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


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\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}











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