\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 124, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/124\hfil Infinite multiplicity] {Infinite multiplicity of positive solutions for singular nonlinear elliptic equations with convection term and related supercritical problems} \author[C. C. Aranda \hfil EJDE-2009/124\hfilneg] {Carlos C. Aranda} % in alphabetical order \address{Carlos C. Aranda \newline Laboratorio de modelizaci\'on, c\'alculo num\'erico y diseno experimental\\ Facultad de Recursos Naturales, Universidad Nacional de Formosa, Argentina} \email{carloscesar.aranda@gmail.com} \thanks{Submitted August 24, 2009. Published October 1, 2009.} \thanks{Supported by Secretar\'{i}a de Ciencia y Tecnolog\'{i}a, UNaF} \subjclass[2000]{35J25, 35J60} \keywords{Bifurcation; degree theory, nonlinear eigenvalues and eigenfunctions} \begin{abstract} In this article, we consider the singular nonlinear elliptic problem \begin{gather*} -\Delta u = g(u)+h(\nabla u)+f(u) \quad\text{in }\Omega, \\ u = 0 \quad\text{on }\partial\Omega. \end{gather*} Under suitable assumptions on $g$ , $h$, $f$ and $\Omega$ that allow a singularity of $g$ at the origin, we obtain infinite multiplicity results. Moreover, we state infinite multiplicity results for related boundary blow up supercritical problems and for supercritical elliptic problems with Dirichlet boundary condition. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} %\usepackage{amssymb,latexsym} \section{Introduction and statement of main results} In 1869, Lane \cite{l} introduced the equation \begin{equation}\label{lane} -\Delta u=u^p \end{equation} for $p$ a nonnegative real number and $u>0$ in a Ball of radius $R$ in $\mathbb{R}^3$, with Dirichlet boundary conditions. Lane was interested in computing both the temperature and the density of mass on the surface of the sun. Today the problem (\ref{lane}) is named Lane-Emden-Fowler equation \cite{e,f}. Singular Lane-Emden-Fowler equations ($p<0$) has been considered in a remarkable pioneering paper by Fulks and Maybe \cite{fm}. Nonlinear singular elliptic equations arise in applications, for example in glacial advance \cite{w}, ecology \cite{gl}, in transport of coal slurries down conveyor belts \cite{c}, micro-electromechanical system device \cite{egg} etc. Nonlinear singular elliptic equations have been studied intensively during the last 40 years, for a detailed review out of our scope in this article, see Hern\'andez and Mancebo \cite{hm}, and the recent book by Ghergu and R\u adulescu \cite{gr2}. Multiplicity is a question with few results. Apparently, the first multiplicity result for the problem \begin{equation}\label{queso} \begin{gathered} -\Delta u = K(x)u^{-p}+u^q \quad\text{in } \Omega, \\ u = 0 \quad\text{on } \partial\Omega, \end{gathered} \end{equation} where $\Omega$ is smooth bounded domain and \[ 0
0}f(s) /s>0$ and
$\lim_{s\to\infty}f(s) /s^{p}<\infty$ for some
$p\in(1,\frac{N}{N-2}]$;
\item $\Omega$ is a strictly convex domain in $\mathbb{R}^{N}$.
\end{enumerate}
Then the problem
\begin{gather*}
-\Delta u = g(u)+\lambda f(u) \quad\text{in }\Omega,\\
u = 0 \text{ on }\partial\Omega,
\end{gather*}
has at least two positive solutions for $\lambda$ positive
and small enough and that $\lambda=0$ is a bifurcation point
from infinity for this problem.
\end{theorem}
Our first result in this article is as follows.
