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\headline={\ifnum\pageno=1 \hfill\else%
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\def\rightheadline{EJDE--2001/29\hfil A priori estimates for global solutions
\hfil\folio}
\def\leftheadline{\folio\hfil Pavol Quittner
 \hfil EJDE--2001/29}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol. {\eightbf 2001}(2001), No. 29, pp. 1--17.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break 
ftp ejde.math.swt.edu (login: ftp)\bigskip} }

\topmatter
\title
A priori estimates for global solutions and multiple equilibria
of a superlinear parabolic problem involving measures
\endtitle

\thanks 
{\it 2000 Mathematics Subject Classifications:} 35B45, 35J65, 35K60.
\hfil\break\indent
{\it Key words:} superlinear parabolic equation, semilinear elliptic equation,
   multiplicity, \hfil\break\indent singular solutions.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted December 14, 200. Published May 2, 2001.
\endthanks

\author  Pavol Quittner  \endauthor

\address 
  Pavol Quittner \hfill\break
  Institute of Applied Mathematics, Comenius University, 
  Mlynsk\'a dolina, SK -- 84248 Bratislava, Slovakia 
   \endaddress
\email quittner\@fmph.uniba.sk \endemail

\abstract
We consider a noncoercive elliptic problem in a bounded domain
with a power nonlinearity and measure data.
It is known that the problem possesses a stable solution
and we prove existence of three further solutions.
The proof is based on uniform bounds of global solutions of
the corresponding parabolic problem and 
on a topological degree argument.
\endabstract
\endtopmatter

\document

\def\eq#1{\ifmmode \eqno{(#1)}\else \hbox{\rm(#1)}\fi}
\def\W#1#2{W^{#1}_{#2,\gamma}}
\def\Tmax{T_{\hbox{\sevenrm max}}}


\head{1. INTRODUCTION} \endhead

In this paper we consider the problem
$$ \gathered
  u_t = \Delta u+|u|^{p-1}u+\mu,
                    \quad x\in\Omega,\ t>0, \\
  u =0,            \quad x\in\partial\Omega,\ t>0,\\
  u(x,0) = u_0(x), \quad x\in\bar\Omega,
\endgathered \eq{1.1}
$$
where $\Omega\subset{\Bbb R}^n$ is a smoothly bounded domain with $n\geq2$,
$\mu$ is a positive bounded Radon measure on $\Omega$ and
$$ p>1,\quad  p<\frac{n}{n-2}\ \hbox{ if }n>2.\eq{1.2}
$$
The restriction $p(n-2)<n$ is not of technical nature,
it is necessary for the local existence of the solution
(see \cite{7} or \cite{26} and the references therein).

If $\mu=0$ and $p>1$, $p(n-2)<n+2$, then
the Ljusternik-Schnirelman theory guarantees the existence of
infinitely many stationary solutions of problem \eq{1.1}.
A generalization of this result for $\mu\ne0$, $\mu$ regular,
was obtained under various additional assumptions on $\mu$ and $p$
using perturbation methods in  \cite{25}, \cite{5}, \cite{23},
\cite{6} (see \cite{9} and the references therein
for the case of non-homogeneous boundary conditions and related
problems).
Variational methods were used also in \cite{27} for the proof of existence of
at least four solutions of the homogeneous Dirichlet problem
for the equation $0=\Delta u+f(u)$, where $f$ was a superlinear
(non-symmetric)  $C^1$-function with subcritical growth
(see \cite{8} for additional properties of these solutions
and further references).

In the present paper, we assume \eq{1.2} and 
we consider a general measure $\mu$ of the form
$$\mu=a\mu_0,\quad
\hbox{ where $a\in{\Bbb R}^+$ and
 $\mu_0$ is a bounded positive Radon measure on $\Omega$}.
\eq{1.3}$$
Denote
$$ a^* = \sup \{a>0\,:\, \hbox{\eq{1.1} has a positive equilibrium}\}. $$
It follows from \cite{7} (see also \cite{4} for a more general setting)
that $a^*>0$. Assuming
$$ 0<a<a^*, \eq{1.4}$$
we show existence of at least four
stationary solutions of \eq{1.1}.
We use a dynamical method which consists in looking for
stationary solutions in the $\omega$-limit sets of some global
trajectories of \eq{1.1}.
This approach does not require any symmetry of the problem
so that one can use it also for more general problems
(for example, $u_t=\Delta u+f(u)+\mu$, where $f$ is as in \cite{27}).   
In general, our method can yield different solutions from those
obtained by variational methods:
see \cite{19}, where it is used in the study
of the Dirichlet problem for the equation
$0=\Delta u+ u_+^p-u_-^q$ with $0<q<1<p$, $p(n-2)<n+2$.

The crucial prerequisites for our approach are a priori estimates
for global solutions of \eq{1.1}.
In the case $\mu=0$ and $p>1$, $p(n-2)<n+2$,
it is known that all global solutions of \eq{1.1} are
bounded and the corresponding bound depends only on
a suitable norm of the initial function $u_0$, see \cite{20}.
In this paper we generalize
the a priori estimates from \cite{20} to the case $\mu\ne0$
(under assumptions \eq{1.2},\eq{1.3},\eq{1.4})
and then we use these estimates for the dynamical
proof of existence of multiple equilibria.
The main difficulty in this generalization and the subsequent application
consists in the fact that the solutions of \eq{1.1} are not
regular enough for the direct use of the technical tools
exploited in \cite{20},\cite{19} (for example,
the standard Lyapunov functional is not well defined
in our situation).
These difficulties also rule out a straightforward use of variational
methods for the proof of the multiplicity result.

Positive stationary solutions of problem \eq{1.1} were studied
by several authors, see references in \cite{4}.
If problem \eq{1.1} has a positive equilibrium then there exists
a minimal positive equilibrium $v_1$ of this problem (see \cite{4}).
It follows from \cite{4, Theorems~1.2,~1.3}
that assuming \eq{1.2},\eq{1.3},\eq{1.4},
problem \eq{1.1} admits at least two positive equilibria.
The proof of this multiplicity result
was based on a priori estimates of positive
stationary solutions and the computation of the Leary-Schauder index
of the solution $v_1$.
In the present paper, we shall use the local information on
the solution $v_1$ and our a priori estimates
of global solutions of \eq{1.1} in order to prove the existence
of equilibria $v_2,v_3,v_4$ such that $v_2>v_1>v_3$ and $v_4-v_1$ changes sign.

Assumption \eq{1.4} is crucial also for the proof of a priori
estimates of global solutions of \eq{1.1}: instead of
estimating the singular solution $u(t)$ (which need not be even
continuous, in general), we estimate the difference $w(t)=u(t)-v_1$
which turns out to be a H\"older continuous function.

Let us mention that a priori estimates of global or periodic
solutions of similar superlinear parabolic problems with regular data
were already used for the proof of existence
of positive stationary solutions (see \cite{17}, \cite{10}, \cite{24}),
sign-changing stationary solutions
(see \cite{19}, \cite{13}),
infinitely many stationary solutions (see \cite{22}),
periodic solutions
(see \cite{11}, \cite{12}, \cite{16}),
for establishing the blow-up rate of nonglobal solutions
(see \cite{15}, \cite{14}),
and for the study of the boundary of domains of attraction
of stable equilibria (see \cite{18}).

This paper is organized as follows.
Section~2 deals with existence and regularity of solutions of \eq{1.1}.
Main results of the paper are stated in
Theorem~3.1 (a~priori estimates) and Theorem~4.5
(existence of multiple stationary solutions).

%______________________________________________________________________________
\head{2. PRELIMINARIES} \endhead

Let $q\in(1,\infty)$, $q'=q/(q-1)$,
let $W^z_q(\Omega)$, $z\geq0$, denote the usual Sobolev-Slobodeckii
space and $\gamma$ the trace operator,
$\gamma:W^z_q(\Omega)\to W^{z-1/q}_q(\partial\Omega)$ for $z>1/q$.
For $\theta\in I_q:=[-2,2]\setminus\{1/q+m\,:\,m\in{\Bbb Z}\}$ put
$$ W^\theta:=\W\theta q:=\left.\cases
       \{u\in W^\theta_q(\Omega)\,:\,\gamma u=0 \}
            &\hbox{if }1/q<\theta,\\
      W^\theta_q(\Omega) &\hbox{if }0\leq \theta<1/q,\\
      \big(\W{-\theta}{q'}\big)' &\hbox{if }\theta<0,
 \endcases \quad \right\}\eq{2.1}$$
and let $|\cdot|_{\theta,q}$ denote the norm in $\W\theta q$.
The norm in $\W 0q=L_q(\Omega)$ will be denoted simply by $|\cdot|_q$.
The norm in the H\"older space $C^{0,\alpha}(\bar\Omega)$ will be denoted
by $\|\cdot\|_{0,\alpha}$.

