### On global existence and stationary solutions for two classes of semilinear parabolic problems

Pavol Quittner

Abstract. We investigate stationary solutions and asymptotic behaviour of solutions of two boundary value problems for semilinear parabolic equations. These equations involve both blow up and damping terms and they were studied by several authors. Our main goal is to fill some gaps in these studies.

Keywords: global existence, blow up, semilinear parabolic equation, stationary solution Classification: 35K60, 35J65, 35B40

1. Introduction.

Consider the following two problems

(NBC)

ut=△u−au^{p} in (0,∞)×Ω,

∂u

∂n =u^{q} on (0,∞)×∂Ω,
u(0, x) =u0(x) x∈Ω,

(DGT)

u_{t}=△u− |∇u|^{q}+λu^{p} in (0,∞)×Ω,

u= 0 on (0,∞)×∂Ω,

u(0, x) =u0(x) x∈Ω,

where Ω is a smoothly bounded domain inR^{N},p, q >1,a, λ >0 andu_{0}∈W^{1,∞}(Ω)
is a non-negative function. These problems were studied by many authors (see
e.g. [CFQ], [E], [FQ1], [LGMW] in the case of (NBC) and [AW], [C], [CW1], [CW2],
[F], [KP], [Q1] in the case of (DGT)). In both problems there is a blow-up term
(u^{q} andλu^{p}) and a damping term (−au^{p}and−|∇u|^{q}). These terms cause that the
corresponding solutions admit an interesting asymptotic behaviour which strongly
depends on the parameters p, q, a, λ. The main purpose of this paper is to fill
some gaps in the studies of these problems; i.e. to investigate the behaviour of the

This research was done while the author was visiting CEREMADE at the University Paris Dauphine which was possible due to a fellowship from the French Government. The author thanks Profs. J.P. Aubin and H. Frankowska for their hospitality and Prof. M. Struwe for the discussions concerning Theorem 4.2

solutions for those parameters p, q, a, λ or N for which the results in the above mentioned papers are not satisfactory.

In the case of (NBC) (the problem withNonlinear BoundaryConditions), the study was almost completely done in [CFQ] forN = 1. Particularly, it was shown that the exponentp= 2q−1 is critical for the blow up in the following sense:

(i) ifp <2q−1 (orp= 2q−1 anda < q) then there exist solutions of (NBC) which blow up in finite time,

(ii) if p >2q−1 (or p= 2q−1 and a > q) then all solutions of (NBC) exist globally and are globally bounded,

(iii) if p= 2q−1 and a= q then all nontrivial solutions of (NBC) exist glob- ally but they are unbounded; they tend pointwise to a singular stationary solution.

The assertions (i) and (ii) were shown also for Ω being a ball in R^{N}, N > 1.

However, if Ω is a general bounded domain inR^{N},N >1, then [CFQ] or [E] imply
blow up of suitable solutions of (NBC) only forp≤qand the global existence and
boundedness is shown in [CFQ] only forp > c(q),q < ^{N+1}_{N−1}, where

c(q) := N−q(N−2)

N+ 1−q(N−1)(q+ 1)−1>2q−1.

The main result of this paper concerning the global existence for (NBC) in the case of a general domain Ω is the following:

(a) if p > 2q−1 then all solutions of (NBC) exist globally and are globally bounded,

(b) if p < 2q−1 (or p = 2q−1 and a is sufficiently small) then there exist
initial functionsu_{0} such that the corresponding solutions of (NBC) blow up
in L^{∞}(Ω)-norm.

It has to be mentioned that in the case (b) we do not know whether the blow up
occurs in finite or infinite time. We find only a subsolutionu^{+}such that any positive
stationary solution has to intersectu^{+} so that the solution of (NBC) starting at
u^{+} cannot be bounded. In the case (a) we show that a simple substitution leads
to the casep > c(q) which was already solved in [CFQ]. Hence, forp >2q−1 we
obtain global existence, boundedness and also the existence of a positive stationary
solution of (NBC).

Considering the (positive) stationary solutions of (NBC) we are mainly interested
in the caseq < p < 2q−1, N >1. The results of [CFQ] imply that in this case
there existsa_{0}≥0 such that the stationary problem corresponding to (NBC) has

(j) no positive solutions fora < a_{0},

(jj) at least one positive solution fora > a_{0},

(jjj) in the subcritical case (q < _{N−2}^{N} ) at least two positive solutions for a ∈
{a_{1}, a_{2},· · · }, where a_{k}→ ∞.

