\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 118, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/118\hfil Sub-supersolution theorems] {Sub-supersolution theorems for quasilinear elliptic problems: A variational approach} \author[V. K. Le \& K. Schmitt \hfil EJDE-2004/118\hfilneg] {Vy Khoi Le, Klaus Schmitt} % in alphabetical order \dedicatory{Dedicated to Hans Knobloch with much admiration and appreciation} \address{Vy Khoi Le \hfill\break Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65401, USA} \email{vy@umr.edu} \address{Klaus Schmitt \hfill\break Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA} \email{schmitt@math.utah.edu} \date{} \thanks{Submitted July 28, 2004. Published October 7, 2004.} \subjclass[2000]{35B45, 35J65, 35J60} \keywords{Sub and supersolutions; periodic solutions; variational approach} \begin{abstract} This paper presents a variational approach to obtain sub - supersolution theorems for a certain type of boundary value problem for a class of quasilinear elliptic partial differential equations. In the case of semilinear ordinary differential equations results of this type were first proved by Hans Knobloch in the early sixties using methods developed by Cesari. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{definition}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \section{Introduction - Problem setting} We are interested here in a variational sub-supersolution approach to a quasilinear elliptic boundary value problem which, in the one-space dimensional and semilinear case, is a boundary value problem for a second order scalar ordinary differential equation %%@ subject to periodic boundary conditions. The latter problem was first studied by Hans Knobloch \cite{knobloch:enm63} and later by many other authors using various kinds of nonlinear analysis methods (see, e.g., \cite{schmitt:psn67,mawhin:nfa74,knobloch:nbv77,habets:nbv83}). The present paper is a continuation of and extends the results of the recent note \cite{schmitt:pss04}. In the case of boundary value problems subject to Dirichlet boundary conditions, sub-supersolution results, in a variational setting, were first obtained by Peter Hess \cite{hess:sne76}. We here follow closely the approach used in \cite{le:bvp98}, where Dirichlet boundary value problems for degenerate elliptic equations were studied. Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with smooth boundary. We consider the following boundary value %%@ problem: \begin{gather} \label{1} -\mathop{\rm div} [A(x,\nabla u)] + f(x,u) = 0,\quad x\in \Omega , \\ \label{2} u(x) = {\rm constant}, \quad x\in \partial\Omega , \\ \label{3} \int_{\partial\Omega} A(x,\nabla u) \cdot n ,S = 0 . \end{gather} (Note that in condition (\ref{2}) it is understood that the trace of $u$ is a constant function, with the constant not being fixed.) Here, $A : \Omega \times \mathbb{R}^N \to \mathbb{R}^N$ is a Carath\' eodory function satisfying the following conditions: \begin{itemize} \item There exist $p\in (1,\infty)$, $a_1\in L^{p'}(\Omega)$ ($p'$ is the conjugate of $p$), and $b_1 >0$ %%@ such that \begin{equation} \label{4} | A(x,\xi) | \le a_1(x) + b_1 |\xi|^{p-1}, \end{equation} for a.e.