is no potential term (that is $\xi \equiv 0)$, the singular term is the parametric one and has the spcial form $\lambda u^{-\gamma}$ and finally the perturbation $f(z,x)$ is $(p-1)$-linear in $x\in\RR$ near $+\infty$. It will be interesting to see if the results of our work here are still valid if the nonlinearity $f(z,x)$ is $(p-1)$-linear in $x\in\RR$ near $+\infty$.

\section{Mathematical Background-Hypotheses}

In the section we present the main mathematical tools which we will use in the analysis of problem \eqref{eqp}, we also fix our notation and we finally state the hypotheses on the data of the problem.
%acolo e x steluta dupa banach space..... sau, asta e pagina 3

So, let $X$ be a Banach space, $X^*$ its topological dual and let $\varphi \in C^1(X).$ We say that $\varphi(\cdot)$ satisfies the "C-condition", if the following property holds

$$
\begin{array}{ll}
    \mbox{"Every squence}\ \{ u_n \}_{n\geq1}\subseteq X\ \mbox{such that}\\
    \{ \varphi(u_n) \}_{n\geq1}\subseteq \RR\ \mbox{is bounded,}\\
    (1+||u_n||_X)\varphi^{'}(u_n)\rightarrow 0\ \mbox{in} X^*\ \mbox{as}\ n\rightarrow\infty,\\
    \mbox{admits a strongly convergent subsequence".}
\end{array}
$$

This is a compactness-type condition on the functional $\varphi(\cdot)$, which leads to the minimax theory of the critical values of $\varphi(\cdot)$ (see, for example, Papageorgiou-Radulescu-repovs \cite{15}). By $K_\varphi$ we denote the critical set of $\varphi$, that is,

$$ K_{\varphi} = \{ u\in X: \varphi^{'}(u)=0\}. $$

The main spaces in the analysis of problem \eqref{eqp} are the Sobolev space $W^{1,p}_0(\Omega) (1<p<\infty)$ and the Banach space $C_0^1(\underline{\Omega})=\{u\in C^1(\underline{\Omega}): u_{\partial\Omega} = 0\}$. By $||\cdot||$ we denote the norm of $W^{1,p}_0$. From the Poincare inequality we have

$$ ||u| = ||Du||_p\ \mbox{for all}\ u \in W^{1,p}_0(\Omega). $$

The Banach space $C_0^1(\Omega)$ is ordered with positive(order) cone $C_+ = \{u\in C_0^1(\underline{\Omega}): u(z)\geq 0\ \mbox{for all}\ z\in\underline{\Omega} \}.\ \mbox{This cone has a nonempty interior given by}$

$$
\begin{array}{ll}
        int C_+ = \{u\in C_+ : u(z) >0\ \mbox{for all}\ z\in \Omega , \frac{\partial u}{\partial n} |_{\partial\Omega} <0 \} , \\
        \mbox{with}\ n(\cdot)\ \mbox{being the outward unit normal on}\ \partial\Omega.
\end{array}
$$

We will also use two more ordered Banach spaces. The first is

$$ C_0(\overline{\Omega}) = \{ u\in C(\overline{\Omega}):u|_{\partial\Omega} = 0 \}.$$

This is ordered with positive (order) cone $K_+ = \{ u \in C_0 (\overline{\Omega}): u(z) \geq 0$ for all $z\in\overline{\Omega}$. This cone has a nonempty interior given by

$$
\begin{array}{ll}
    intK_+ = \{ u \in K_+: c_u \hat{d} \ leq u \ \mbox{for some}\ c_u>0 \}, \\
    \mbox{where}\ \hat{d}(z) = d(z,\partial\Omega) z\in\overline{\Omega}. \ \mbox{On account of Lemma 14.16, p.355, of Gilbard-Trudinger \cite{6},  we have}
\end{array}
$$

\begin{equation}
    "  c_u \hat{d} \leq u \ \mbox{for some}\ c_u > 0\ \mbox{if and only if}\ \hat{c}_u \hat{u}_1 \leq u\ \mbox{for some} \ \hat{c}_u > 0 ",
    \label{eq1}
\end{equation}
with $\hat{u}_1$ being the positive, $L^p$-normalized (that is, $||\hat{u}_1||_p=1$) eigenfunction corresponding to the principal eigenvalue $\hat{\lambda}_1>0$ of the Dirichlet $p$-Laplacian. The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski-Papageorgiou \cite{4}, pp.737-738), imply that $\hat{u}_1\in int C_+$.