Let $M=M(\Omega)$ be the space of bounded Radon measures on $\Omega$.
The spaces $\W\theta q$ are ordered Banach spaces and $M(\Omega)$ is a
Banach lattice (cf.~\cite{4, Section~5}).
Moreover, $M(\Omega)\hookrightarrow\W\theta q$ provided $\theta<-n/q'$.
For $u,v\in\W\theta q$, we write $u<v$ if $v-u$ belongs to the positive
cone of $\W\theta q$ and $u\ne v$. We denote also
$[u,v]=\{w\,:\,u\leq w\leq v\}$
and we put $a\wedge b:=\min\{a,b\}$, $a\vee b:=\max\{a,b\}$.
By $c$ and $C$ we denote positive constants which may vary from step to step;
by $c_1,C_1,c_2,C_2,\dots$ we denote fixed positive constants.

Let $A_2:W^2\to W^0:u\mapsto-\Delta u$.
It is well known that $A_2$ is an isomorphism of $W^2$ onto $W^0$
and it generates an analytic semigroup in $W^0$.
Moreover, the operator $A_2$ can be extended to an isomorphism
$A_0:W^0\to W^{-2}$ such that the $W^{\theta-2}$-realization
$A_\theta$ of $A_0$
is an isomorphism of $W^\theta$ onto $W^{\theta-2}$ and it generates
an analytic semigroup $e^{-tA_\theta}$ in $W^{\theta-2}$
for any $\theta\in I_q$, $\theta\geq0$,
$$ |e^{-tA_\theta}u|_{\eta,q}
 \leq Ce^{-ct}\big(|u|_{\eta,q}\wedge t^{(\theta-\eta)/2-1}|u|_{\theta-2,q}\big)
 \eq{2.2}$$
for any $\eta\in I_q\cap(\theta-2,\theta)$ (see \cite{2}).

The results of \cite{2, Section~12} imply that problem \eq{1.1}
admits a unique maximal solution $u\in C\big([0,T),\W zq\big)$
satisfying the variation-of-constants formula
$$ u(t) = e^{-tA_z}u_0+\int_0^te^{-(t-\tau)A_z}(|u(\tau)|^{p-1}u(\tau)+\mu)\,d\tau
\eq{2.3}$$
provided   $u_0\in\W zq$ and
$$-\frac{n}{p}\leq z-\frac{n}{q}<2-n,\quad  q>1,\quad z\geq0, \quad z\in I_q.
\eq{2.4}$$

The existence of a unique $u$ satisfying (2.3)
can be proved directly in the following way.
Condition \eq{2.4} guarantees  $\W zq\hookrightarrow L_p(\Omega)$ and
$L_1(\Omega)\hookrightarrow M(\Omega)\hookrightarrow\W{z-2+\varepsilon}q$ for some $\varepsilon>0$,
hence the mapping $F:\W zq\to\W{z-2+\varepsilon}q:u\mapsto |u|^{p-1}u+\mu$
is well defined and Lipschitz continuous.
Now using \eq{2.2} we obtain
$$\aligned
|e^{-tA_z}u_0|_{z,q} &\leq C|u_0|_{z,q}\leq C, \\
|e^{-tA_z}F(u)|_{z,q}
  &\leq Ct^{-1+\varepsilon/2}|F(u)|_{z-2+\varepsilon,q}  \\
  &\leq Ct^{-1+\varepsilon/2}\big(1+|u|_{z,q}^p\big), \\
|e^{-tA_z}\big(F(u)-F(v)\big)|_{z,q}
  &\leq Ct^{-1+\varepsilon/2}\big(1+|u|_{z,q}^{p-1}+|v|_{z,q}^{p-1}\big)
    |u-v|_{z,q}.
\endaligned \eq{2.5}$$
These inequalities easily imply that the operator
$$ R(u)(t) = e^{-tA_z}u_0+
   \int_0^te^{-(t-\tau)A_z} F\big(u(\tau)\big)\,d\tau$$
is a contraction
in an appropriate ball of the Banach space $C\big([0,T],\W zq\big)$
if $T$ is small enough.
The fixed point of $R$ is the solution of \eq{2.3}, hence of \eq{1.1}.

Solutions of \eq{1.1} are not continuous, in general.
Anyhow, if $u,v:[0,T]\to\W zq$ are two solutions of \eq{1.1}
with initial conditions $u_0,v_0$, respectively,
then the difference $w(t)=u(t)-v(t)$ is H\"older continuous for $t>0$
and its $C^{0,\alpha}(\bar\Omega)$-norm
(where $\alpha>0$ is sufficiently small)
can be estimated by the $\W zq$-norm of $w(0)$.
More precisely, the following lemma is true.

\proclaim{Lemma~2.1}
Let $u,v:[0,T]\to\W zq$ be two solutions of \eq{1.1}
with initial conditions $u(\cdot,0)=u_0$, $v(\cdot,0)=v_0$.
Put $w=u-v$ and denote
$$K_u=\sup_{\tau\in[0,T]}|u(\tau)|_{z,q}.\eq{2.6}$$
There exist $r>n$ and $\alpha>0$ such that
$w(t)\in C^{0,\alpha}(\bar\Omega)\cap \W1r$
for any $t>0$ and
$$ |w(t)|_{z,q}+|w(t)|_{1,r}+\|w(t)\|_{0,\alpha}
  \leq c(t_0,T,K_u\vee K_v)|w(0)|_{z,q} \eq{2.7}$$
for any $t\in[t_0,T]$ and $t_0>0$.
Moreover, $w\in C^{0,\tilde\alpha}([t_0,T],C^{0,\alpha}(\bar\Omega)\cap \W1r)$
for some $\tilde\alpha>0$
and the norm of $w$ in this space can be bounded by a constant
depending on $t_0,T,K_u\vee K_v$.
\endproclaim

\demo{Proof}
Let $\tilde z,q$ satisfy \eq{2.4} (with $z$ replaced by $\tilde z$),
$\tilde z>z$.
Estimating the $\W {\tilde z}q$-norm in \eq{2.3} we obtain
$$ \align
|u(t)|_{\tilde z,q}
 &\leq Ct^{z-\tilde z}|u_0|_{z,q}
 +C\int_0^t e^{-c(t-\tau)}(t-\tau)^{-1+\tilde\varepsilon/2}
     \big|F\big(u(\tau)\big)\big|_{\tilde z-2+\tilde\varepsilon}\,d\tau \\
 &\leq C(K_u)(1+t^{z-\tilde z}),
\endalign $$
where $\tilde\varepsilon>0$ is small enough.
Using the imbedding
$\W{z_1}{q_1}\hookrightarrow\W{z_2}{q_2}$ if $z_1-\frac{n}{q_1}>z_2-\frac{n}{q_2}$
and $z_1\geq z_2$ and repeating the estimate above with different $z,\tilde z,q$,
if necessary, we get
$$ |u(t)|_{\tilde z,\tilde q} \leq C(\delta,K_u)
   \quad \hbox{for any } t\in[\delta,T],\eq{2.8}$$
whenever $\tilde z,\tilde q$ satisfy \eq{2.4}.
Analogous estimates and the generalized Gronwall inequality
\cite{1, Theorem~II.3.3.1} imply
$$ |w(t)|_{\tilde z,\tilde q} \leq C(\delta,T,K_u,K_v)|w(0)|_{z,q}
   \quad \hbox{for any } t\in[\delta,T].\eq{2.9}$$
Estimates \eq{2.8} and \eq{2.9} imply that
we may assume both $z=1$ and
the boundedness of $u(\tau),v(\tau)$, $\tau\in[0,T]$,
in $\W{\tilde z}{\tilde q}$
for any $\tilde z,\tilde q$ satisfying \eq{2.4}.
In particular, $u(\tau),v(\tau)$ are bounded in $\W1q$ for any $q<n/(n-1)$,
hence in $L_r(\Omega)$ for any $r<n/(n-2)$.