\ $x\in \Omega$, all $\xi\in \mathbb{R}^N$. \item $A(x,\xi)$ is monotone in $\xi$, that is \begin{equation} \label{5} [A(x,\xi) - A(x,\xi ')]\cdot (\xi -\xi ') \ge 0, \quad \mbox{ for a.e.\ $x\in \Omega$, all $\xi, \xi '\in \mathbb{R}^N$}. \end{equation} \item $A$ has the following coercivity property: There exist $a_2\in L^1(\Omega)$ and $b_2 >0$ such that \begin{equation} \label{6} A(x,\xi) \cdot \xi \ge b_2|\xi|^p - a_2(x) ,\quad \mbox{ for a.e.\ $x\in \Omega$, all $\xi\in \mathbb{R}^N$}. \end{equation} \end{itemize} \begin{rem}\rm (a) When $N=1$ and $\Omega = (a,b)$, the boundary condition (\ref{2})-(\ref{3}) becomes the boundary condition on $(a,b)$: $$ u(a) = u(b), \; A(a,u'(a)) = A(b,u'(b)) , $$ which, when $A(x,v)=v$ is the usual set of periodic boundary conditions $$ u(a) = u(b), \quad u'(a) = u'(b) . $$ (b) An example of the operator $A$ above is the $p$-Laplacian, i.e., $$ A(x,\nabla u) = |\nabla u |^{p-2}\nabla u ,\quad p>1. $$ It is easy to check that $A$ satisfies conditions (\ref{4}), (\ref{5}), and (\ref{6}) above. In this case, the boundary %%@ condition (\ref{3}) becomes $$ \int_{\partial\Omega} |\nabla u|^{p-2}\frac{\partial u}{\partial n} \,dS = 0\,. $$ \end{rem} Assume that $f : \Omega \times \mathbb{R} \to \mathbb{R}$ is a Carath\'eodory function with some appropriate growth condition to be specified later. We denote by $W^{1,p}(\Omega)$ the usual Sobolev space, equipped with the norm \begin{equation} \label{7} \| u\| = \| u\|_{W^{1,p}(\Omega)} = \left(\| u\|^p_{L^p(\Omega)} +\| |\nabla u | \|^p_{L^p(\Omega)} \right)^{1/p} . \end{equation} Let $\mathcal{A} , F : W^{1,p}(\Omega) \to [W^{1,p}(\Omega)]^*$ be defined by $$ \langle Fu , v \rangle = \int_\Omega f(x,u) v \,dx , $$ and $$ \langle \mathcal{A} u , v \rangle = \int_\Omega A(x, \nabla u) \cdot\nabla v \,dx , \quad \forall u, v\in W^{1,p}(\Omega) . $$ From (\ref{4})-(\ref{6}), we see that $\mathcal{A}$ is continuous, bounded, monotone, and coercive in the following sense: \begin{equation} \label{8} \langle \mathcal{A} u , u\rangle \ge b_2 \| |\nabla u |\|_{L^p(\Omega)}^p - \| a_2\|_{L^1(\Omega)} , \quad\forall u \in W^{1,p}(\Omega). \end{equation} Let $$ V_c = \{u\in W^{1,p}(\Omega) : u\big|_{\partial\Omega} = \mbox{constant}\}. $$ Then $V_c$ is a closed subspace of $W^{1,p}(\Omega)$ and thus a reflexive Banach space with the restricted %%@ norm of (\ref{7}). The weak (variational) formulation of the boundary value problem (\ref{1})-(\ref{3}) is the %%@ following variational equality: \begin{equation} \label{9} \begin{gathered} \int_\Omega A(x,\nabla u) \cdot \nabla v \,dx + \int_\Omega f(x,u) v \,dx =0,\quad \forall v\in V_c \\ u\in V_c . \end{gathered} \end{equation} To check this, note that if $u$ satisfies (\ref{1})-(\ref{3}) and $v\in V_c$ then \begin{align*} 0 & = -\int_\Omega \mathop{\rm div} A(x,\nabla u) v \,dx + \int_\Omega f(x,u) v \,\,dx \\ & = \int_\Omega A(x,\nabla u)\cdot \nabla v \,dx - (v|_{\partial\Omega}) \int_{\partial\Omega}A(x,\nabla u)\cdot n \,dS + \int_\Omega f(x,u) v \,dx \\ & = \int_\Omega A(x,\nabla u)\cdot \nabla v \,dx+ \int_\Omega f(x,u) v \,dx \,. \end{align*} Hence, we have (\ref{9}). Conversely, if $u\in