Moreover, if Ω is a ball thena0>0 and (NBC) has at least one positive stationary
symmetric solution fora=a_{0} and at least two positive stationary symmetric solu-
tions fora > a_{0} (see [CFQ] for a more precise information forN = 1). The main

difficulty in proving this additional property for a general domain is the absence of apriori estimates for stationary solutions. In this paper we show that for a general domain Ω

(α) a_{0}>0,

(β) in the subcritical case, (NBC) has at least two positive stationary solutions
for almost alla > a_{0}.

Moreover, for Ω being a ball in R^{2} we find apriori estimates for all positive sta-
tionary solutions (note that the apriori estimates in [CFQ] forN >1 concern only
symmetric solutions).

The proof of (α) is based on the apriori estimate of min{u(x) ;x∈∂Ω}, whereu is any positive stationary solution. The proof of (β) is based on a trick of M. Struwe [S1].

Concerning the problem (DGT) (the problem withDampingGradientTerm), it
is known that forp > q >1 blow-up of solutions inL^{∞}-norm in finite time can occur
(see [CW1], [KP], [F], [Q1]) while forp≤qany solutionuis bounded in [0, T)×Ω,
whereT is the maximal existence time foru(see [F]). In this paper we show that if
the existence timeTof a solutionuof (DGT) is finite then lim_{t→T}_{−}ku(t,·)kL^{∞}(Ω)=
+∞. Consequently, the solution exists globally ifp≤q.

Our main results concerning the stationary solutions of (DGT) are the following:

(k) if q ≥p then there exists λ_{0} >0 such that the stationary problem corre-
sponding to (DGT) has

(k1) no positive solutions forλ < λ0,

(k2) at least one positive solution forλ=λ_{0},
(k3) at least two positive solutions forλ > λ_{0},

(kk) ifp <(N+ 2)/(N−2) (in the caseN >2) andq <min{2,(N+ 2)/N}then
there exists λ_{0} ≥ 0 such that (DGT) has at least one positive stationary
solution for anyλ > λ_{0} (see Theorem 6.2 and Remark 6.2).

The stationary problem for (DGT) was studied also in [AW], [C], [CW1], [CW2],
[FQ2], [SZ]. However, these studies concern mostly the case where Ω is a ball inR^{N}
or Ω =R^{N} when one can make use of the symmetry of the solution and apply time
map technique (shooting method). Let us also emphasize that in the case (k) we
do not need any subcriticality condition for por q since we work with dynamical
methods in this case. The proof of (kk) is based on the use of Leray-Schauder degree
and the apriori estimates from [BT] and [FLN]. Finally, let us mention that in the
case (k), i.e. q≥p,

(kα) ifp=qthen λ_{0} ≥pdiam (Ω)^{−p} (see [C, Theorem 4]),

(kβ) ifq= 2 and N= 1 then there are exactly two positive solutions forλ > λ_{0}
(see [S, Example 3.2.2])

and in the casep <(N+ 2)/(N−2) and Ω being a ball in R^{N}

(kkα) ifq <2p/(p+ 1) ( ⇒ q <min(2,(N+ 2)/N)) then there exists a positive solution for anyλ >0. Moreover, this solution is unique if N = 1 (see [C, Theorem 3]).

(kkβ) If q = 2p/(p+ 1) and λ ≤ (2p)^{p}/(p+ 1)^{2p+1} then (DGT) does not have

positive stationary solutions. The estimate on λ is precise if and only if N = 1 (see [C, Theorem 3], [FQ2]).

(kkγ) If q > 2p/(p+ 1) and λ >0 is small then (DGT) does not have positive stationary solutions (see the proof of Theorem 3 (iii) in [C]).

2. Global existence for (NBC).

In this section we show that the assumptionp >2q−1 implies the global existence and boundedness of solutions of (NBC). Our results also imply the existence of a positive stationary solution for (NBC), since the zero solution is unstable.

Due to the results of [CFQ], it is sufficient to consider the caseN >1. As shown
in [CFQ], (NBC) generates a local semiflow in{u∈W^{1,r}(Ω) ;u≥0}for anyr > N.
Hence, we shall supposeu_{0}∈W^{1,r}(Ω) for somer > N,u_{0}≥0.

In [CFQ], the estimates from [FK] were used to get the global existence and boundedness results for (NBC) under the assumption

(2.1) q < N+ 1

N−1, p > c(q) := N−q(N−2)

N+ 1−q(N−1)(q+ 1)−1.

We use this information and a simple substitution to get the desired result.

Theorem 2.1. Ifp > 2q−1 >1then any solution of (NBC)exists globally and stays uniformly bounded.