The function $w$ solves the equation
$w_t=\Delta w+\Phi(u,v)$ in $\Omega$ with
$$ |\Phi(u,v)|=\big||u|^{p-1}u-|v|^{p-1}v\big|\leq C|w|\varphi, \quad 
  \varphi:=\big(|u|+|v|\big)^{p-1},$$
where the function $\varphi(t)$ is bounded in $L_s(\Omega)$ for some $s>\frac n2$.
Put $Q=q$, $R=1$ and choose $\beta>0$ and $\varepsilon>0$ small.
Then
$$ L_R(\Omega)\hookrightarrow \W{\beta-1+\varepsilon}Q
   \quad \hbox{and}\quad 
   \W{1+\beta}Q\hookrightarrow L_{sR/(s-R)} $$
due to
$$ \frac1R<\frac1Q+\frac{1-\beta}n
\quad \hbox{and}\quad 
 \frac1R\geq\frac1Q+\frac1s-\frac{1+\beta}n, \quad s>R,
 \eq{2.10}$$
hence
$$\align
|\Phi\big(u(\tau),v(\tau)\big)|_{\beta-1+\varepsilon,Q}
  &\leq C|\Phi\big(u(\tau),v(\tau)\big)|_R
  = C|w(\tau)\varphi(\tau)|_R \\
  &\leq C|w(\tau)|_{sR/(s-R)}|\varphi(\tau)|_s
  \leq C|w(\tau)|_{1+\beta,Q} .
\endalign  $$
This and the variation-of-constants formula imply
$$\align
    |w(t)|_{1+\beta,Q}
    &\leq Ct^{-\beta}|w_0|_{1,Q}+C\int_0^t(t-\tau)^{-1+\varepsilon/2}
        |\Phi\big(u(\tau),v(\tau)\big)|_{\beta-1+\varepsilon,Q}\,d\tau \\
    &\leq Ct^{-\beta}|w_0|_{1,Q}
      + C\int_0^t(t-\tau)^{-1+\varepsilon/2} |w(\tau)|_{1+\beta,Q},
\endalign$$
so that the Gronwall inequality implies
$$|w(t)|_{1+\beta,Q}\leq C_1(t)|w_0|_{1,Q},$$
where $C_1(t)$ is bounded for $t$ lying in compact subsets of $(0,T]$.
Now $\W{1+\beta}Q \hookrightarrow\W1{\tilde Q}$ for some $\tilde Q>Q$.
If $\tilde Q\leq n$ then
repeating the estimates above with $Q$ replaced by $\tilde Q$,
$R$ by $\tilde R$ and $\beta$ by $\tilde\beta$ such that \eq{2.10}
remains true, we obtain
$$|w(2t)|_{1+\tilde\beta,\tilde Q}\leq C_2(t)|w(t)|_{1,\tilde Q}
 \leq CC_1(t)C_2(t)|w_0|_{1,Q}.$$
A standard bootstrap argument yields the estimate of $w$
in $\W1{Q}$ for some $Q>n$,
hence in $C^{0,\alpha}(\bar\Omega)$ for some $\alpha>0$.
This shows \eq{2.7}.
Notice that an upper bound for the bootstrap procedure is given by
$1/Q>1/s-1/n$.

The H\"older continuity of $w:[t_0,T]\to \W1Q$ for some $Q>n$
follows from the variation-of-constants formula,
the estimates above and the estimates
$$\align
\Big|\int_{t_1}^{t_2}e^{-(t_2-\tau)A_z}\Phi(u(\tau),v(\tau))\,d\tau\Big|_{1,Q}
 &\leq C\int_{t_1}^{t_2}(t_2-\tau)^{-1+\varepsilon/2}\,d\tau
  \leq C(t_2-t_1)^{\varepsilon/2}, \\
|e^{-(t_2-t_1)A_z}f-f|_{1,Q} &\leq C(t_2-t_1)^{\beta/2}|f|_{1+\beta,Q},
\endalign $$
where $T\geq t_2>t_1\geq t_0$ and
$f:=\int_0^{t_1}e^{-(t_1-\tau)A_z}\Phi(u(\tau),v(\tau))\,d\tau$
(cf.~also \cite{1, Theorem~II.5.3.1}).
\qed\enddemo

\proclaim{Remark~2.2}\rm
Let $u$ and $v$ be solutions of \eq{1.1} on $[0,T]$
with initial conditions $u(\cdot,0)=u_0$ and $v(\cdot,0)=v_0$,
where $u_0,v_0\in\W zq$ and $z,q$ satisfy \eq{2.4}.
Let $\mu_k\to\mu$ in $M(\Omega)$ and
$u_{0,k}\to u_0$, $v_{0,k}\to v_0$ in $\W zq$.
Let $u_k,v_k$  be solutions
of \eq{1.1} with $\mu$ replaced by $\mu_k$
and initial conditions $u_k(\cdot,0)=u_{0,k}$, $v_k(\cdot,0)=v_{0,k}$.
Put $w=u-v$, $w_k=u_k-v_k$.
Then the variation-of-constants formula, estimates \eq{2.5},
Gronwall's inequality and obvious modifications of estimates
in the proof of Lemma~2.1
imply that $u_k,v_k$ are well
defined on $[0,T]$ for $k$ large enough,
$\sup_{t\in[0,T]}|u(t)-u_k(t)|_{z,q}\to0$ as $k\to\infty$
and
$\sup_{t\in[t_0,T]}|w(t)-w_k(t)|_{1,r}\to0$ as $k\to\infty$
for some $r>n$ and any $t_0>0$.
\endproclaim

The following theorem follows from \cite{21, Theorem~3.1}.

\proclaim{Theorem~2.3}
Let  $z,q$ satisfy \eq{2.4} and
$u\in C\big([0,T),\W zq\big)$ be the maximal solution of \eq{1.1}.
Let $u(t)$ be bounded in $L_r(\Omega)$ for $t\in[0,T)$, where
$r>\frac{n}{2}(p-1)$, $r>1$.
Then $T=+\infty$ and $u(t)$ is bounded in $\W zq$ for $t\in[0,\infty)$.
\endproclaim

%______________________________________________________________________________
\head{3. A PRIORI ESTIMATES} \endhead

The main result of this section is the following

\proclaim{Theorem~3.1}
Assume \eq{1.2},\eq{1.3},\eq{1.4}.
Let $u$ be a global solution of \eq{1.1} and
let $z,q$ satisfy \eq{2.4}.
Then $|u(t)|_{z,q}\leq c$, where $c$ depends only
on the norm of $u_0$ in $\W zq$.
\endproclaim

\demo{Proof}
Assumption \eq{1.4} guarantees existence of
the minimal positive stationary solution $v_1$.
Let $u$ be a global solution of \eq{1.1} and put
$w(t):=u(t)-v_1$.

The functions $v_1$ and $w$ are solutions of the following problems
$$ \gathered
  0 = \Delta v_1 + v_1^p + \mu, \quad  x\in\Omega, \\
  v_1 = 0,                     \quad  x\in\partial\Omega,
\endgathered \eq{3.1} $$
and
$$ \gathered 
   w_t = \Delta w+h(w), \quad  x\in\Omega,\ t>0, \\
   w   = 0,  \quad  x\in\partial\Omega,\ t>0, \\
   w(x,0) = u_0(x)-v_1(x), \quad  x\in\bar\Omega
\endgathered \eq{3.2} $$
where
$$ h(w):=|w+v_1|^{p-1}(w+v_1)-v_1^p.$$