The second ordered space is $C_1(\overline{\Omega})$ with positive (order) cone

%jos acolo este u^-1 (0) ? este zero ? pagina 5 pe la juma
$$ \hat{C}_+ = \{ u \in C^1(\overline{\Omega}): u(z) \geq 0 \ \mbox{for all}\ z\in\overline{\Omega}, \frac{\partial u}{\partial n}|_{\partial\Omega \cap u^{-1}(0)} <0 \}.$$
Clearly this cone has a nonempty interior.

Concerning ordered Banach spaces with an order cone which has a nonempty interior (solid order cone), we have the following result which will be useful in our analysis (see Papageorgiou-Radulescu-Repovs \cite{15}, Proposition 4.1.22, p.226).

%verifica daca se termina inainte de remarks sau dupa remark prop asta
\begin{prop}\label{prop1}
    If X is an ordered Banach space with positive (order) cone K, intK $\neq \emptyset$, and $e \in intK$, then for every $u\in X$ we can find $\lambda_u >0$ such that $\lambda_u e - u\in K.$
\end{prop}

Let $l\in C^1(0,\infty)$ with $l(t)>0$ for all $t>0$. We assume that

\begin{equation}
    \begin{array}{ll}
        0<\hat{c}\leq\frac{l^{'}(t)t}{l(t)}\leq c_0, c_1 t^{p-1}\leq l(t) \leq c_2[t^{s-1}+ t^{p-1}] \\
        \mbox{for all}\ t>0,\ \mbox{some}\ c_{1},c_{2}>0, 1\leq s\leq p.
    \end{array}
    \label{eq2}
\end{equation}

Then the conditions on the map $a(\cdot)$ are the following:

$H(\alpha): a(y) = a_0(|y|)y$ for all $y\in\RR^{N}$ is strictly increasing on $(0,+\infty), a_0(t) t\rightarrow 0^+$ as $t\rightarrow 0^+$ and

%e sigur bigtriangledown? sau nabla pagina 6
\begin{itemize}
    \item [(i)] $a_0\in C^1(0,+\infty), t\rightarrow a_0 (t)$ is strictly increasing on $(0,+\infty), a_0(t) t\rightarrow 0^+$ as $t\rightarrow 0^+$ and
        $$ lim_{t\to 0^+}\frac{a^{'}_0(t)t}{a_0(t)} > -1; $$
    \item [(ii)] there exists $c_3>0$ such that
        $$ |\nabla a(y)| c_3 \frac{l(|y|}{|y|}\ \mbox{for all}\ y\in \RR^{N} \\ \{0\}; $$
    \item [(iii)] $(\nabla a(y)\xi,\xi)_{\RR^{N}} \geq \frac{l(|y|)}{|y|}|\xi|^2$ for all $y\in\RR^{n} \\ {0}$, all $\xi\in\RR^{N}$;
    \item [(iv)] if $G_0(t) = \int_{0}^{t} a_(s)s ds $, then there exists $\tau\in (q,p]$ such that
        $$ \limsup_{t\to 0^+} \frac{\tau(G_0(t)}{t^{\tau}} \leq c^* $$
\end{itemize}
and $0\leq p G_0(t) - a_0(t) t^2$ for all $t>0$.

\begin{remark}\label{rem1}
Hypotheses $H(a)(i),(ii),(iii)$ are dictated by the nonlinear regularity theory of Lieberman [10] and the nonlinear maximum principle of Pucci-Serrin \cite{21}. Hypothesis $H(a)(iv)$ serves the needs of our problem, but in fact it is a mild condition and it is satisfied in all cases of interest (see the Examples below). These conditions were used by Papageorgiou-Radulescu \cite{13} and by Papageorgiou-Radulescu-Repovs \cite{16}.
\end{remark}