Proof: Letube a maximal solution of (NBC),m≥1. Then v :=u^{m} solves the
problem

(2.2)

v_{t}=△v−m−1
m

1

v|∇v|^{2}−mav^{p}^{∗} in (0, T)×Ω,

∂v

∂n=mv^{q}^{∗} on (0, T)×∂Ω,

v(0, x) =u^{m}_{0}(x) x∈Ω,

where p^{∗} = (p+m−1)/m, q^{∗} = (q+m−1)/mand T is the maximal existence
time foru. Using the comparison principle one simply getsv ≤w, wherewsolves
the problem

(2.3)

w_{t}=△w−maw^{p}^{∗} in (0, T)×Ω,

∂w

∂n =mw^{q}^{∗} on (0, T)×∂Ω,
w(0, x) =u^{m}_{0} (x) x∈Ω.

Now it is sufficient to verify that the couple (p^{∗}, q^{∗}) fulfils the condition (2.1) iff
2m >(q−1)(N−1) and 2m(p+ 1−2q)>(q−1) (p−1)(N−1)−(q−1)(N−2)

, which is clearly true if m is sufficiently large. Hence, for m large one can apply the results of [CFQ] for the solutionwto get its global existence and boundedness and, consequently, also the global existence and boundedness for v and u (it is obvious that the linear factormin (2.3) does not play any significant role in [CFQ,

Theorem 4.6]).

Corollary 2.1. Ifp >2q−1then(NBC) has a positive stationary solution.

Proof: Put u_{0} ≡ε, whereε >0 is small enough. Since (NBC) possesses a Lya-
punov function Φ (see [CFQ]) and Φ(u_{0})<0 = Φ(0), theω-limit set of the solution
starting atu_{0} consists of (nonnegative) equilibria which are different from 0. Due
to the maximum principle, these equilibria are positive.

3. Blow up for (NBC).

In this section we shall suppose thatp≤2q−1 (andais small enough ifp= 2q−1) and we shall show that there exists a solution of (NBC) which blows up (in finite or infinite time). As a by-product of our considerations we obtain also an apriori bound for min

x∈∂Ωu(x), whereuis any positive stationary solution of (NBC).

Lemma 3.1. Let α > 2 be fixed and u_{δ}(x) := _{1}

ε δ−dist(x, ∂Ω)+α

, where
δ >0,αε^{α(q−1)}=δ^{α(q−1)+1} andv^{+}:= max(v,0). Ifδ is sufficiently small thenu_{δ}
is a subsolution for(NBC) and any positive stationary solutionuof (NBC) fulfils
min∂Ω(u−u_{δ})<0.

Proof: One can easily verify that u_{δ} fulfils the boundary condition in (NBC)
for any δ > 0. Further suppose that dist (x, ∂Ω) ≤ δ and δ is sufficiently small.

Denotingd(x) := dist (x, ∂Ω) and ϕ(d) :=_{1}

ε(δ−d)^{+}α one hasu_{δ}(x) =ϕ d(x)
and

△u_{δ}= (ϕ^{′′}◦d)|∇d|^{2}+ (ϕ^{′}◦d)△d.

Let y =y(x) ∈∂Ω be the closest point to x in ∂Ω and let n=n(x) be the unit (outward) normal to∂Ω aty(x). Then we have

|∇d|^{2}=∂d

∂n 2

= 1, ∂^{2}d

∂n^{2} = 0, |△d| ≤C,

where C is some constant depending only on the curvature of ∂Ω (cf. [GT, Lem- mas 14.16 and 14.17]). Using these estimates and the inequality p≤2q−1 (and a≪1 ifp= 2q−1) one can easily check that

△u_{δ}≥ 1

2ϕ^{′′}◦d≥aϕ^{p}◦d=au^{p}_{δ}

for δ sufficiently small, where the inequalities are strict if d(x) < δ. Henceu_{δ} is
a (strict) subsolution forδ≤δ_{0} andu_{δ}

∂Ω = ^{δ}_{ε}α

→+∞asδ→0+.

Now suppose that u is a positive stationary solution, u ≥ u_{δ}_{0} on ∂Ω. Put
Ω^{−} :={x∈Ω ; u(x)< u_{δ}_{0}(x)}. If Ω^{−} 6=∅ then the functionw :=u−u_{δ}_{0} fulfils
w= 0 on∂Ω^{−} and△w=△u− △u_{δ}_{0} ≤au^{p}−au^{p}_{δ}

0 <0 in Ω^{−}, i.e. w >0 in Ω^{−}
which is a contradiction. Hence Ω^{−}=∅andu≥u_{δ}_{0} in Ω.

Choose δ≤δ_{0} such that u≥ u_{δ} in Ω andu(x_{0}) = u_{δ}(x_{0}) at some x_{0} ∈ Ω. This
choice leads to a contradiction with the maximum principle:

ifx_{0}∈Ω thenu_{δ}(x_{0})6= 0 and△u(x_{0})≥ △u_{δ}(x_{0})> au^{p}_{δ}(x_{0}) =au^{p}(x_{0});

ifx0∈∂Ω then△u_{δ}(x)≥au^{p}_{δ}(x) +η > au^{p}(x) =△u(x) for someη >0 and all
x∈Ω close tox_{0}which gives a contradiction with^{∂(u}_{∂n}^{δ}^{−u)}(x_{0}) = 0, (u_{δ}−u)(x_{0}) = 0,

u_{δ}≤u.