Due to Lemma~2.1, we have
$$w(t)\in \W1r\hookrightarrow C^{0,\alpha}(\bar\Omega)$$
for some $r>n$, $\alpha>0$ and any $t>0$.
Moreover, putting
$$ f(w):= \frac1{p+1}\big(|w+v_1|^{p+1}-v_1^{p+1}\big)-wv_1^p, $$
the regularity of $w$ and the mean value theorem imply
$$ |f(w)|\leq p\big(|w|+|v_1|\big)^{p-1}|w|^2 \leq C(w)\big(1+|v_1|^{p-1}\big)
   \in L_s(\Omega)$$
for some $s>\frac{n}{2}$, since $v_1\in L_r(\Omega)$ for any $r<\frac n{n-2}$
(see \cite{4}).
Consequently, the energy functional
$$ E(w):=\frac12\int_\Omega|\nabla w|^2\,dx-\int_\Omega f(w)\,dx\eq{3.3}$$
is well defined along the solution $w=w(t)$.
Multiplying the equation in \eq{3.2} by $w$ and integrating by parts yields
$$ \frac12\frac{d}{dt}\int_\Omega w(t)^2\,dx
 =-2E\big(w(t)\big)+\int_\Omega g\big(w(t)\big)
 \,dx,\eq{3.4}$$
where
$$ g(w) =
 \frac{p-1}{p+1}|w+v_1|^{p+1}-|w+v_1|^{p-1}(w+v_1)v_1+wv_1^p+\frac2{p+1}v_1^{p+1}.$$

We shall show in Lemma~3.2 that there exist positive constants
$c_0,c_1,\dots,c_4$ such that
$$ \gathered
g(w) \geq c_0|w|^{p+1}-c_1w^2v_1^{p-1}, \quad \\
 c_2|w|^{p+1}+c_3w^2v_1^{p-1}\geq f(w) \geq c_4|w|^{p+1}.
\endgathered \eq{3.5} $$
Assume that $\varepsilon>0$.
Integrating inequalities in \eq{3.5} and using the estimate
$$ \int_\Omega w^2v_1^{p-1}\,dx \leq |w|_{r_1}^2|v_1|_{r_2}^{p-1}
  = C|w|_{r_1}^2 \leq \varepsilon|\nabla w|_2^2+C_\varepsilon|w|_2^2 \eq{3.6}$$
(where $r_1<\frac{2n}{n-2}$ and $r_2<\frac{n}{n-2}$ are suitable exponents
required by the corresponding H\"older inequality) one obtains
$$ \gathered 
 \int_\Omega g(w)\,dx \geq \int_\Omega\big(C_0 |w|^{p+1} -C_1 w^2-\varepsilon|\nabla w|^2\big)\,dx , \\
 \int_\Omega\big(C_2 |w|^{p+1}+C_3 w^2+\varepsilon|\nabla w|^2\big)\,dx \geq
 \int_\Omega f(w)\,dx\geq  \int_\Omega C_4 |w|^{p+1}\,dx ,
 \endgathered \eq{3.7} $$
where $C_0,C_1,\dots,C_4$ are positive constants
(and $C_1,C_3$ depend on $\varepsilon$).
The choice of $r_1,r_2$ in \eq{3.6} is possible due to
$$ 2\frac{n-2}{2n}+(p-1)\frac{n-2}n<1. $$

Now \eq{3.4}, \eq{3.7} and the choice
$\varepsilon\leq1/4\wedge C_0/(8C_2)$ imply
$$ \aligned
  \frac12\frac{d}{dt}\int_\Omega w(t)^2\,dx
     &\geq -2(1+2\varepsilon)E\big(w(t)\big)+\tilde C_0\int_\Omega|w|^{p+1}\,dx
       - \tilde C_1\int_\Omega w^2\,dx \\
    &\geq -2(1+2\varepsilon)E\big(w(t)\big)+\hat C_0\Bigl(\int_\Omega w^2\,dx\Bigr)^{(p+1)/2}
       - \hat C_1.
  \endaligned \eq{3.8}$$
Let $t_1<t_2$ be fixed positive numbers
and let $\mu_k$ be smooth positive functions, $\mu_k\to\mu$ in $M(\Omega)$.
Denote by \eq{1.1}${}_k$ problem \eq{1.1} with $\mu$ replaced by $\mu_k$.
Then problem \eq{1.1}${}_k$ admits a classical solution $u_k$
defined on $[0,t_2]$ for $k$ large enough (cf.~Remark~2.2).
Moreover, for $k$ large enough,
\cite{4, Theorem~6.3} implies existence of positive stationary
solutions $v_{1,k}$ of \eq{1.1}${}_k$
such that $v_{1,k}\to v_1$ in $\W zq$.
Set $w_k=u_k-v_{1,k}$ and denote by \eq{3.2}${}_k$ problem \eq{3.2}
with $w$ replaced by $w_k$
and $h(w)$ by $h_k(w_k):=|w_k+v_{1,k}|^{p-1}(w_k+v_{1,k})-v_{1,k}^p$.
Multiplying \eq{3.2}${}_k$
by $\partial_tw_k$ and integrating over
$(x,t)\in Q:=\Omega\times(t_1,t_2)$ yields
$$
 \int_{t_1}^{t_2}\int_\Omega (w_k)_t^2\,dx\,dt
 = E_k\big(w_k(t_1)\big)-E_k\big(w_k(t_2)\big),
 \eq{3.9}
$$
where
$$\gathered
E_k(w_k) :=\frac12\int_\Omega|\nabla w_k|^2\,dx-\int_\Omega f_k(w_k)\,dx, \\
f_k(w_k) :=\frac1{p+1}\big(|w_k+v_{1,k}|^{p+1}-v_{1,k}^{p+1}\big)-w_kv_{1,k}^p.
\endgathered$$
Since the right-hand side of \eq{3.9} is uniformly bounded
due to Remark~2.2, we may assume that $(w_k)_t$ converges weakly
in $L_2(Q)$ to some function $\tilde w$.
Remark~2.2 implies the pointwise convergence of $w_k$ to $w$
in $\bar Q$, hence
$h(w_k)\to h(w)$ in
$L_r(Q)$ for any $r<n/[p(n-2)]$
(recall that $v_1\in L_R(\Omega)$ for any $R<n/(n-2)$).
Passing to the limit in the weak formulation of \eq{3.2}${}_k$ shows
$\tilde w=w_t$. Thus, passing to the limit in \eq{3.9} gives
$$
 \int_{t_1}^{t_2}\int_\Omega w_t^2\,dx\,dt \leq E\big(w(t_1)\big)-E\big(w(t_2)\big).
 \eq{3.10}
$$
Consequently, the function $t\mapsto E\big(w(t)\big)$ is nonincreasing.
Now \eq{3.8} and the global existence of $w$ imply both
$$ \big|E\big(w(t)\big)\big| \leq c \eq{3.11}$$
and
$$ \int_\Omega w^2(t)\,dx \leq c \eq{3.12}$$
(otherwise the function $y(t):=\int_\Omega w^2(t)\,dx$ has to
blow up in finite time).
Estimates \eq{3.11} and \eq{3.10} entail
$$ \int_0^\infty\int_\Omega w_t^2\,dx\,dt \leq c. \eq{3.13}$$
Now \eq{3.3},\eq{3.7},\eq{3.11}
and \eq{3.4},\eq{3.7},\eq{3.11} show
that
$$ \int_\Omega|\nabla w|^2\,dx
  \leq c\Bigl(1+\int_\Omega|w|^{p+1}\,dx\Bigr)
  \leq c\Bigl(1+\frac12\frac{d}{dt}\int_\Omega w^2\,dx\Bigr), $$
hence
$$ \int_\Omega|w|^{p+1}\,dx+\int_\Omega|\nabla w|^2\,dx
  \leq c\Bigl(1+\Big|\int_\Omega ww_t\,dx\Big|\Bigr) .\eq{3.14} $$
Squaring \eq{3.14} and integrating over time yields
$$ \int_t^{t+1}\Bigl(\int_\Omega|w|^{p+1}\,dx\Bigr)^2\,dt
 + \int_t^{t+1}\Bigl(\int_\Omega|\nabla w|^2\,dx\Bigr)^2\,dt
  \leq c, \eq{3.15}$$
where we have used
$$\int_t^{t+1}\Big|\int_\Omega ww_t\,dx\Big|^2dt
  \leq \int_t^{t+1}|w|_2^2|w_t|_2^2\,dt
  \leq c\int_t^{t+1}|w_t|_2^2\,dt \leq c $$
(see \eq{3.12} and \eq{3.13}).
Estimates \eq{3.13}, \eq{3.15} and
\cite{10, the proof of Proposition~2}
imply uniform bounds
(depending only on $|u_0|_{z,q}$)
for $|w(t)|_r$ if $r<6n/(3n-4)$.
Since $v_1\in L_s(\Omega)$ for any $s<n/(n-2)$,
the last estimate, \eq{1.2}
and Theorem~2.3 imply the a priori bound
for $|u(t)|_{z,q}$ if $n>2$.