Hypotheses $H(a)$ imply that the primitive $G_0(\cdot)$ is strictly increasing and strictly convex. We set $G(y) = G_0(|y|)$ for all $y\in \RR^{N}.$ Evidently $G(\cdot)$ is convex, $G(0) = 0$ and

$$
\begin{array}{ll}
    \nabla G(y) = G_0^{'}(|y|)\frac{y}{|y|} = a_0 (|y|)y = a(y)\ \mbox{for all}\ y\in\RR^{N}\\{0},\nabla G(0)=0,\\
    \mbox{that is},\ G(\cdot) \ \mbox{is the primitive of}\ a(\cdot).\ \mbox{From the convexity of}\ G(\cdot) we have
\end{array}
$$

\begin{equation}
    G(y)\leq(a(y),y)_{\RR^{N}}\ \mbox{for all} y\in\RR^{N}.
    \label{eq3}
\end{equation}

Using hypotheses $H(a)(i),(ii),(iii)$ and \eqref{eq2}, we obtain easily the following lemma. which summarizes the main properties of the map $a(\cdot)$:

%the map? pagina 7
\begin{lemma}\label{lem1}
If hypotheses $H(a)(i),(ii),(iii)$ hold, then
\begin{itemize}
    \item [(a)] the map $y\rightarrow a(y)$ is continuous, strictly monotone (hence maximal monotone too);
    \item [(b)] $|a(y)| \leq c_4 [|y|^{s-1} +|y|^{p-1}]$ for some $c_4>0$, all $y\in\RR^{N}$;
    \item [(c)] $(a(y),y)_{\RR^{N}} \geq \frac{C_1}{p-1} |y|^p$ for all $y\in\RR^{N}$.
\end{itemize}
\end{lemma}

Using this lemma and \eqref{3}, we obtain the following growth estimates for the primitive $G(\cdot)$

%aici trebuie Label?
\begin{corollary}\label{cor1}
If hypotheses $H(a)(i),(ii),(iii),$ then $\frac{a}{p(p-1)} |y|^{p} \leq G(y) \leq c_5 [1+|y|^p] $ for some $c_5>0$, all $y\in\RR^{N}$.
\end{corollary}

The examples that follow confirm that the framework provided by hypotheses $H(a)$ above , is broad and includes many differential operators of interest.

%pagina 8 este D_p sau Delta pe acolo ca nu inteleg
\begin{ex}\label{ex1}
        (a) $a(y) = |y|^{p-2}y$ with $1<p<\infty$.
        This map corresponds to the $p$-Laplace differential operator defined by
        $$ D_p u = div(|Du|^{p-2} Du)\ \mbox{for all}\ u\in W^{1,p}_0 (\Omega). $$
        (b) $a(y)=|y|^{p-2} y + \mu|y|^{q-2} y$ with $1<q<p<\infty,\mu \geq 0$.
        This map corresponds to the $(p,q)$-Laplace differential operator defined by
        $$ D_p u + D_q u \ \mbox{for all}\ u\in W^{1,p}_0(\Omega). $$
        Such operators arrise in models of physical processes. We mention the works of Cherfils-Ilyasov \cite{1} (reaction-diffusion systems) and Zhikov \cite{22} (homogenization of composites consisting of two materials with distinct hardening exponents, in elasticity theory).
        (c) $a(y) =(1+|y|^2)^{\frac{p-2}{2}}y$ with $1<p<\infty$.
        This map corresponds to the modified capilarry operator.
        (d) $a(y) = |y|^{p-2}y[1+ \frac{1}{1+ |y|^p}]$ with $1<p<\infty>$
\end{ex}

The hypotheses on the potential term $\xi(\cdot)$ and on the singular part $\vartheta(\cdot)$ of the reaction are the following:

$H(\xi): \xi\in L^\infty(\Omega).$
$H(\vartheta): \vartheta:(0,+\infty)\rightarrow (0,+\infty)$ is a locally Lipschitz function such that
\begin{itemize}
    \item [(i)] for some $\gamma\in(0,1)$ we have
        $$ 0 < c_6\leq \liminf_{x\to 0^+}\vartheta(x)x^{\gamma} \leq \limsup_{x\to 0^+}\vartheta(x) x^\gamma \leq c_7; $$
    \item [(ii)] $vartheta(\cdot)$ is nonincreasing.
\end{itemize}

\begin{remark}\label{rem2}
 In the literature we almost always encounter the following particular singular term
$$ \vartheta(x) = x^{-\gamma}\ \mbox{for all}\ x>0, \ \mbox{with}\ 0<\gamma<1. $$
\end{remark}
Of course hypotheses $H(\vartheta)$ provide a much more general framework and can accomodate also singularities like the ones that follow:

\begin{eqnarray*}
    &&\vartheta_1(x)=x^{-\gamma}[1+ln(1+x)] x>0,  \\
    &&\vartheta_2(x)=x^{-\gamma}e^{-x} x>0 \\
    &&\vartheta_3(x)= \left\{
            \begin{array}{ll}
                    x^{-\gamma}(1-\eta sin x )& 0<x\leq \frac{\pi}{2}\\
                    x^{-\gamma}(1-\eta)& \ \mbox{if}\ \frac{\pi}{2} <x
            \end{array}
            \ \mbox{with}\  0<\gamma<1.
        \right.
\end{eqnarray*}

The following strong comparison principle can be found in Papageorgiou-Radulescu-Repovs \cite{16}, Proposition \ref{prop6} ( see also Papageorgiou-Smyrlis \cite{17}, Proposition \ref{prop4}

\begin{prop}\label{prop4}
    If hypotheses $H(a),H(\vartheta)$ hold, $\hat{\xi}\in L^{\infty}(\Omega), \hat{\xi}(z) \geq 0$  for almost all $z\in \Omega, h_1,h_2 \in L^{\infty}(\Omega)$ satisfy
    $$ 0<c_8\leq h_2(z) - h_1(z) \ \mbox{for almost all}\ z\in\Omega$$
    and $u,v\in C^{1,\alpha}(\hat{\Omega})$ satisfy $0<u(z)\leq v(z)$ for all $z\in\Omega$ and

    for almost all $z\in\Omega$ we have
    \begin{itemize}
        \item [-] diva$(Du(z)) - \vartheta(u(z)) + \xi(z) u (z)^{p-1} = h_1(z)$
        \item [-] diva$(Dv(z) - \vartheta(v(z)) + \xi(z) v(z)^{p-1} = h_2(z)$,
    \end{itemize}
    then $v-u\in int\hat{C}_+.$
\end{prop}

In what follows $p^*$ is the critical Sobolev exponent corresponding to $p$, that is,
$$
p^* =\left\{
        \begin{array}{ll}
            \frac{Np}{N-p}& \mbox{if}\ p<N.\\
            +\infty& \mbox{if}\ N\leq p
        \end{array}
    \right.
    .
$$
Now we introduce our hypotheses on the nonlinearity f(z,x).

%este rho sau p la (v)
$H(f): f:\Omega\times\RR\rightarrow\RR_+$ is a Caratheodory function such that $f(z,0)=0$ for almost all $z\in\Omega$ and
\begin{itemize}
    \item [(i)] $f(z,x) \leq a(z) [1+x^{p-1}]$ for almost all $z\in\Omega$, all $x\geq 0$, with $a\in L^{\infty}(\Omega),p<r<p^*;$
    \item [(ii)] if $F(z,x)=J_0^x f(z,s) ds,$
    \item [] then $\lim_{x\to+\infty}\frac{F(z,x)}{x^{p}} = +\infty$ uniformly for almost all $z\in\Omega$;
    \item [(iii)] there exists $\sigma\in((r-p) max\{ \frac{N}{p}, 1 \}.p^*),\sigma>q$ such that
    \item [(iv)] $\limsup_{x\rightarrow 0^+} \frac{f(z,x)}{x^{r-1}} \leq \eta_0 $ uniformly for almost all $z\in\Omega$;
    \item [(v)] for every $\rho$, there exists $\hat{\xi}_\rho>0$ such that for almost all $z\in\Omega$ the function
    $$ x\rightarrow f(z,x) + \hat{\xi}_\rho x^{\rho-1} $$
    is nondecreasing on $[0,\rho]$.
\end{itemize}

\begin{remark}\label{rem3}
    Since our aim is to find positive solutions and the above hypotheses concern the positive semiaxis $\RR_+ = [0,+\infty)$, we may assume that
    \begin{equation}
        f(z,x) = 0,\ \mbox{for almost all}\ z\in\Omega, \ \mbox{all}\ x\leq 0.
        \label{eq4}
    \end{equation}
\end{remark}