Corollary 3.1. The solution of(NBC)starting atu_{δ}blows up.

Proof: Letu be the solution starting atu_{δ}. Then u_{t} ≥0 due to the maximum
principle. Ifuis bounded, thenu(t,·) has to converge to a stationary solutionw≥
u_{δ} since the orbit{u(t,·) ;t≥0} is relatively compact in the appropriate Sobolev
space (see [CFQ]). However, this gives us a contradiction with min

∂Ω(w−u_{δ})<0.

4. Stationary solutions for (NBC).

Supposeq < p <2q−1,N >1 and put

a_{0}:= inf{a >0 ; there exists a positive stationary solution of (NBC)}.
It follows from [CFQ] that a0 < ∞. First we prove the assertion (α) from the
introduction.

Theorem 4.1. If a > 0 is small enough then (NBC) does not have positive sta- tionary solutions.

Proof: By contradiction. Suppose that fora_{m}↓0 there exist positive stationary
solutionsu_{m}. By Lemma 3.1 we have min_{∂Ω}u_{m} =u_{m}(x_{m})≤K for some x_{m} ∈

∂Ω and a positive constant K. Let Ω_{m} be the component of the set {x ∈ Ω ;
u_{m}(x) < 2u_{m}(x_{m})} containing x_{m} in its closure. Let v_{m} be the solution of the
problem△vm= 0 in Ωm,vm =um on∂Ωm. Then ^{∂v}_{∂n}^{m}(xm)≤0 since vm attains
its minimum at xm (and ∂Ωm∩Um =∂Ω∩Um for some neighbourhoodUm of
xm). On the other hand, putting wm := um −vm we have wm = 0 on ∂Ωm,
0 ≤ △wm =amu^{p}_{m} ≤am2^{p}u^{p}_{m}(xm) in Ωm ⊂ Ω. The standard regularity theory
implies now

(4.1) u^{q}_{m}(x_{m}) =∂u_{m}

∂n (x_{m})≤∂w_{m}

∂n (x_{m})≤Ca_{m}2^{p}u^{p}_{m}(x_{m})≤Ca_{m}2^{p}K^{p}
for suitable C > 0, hence um(xm) → 0 as m → ∞. Using (4.1) again, we get
1≤Cam2^{p}u^{p−q}m (xm)→0, which is a contradiction.

Now supposeq < p <2q−1,q < _{N−2}^{N} ifN >2, and a > a_{0}. Then it follows
from [CFQ] that there exists a positive stationary solution u of (NBC) which is
a local minimizer of the corresponding functional

Φ(u) = Φ_{a}(u) = 1
2

Z

Ω|∇u|^{2}dx+ a
p+ 1

Z

Ω|u|^{p+1}dx− 1
q+ 1

Z

∂Ω|u|^{q+1}dS
in the Sobolev spaceW^{1,2}(Ω).

Puttingw_{ε}(x) :=ε^{−}^{q+δ}^{q}^{−}^{1} ε−dist (x, ∂Ω)+

, where 0< δ < ^{2q−1−p}_{p−q} , one can straight-
forwardly check that Φ(w_{ε}+u)→ −∞asε→0+. Hence to obtain a second critical
point of Φ (lying aboveu) one can use the mountain pass theorem for Φ with respect
to the convex set{w∈W^{1,2}(Ω) ;w≥u}similarly as in the proof of Theorem 2.1 (i)
in [CFQ]. The difficulty consists in verifying the corresponding Palais-Smale condi-
tion (cf. also Remark 2.4 in [CFQ]). Using a trick of M. Struwe we are able to do
this only for almost alla≥a_{0}.

Theorem 4.2. Letq < p <2q−1, q < _{N−2}^{N} . Then for a.a. a≥a_{0}, the problem
(NBC)has at least two positive stationary solutions.

Proof: Fixa_{2}> a_{1}> a_{0}and letua1 be a positive solution corresponding toa_{1}. As
shown in [CFQ], choosingu_{a}_{2} a global minimizer of Φ_{a}_{2} inK_{1} :={u∈W^{1,2}(Ω) ;
0 ≤ u ≤ ua1} we get a stationary solution of (NBC) with 0 < ua2 < ua1 in Ω,
Φa2(ua2)<Φa2(0) = 0. PutS :={u∈K1; Φa2(u) = Φa2(ua2)}. Then Φ^{′}_{a}_{2}(u) = 0
for anyu∈S and the setS is compact since Φ^{′}_{a}_{2} has the form identity +F, where
F mapsK_{1} into a compact set. Moreover,ν_{0}:= ^{1}_{2}dist (S,{u;u≥ua1})>0. Next
we show by contradiction that there existsν >0 (ν≤ν_{0}) such that

δ:=δ(ν) := inf{Φa2(u) ; dist(u, S) =ν} −Φa2(ua2)>0.