\begin{theorem}\label{11}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Suppose the following conditions hold:
\begin{enumerate}
\item $g:(0,\infty) \to(0,\infty)$ is non increasing
locally H\"{o}lder continuous function (that may be singular
at the origin);
\item $f$ is continuous, nonnegative and non decreasing
function with $f(0)=0$;
\item $f(\xi_{i})\geq \beta\xi_{i}$, $f(\eta_{i})\leq \alpha\eta_{i}$
with
\[
\xi_{1}<\eta_{1}<\dots<\xi_{i}<\eta_{i}<\xi_{i+1}<\dots<\xi_{m},
\quad m\leq\infty ;
\]
\item $ \beta C(\Omega)(\int_{K}\varphi_{1})\varphi_{1}\geq 1 $, on
$K\subset\Omega$ compact where $\varphi_1$, $\lambda_1$ are
the principal eigenfunction an principal eigenvalue of the operator
$-\Delta$ ($-\Delta\varphi_1=\lambda_1\varphi_1$) with
Dirichlet boundary conditions;
\item $v+\alpha\eta_{i}e\leq\eta_{i}$, where
\begin{gather*}
-\Delta v = g(v) \quad\text{in }\Omega, \\
v = 0 \quad\text{on }\partial\Omega,
\end{gather*}
and
\begin{gather*}
-\Delta e = 1 \quad\text{in }\Omega, \\
e = 0 \quad\text{on }\partial\Omega.
\end{gather*}
\end{enumerate}
Then the problem
\begin{equation}\label{1}
\begin{gathered}
-\Delta u = g(u)+f(u) \quad\text{in }\Omega, \\
u = 0 \quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
has $m\leq\infty$ nonnegative classical solutions.
Moreover the problem
\begin{equation}\label{sumo1}
\begin{gathered}
-\Delta u = g(u)+f(u) \quad\text{in }\Omega, \\
u = \epsilon \quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
has $2m-1\leq\infty$ nonnegative classical solutions for
all $\epsilon >0$.
\end{theorem}
The behavior of the function $f$ in Theorem \ref{11} is closely
related to a similar nonlinearity studied by Kielh\"ofer and Maier
in \cite{km}. Under our best knowledge this is the first result
on infinite multiplicity for nonlinear singular equations.
Hern\'andez, Mancebo and Vega obtained the following theorem.
\begin{theorem}[\cite{hmv}]
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Suppose the following conditions hold: $-1 0$
small enough.
\end{theorem}
Existence and nonexistence results for singular nonlinear elliptic
equations with convection term have been stated by
Zhang \cite{z1}, Zhang and Yu \cite{zy}, Ghergu and
R\u adulescu \cite{gr,gr1}. Multiplicity for singular
Lane-Emden-Fowler equation with convection term is a topic
essentially open. A result was stated by Aranda and
Lami Dozo in \cite{al}:
\begin{theorem}[\cite{al}]\label{williams}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Suppose the following conditions hold:
\begin{enumerate}
\item $0 0$;
\item $ 0\leq\nu 1$, satisfying:
\begin{itemize}
\item[(a)] Let $P$ be the positive cone in $L^\infty(\Omega)$.
Let $\mathcal{S}_{\epsilon,\delta} :P\to P$ be the solution operator
for the problem (\ref{poison}), gives
$\mathcal{S}_{\epsilon,\delta}(w)=u_{\epsilon,\delta}$.
Then $\mathcal{S}_{\epsilon,\delta} :P\to P$ is continuous,
compact and non decreasing map.
\item[(b)] If $\epsilon_0<\epsilon_1$, then
$\mathcal{S}_{\epsilon_0,\delta}(w)\leq \mathcal{S}_{\epsilon_1,
\delta}(w)$.
\item[(c)] If $\delta_0<\delta_1$, then
$\mathcal{S}_{\epsilon,\delta_0}(w)\leq \mathcal{S}_{\epsilon,
\delta_1}(w)$.
\item[(d)] $\inf_{\Omega'}u_{\epsilon,\delta}\geq C$,
$\overline\Omega'\subset\Omega$, where
$C=C(p,\Omega',w(x))$ is a constant independent of $\epsilon$,
$\delta$ and $C(\alpha,\Omega',0)>0$.
\item[(e)] For $0<\epsilon,\delta<1$, we have
$\| u_{\epsilon,\delta} -\epsilon\|_{H^{1,2}_0(\Omega)}\leq C$,
where $C$ is a constant independent of $\epsilon$ and $\delta$.
\end{itemize}
\end{lemma}
\begin{remark} \label{rmk46} \rm
Items (a)--(c) contain the monotone and compactness properties
of approximate solutions.
Item (d) is a uniform Harnack inequality.
Item (e) contains a uniform bound necessary for the compensated
compactness technique.