V_c$ is a solution of (\ref{9}) then by choosing $v\in C_0^\infty (\Omega) \subset V_c$ in (\ref{9}) and applying the divergence theorem as above, we see that %%@ (\ref{1}) holds. Choosing $v=1$ in (\ref{9}), we have $ \int_\Omega f(x,u) \,dx =0$. On the other hand, integrating %%@ (\ref{1}) over $\Omega$ and using once more the Divergence theorem yield $$ 0 = - \int_\Omega \mathop{\rm div} A(x,\nabla u) \,dx + \int_\Omega f(x,u) \,dx = -\int_{\partial\Omega} A(x,\nabla u)\cdot n \,dS . $$ Hence, we have the boundary condition (\ref{3}). \section{Sub-Supersolutions} We shall study the existence of solutions of (\ref{9}) by first defining appropriate concepts of sub- and supersolutions. \begin{definition}\label{def10a} \rm A function $\underline{u}$ (resp.\ $\overline{u}$) in $V_c$ is called a subsolution (resp.\ supersolution) of (\ref{9}) if \begin{equation} \label{12} \int_\Omega A(x, \nabla \underline{u}) \cdot \nabla v \,dx + \int_\Omega f(x, \underline{u}) v \,dx \le 0 \quad (\mbox{resp.} \ge 0) , \end{equation} for all $v\in V_c$, $v\ge 0$ a.e.\ in $\Omega$. \end{definition} \begin{rem}\rm When $A$ is the Laplacian, i.e., $A(x,\nabla u) = \nabla u$ and $p=2$, or when $N=1$ (ODE case), the above definition of sub- and supersolutions is the variational form of that given in \cite{schmitt:pss04}, without imposing additional smoothness assumptions. \end{rem} As is the case with solutions satisfying additional smoothness conditions, sub- and supersolutions, when smooth enough, satisfy additional boundary conditions. Let us see this in the case of the $p-$Laplacian. For assume that $\alpha\in V_c\cap W^{2,p}(\Omega)$ satisfies (cf.\ (17) of \cite{schmitt:pss04}): \begin{equation} \label{28} \int_\Omega |\nabla \alpha|^{p-2}\nabla \alpha \cdot \nabla \phi \,dx + \int_\Omega f(x,\alpha) \phi \,dx \le 0, \quad \forall \phi\in C_0^\infty(\Omega), \phi \ge 0 , \end{equation} and \begin{equation} \label{29} \int_{\partial\Omega}|\nabla \alpha |^{p-2} \nabla \alpha \cdot n \,dS \le 0 . \end{equation} Since $\alpha\in W^{2,p}(\Omega)$, Green's theorem (or the Divergence theorem) implies that $$ \int_\Omega [-\mathop{\rm div}\left (|\nabla \alpha |^{p-2}\nabla \alpha \right ) + f(x,\alpha)]\phi \,dx \le 0, \quad \forall \phi\in C_0^\infty(\Omega),~ \phi \ge 0 , $$ i.e., (in the sense of distributions), \begin{equation} \label{30} -\mathop{\rm div}\left (|\nabla \alpha |^{p-2}\nabla \alpha\right ) + f(x,\alpha) \le 0 \quad\mbox{ a.e.\ on } \Omega . \end{equation} Let $v\in V_c$, $v\ge 0$ a.e.\ on $\Omega$. It follows from (\ref{30}) that \begin{align*} 0 & \ge \int_\Omega [ -\mathop{\rm div}\left (|\nabla \alpha |^{p-2}\nabla \alpha\right ) + f(x,\alpha)]v \,dx \\ & = \int_\Omega |\nabla \alpha |^{p-2}\nabla \alpha \cdot \nabla v \,dx - \int_{\partial\Omega} |\nabla \alpha |^{p-2}\frac{\partial\alpha}{\partial\nu} v \,dS + \int_\Omega f(x,\alpha) v \,dx . \end{align*} Hence, $$ \int_\Omega |\nabla \alpha |^{p-2}\nabla \alpha \cdot \nabla v \,dx + \int_\Omega f(x,\alpha) v \,dx \le (v|_{\partial\Omega})\int_{\partial\Omega}|\nabla \alpha |^{p-2} \frac{\partial\alpha}{\partial\nu}\, dS \le 0 \,, $$ that is, $\alpha$ satisfies (\ref{12}). Conversely, assume $\alpha\in V_c\cap W^{2,p}(\Omega)$ satisfies (\ref{12}). Since $C_0^\infty(\Omega)\subset V_c$, we have (\ref{28}). To prove that $\alpha$ satisfies (\ref{29}), we choose a %%@ sequence $\{\Omega_n\}$ of subdomains of $\Omega$ such that \begin{equation} \label{31} \overline{\Omega_n} \subset \Omega_{n+1},\;\forall n, \quad\mbox{and} \quad \Omega = \bigcup_{n=1}^\infty \Omega_n . \end{equation} For each $n\in \mathbb N$, choose $\phi_n\in C_0^\infty(\Omega)$ such that $0 \le \phi_n(x)\le 1$, $\forall x\in\Omega$, and $\phi_n (x) = 1,\;\forall x\in \Omega_n$. Let $v_n = 1-\phi_n \; (n\in \mathbb N)$. Then $v_n\in V_c$, $v_n =1$ on $\partial\Omega$, and $0 \le v_n \le 1$ on $\Omega$. Letting $v=v_n$ in (\ref{12}), one gets \begin{equation} \label{32} \begin{aligned} 0 & \ge \int_\Omega |\nabla \alpha |^{p-2}\nabla \alpha\cdot\nabla v_n \,dx + \int_\Omega f(x,\alpha) v_n \,dx \\ & = \int_\Omega [-\mathop{\rm div}\left (|\nabla \alpha |^{p-2} \nabla\alpha\right ) + f(x,\alpha)] v_n \,dx + \int_{\partial\Omega} |\nabla \alpha |^{p-2}\frac{\partial\alpha}{\partial\nu} v_n \,dS \\ & = \int_\Omega [-\mathop{\rm div}\left (|\nabla \alpha |^{p-2} \nabla\alpha\right ) + f(x,\alpha)] v_n \,dx + \int_{\partial\Omega}|\nabla \alpha |^{p-2} \frac{\partial\alpha}{\partial\nu}\, dS . \end{aligned} \end{equation} Because $v_n =0$ on $\Omega_n$, from (\ref{31}) and the Dominated convergence theorem, one obtains $$ \lim_{n\to\infty} \int_\Omega [-\mathop{\rm div}\left (|\nabla \alpha |^{p-2} \nabla \alpha\right ) + f(x,\alpha)] v_n \,dx = 0 . $$ Letting $n\to\infty$ in (\ref{32}), we obtain $ \int_{\partial\Omega} |\nabla \alpha |^{p-2}\frac{\partial\alpha}{\partial\nu} \, dS \le 0$. \section{Existence Results} Our main existence result is the following theorem. \begin{thm}\label{thm14} Assume there exists a pair of sub- and supersolution $\underline{u}$ and $\overline{u}$ of (\ref{9}) such that $\underline{u} \le \overline{u}$ and that $f$ satisfies the following growth condition: \begin{equation} \label{14a} | f(x, u) | \le a_3 (x) , \end{equation} for a.e.\ $x\in \Omega$, all $u\in [\underline{u}(x), \overline{u}(x)]$, with $a_3\in L^{p'}(\Omega)$. Then, (\ref{9}) has a solution $u\in V_c$ such that $\underline{u}\le u \le \overline{u}$. \end{thm} \begin{proof} We define \begin{equation} \label{15} b(x,u) = \left \{\begin{array}{rcl} [u-\overline{u}(x)]^{p-1} & \mbox{if }& u > \overline{u}(x) \\ 0 & \mbox{if } & \underline{u}(x) \le u \le \overline{u}(x) \\ - [\underline{u}(x) - u]^{p-1} & \mbox{if }& u < \underline{u}(x) , \end{array} \right . \end{equation} for $x\in \Omega$, $u\in \mathbb{R}$, and \begin{equation} \label{16} (Tu) (x) = \left \{ \begin{array}{rcl} \overline{u}(x) & \mbox{if }& u(x) > \overline{u}(x) \\ u(x) & \mbox{if } & \underline{u}(x) \le u(x) \le \overline{u}(x) \\ \underline{u} (x) & \mbox{if }& u(x) < \underline{u}(x) , \end{array} \right . \end{equation} for $x\in \Omega$ and $u\in W^{1,p}(\Omega)$. Straightforward calculations show that $$ |b(x,u) | \le a_4(x) + b_4|u|^{p-1} , $$ for a.e.