Hence assume thatδ(νn)≤0 for some νn ↓0. Let nbe fixed andν :=νn. Then
there existum such that dist (um, S) = ν and lim sup_{m→∞}Φa2(um)≤Φa2(ua2).

Consequently, um = u^{S}_{m}+vm, where u^{S}_{m} ∈ S and kvmk = ν. We may suppose
u^{S}_{m}→u^{S} ∈S andv_{m}⇀ v,kvk ≤ν.

Ifv_{m}→v then dist (u^{S}+v, S) =ν, Φ_{a}_{2}(u^{S}+v)≤Φ_{a}_{2}(u_{a}_{2}).

Ifv_{m} 6→v then Φ_{a}_{2}(u^{S}+v)<lim sup_{m→∞}Φ_{a}_{2}(u^{S}_{m}+v_{m})≤Φ_{a}_{2}(u_{a}_{2}) so that
u^{S}+v /∈S, 0<dist (u^{S}+v, S)≤ kvk ≤ν.

Let w^{S} be a local minimizer of Φa2 in {u;ku−u^{S}k ≤ dist (u^{S}+v, S)} such
thatw^{S} ∈/ S. By the definition ofS we havew^{S} ∈/K_{1}. By the same way as in the
end of the proof of [CFQ, Lemma 2.4] one getsw^{S} ∈C^{1}(Ω), kw^{S}−u^{S}k_{C}^{1}_{(Ω)}→0
forν=ν_{n}→0. Since dist_{C}1(Ω)(S, C^{1}(Ω)\K_{1})>0 by the maximum principle and
w^{S}∈/S, we get a contradiction.

Now choose ν and δ= δ(ν) with the properties above and fix ε > 0 such that
Φa2(ua1+wε)<Φa2(ua2). Further fixα∈(0, a2−a1) such that_{p+1}^{α} R

Ωu^{p+1}dx≤ ^{δ}_{3}
for any u ∈ {v; dist (v, S) ≤ ν} ∪ {ua1 +wε} and let ua2+α be a fixed positive
stationary solution fora=a2+αlying belowua2. Put

K_{2}:={u∈W^{1,2}(Ω) ;u≥ua2+α}

P :={p˜∈C([0,1], K_{2}) ; ˜p(0) =ua2,p(1) =˜ ua1+wε}
γ_{a}:= inf

˜

p∈P sup

u∈˜p([0,1])

Φ_{a}(u) for |a−a_{2}|< α.

Then, obviously,γ: (a_{2}−α, a_{2}+α)→R is a nondecreasing function so thatγ is
differentiable almost everywhere. Choosea∈(a_{2}−α, a_{2}+α) such that there exists
γ^{′}_{a}. We shall show that there exists a positive stationary solutionuof (NBC) with
Φa(u) =γa. Since any global minimizeruaof Φa inK1 fulfils

Φ_{a}(u_{a}) ≤Φ_{a}(u_{a}_{2})≤Φ_{a}_{2}(u_{a}_{2}) +δ

3 = inf{Φ_{a}_{2}(u) ; dist (u, S) =ν} −2δ
3

<inf{Φa(u) ; dist (u, S) =ν} ≤γa,

we find two positive solutions for (NBC) and we are done.

We shall proceed similarly as in [S1, Lemma 6.3]. Leta_{m}∈(a_{2}−α, a),a_{m}↑a,
and letp_{m}∈Pbe such that sup_{u∈p}_{m}Φ_{a}(u)≤γ_{a}+(a−a_{m}) (wherep_{m}=p_{m}([0,1])).

The definition ofγamimplies now thatSm:={u∈pm; Φam(u)≥γam−(a−am)} 6=

∅. Since Φa(u)≥Φam(u) we get also that

W_{m}_{0} :={u∈K_{2};γ_{a}_{m}−(a−a_{m})≤Φ_{a}_{m}(u)≤Φ_{a}(u)≤γ_{a}+ (a−a_{m})
for somem≥m_{0}}

is nonempty,W_{m+1}⊂W_{m}. It is easy to see that foru∈W_{m}_{0} we have
1

p+ 1 Z

Ω

u^{p+1}dx≤ γa−γam

a−a_{m} + 2 for suitablem≥m_{0},
so thatW_{m}_{0} is bounded inL^{p+1}(Ω).