Hypotheses $H(f)(ii),(iii)$ imply that
\begin{equation}
    \lim_{x\to+\infty} \frac{f(z,x)}{x^{p-1}} = +\infty \ \mbox{uniformly for almost all}\ z\in\Omega.
    \label{eq5}
\end{equation}

So, the nonlinearity $f(z,\cdot)$ is $(p-1)$-superlinear near $+\infty$. Howerver, this superlinearity of $f(z,\cdot)$ is not formulated using the AR-condition. We recall that the AR-condition (unilateral version due to \eqref{4}), says that there exist $\gamma>p$ and $M>0$ such that


\begin{subequations}
    \begin{align}
        &0<\gamma F(z,x) \leq f(z,x) x \ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq M,    \label{eq6a}\\
        &0<\essinf_{\Omega} F(\cdot,M).
        \label{eq6b}
    \end{align}
\end{subequations}

If we integrate \eqref{eq6a} and use eqref{6b}, then we obtain the weaker condition


\begin{equation}
    \begin{array}{ll}
    &c_9 x^{\gamma} \leq F(z,x)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq M,\ \mbox{some}\ c_9>0, \\
    \Rightarrow &c_9 x^{\gamma-1}\leq f(z,x)\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq M.
    \end{array}
    \label{eq7}
\end{equation}

Therefore the AR-condition implies that $f(z,\cdot)$ exhibits at least $(\gamma-1)$-polynomial growth. Evidently \eqref{eq7} implies the much weaker condition \eqref{eq5}. In this work instead of the AR-condition, we employ the less restrictive hypothesis $H(f)(iii)$. This way we incorporate in our framework also $(p-1)$-superlinear terms with "slower" growth near $+\infty$, which fail to satisfy the AR-condition. The following function satisfies hypotheses $H(f)$ but fials to satisfy the AR-condition (for the sake of simplicity we drop the z-dependence)

$$ f(x) = x^{p-1}ln(1+x) \ \mbox{for all}\ x\geq 0. $$

Finally let us fix the notation which we will use throughout this work. For $x\in\RR$ we set $x^{\pm} = max\{ \pm x,0\}.$ Then for $u\in W^{1,p}_0(\Omega)$ we define $u^{\pm}(z) = u(z)^{\pm}$ for almost all $z\in\Omega$. Then

$$ u^{\pm}\in W^{1,p}_0(\Omega),u=u^{+}-u^{-},\ |u|=u^{+} + u^{-}. $$

If $u,v\in W^{1,p}_0(\Omega)$ and $u\leq v$, then we define.


$$
\begin{array}{ll}
&[u,v] = \{ y\in W^{1,p}_0(\Omega): u(z) \leq y(z) \leq v(z) \ \mbox{for almost all}\ z\in\Omega ,\} \\
&[u) = \{ y\in W^{1,p}_0(\Omega): u(z)\leq y(z)\ \mbox{for almost all}\ z\in\Omega \}.
\end{array}
$$

Also by $int_{c_0^1(\overline)}[u,v]$ we denote the interior in the $c_0^1(\overline{\Omega})$ - norm topology of the set $[u,v]\cap C_0^1(\overline{\Omega})$.