\end{remark}
\begin{proof}[Proof of Lemma \ref{adel}]
Let $g_{j}:\mathbb{R}\to\mathbb{R}$ be a non increasing and
locally H\"older continuous function defined by
\[
g_{j}(s) = \begin{cases}
s^{-p} & \text{if } s\geq\frac{1}{j},\\
C_{j} & \text{if } s\leq \frac{1}{j+1}.
\end{cases}
\]
Using a standard argument involving $L^r$ estimates
\cite[Theorem 9.10]{gt},
Sobolev imbedding \cite[Theorem 7.26]{gt}, \cite[Theorem 10.1]{gt}
and the Schauder fixed point Theorem, we deduce that the problem
\begin{gather*}
-\Delta u_{j,\epsilon,\delta} +\frac{u_{j,\epsilon,
\delta}|\nabla u_{j,\epsilon,\delta}|^2 }{1+\delta u_{j,\epsilon,\delta}|
\nabla u_{j,\epsilon,\delta}|^2}
= g_j(u_{j,\epsilon,\delta})+w(x) \quad\text{in } \Omega, \\
u_{j,\epsilon,\delta} = \epsilon \quad\text{on } \partial\Omega,
\end{gather*}
has a unique solution
$u_{j,\epsilon,\delta}\in\mathcal{W}^{2,r}(\Omega)\cap C(\overline\Omega)$
for all $r>1$. If $w\in L^r(\Omega)$, $r>N$, then by
\cite[Theorem 7.26]{gt},
$u_{j,\epsilon,\delta}\in\mathcal{W}^{2,r}(\Omega)
\hookrightarrow C^{1,\gamma}(\overline\Omega)$ for some $\gamma>0$.
Calling
\[
b_\delta(u,\nabla u)=\frac{u|\nabla u|^2}{1+\delta u|\nabla u|^2},
\]
we deduce
\begin{align*}
&-\Delta u_{j,\epsilon,\delta}+b_\delta(u_{j,\epsilon,\delta},
\nabla u_{j,\epsilon,\delta})-g_{j+1}(u_{j,\epsilon,\delta}) \\
&\leq -\Delta u_{j,\epsilon,\delta}+b_\delta(u_{j,\epsilon,\delta},
\nabla u_{j,\epsilon,\delta})-g_{j}(u_{j,\epsilon,\delta}) \\
&= w(x)\\
&= -\Delta u_{j+1,\epsilon,\delta}+b_\delta(u_{j+1,\epsilon,\delta},
\nabla u_{j+1,\epsilon,\delta})-g_{j+1}(u_{j+1,\epsilon,\delta})
\end{align*}
in $\Omega$ and $u_{j+1,\epsilon,\delta}=u_{j,\epsilon,\delta}=\epsilon$
on $\partial\Omega$. Using Theorem 10.1 \cite{gt}, we obtain that
$u_{j+1,\epsilon,\delta}>u_{j,\epsilon,\delta}$ in $\Omega$.
Moreover from
\[
-\Delta u_{j,\epsilon,\delta} +b_\delta (u_{j+1,\epsilon,\delta},
\nabla u_{j+1,\epsilon,\delta})
= g_j(u_{j,\epsilon,\delta})+w(x)
\geq -\Delta\epsilon+b_\delta(\epsilon,\nabla\epsilon)\quad
\text{in }\Omega,
\]
and $u_{j,\epsilon,\delta}=\epsilon$ on $\partial\Omega$, using
again \cite[Theorem 10.1]{gt}, we conclude
$u_{j,\epsilon,\delta}>\epsilon$ on $\Omega$. Letting
$u_{\epsilon,\delta}=\lim_{j\to\infty}u_{j,\epsilon,\delta}$, we have
\begin{gather*}
-\Delta u_{\epsilon,\delta} +b_\delta(u_{\epsilon,\delta},
\nabla u_{\epsilon,\delta}) =
u_{\epsilon,\delta}^{-p}+w(x) \quad\text{in } \Omega, \\
u_{\epsilon,\delta} = \epsilon \quad\text{on } \partial\Omega.
\end{gather*}
Using standard Nemytskii mappings properties and Sobolev Imbedding
Theorems, we demonstrate the continuity and compacity of the map
$\mathcal{S}_{\delta,\epsilon}$. This states ($\mathfrak{a}$).