If $n=2$ then one can make a bootstrap argument based on maximal
regularity as in \cite{20} to get a priori bound in $L_r(\Omega)$
for any $r>1$. Since it is not completely clear which estimate
corresponds to \cite{20, (16)} in our case (and also for the
reader's convenience) we repeat the whole argument from \cite{20}.

We already know by \eq{3.11} that
$$ -C\leq \frac12\int_\Omega|\nabla w(t)|^2\,dx-\int_\Omega f\big(w(t)\big)\,dx
       \leq C. \eq{3.16} $$
Moreover, \eq{3.15} implies
$$ \sup_{t\geq t_0} \int_t^{t+1}|w(s)|_{p+1}^{(p+1)r}\,ds < C \eq{3.17} $$
for any $t_0>0$ and $r=2$.
The interpolation theorem in \cite{10, Appendice}, \eq{3.13}
and \eq{3.17} imply
$$ \sup_{t\geq t_0}|w(t)|_\lambda < C \eq{3.18} $$
for any
$$ \lambda<\lambda(r):=p+1-\frac{p-1}{r+1}. $$
Due to Theorem~2.3 and the definition of $w$, estimate \eq{3.18}
guarantees the required bound in $\W zq$ if
$$ \lambda(r)>\frac n2(p-1)=p-1, $$
or, equivalently,
$$ p<p(r):=2r+3.$$
Fix $t_0\in(0,1)$.
Our bootstrap argument is as follows: assuming \eq{3.17}
for some $r\geq2$, we shall show the same estimate for some
$\tilde r>r$ (with the difference $\tilde r-r$ bounded away from zero).
Thus, after finitely many steps we prove \eq{3.17}
with some $r$ satisfying $2r+3>p$ which will conclude the proof.

Hence, let \eq{3.17} be true for some $r\geq2$.
Then \eq{3.18} is true for $\lambda<\lambda(r)$.
Choose $\lambda\in\big(2,\lambda(r)\big)$ and denote
$$ \theta=\frac{p+1}{p-1}\frac{\lambda-2}\lambda,
  \quad  \lambda'=\frac\lambda{\lambda-1}
  \quad \hbox{ and }\quad  p_1=\frac{p+1}p.$$
Then $\theta\in(0,1)$ and  $\lambda'\in(p_1,2)$ due to $\lambda<p+1$.
Moreover,
$$\frac\theta{p_1}+\frac{1-\theta}2=\frac1{\lambda'}.$$
Using \eq{3.14},
H\"older's inequality, \eq{3.18} and
interpolation, we obtain
$$\eqalign{
 |w(t)|_{1,2}^2 &\leq C|\nabla w(t)|_2^2 \leq C\big(1+|w(t)w_t(t)|_1\big) \cr
  &\leq C\big(1+|w_t(t)|_{\lambda'}\big)
   \leq C\big(1+|w_t(t)|_{p_1}^\theta |w_t(t)|_2^{1-\theta}\big).}\eq{3.19}$$

Let $t\geq t_0$ and $\delta\in(0,t_0/2)$ be given. Then
\eq{3.15} implies
$$ \int_{t-\delta}^{t-\delta/2}|w(s)|_{1,2}^2\,ds < C, $$
hence there exists $\tau_1\in(t-\delta,t-\delta/2)$ such that
$$ |w(\tau_1)|_{1,2}<C_5, $$
where $C_5$ depends on $\delta$ but it does not depend on $t$.
Given $\tilde q<n/(n-1)=2$, the last estimate and
$v_1\in W^1_{\tilde q}(\Omega)$ (cf.~\cite{4})
imply
$|u(\tau_1)|_{1,\tilde q}<{\tilde C}_5$, where
${\tilde C}_5={\tilde C}_5(C_5,v_1,\tilde q)$.
The existence proof for \eq{1.1} based on \eq{2.5}
shows that there exists $\delta_1=\delta_1(C_5,{\tilde C}_5)>0$ small
($\delta_1\leq\delta/4$) such that $u$ and $w$ stay bounded
on $(\tau_1,\tau_1+2\delta_1)$
in $\W 1{\tilde q}$
by a constant $C_6=C_6(C_5,{\tilde C}_5,\delta_1)$.
By Lemma~2.1, $w(t)$ stays bounded in $C(\bar\Omega)$
on $(\tau_1+\delta_1,\tau_1+2\delta_1)$ by a constant
$C_7=C_7(C_6,\delta_1)$.
Since $v_1\in L_s(\Omega)$ for any $s$
and $|h(w)|\leq\tilde C|v_1|^{p-1}$ if $|w|\leq C$,
the function $h$ stays bounded in $L_\rho(\Omega)$
(for some $\rho>p_1$) on $(\tau_1+\delta_1,\tau_1+2\delta_1)$
by a constant $C_8=C_8(C_7,v_1,\rho)$.
Now standard estimates in the variation-of-constants formula
for $w$ on $(\tau_1+\delta_1,\tau_1+2\delta_1)$
imply
$$ |w(\tau_1+2\delta_1)|_{2-\varepsilon,\rho} \leq C_9, $$
where $C_9=C_9(C_8,\delta_1,\varepsilon)$ and $\varepsilon>0$ is small.
Choose $\varepsilon<2/p_1-2/\rho$. Then $W^{2-\varepsilon}_\rho(\Omega)\hookrightarrow X_P$,
where $X_P:=(E_0,E_1)_{1-1/P,P}$ is
the real interpolation space between
$E_0=L_{p_1}(\Omega)$ and $E_1=W^2_{p_1}(\Omega)$,
and $P>1$ is arbitrary.
Consequently,
$$ \|w(\tau)\|_{X_P} \leq C_{10}, \eq{3.20} $$
where $\tau:=\tau_1+2\delta_1\in(t-\delta,t)$.
Notice that given $t\geq t_0$ and $\delta\in(0,t_0/2)$
we have found $\tau\in(t-\delta,t)$ and
$C_{10}=C_{10}(\delta,v_1,|u_0|_{z,q},P)$
such that \eq{3.20} is true and
$C_{10}$ is independent of $w$ and $t$.

We have
$1-\theta=\frac2{p-1}\big(\frac{p+1}\lambda-1\big)<\frac2r$
for $\lambda$ sufficiently close to $\lambda(r)$
since the last inequality is satisfied for $\lambda=\lambda(r)$.
Now choose $\tilde r>r$ such that
$$ \beta:=\frac2{(1-\theta)\tilde r}>1$$
and notice that $\theta\tilde r\beta'>1$ where $\beta'=\beta/(\beta-1)$.
Next we use \eq{3.7} and \eq{3.16},
then \eq{3.19}, H\"older's inequality, \eq{3.13},
maximal Sobolev regularity
(see \cite{1, Theorem~III.4.10.7}),
\eq{3.20}
and inequality $|h(w)|\leq C(|w|^p+|v_1|^p)$
to get
$$ \eqalign{
\int_\tau^{t+1}|w(s)|_{p+1}^{\tilde r(p+1)}\,ds
 &\leq C\Bigl(1+\int_\tau^{t+1}|w(s)|_{1,2}^{2\tilde r}\,ds\Bigr) \cr
   &\leq C\Bigl(1+\int_\tau^{t+1}|w_t(s)|_{p_1}^{\theta\tilde r}
                            |w_t(s)|_2^{(1-\theta)\tilde r}\,ds\Bigr) \cr
  &\leq C\Bigl(1+
     \Bigl(\int_\tau^{t+1}|w_t(s)|_{p_1}^{\theta\tilde r\beta'}\,ds
         \Bigr)^{1/\beta'}
     \underbrace{\Bigl(\int_\tau^{t+1}|w_t(s)|_2^2\,ds
                 \Bigr)^{1/\beta}}_{\leq C}\Bigr) \cr
  &\leq C\Bigl(1+\Bigl(
       \int_\tau^{t+1}|h(w(s))|_{p_1}^{\theta\tilde r\beta'}\,ds
     \Bigr)^{1/\beta'} + \|w(\tau)\|_{X_P}^{\theta\tilde r}\Bigr) \cr
  &\leq C\Bigl(1+\Bigl(
       \int_\tau^{t+1}|w(s)|_{p+1}^{p\theta\tilde r\beta'}\,ds
     \Bigr)^{1/\beta'} \Bigr),
}$$
where $P=\theta\tilde r\beta'$.
Now we see that the last estimate implies \eq{3.17} with $\tilde r$
instead of $r$ provided $p\theta\tilde r\beta'\leq \tilde r(p+1)$,
that is if $\theta\beta'\leq p_1$.
This condition is equivalent to
$$ p\leq \frac{\lambda(\tilde r-1)-\tilde r}{\tilde r-2}. \eq{3.21}$$
Considering $\tilde r\to r+$ and $\lambda\to\lambda(r)-$
we see that it is sufficient to verify
$$ p(r-2)< \lambda(r)(r-1)-r,$$
which is equivalent to $(p-1)2r>0$.
Consequently, the sufficient condition for bootstrap is satisfied
and we are done.
Note that the possibility of choosing $\tilde r-r$ bounded away from
zero follows by an easy contradiction argument.
\qed\enddemo