By $A:W^{1,p}_0(\Omega)\rightarrow W^{-1,p^{'}}(\Omega)=W^{1,p}_0(\Omega)^*(\frac{1}{p} + \frac{1}{p^{'}} = 1)$ we denote the nonlinear operator defined by

$$\langle A(u),h\rangle =\int_{\Omega} (a(Du),Dh)_{\RR^{N}} dz \ \mbox{for all}\ u,h\in W^{1,p}_0(\Omega). $$

We know (see Gasinski-Papageorgiou \cite{4}), that $A(\cdot)$ is continuous, strictly monotone (hence maximal monotone too) and of type $(S)_+$, that is

$$
\begin{array}{ll}
"\mbox{If}\ u_n\stackrel{w}{\rightarrow} u \ \mbox{in}\ W^{1,p}_0(\Omega) \ \mbox{and}\ \limsup_{n\to\infty}\langle A(u_n),u_n -u\rangle \leq 0, \\
\mbox{then}\ u_n\rightarrow u \ \mbox{in}\ W^{1,p}_0(\Omega)."
\end{array}
$$


We introduce the following two sets related to problem \eqref{eqp}:

$$
\begin{array}{ll}
    \mathcal{L} = \{ \lambda >0 :\ \mbox{problem \eqref{eqp} admits a positive solution} \},\\
    S_{\lambda} = the set of positive solutions for problem \eqref{eqp}.
\end{array}
$$

We let $\lambda^* = \sup\mathcal{L}. $

\section{Positive Solutions}

We start by considering the following purely singular problem:
\begin{equation}
    -diva(Du(z))) + \xi^+(z)u(z)^{p-1} = \vartheta(u(z))\ \mbox{in}\ \Omega, u|_{\partial\Omega} = 0, u>0.
    \label{eq8}
\end{equation}

From Papageorgiou-Radulescu-repovs \cite{16} (Proposition \ref{prop10}), we have:

%prop 51 wtf? si se termina unde zic eu sau mai departe?
\begin{prop}\label{prop5}
If hypotheses $H(a),H(\xi),H(\vartheta)$ hold, then problem \eqref{eq8} admits a unique positive solution $v\in int C_+$
Let $\beta > ||\xi||_\infty$. Then hypotheses $H(f)(i),(iv)$ and since $1<q<p<r$,imply that we can find $c_{10},c_{11}>0 such that$
\end{prop}

\begin{equation}
    \lambda x^{q-1} + f(z,x) \leq \lambda c_{10} x^{q-1} + c_{11}x^{r-1} -\beta x^{p-1}\ \mbox{for almost all}\ z\in\Omega,\ \mbox{all}\ x\geq 0.
    \label{eq9}
\end{equation}

Let $K_\lambda(x) = \lambda c_{10} x^{q-1} + c_{11}x^{r-1} -\beta x^{p-1}$ for all $x\geq 0$ and with $v\in intC_+$ from Proposition \ref{prop4}, we consider the following auxiliary Dirichlet problem

\begin{equation}
\left\{
        \begin{array}{ll}
            -diva(Du(z)) + \xi(z) u(z)^{p-1} = \vartheta(v(z)) + k_\lambda(u(z))\ \mbox{in}\ \Omega.\\
            u|_{\partial\Omega} = 0 ,>0.
        \end{array}
    \right\}\tag*{$(10)_\lambda$}\label{eq10}
\end{equation}

For this problem we prove the following result.

\begin{prop}\label{prop6}
    If hypotheses $H(a),H(\xi),H(\vartheta) hold$, then for all $\lambda>0$ small problem \ref{eq10} has a smallest positive solution
    $$ \overline{u}_\lambda \in intC_+ $$
\end{prop}

\begin{proof}
Recall that $v\in intC_+$ (see Proposition \ref{prop4}). Hence $v\in intK_+$ (see \eqref{eq1}). For $s>N$ we consider the function $\hat{u}^{\frac{1}{s}}_1\in K_+$. According to Proposition \ref{prop1}, we can find $\mu>0$ such that
\addtocounter{equation}{+1}
\begin{equation}
    \begin{array}{ll}
        &\hat{u}^{\frac{1}{s}}_1\leq \mu v,\\
        \Rightarrow &v^{-\gamma}\leq \mu^{\gamma} \hat{u}_1^{-\frac{\gamma}{s}} .
    \end{array}
    \label{eq11}
\end{equation}
\end{proof}