Comparison \cite[Theorem 10.1]{gt} implies if
$\epsilon_0<\epsilon_1$ then
$\mathcal{S}_{\epsilon_0,\delta}(w)=u_{\epsilon_0,
\delta}0$.
Then the problem
\begin{equation}\label{e5}
\begin{gathered}
-\Delta u = \lambda u^{-q}-u^{-p} \quad\text{in }\Omega, \\
u = 0 \quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
has a unique nonnegative classical solution.
\end{theorem}
Our second Theorem is related to multiplicity of a nonlinear
eigenvalue problem.
\begin{theorem}\label{drno5}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Suppose the following conditions hold:
\begin{enumerate}
\item $0
N$ with $\mathfrak{u}\geq \xi_i\kappa$;
\item $v(x)+\alpha\eta_{i}e(x)<\eta_{i}$ for all $x\in\overline{\Omega}$,
where
\begin{gather*}
-\Delta v = v^{-p} \quad\text{in }\Omega, \\
v = 0 \quad\text{on }\partial\Omega,
\end{gather*}
and
\begin{gather*}
-\Delta e = 1 \quad\text{in }\Omega, \\
e = 0 \quad\text{on }\partial\Omega.
\end{gather*}
\end{enumerate}
Then the problem
\begin{equation}\label{alice}
\begin{gathered}
-\Delta u +u|\nabla u|^2 = u^{-p}+f(u) \quad\text{in } \Omega, \\
u = 0 \quad\text{on } \partial\Omega,
\end{gathered}
\end{equation}
has $m$ solutions in $H^{1,2}_0(\Omega)$.
\end{theorem}
\begin{remark} \label{rmk2} \rm
Condition (3) indicates again a complex relation between domain,
convection term and multiplicity.
\end{remark}
For large solutions Ghergu et al. stated the following result.
\begin{theorem}[\cite{gnr}]
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Suppose the following conditions hold:
\begin{enumerate}
\item $f\in C^1[0,\infty)$, $f'\geq 0$, $f(0)=0$ and $f>0$ on
$(0,\infty)$;
\item $\int_1^\infty\left[F(t)\right]^{-2/a}dt<\infty$, where
\item $\frac{F(t)}{f^{2/a}}\to 0$ as $t \to 0$;
\item $\mathfrak{p}$, $\mathfrak{q}\in C^{0,\gamma}(\overline\Omega)$
are nonnegative functions such
that for every $x_0\in\Omega$ with $\mathfrak{p}(x_0)=0$,
there exists a domain $\Omega_0\ni x_0$ such that
$\overline\Omega_0\subset\Omega$ and $\mathfrak{p}>0$ on
$\partial\Omega_0$;
\item $02$.
Then the problem
\begin{equation} \label{e11}
\begin{gathered}
\Delta v = \frac{2}{v}|\nabla v|^2+v^p+\tilde{f}(u)
\quad\text{in }\Omega, \\
u = \infty \quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
has $m\leq\infty$ nonnegative classical solutions.
Moreover the problem
\begin{equation} \label{e12}
\begin{gathered}
\Delta v = \frac{2}{v}|\nabla v|^2+v^p+\tilde{f}(u)
\quad\text{in }\Omega, \\
u = M \quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
has $2m-1\leq\infty$ nonnegative classical solutions for all
$M >0$ big enough.
\end{theorem}
\begin{theorem}\label{supercritico}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Suppose the following conditions hold:
\begin{enumerate}
\item $f(s)=s^2\tilde{f}(\frac{1}{s})$ satisfies
{\rm (2)--(4)} of Theorem \ref{11};
\item $2 0 \quad\text{in } \Omega,\\
\varphi_{1} = 0 \quad\text{on } \partial\Omega.
\end{gather*}
\end{lemma}
\begin{remark} \label{rmk3} \rm
The proof of Lemma \ref{hopf} given by Brezis and Cabre in \cite{bc}
relies on the superharmonicity of the laplacian operator.
\end{remark}
\begin{theorem}[\cite{ag}]\label{H}
Let $P$ be the positive cone in $L^\infty(\Omega)$.