\ $x\in \Omega$, all $u\in \mathbb{R}$, where $b_4>0$ and $a_4\in L^{p'}(\Omega)$. Therefore, the operator $B: W^{1,p}(\Omega) \to [W^{1,p}(\Omega)]^*$ given by $$ \langle B u , v\rangle = \int_\Omega b(x,u) v \,dx \; (u,v\in W^{1,p}(\Omega)) $$ is well defined, completely continuous, and bounded. Moreover, there are $a_5, b_5 >0$ such that \begin{equation} \label{17} \langle Bu, u\rangle \ge b_5 \| u\|_{L^p(\Omega)}^p - a_5 ,\; \forall u\in W^{1,p}(\Omega) . \end{equation} Let us consider the following variational equality in $V_c$: \begin{equation} \label{18} \begin{gathered} \langle \mathcal{A} u + Bu + F(Tu) , v\rangle =0,\;\forall v\in V_c \\ u\in V_c . \end{gathered} \end{equation} It follows from (\ref{14a}) that $F\circ T$ is well defined and completely continuous from $W^{1,p}(\Omega)$ to its dual space. Because $\mathcal{A}$ is monotone, $\mathcal{A} + B + F\circ T$ is pseudo-monotone. Next, let us show that $\mathcal{A} + B + F\circ T$ is coercive on $W^{1,p}(\Omega)$ in the following sense: \begin{equation} \label{18a} \lim_{\|u\|\to\infty}\frac{\langle \mathcal{A} u + Bu + F(Tu) , u\rangle}{\| u\|} = \infty . \end{equation} In fact, from (\ref{16}) and (\ref{14a}), \begin{equation} \label{19} |\langle F(Tu),u\rangle| = \big|\int_\Omega f(x,Tu) u \,dx\big| \le \int_\Omega a_3 |u| \,dx \le \| a_3\|_{L^{p'}(\Omega)} \| u\|_{L^{p}(\Omega)} . \end{equation} Combining (\ref{19}) with (\ref{17}) and (\ref{8}), we get \begin{align*} & \langle \mathcal{A} u + Bu + F(Tu) , u\rangle \\ &\ge b_2\| |\nabla u |\|^p_{L^{p}(\Omega)} - \| a_2\|_{L^{1}(\Omega)} + b_5 \| u \|^p_{L^{p}(\Omega)} - a_5 - \| a_3\|_{L^{p'}(\Omega)} \| u\|_{L^{p}(\Omega)} \\ &\ge \min\{b_2 , b_5\} (\| u \|^p_{L^{p}(\Omega)} + \| |\nabla u |\|^p_{L^{p}(\Omega)} ) - \| a_3\|_{L^{p'}(\Omega)} \| u\| - \| a_2\|_{L^{1}(\Omega)} - a_5 \\ &= b_6 \| u\|^p - a_6 \| u\| - a_7, \quad \forall u\in W^{1,p}(\Omega) , \end{align*} with $a_6, a_7, b_6 >0$. Because $p>1$, this estimate implies (\ref{18a}). Since $V_c$ is a closed subspace of $W^{1,p}(\Omega)$, the existence of solutions of (\ref{18}) follows from classical %%@ existence theorems for elliptic variational inequalities (cf.\ e.g.\ \cite{lions:qmr69}). Assume that $u$ is any solution %%@ of (\ref{18}). We prove that \begin{equation} \label{20} \underline{u} \le u \le \overline{u} \quad \mbox{ a.e.\ in } \Omega , \end{equation} and thus $u$ is also a solution of (\ref{9}). Let us verify the first inequality in (\ref{20}). Since $u,\underline{u}\in W^{1,p}(\Omega)$, we have $(\underline{u}-u)^+\in %%@ W^{1,p}(\Omega)$. Moreover, since $u|_{\partial\Omega}$ and $\underline{u}|_{\partial\Omega}$ are constants, $$ [(\underline{u}-u)^+]|_{\partial\Omega} = \left(\underline{u}|_{\partial\Omega} - u|_{\partial\Omega}\right)^+ = \mbox{constant}, $$ i.e., \begin{equation} \label{21} (\underline{u}-u)^+ \in V_c . \end{equation} Choosing $v = (\underline{u}-u)^+$ in (\ref{18}), one obtains \begin{equation} \label{22} \int_\Omega A(x,\nabla u)\cdot \nabla [(\underline{u}-u)^+] \,dx + \int_\Omega [b(x,u) + f(Tu)] (\underline{u}-u)^+\, dx = 0. \end{equation} On the other hand, letting $v=(\underline{u}-u)^+ (\ge 0)$ in (\ref{12}) gives us \begin{equation} \label{23} \int_\Omega A(x,\nabla \underline{u})\cdot \nabla [(\underline{u}-u)^+] \,dx + \int_\Omega f(\underline{u})(\underline{u}-u)^+ \,dx \le 0. \end{equation} Subtracting (\ref{22}) from (\ref{23}) yields \begin{equation} \label{24} \begin{aligned} & \int_\Omega [A(x,\nabla \underline{u}) - A(x,\nabla u) ] \cdot \nabla [(\underline{u}-u)^+] \,dx + \int_\Omega [ f(\underline{u}) - f(Tu)] (\underline{u}-u)^+ \,dx \\ &\le \int_\Omega b(x,u) (\underline{u}-u)^+ \,dx . \end{aligned} \end{equation} Note that from (\ref{5}) and Stampacchia's theorem (cf.