Foru∈K_{2}, put
g(u) := sup

v∈K2

ku−vk≤1

hΦ^{′}_{a}(u), u−vi, g_{m}(u) := sup

v∈K2

ku−vk≤1

hΦ^{′}_{a}_{m}(u), u−vi.

Let K(u) := u−Φ^{′}_{a}(u) and let P_{2} be the orthogonal projection in W^{1,2}(Ω) onto
K_{2}. ThenK is a compact map and

hu−K(u), u−P_{2}K(u)i ≤g(u) max(1,ku−P_{2}K(u)k).

Using the characterization of the projectionP_{2} we get

hK(u)−P_{2}K(u), u−P_{2}K(u)i ≤0 for anyu∈K_{2}
and adding the last two inequalities we obtain

(4.2) ku−P2K(u)k ≤max(g(u),p

g(u)) for anyu∈K2.

Suppose that there existum ∈Wm such thatg(um)→0. Choosingv =um+

um

kumk in the definition ofg(um) we get−hΦ^{′}_{a}(um), umi ≤g(um)kumk. Adding this
inequality to the inequality (q+ 1)Φ_{a}(u_{m})≤C and using the boundedness ofW_{m}
inL^{p+1}(Ω) we get

Z

Ω|∇u_{m}|^{2}dx≤ 1

q−1g(u_{m})ku_{m}k+ ˜C,

which gives the boundedness of {um} in W^{1,2}(Ω). Hence we may suppose that
(a subsequence of){um} converges weakly to someu∈K2. Now the compactness
of K and (4.2) give us u_{m} →u = P_{2}K(u), Φ_{a}(u) = γ_{a}. Since u_{a}_{2}_{+α} is a strict
subsolution for (NBC) we get Φ^{′}_{a}(u) = 0 (cf. the proof of Lemma 2.4 in [CFQ]).

Now assume that the sequence{u_{m}} above does not exist, i.e. g(u) ≥ 4κ for
someκ > 0 and anyu∈ W_{m}_{1}. We may suppose that u_{a}_{2}, u_{a}_{1}+w_{ε} ∈/ W_{m}_{1} and
that g(u)≥3κ,g(u)−g(u_{m})≤κfor some neighbourhood ˜W of W_{m}_{1} in K_{2} such
thatua2, ua1 +wε∈/W˜ and ˜W is bounded in L^{p+1}(Ω). By [S2, Lemma 1.6], there
exists a Lipschitz continuous vector field ˜e: ˜W →W^{1,2}(Ω) such that

˜

e(u) +u ∈K_{2},
ke(u)˜ k <1,

hΦ^{′}_{a}(u),˜e(u)i <−minng(u)^{2}
C ,1o

for anyu∈W˜, whereC >0 is a fixed constant. Consequently, ifm is sufficiently
large thenhΦ^{′}_{a}_{m}(u),˜e(u)i<−β for someβ >0 and anyu∈W˜.

Now letη :W^{1,2}(Ω)→[0,1] be a Lipschitz function such thatη = 1 onW_{m}_{1} and
η= 0 outside ˜W. Extend ˜etoK2 by lettinge(u) :=η(u)˜e(u) foru∈W˜,e(u) := 0
foru /∈W˜. The functioneis Lipschitz and

hΦ^{′}_{a}_{m}(u), e(u)i

<−β foru∈Wm1,

≤0 foru∈K_{2},

= 0 foru /∈W .˜

Letψ: [0,∞)×K_{2} →K_{2}be the solution of the initial value problem

∂

∂tψ(t, u) =e ψ(t, u) , ψ(0, u) =u.

Letp^{t}_{m}:=ψ(t, pm),q_{m}^{t} :={u∈p^{t}_{m}; Φam(u)≥γam−(a−am)}. Since_{dt}^{d}Φam ψ(t, u)

≤0 for anyuand _{dt}^{d}Φ_{a}_{m} ψ(t, u)

t=0≤ −β foru∈q_{m}^{t} , we get inf_{u∈p}^{t}

mΦ_{a}_{m}(u)<

γam fort large enough which gives us a contradiction with the definition ofγam. In the rest of this section suppose thatN = 2,q < p <2q−1.

Lemma 4.1. Letunbe positive stationary solutions of(NBC)witha=an≤A <

∞such thatUn:= max

Ω un→+∞asn→ ∞. PutVn:= max

Ω |∇un|and letε >0.

Then

n→∞lim
Un^{q}

Vn = lim

n→∞

Vn

Un^{q+ε}

= 0.