\proclaim{Lemma~3.2}
The functions $f,g$ from the proof of Theorem~3.1
satisfy \eq{3.5} for any $w\in{\Bbb R}$ and $v_1>0$.
\endproclaim

\demo{Proof}
Since $f$ and $g$ can be viewed as positively homogeneous functions
of two variables $w,v_1$ and $v_1>0$ one can put $v_1=1$.
Consequently, we have to show
$$ \align
g_1(w) &\geq c_0|w|^{p+1}-c_1w^2, \\
 c_2|w|^{p+1}+c_3w^2\geq f_1(w) &\geq c_4|w|^{p+1},
\endalign $$
where
$$ \align
 f_1(w) &= \frac1{p+1}\big(|w+1|^{p+1}-1\big)-w, \\
 g_1(w) &=
 \frac{p-1}{p+1}|w+1|^{p+1}-|w+1|^{p-1}(w+1)+w+\frac2{p+1}.
\endalign $$

First let us show $f_1(w)\geq c_4|w|^{p+1}$.
If $w>-1$ then $f_1'(w)=(w+1)^p-1$ has the same sign as $w$ and $f_1(w)=0$,
hence $f_1(w)>0$ if $w>-1$, $w\ne0$.
Obviously, $f_1(w)\geq-\frac1{p+1}-w>0$ if $w\leq-1$.
Since $f_1(w)\approx\frac p2w^2$ as $w\to0$,
there exists $\delta_1>0$ such that
$f_1(w)\geq |w|^{p+1}$ for $|w|\leq\delta_1$.
Since $f_1(w)/|w|^{p+1}\to\frac1{p+1}$ as $|w|\to\infty$,
there exists $K_1>\delta_1$ such that
$f_1(w)\geq\frac1{2(p+1)}|w|^{p+1}$ for $|w|\geq K_1$.
The function $f_1$ is positive and continuous on the
compact set $M_1:=[-K_1,-\delta_1]\cup[\delta_1,K_1]$,
hence there exists $\varepsilon>0$ such that
$f_1(w)\geq\varepsilon K_1^{p+1}\geq\varepsilon|w|^{p+1}$ for $w\in M_1$.
Consequently, it is sufficient to choose
$c_4=1\wedge\varepsilon\wedge\frac1{2(p+1)}$.

The same arguments as above show
$f_1(w)\leq c_2|w|^{p+1}+c_3w^2$
if $c_2,c_3$ are sufficiently large.

The inequality for $g_1$ is equivalent to $G_1(w)+c_1w^2\geq G_2(w)$,
where
$$ \align
 G_1(w) &= \frac{p-1}{p+1}|w+1|^{p+1}+w+\frac2{p+1}, \\
 G_2(w) &= |w+1|^{p-1}(w+1)+c_0|w|^{p+1}.
 \endalign $$
Fix $c_0<\frac{p-1}{p+1}$ and assume $c_1\geq1$.
Since $G_1(w)-G_2(w)=o(w^2)$ as $w\to0$,
there exists $\delta_2>0$ such that $G_1(w)+c_1w^2\geq G_2(w)$
for $|w|\leq\delta_2$ (and $\delta_2$ does not depend on $c_1\geq1$).
Since $G_1(w)/G_2(w)\to\frac{p-1}{(p+1)c_0}>1$ as $|w|\to\infty$,
there exists $K_2>\delta_2$ such that
$G_1(w)\geq G_2(w)$ for $|w|\geq K_2$.
Since the function $G_2$ is bounded on the compact set
$M_2:=[-K_2,-\delta_2]\cup[\delta_2,K_2]$ by some constant $D_2$,
the choice $c_1>D_2/\delta_2^2$ guarantees
$c_1w^2\geq D_2\geq G_2(w)$ on $M_2$.
\qed\enddemo

%______________________________________________________________________________
\head{4. STATIONARY SOLUTIONS} \endhead

In this section we consider the problem
$$ \gathered
  0 = \Delta u + |u|^{p-1}u + \mu, \quad  x\in\Omega, \\
  u = 0,                     \quad  x\in\partial\Omega,
\endgathered \eq{4.1} $$
where $\Omega\subset{\Bbb R}^n$ is a smoothly bounded domain, $n\geq2$,
$p$ satisfies \eq{1.2}, and $\mu$ satisfies \eq{1.3} and \eq{1.4}.
Recall that assumption \eq{1.4} guarantees the existence
of the minimal positive solution $v_1$ of \eq{4.1}.

We fix $z=1$ and $q$ satisfying \eq{2.4} and denote
$$X=\W zq,\quad Y=\W{z-2}{q}\quad\hbox{and}\quad Z=L_p(\Omega).$$
Notice that $X\hookrightarrow Z\hookrightarrow L_1(\Omega)\hookrightarrow M(\Omega)\hookrightarrow Y$.
Recall also from Section~2
that $A:=A_z:X\to Y$
is a linear isomorphism and
denote
$$ F(u)=|u|^{p-1}u+\mu \quad \hbox{and}\quad  S=A^{-1}F.$$
The results of \cite{4} imply that $A^{-1}\geq0$,
$F:Z\to Y$ and $S:Z\to X$ are nondecreasing, $S$ is compact.
The solutions of \eq{4.1} correspond to the fixed points
of the operator $S|_X:X\to X$.
We denote by ${\Cal E}$ the set of all solutions of \eq{4.1}.

In our study we shall use also the semiflow generated
by problem \eq{1.1}.
The considerations in Section~2 imply that this semiflow can
be considered both in $X$ and in $Z$.
Due to \cite{4, Theorem~5.1} and \cite{1, Theorem~II.6.4.1},
this semiflow is order preserving.

We call $u\in Z$ a {\bf supersolution} of \eq{4.1}
if $u\geq S(u)$ and $(1-e^{-tA})(u-S(u))\geq 0$ for all $t>0$.
If $u\in X$ then these conditions
may be replaced by a single condition $Au\geq Fu$:
this follows from the following facts:
$A^{-1}\geq0$, $e^{-tA}\geq0$,
$\frac{1-e^{-tA}}t w\to Aw$ if $t\to0$, $w\in X$, and
$(1-e^{-tA})w=\int_0^te^{-sA}Aw\,ds\geq0$ if $w\in X$, $Aw\geq0$.
The subsolution is defined in an analogous way.
One of the basic properties of sub- and supersolutions is
formulated in the following

\proclaim{Proposition 4.1}
If $u^+\in Z$ is a supersolution of \eq{4.1} and
$u_0\in X$, $u_0\leq u^+$, then
the solution $u:[0,\Tmax)\to X$ of \eq{1.1} satisfies $u(t)\leq u^+$
for any $t\in[0,\Tmax)$, where $\Tmax$ is the maximal existence
time of this solution.
Analogous assertion is true for subsolutions.
\endproclaim

\demo{Proof}
The solution $u$ can be (locally) obtained as the limit of
the sequence $\{u_k\}$, where $u_1(t)\equiv u_0$
and
$$ u_{k+1}(t) = e^{-tA}u_0+\int_0^te^{-(t-s)A}F\big(u_k(s)\big)\,ds.$$
(cf.~the existence proof in Section~2).
We shall show by induction that $u_k(t)\leq u^+$.
Obviously, $u_1(t)\leq u^+$. Hence assume $u_k(t)\leq u^+$.
Then $F(u_k(s))\leq F(u^+) = AS(u^+)$, so that
$$ \eqalign{
u_{k+1}(t)
&\leq e^{-tA}u^+ + \int_0^te^{-(t-s)A}AS(u^+)\,ds \cr
&= e^{-tA}u^+ + S(u^+) - e^{-tA}S(u^+) \leq u^+.\ \qed
}$$
\enddemo

In what follows, we shall construct a subsolution $v_\varepsilon$
and a supersolution $v^\varepsilon$ such that
$$ v^\varepsilon\geq v_1+\varepsilon,\quad  v_\varepsilon\leq-\varepsilon\quad 
   \hbox{and}\quad  [v_\varepsilon,v^\varepsilon]\cap{\Cal E}=\{v_1\}.$$

Due to \cite{4, Section~12}, the operator
$F:Z\to Y$ is of the class $C^1$
and the operator $u\mapsto u-S'(v_1)u$ is an isomorphism
considered both as an operator $X\to X$ and $Z\to Z$.
Consequently, $v_1$ is an isolated stationary solution of \eq{1.1}
both in $X$ and in $Z$.