From the Lemma in Lazer-McKenna \cite{9}, we have

\begin{equation}
    \begin{array}{ll}
        &\hat{u}_1^{\frac{\gamma}{s}} \in L^s (\Omega),\\
        \Rightarrow&v^{-\gamma}\in L^s(\Omega)(see \eqref{eq11}).
    \end{array}
    \label{eq12}
\end{equation}

Hypotheses $H(\theta)$ imply that we can find $c_{12}>0$ and $\delta>0$ such that

\begin{equation}
    0\leq\vartheta(x)\leq c_{12}x^{-\gamma} \ \mbox{for all}\ 0\leq x\leq\delta\ \mbox{and}\ 0\leq\vartheta(x)  \leq\vartheta(\delta)\ \mbox{for all}\ x>\delta.
    \label{eq13}
\end{equation}

From \eqref{eq12},\eqref{eq13} it follows that

$$ \vartheta (v(\cdot))\in L^s(\Omega) (s>N).$$

Let $\hat{k}_\lambda(x) = \lambda c_10 x^{q-1} + c_{11}x^{r-1}$ for all $x\geq 0$ and set $\hat{K}_\lambda(x) = \int_{0}^{x} \hat{k}_\lambda(s)ds$. We consider the $C^1$- functional $\psi_\lambda: W^{1,p}_0(\Omega)\rightarrow \RR$ defined by

\begin{eqnarray}
    &&\psi_\lambda(u) = \int_{\Omega} G(Du)dz+\frac{1}{p}\int_{\Omega}[\xi(z)+\beta]|u|^p dz-\int_{\Omega}\hat{K}_\lambda(u^+)dz - \int_{\Omega}\vartheta(v) u^+ dz \nonumber\\
    &&\mbox{for all}\ u\in W^{1,p}_0(\Omega)\nonumber\\
    &&\geq\frac{u}{p(p-1)}||Du||_p^p +\frac{1}{p}\int_{\Omega} [\xi(z)+\beta]|u|^p dz - \frac{\lambda c_{10}}{q}||u||_q^q - \frac{c_{11}}{r}||u||_r^r - \int_{\Omega}\vartheta(v)|u|dz  \nonumber\\
    &&\mbox{ (see Corollary \ref{cor3})}\nonumber\\
    &&\geq c_{12}||u||^p -c_{13}[\lambda||u||+||u||^r]\nonumber\\
    &&\mbox{for some}\ c_{12},c_{13}>0\ \mbox{and all}\ 0<\lambda\leq 1\ (\mbox{recall}\ \beta>||\xi||_\infty \ \mbox{and}\ 1<q<r)\nonumber\\
    &&=[c_{12} - c_{13}(\lambda||u||^{1-p}+||u||^{r-p})]||u||^p.
    \label{eq14}
\end{eqnarray}

%acolo e \Im sau altceva
We introduce the function $\Im_\lambda(t)=\lambda t^{1-p} + t^{r-p},t>0. $ Evidently $\Im_\lambda \in C^{1}(0,+\infty)$ and since $1<p<r$, we see that

$$ \Im_\lambda(t)\rightarrow+\infty \ \mbox{as}\ t\rightarrow 0^+ \ \mbox{and as}\ t\rightarrow+\infty.$$

So, we can find $t_0>0$ such that

$$
\begin{array}{ll}
    &\Im_\lambda(t_0)=inf[\Im_\lambda(t):t>0],\\
    \Rightarrow&\Im_\lambda^{'}(t_0) = 0,\\
    \Rightarrow&\lambda(p-1)t^{-p}_0 = (r-p) t^{r-p-1}_0\\
    \Rightarrow&t_0 = \left[ \frac{\lambda(p-1}{r-p} \right]^{\frac{1}{r-1}}
\end{array}
$$

Since $\frac{p-1}{r-1}$, it follows that

$$ \Im_\lambda(t_0)\rightarrow 0\ \mbox{as}\ \lambda\rightarrow 0^+ $$

So, we can find $\lambda_0\in(0,1]$ such that

$$ \Im_\lambda(t_0)\leq \frac{c_{12}}{c_{13}} \ \mbox{for all}\ \lambda\in(0,\lambda_0]. $$