Proof: Ifuis a positive stationary solution of (NBC) thenw:=|∇u|^{2} fulfils

△w= 2pu^{p−1}w+ 2X

i,j

∂^{2}u

∂x_{i}∂x_{j}
2

>0 in Ω,

hencewattains its maximum on the boundary∂Ω. Consequently,V_{n}=|∇u_{n}(˜x_{n})|
and U_{n}=u_{n}(x_{n}) for some x_{n},x˜_{n}∈∂Ω. Putα:= q−1

q−1 +ε/2 and choose a unit
vectorν_{n}such thatν_{n}is not tangential to∂Ω at ˜x_{n}and

∂un

∂νn(˜x_{n})
≥ 1

2V_{n}. We may
suppose that ˜x_{n}+tν_{n}∈Ω fort >0 small (t <4U_{n}^{1−q}). The estimate 0< u_{n}≤U_{n}
implies that there existtn∈

0,4Un

V_{n}

i such that

∂u

∂ν_{n}(˜xn+tnνn)
≤ 1

4Vn so that
theC^{1,α}(Ω)-norm ofu_{n} can be estimated below by

(4.3) ku_{n}kC^{1,α} ≥ 1

4^{α+1}V_{n}^{1+α}U_{n}^{−α}.
On the other hand, theL^{r}-estimates (withr > _{1−α}^{N} ) imply

(4.4) ku_{n}kC^{1,α} ≤C_{1}ku_{n}kW^{2,r}≤C_{2} ka_{n}u^{p}_{n}kL^{r}+ku^{q}_{n}kW^{1,r}

≤C_{3}(U_{n}^{p}+U_{n}^{q−1}V_{n})≤C_{4}U_{n}^{q−1}V_{n},

sinceVn≥Un^{q}and p <2q−1. Using (4.3) and (4.4) we getVn≤C Un^{(q−1+α)/α}=
C U_{n}^{q+ε/2}, so that lim

n→∞

V_{n}
U_{n}^{q+ε} = 0.

To showU_{n}^{q}/Vn →0, suppose the contrary, i.e. Vn ≤C U_{n}^{q} for suitable C >0
(and a suitable subsequence of{V_{n}}). Puty:=A_{n}(x−x_{n})U_{n}^{q−1} andv_{n}=v_{n}(y) :=

u_{n}(x)/U_{n}, where A_{n} is an orthogonal 2×2 matrix such that the transformation
x7→y maps the tangent to∂Ω atxn to the line {y= (y1, y2)∈R^{2};y2 = 0} and
the pointx_{n}−ν_{n} (whereν_{n} is the unit outward normal to∂Ω atx_{n}) to the point
(0,1). Thenv_{n}fulfils

△vn = an

Un^{2q−p−1}

v_{n}^{p} in Ω^{n},

∂v_{n}

∂y_{2} =−v^{q}_{n} on∂Ω^{n},

where Ω^{n} := {y =y(x) ;x ∈ Ω}. Moreover,v_{n} >0, max_{Ω}nv_{n} = v_{n}(0) = 1 and

|∇vn| ≤C. Passing to the limit we getvn→v, wherevis a nonnegative harmonic
function in the halfspace [y2>0] fulfilling the boundary condition ∂v/∂y2=−v^{q}.
Moreover,v(0) = 1,v≤1 and|∇v| ≤C. Hence,w:=−∂v

∂y2 is harmonic, bounded
byC andw=v^{q} on [y_{2}= 0]. The Poisson’s formula ([SW, Theorem II.2.1]) gives
us

w(0, λ) =c Z

[y2=0]

v^{q}(y)λ

λ^{2}+|y|^{2}dy≥c
Z

[y2=0]∩[|y1|≤1/(2C)]

1 2

q λ

λ^{2}+|y|^{2}dy≥˜c/λ,
since v(y) ≥ ^{1}_{2} for |y| ≤ 1/(2C). This estimate gives us a contradiction, since
ϕ(·) :=v(0,·) :R^{+}→[0,1] fulfilsϕ^{′}(λ) =−w(0, λ)≤ −˜c/λ.

Theorem 4.3. LetΩ ={x∈R^{2};|x|<1}and q < p <2q−1. Then all positive
stationary solutions of(NBC) are uniformly bounded for a varying in a bounded
subset ofR^{+}.

Proof: Suppose the contrary and let u=u_{n} be as in Lemma 4.1 (we shall fix n
and omit the index n). Let (r, ϕ) be the polar coordinates inR^{2} and let ˜ube the
solution of the problem

△u˜ = 0 in Ω,

˜

u =u on∂Ω.

Then ˜u≥u, hence ˜ur:= ∂˜u

∂r ≤ ∂u

∂r =u^{q} on∂Ω.