Similarly, the operator $\tilde F:Z\to Y:u\mapsto |u|^p+\mu$
is $C^1$ and $\tilde F'(v_1)=F'(v_1)$, hence
the implicit function theorem guarantees
the unique solvability of the equation
$ u=A^{-1}\tilde F(u+\varepsilon) $
in the neighbourhood of $v_1$ if $\varepsilon>0$ is small enough.
Denoting this solution by $u^\varepsilon$, the function $v^\varepsilon:=u^\varepsilon+\varepsilon$ satisfies
$0=\Delta v^\varepsilon+|v^\varepsilon|^p+\mu$ in $\Omega$
and $v^\varepsilon=\varepsilon$ on $\partial\Omega$.
Since $\Delta v^\varepsilon\leq0$, we have $u^\varepsilon=v^\varepsilon-\varepsilon\geq0$, hence
$u^\varepsilon=S(u^\varepsilon+\varepsilon)$ and
$v^\varepsilon-S(v^\varepsilon)=\varepsilon$.
Consequently, $v^\varepsilon$ is a supersolution of problem \eq{4.1}.

Next we show that
$$ \hbox{any positive supersolution $v^+$ of \eq{4.1} fulfils $v^+\geq v_1$.}
\eq{4.2}$$
Indeed, assuming the contrary and denoting $y=v_1\wedge v^+$ we have
$S(y)\leq S(v_1)=v_1$ and $S(y)\leq S(v^+)\leq v^+$, hence $S(y)\leq y$.
Since $0$ is a subsolution of \eq{4.1},
and the operator $S:[0,y]\to[0,y]$ is nondecreasing and compact,
there exists a solution of the problem $u=S(u)$ in the order
interval $[0,y]$ (see \cite{3, Corollary~6.2}).
Since $y<v_1$, we obtain a contradiction with the minimality of $v_1$.

We have $v^\varepsilon\geq v_1$, $v^\varepsilon-v_1=\varepsilon$ on $\partial\Omega$
and $\Delta(v^\varepsilon-v_1)=v_1^p-(v^\varepsilon)^p\leq 0$ in $\Omega$,
therefore $v^\varepsilon\geq v_1+\varepsilon$.
Finally, choosing $\varepsilon$ small enough we may assume
$[v_1,v^\varepsilon]\cap{\Cal E}=\{v_1\}$.

Similarly as above, the implicit function theorem
used for the problem
$$ \gathered 
  0 = \Delta u + |u|^{p-1}u , \quad  x\in\Omega, \\
  u = 0,                     \quad  x\in\partial\Omega
\endgathered \eq{4.3} $$
yields the existence of a unique solution $v_\varepsilon$
of the problem
$0 = \Delta v + |v|^{p-1}v$ in $\Omega$,
$v=-\varepsilon$ on $\partial\Omega$,
in the neighbourhood of the zero solution of \eq{4.3}.
Obviously, $v_\varepsilon$ is a subsolution of both \eq{4.3} and \eq{4.1}.
Standard regularity estimates imply that the $C(\bar\Omega)$-norm of $v_\varepsilon+\varepsilon$
can be bounded by $C\varepsilon^p$ (where $C$ is a given constant),
hence $v_\varepsilon<0$ if $\varepsilon>0$ is small enough.
Since $\Delta v_\varepsilon=-v_\varepsilon|v_\varepsilon|^{p-1}>0$ in $\Omega$
and $v_\varepsilon=-\varepsilon$ on $\partial\Omega$,
we have $v_\varepsilon\leq-\varepsilon$ in $\bar\Omega$.
As before, the order interval $[v_\varepsilon,0]$ does not contain
any solution of \eq{4.3} different from $0$ if $\varepsilon$ is small enough.

Next we show that
$$ \hbox{any supersolution $v^+$ of \eq{4.1}
satisfying $v^+\geq v_\varepsilon$ fulfils $v^+\geq v_1$.}
\eq{4.4}$$
Assume the contrary and let
$v^+\geq v_\varepsilon$ be a supersolution of \eq{4.1}, $v^+\not\geq v_1$.
Then $v^+$ cannot be positive due to \eq{4.2}. Since $v^+$ is
also a supersolution of \eq{4.3},
the function $y:=v^+\wedge0<0$ is a supersolution of \eq{4.3} as well.
Consequently,
we can find a solution of \eq{4.3} between the subsolution $v_\varepsilon$
and the supersolution $y$, a contradiction.

Notice that \eq{4.4} and $[v_1,v^\varepsilon]\cap{\Cal E}=\{v_1\}$
imply $[v_\varepsilon,v^\varepsilon]\cap{\Cal E}=\{v_1\}$.

Now denote $D_A$ the domain of attraction of the equilibrium $v_1$
(that is $D_A$ is the set of all initial conditions $u_0\in X$
for which the solution $u(t)$ of \eq{1.1} exists globally and tends
to $v_1$ in $X$ as $t\to+\infty$).
Summarizing the considerations above we obtain the following

\proclaim{Lemma~4.2}
The sets $\{v\in X\,:\,v\geq v_\varepsilon\}$ and $\{v\in X\,:\,v\leq v^\varepsilon\}$
are invariant under the semiflow defined by \eq{1.1}.
The set $[v_\varepsilon,v^\varepsilon]\cap X$ is a subset of $D_A$.
The set $D_A$ is open in $X$.
\endproclaim

\demo{Proof}
The invariance follows from Proposition~4.1.

If $u_0\in[v_\varepsilon,v^\varepsilon]$ then the corresponding solution $u(t)$ of \eq{1.1}
stays in the same order interval, hence it is global due to Theorem~2.3.
The existence of the Lyapunov functional (see \eq{3.10}),
the boundedness of $u(t)$ and the compactness of the semiflow
imply that the $\omega$-limit set $\omega(u_0)$
of this solution is a nonempty compact set consisting of equilibria.
Since $\omega(u_0)\subset[v_\varepsilon,v^\varepsilon]$ and $[v_\varepsilon,v^\varepsilon]\cap{\Cal E}=\{v_1\}$,
we have $\omega(u_0)=\{v_1\}$.

Now let  $u_0\in D_A$, $\delta>0$, $K:=\|v_1\|_X+1$
and $\eta=\frac\varepsilon{2c}$,
where $c=c(\delta,\delta,K)$ is the constant from \eq{2.7}.
Since $u(t)\to v_1$ in $X$ as $t\to\infty$,
there exists $t_1>0$ such that $\|u(t)-v_1\|_X\leq\eta$ if $t\geq t_1$.
Put $T=t_1+\delta$. Then \eq{2.7} implies
$$\|u(T)-v_1\|_{0,\alpha}\leq c(\delta,\delta,K)\|u(t_1)-v_1\|_X\leq\frac\varepsilon2.$$
If $y_0\in X$, $\|y_0-u_0\|_X\leq\frac12$,
and $y$ is the solution of \eq{1.1}
with the initial condition $y_0$,
then there exists $\beta>0$ (independent of $y_0$)
such that $y(t)$ exists and
$\|y(t)-u(t)\|_X\leq1$ for $t\leq\beta$.
If we require
$\|y_0-u_0\|_X\leq\frac\varepsilon{2c}$, where $c=c(\beta,T,K_u+2)$
is the constant from \eq{2.7},
then \eq{2.7} implies existence of $y(t)$ for $t\leq T$ and the estimate
$$\|y(T)-u(T)\|_{0,\alpha}\leq c(\beta,T,K_u+2)\|u_0-y_0\|_X\leq\frac\varepsilon2,$$
hence
$$\|y(T)-v_1\|_{0,\alpha}\leq
  \|y(T)-u(T)\|_{0,\alpha}+\|u(T)-v_1\|_{0,\alpha}\leq\varepsilon,$$
so that
$y(T)\in[v_\varepsilon,v^\varepsilon]\cap X\subset D_A$.
This implies that the set $D_A$ is open in $X$.
\qed\enddemo