\ e.g.\ \cite{kinderlehrer:ivi80, gilbarg:epd83}), we have \begin{equation} \label{25} \begin{aligned} & \int_\Omega [A(x,\nabla \underline{u}) - A(x,\nabla u) ] \cdot \nabla [(\underline{u}-u)^+] \,dx \\ &= \int_{\{x\in \Omega : \underline{u}(x) > u(x)\}} [A(x,\nabla \underline{u}) - A(x,\nabla u) ] \cdot (\nabla\underline{u}- \nabla u) \,dx \\ &\ge 0 . \end{aligned} \end{equation} From the definition of $Tu$ in (\ref{16}), we have $T u(x) = \underline{u} (x) $ on $\{x\in \Omega : \underline{u}(x) > u(x)\}$ and thus \begin{equation} \label{26} \int_\Omega [ f(\underline{u}) - f(Tu)] (\underline{u}-u)^+ \,dx = \int_{\{x\in \Omega : \underline{u}(x) > u(x)\}} [ f(\underline{u}) - f(Tu)] (\underline{u}-u) \,dx = 0 . \end{equation} Using (\ref{25}) and (\ref{26}) in (\ref{24}), one obtains $$ 0 \le \int_\Omega b(x,u) (\underline{u}-u)^+ \,dx = - \int_{\{x\in \Omega : \underline{u}(x) > u(x)\}} (\underline{u}-u)^p \,dx \le 0 . $$ This implies that $$ \int_{\{x\in \Omega : \underline{u}(x) > u(x)\}} (\underline{u}-u)^p \,dx = 0, $$ i.e., $\underline{u}-u = 0$ a.e.\ on ${\{x\in \Omega : \underline{u}(x) > u(x)\}}$, or, ${\{x\in \Omega : \underline{u}(x) > u(x)\}}$ has measure 0. This shows the first inequality in (\ref{20}). The other inequality there is established in the same way. From (\ref{20}) and (\ref{15})-(\ref{16}), we immediately have $b(x,u(x)) =0$ and $Tu(x) = u(x)$ for a.e.\ $x\in \Omega$. (\ref{18}) thus becomes (\ref{9}). \end{proof} \begin{rem}\rm By modifying the proof of Theorem \ref{thm14} appropriately, we can extend that theorem to the existence of solutions of (\ref{9}) between a finite number of sub- and supersolutions. In fact, we can show that if $\underline{u}_1, \dots, \underline{u}_k$ (resp.\ $\overline{u}_1, \dots, \overline{u}_m$) are subsolutions (resp.\ supersolutions) of (\ref{9}) such that $$ \max\{\underline{u}_1, \dots, \underline{u}_k\} \le\min \{\overline{u}_1, \dots, \overline{u}_m\} , $$ and that $f$ satisfies an appropriate growth condition between these sub- and supersolutions, then there exists a solution $u$ of (\ref{9}) such that $\max\{\underline{u}_1, \dots, \underline{u}_k\} \le u \le\min \{\overline{u}_1, \dots, \overline{u}_m\}$ ( see, for example, \cite{le:bvp98} for more details). \end{rem} \begin{rem} \rm We note that in order for our method of proof of Theorem \ref{thm14} to work the important property of the subspace $V_c$ that was needed was that $u^+\in V_c$ for any $u\in V_c.$ We therefore see that Theorem \ref{thm14} remains valid, if $V_c$ is replaced by any subpsce $V$ which has this property (and, of course, the definitions of sub- and supersolutions are appropriately modified). 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