For $ \rho=t_0 $, from \eqref{eq14} we see that
\begin{equation}
    \psi_\lambda|_{\partial\hat{\beta}_\rho}>0
    \label{eq15}
\end{equation}

where $\hat{\beta}_{\rho} = \{u_\in W^{1,p}_0(\Omega):||u||\leq\rho \}$ and $\partial\overline{\beta}_\rho = \{u\in W^{1,p}_0 (\Omega):||u||=\rho\}$

We fix $\lambda\in(0,\lambda_0]$. Hypothesis $H(a)(iv)$ implies that we can find $c^*_0 > c^*$ and $\delta>0$ such that

$$ G(y)\leq \frac{c_0^*}{\tau}|y|^{\tau}\ \mbox{for all}\ |y|\leq\delta. $$

Let $u\in intC_+$ and choose $t\in(0,1)$ small such that

$$ t|Du(z)|\leq\delta\ \mbox{for all} z\in\overline{\Omega}.  $$

Then we have

$$ \psi_\lambda(tu)\leq\frac{t^{\tau}c_0^*}{\tau}||Du||_\tau^\tau + \frac{t^p}{p}\int_{\Omega}[\xi(z)+\beta]|u|^p dz- \frac{\lambda t^q}{q}||u||_q^q $$

Since $q<\tau\leq p$, choosing $t\in(0,1)$\ even smaller if necessary, we have

\begin{eqnarray}
    &\psi_\lambda(tu)0,\nonumber\\
    \Rightarrow&\inf_{\overline{\beta}_\rho}\psi_\lambda<0
    \label{eq16}
\end{eqnarray}

The functional $\psi_\lambda(\cdot)$ is sequentially weakly lower semicontinuous and by the Eberlein-Smulian theorem and the reflexivity of $W^{1,p}_0(\Omega)$, the set $\overline{B}_p$ is sequentially weakly compact. So, by the Weierstrass-Tonelli theorem, we can find $\overline{u}\in W^{1,p}_0(\Omega)$ such that

\begin{equation}
    \psi_\lambda(\overline{u}) = inf[\psi_\lambda(u): u\in W^{1,p}_0(\Omega)] (\lambda\in(0,\lambda_0])
    \label{eq17}
\end{equation}

From \eqref{eq15},\eqref{eq16} and \eqref{17} it follows that

\begin{eqnarray}
    &&0<||\overline{u}||<p,\nonumber\\
    &&\Rightarrow \psi^{'}_\lambda (\pi) = 0(\mbox{see}\ \eqref{eq17}),\nonumber\\
    &&\Rightarrow <\langle A(\overline{u}h\rangle + int_{\Omega}[E(z)+\beta]|\overline{u}|^{p-2}\hat{u}hdz = \int_{\Omega} [\vartheta(v)+\hat{K}_\lambda(\overline{u}^+)] hdz\\
    &&\mbox{for all}\ h\in W^{1,p}_0(\Omega).\nonumber
    \label{eq18}
\end{eqnarray}

%lem2?
In \eqref{eq18} we choose $h=-\overline{u}^{-}\in W^{1,p}_0(\Omega).$ Using Lemma \ref{lem2}$(c)$ and since $\beta>||\xi||_\infty$ we obtain

\begin{eqnarray*}
    &&c_{14}||\overline{u}^{-}||^p \leq 0\ \mbox{for some}\ c_{14}>0,\nonumber\\
    &&\Rightarrow \overline{u}\geq 0,\overline{u}\neq 0\nonumber
\end{eqnarray*}

Then from \eqref{eq18} we have

\begin{eqnarray}
-diva(D\hat{u}(z)) + \xi(z)\hat{u}(z)^{p-1} = \vartheta(v(z)) + k_\lambda(\hat{u}(z)) \ \mbox{for almost all}\ z\in\Omega,\\
\Rightarrow \hat{u}\in W^{1,p}_0 (\Omega)\ \mbox{is a positive of problem \ref{eq10} for}\ \lambda\in(0,\lambda_0].
\end{eqnarray}
\end{document}