\proclaim{Lemma~4.3}
Let $V$ be a finite dimensional subspace of $X$.
Then the set $D_A\cap V$ is bounded.
\endproclaim

\demo{Proof}
We shall proceed by contradiction.
Assume $u_k\in D_A\cap V$, $\|u_k\|_X\to\infty$ as $k\to\infty$.
Then $\|u_k-v_1\|_X\to\infty$ as well.
Since the solution $U_k(t)$ of \eq{1.1} with the initial condition $u_k$
fulfils $U_k(t)-v_1\in \W12$ for $t>0$
and $\W12\hookrightarrow X$, we may assume $u_k-v_1\in\W12$ and
$\alpha_k:=|u_k-v_1|_{1,2}\to\infty$.
Denote $w_k=(u_k-v_1)/\alpha_k$,
$A_k=\int_\Omega|\nabla w_k|^2\,dx$,
$B_k=\int_\Omega|w|_k^{p+1}\,dx$.
The sequence $w_k$ is bounded in $\W12$ and belongs to a finite dimensional
subspace, hence it is relatively compact and
we may assume $w_k\to w$ in $\W12$, $|w|_{1,2}=1$.
Moreover, we have $A_k\leq1$, $B_k\to \int_\Omega|w|^{p+1}\,dx>0$.
Using \eq{3.3},\eq{3.7} we obtain
$$  E(u_k-v_1) = E(\alpha_k w_k) \leq \alpha_k^2A_k-\alpha_k^{p+1}C_4B_k
    \leq -\hat C_1 $$
for $k$ sufficiently large, where $\hat C_1$ is the constant from \eq{3.8}.
Since $t\mapsto E\big((U_k-v_1)(t)\big)$ is nonincreasing, \eq{3.8} implies
$$ \frac12\frac{d}{dt}\int_\Omega(U_k-v_1)^2\,dx
   \geq \hat C_0\Bigl(\int_\Omega(U_k-v_1)^2\,dx\Bigr)^{(p+1)/2}+\hat C_1,$$
so that $U_k$ cannot exist globally, a contradiction.
\qed\enddemo

In what follows put
$$ V=\{v_1+\alpha_1y_1+\alpha_2y_2\,:\,\alpha_1,\alpha_2\in{\Bbb R}\}, $$
where $y_1,y_2\in X$ are continuous functions such that $y_1>0$ and
$y_2$ changes sign.
Denote ${\Cal E}^+=\{v\in{\Cal E}\,:\,v>v_1\}$ and ${\Cal E}^-=\{v\in{\Cal E}\,:\,v<v_1\}$.
Let $\partial D_A$ be the boundary of $D_A$ in $X$
and $\partial_V D_A$ be the boundary of $D_A\cap V$ in $V$.
Obviously, $\partial_VD_A\subset\partial D_A$.
Our a priori assumptions imply that any solution of \eq{1.1} with the
initial condition belonging to $\partial D_A$
exists globally and is bounded in $X$. Consequently, its $\omega$-limit set
consists of equilibria.
We put
$$ \partial_VD_A^\pm=\{u\in\partial_VD_A\,:\, \omega(u)\cap{\Cal E}^\pm\ne\emptyset\}.$$

\proclaim{Lemma~4.4}
If $u_0\in\partial_VD_A^\pm$ then $\omega(u_0)\subset{\Cal E}^\pm$.
The sets $\partial_VD_A^\pm$ are open in $\partial_VD_A$.
\endproclaim

\demo{Proof}
Let $u_0\in\partial_VD_A^+$. Then there exists
$v^+\in{\Cal E}^+$ and $t_k\to\infty$
such that the solution $u$ of \eq{1.1} fulfils
$u(t_k)\to v^+$ in $X$.
Choosing $\delta>0$,
inequality \eq{2.7} implies $\|u(t_k+\delta)-v^+\|_{0,\alpha}\to0$
as $k\to\infty$, hence
$$ u(t_k+\delta)\geq v^+-\frac\varepsilon2\geq v_\varepsilon+\frac\varepsilon2\geq v_\varepsilon \eq{4.5}$$
for $k$ large enough. Fix $k$ and put $T=t_k+\delta$. We have
$u(t)\geq v_\varepsilon$ for $t\geq T$, hence $\omega(u_0)\subset{\Cal E}^+$.
Moreover, if $y$ denotes the solution of \eq{1.1} with the initial condition
$y_0\in\partial_VD_A$ then \eq{2.7} implies
$$ \|(u-y)(T)\|_{0,\alpha}\leq \frac\varepsilon2 \quad\hbox{provided}\quad
  \|u_0-y_0\|_X<\eta,\eq{4.6}$$
where $\eta>0$ is small enough (cf.~the proof of Lemma~4.2).
Estimates \eq{4.5} and \eq{4.6} imply
$y\geq v_\varepsilon$, hence $\omega(y_0)\subset{\Cal E}^+$.
Consequently,
the set $\partial_VD_A^+$ is open in $\partial_VD_A$.

The proofs in the case $\partial_VD_A^-$ are analogous.
\qed\enddemo


\proclaim{Theorem~4.5}
Assume \eq{1.2}, \eq{1.3}, \eq{1.4}
and let $v_1$ be the minimal positive solution of \eq{4.1}.
Then there exist solutions $v_2,v_3,v_4$ of \eq{4.1}
such that $v_2>v_1>v_3$ and the function $v_4-v_1$ changes
sign.
\endproclaim

\demo{Proof}
The proof is similar to the corresponding proof in \cite{19}.

Due to Lemmata~4.2-4.3, there exists $u_0\in\partial D_A$, $u_0>v_1$.
The a priori estimates from Section~3 guarantee that the $\omega$-limit
set $\omega(u_0)$ is a nonempty compact subset consisting of
equlibria (cf.~the proof of Lemma~4.2).
Proposition~4.1 and \eq{4.4} imply $\omega(u_0)\subset{\Cal E}^+$,
hence there exists $v_2\in{\Cal E}^+$.
Similarly one obtains the existence of $v_3\in{\Cal E}^-$.

The existence of $v_4$ will be shown by contradiction.
Hence, assume the contrary; then
the compact set $\partial_VD_A$ can be written as
a union of two open disjoint subsets $\partial_VD_A^\pm$.
Consequently, both
$\partial_VD_A^+=\partial_VD_A\setminus\partial_VD_A^-$
and $\partial_VD_A^-$ are compact and their distance is positive.
Moreover,
$\partial_VD_A^\pm\cap\{v_1\mp\lambda y_1\,:\,\lambda>0\}=\emptyset$
so that the following homotopy
$$ H(t,u) =\cases
    v_1+(1-2t)(u-v_1)+2ty_1,& u\in \partial_VD_A^+,\ t\in[0,1/2]\\
    v_1+(1-2t)(u-v_1)-2ty_1,& u\in \partial_VD_A^-,\ t\in[0,1/2]\\
    v_1+(2-2t)y_1   +(2t-1)y_2,& u\in \partial_VD_A^+,\ t\in[1/2,1]\\
    v_1+(2-2t)(-y_1)+(2t-1)y_2,& u\in \partial_VD_A^-,\ t\in[1/2,1],
    \endcases $$
fulfils $H(t,u)\ne v_1$ for $u\in\partial_VD_A$.
The homotopy invariance property of the topological degree in $V$
yields
$$ 1 = \hbox{deg}\,(H(0,\cdot),v_1,D_A\cap V)
     = \hbox{deg}\,(H(1,\cdot),v_1,D_A\cap V) = 0,$$
which is a contradiction.
\qed\enddemo
%______________________________________________________________________________

\proclaim{\bf Acknowledgement} \rm The author was partially supported
by the Swiss National Science Foundation and
by VEGA Grant 1/7677/20.
\endproclaim

%______________________________________________________________________________

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%______________________________________________________________________________