Let $S_{\epsilon}:P\to P$ be the solution operator
for the problem
\begin{gather*}
-\Delta u = g(u)+w \quad\text{in }\Omega, \\
u = \epsilon \quad\text{on }\partial\Omega,
\end{gather*}
gives $S_{\epsilon}(w)=u$ where $\epsilon\geq 0$ and
$g:(0,\infty) \to(0,\infty)$ is nonincreasing
locally H\"{o}lder continuous function (that may be singular
at the origin). Then $S_{\epsilon}:P\to P$ is a continuous,
non decreasing and compact map with
$S_{\epsilon_0}(w)\leq S_{\epsilon_1}(w)$ for $\epsilon_0<\epsilon_1$.
\end{theorem}
\begin{lemma}\label{comparacion}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Let $u,v\in C^{2}(\Omega)\cap C(\overline{\Omega})$
be solutions of the problem
\begin{gather*}
-\Delta u -g(u)-h(\nabla u) \geq -\Delta v -g(v)-h(\nabla v)
\quad\text{in }\Omega, \\
u \geq v\geq 0 \quad\text{on }\partial\Omega.
\end{gather*}
Then $u \geq v $ on $\Omega$.
\end{lemma}
\begin{proof}
Indeed suppose $v>u$ somewhere and consider the non empty open set
\[
\Omega_{\delta}=\{x\in\Omega |v(x)>u(x)+\delta, \; \delta >0\}.
\]
Since $u,v\in C^{2}(\Omega)$, we have
\begin{align*}
-\Delta (u+\delta) -h(\nabla (u+\delta))
& = g(u)+q\\
& \geq g(v)+r \\
& = -\Delta v -h(\nabla v) \quad\text{on }\Omega_{\delta},
\end{align*}
with $q,r\in C(\overline{\Omega_{\delta}})$
and $\overline{\Omega_{\delta}}\subset\Omega$.
Also $u+\delta=v $ on $\partial\Omega_{\delta}$ and
so the comparison Theorem 10.1 \cite{gt} implies
$u+\delta\geq v $ on $\Omega_{\delta}$.
It follows $\Omega_{\delta}=\emptyset$ a contradiction.
\end{proof}
\begin{lemma}\label{m29}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Suppose the following conditions hold:
\begin{enumerate}
\item $g:(0,\infty) \to(0,\infty)$ is non increasing
locally H\"{o}lder continuous function (that may be singular
at the origin);
\item $h$ is locally H\"older continuous function on $\mathbb{R}^N$
with $0\leq h( \nabla u )\leq b_{1} | \nabla u |^{s}+b_{0}$,
$ 0
0 \quad\text{in }\partial\Omega,
\end{gather*}
has a classical solution. From
\begin{align*}
\Delta u_{j-1} + g_{j}(u_{j-1})+h(\nabla u_{j-1})+w(x)
& \geq \Delta u_{j-1} + g_{j-1}(u_{j-1})+h(\nabla u_{j-1})+w(x) \\
& = 0 \\
& = \Delta u_{j} + g_{j}(u_{j})+h(\nabla u_{j})+w(x)
\quad\text{in }\Omega,
\end{align*}
$u_{j-1}=u_{j}=\epsilon $ on $\partial\Omega$,
using \cite[Theorem 10.1]{gt}, we infer $u_{j-1}\leq u_{j} $ in
$\Omega$. Therefore for $j$ big enough there exists an unique
$u_{\epsilon}=u_{j}$ solution of
\begin{gather*}
-\Delta u_{\epsilon}
= g(u_{\epsilon})+h(\nabla u_{\epsilon})+w(x) \quad\text{in }\Omega , \\
u_{\epsilon} = \epsilon \quad\text{on }\partial\Omega.
\end{gather*}
If $\epsilon_{0}<\epsilon_{1}$, for $j$ big enough, we have
\[
\Delta u_{\epsilon_{0}} + g_{j}(u_{\epsilon_{0}})
+h(\nabla u_{\epsilon_{0}})+w(x)
= \Delta u_{\epsilon_{1}} + g_{j}(u_{\epsilon_{1}})
+h(\nabla u_{\epsilon_{1}})+w(x),
\]
on $\Omega$, $u_{\epsilon_{0}}0$.
\end{proof}
\begin{lemma}\label{adel}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$.
Suppose that $0