Putw:=ru˜_{r}. Thenwis a harmonic function in Ω,w≤u^{q}≤U^{q} on∂Ω (where
U := max_{Ω}u). Hencew≤U^{q} in Ω and ˜ur=w/r≤2U^{q}in{x∈R^{2}; ^{1}_{2} ≤ |x| ≤1}.
Since ˜uis harmonic in Ω, we have|∇u(x)˜ | ≤U/dist (x, ∂Ω)≤2U ≤2U^{q}for|x| ≤ ^{1}_{2}.
Hence,

(4.5) u˜_{r}≤2U^{q} in Ω.

Chooseα∈(0,1) andε >0 such that

(4.6) p <2q−1−(1−α)(q−1)−αε.

Since

△(u−u) =˜ au^{p}≤AU^{p} in Ω,
u−u˜ = 0 on∂Ω,
theL^{r}-estimates imply

(4.7) ku−u˜kC^{1,α}≤Cku−˜ukW^{2,r}≤C U˜ ^{p}
for anyr >2/(1−α). Using (4.5)–(4.7) we obtain the estimate

∂u

∂r(x)≤C_{1}U^{q} if|x|>1−U^{1−q+ε}.

Now our assumptions and Lemma 4.1 imply U_{n}^{q}/V_{n} → 0, hence V = |∇u(˜x)| =
(K+ 1)U^{q} for some ˜x = ˜x_{n} ∈ ∂Ω and K = K_{n} → ∞. Consequently, denoting
uϕ := _{∂ϕ}^{∂u} we have|uϕ(˜x)| ≥KU^{q} and we may suppose uϕ(˜x)≥KU^{q}. Let (1,ϕ)˜
be the polar coordinates of ˜xand choose ˆϕ:= sup{ϕ <ϕ˜;uϕ(1, ϕ)≤ ^{K}4U^{q}}(using
obvious identification 0≡2π). Then uϕ(1,ϕ) =ˆ ^{K}_{4}U^{q} andω:=|ϕ˜−ϕˆ|< _{K}^{4}U^{1−q}
sinceuis bounded byU. Now the Schauder estimates imply

ku−u˜kC^{2,µ} ≤Ckau^{p}kC^{0,µ}≤CaU^{p}+Ca(2U^{p})^{1−µ} pU^{p−1}(K+ 1)U^{q}µ

≤q(K+ 1)U^{2q−1}

for µ sufficiently small and U large. Since |u_{ϕr}| = |qu^{q−1}u_{ϕ}| ≤ q(K+ 1)U^{2q−1}
on ∂Ω, we have also|u˜_{ϕr}| ≤ 2q(K+ 1)U^{2q−1} on ∂Ω. Now ˜u_{ϕ} is harmonic and,
similarly as in the case of ˜u, the last estimate implies |u˜ϕr| ≤4q(K+ 1)U^{2q−1} in
Ω. Consequently,|uϕr| ≤5q(K+ 1)U^{2q−1} in Ω.

Put S :={(r, ϕ)∈Ω ; ˆϕ < ϕ <ϕ,˜ 1−κ < r <1}, whereκ:= _{20q(K+1)}^{K} U^{1−q}.
Then

uϕ(r,ϕ)˜ ≥KU^{q}−(1−r)5q(K+ 1)U^{2q−1}≥3

4KU^{q} forr≥1−κ,
u_{ϕ}(r,ϕ)ˆ ≤K

4 U^{q}+ (1−r)5q(K+ 1)U^{2q−1} ≤1

2KU^{q} forr≥1−κ.

Hence, (4.8)

Z

S

1

r^{2}u_{ϕϕ}dϕ dr=
Z 1

1−κ

1

r^{2} u_{ϕ}(r,ϕ)˜ −u_{ϕ}(r,ϕ)ˆ

dr≥ κKU^{q}
4(1−κ)^{2} ≥4U
ifK (orU) is sufficiently large. On the other hand, we know thatur≤C_{1}U^{q}in S,
hence

(4.9)

Z

Surrdr dϕ≥ − Z ϕ˜

ˆ

ϕ C1U^{q}dϕ=−ωC1U^{q}≥ −4C1

K U ≥ −U

for K sufficiently large. By Lemma 4.1 we have |ur| ≤ |∇u| ≤ U^{2q−1} for nsuffi-
ciently large so that

(4.10)

Z

S

1

ru_{r}dr dϕ
≤ 1

1−κU^{2q−1}ωκ= 1

2q(1−κ)K^{2}U ≤U
forKlarge enough. Using (4.8)–(4.10) we get R

S△u dx≥2U. However, Z

S△u dx=a Z

Su^{p}dx≤aU^{p}κω≤U

forU and/orK large enough, which gives a contradiction.

5. Global existence for (DGT).

In this section we shall suppose that Ω is a smoothly bounded domain in R^{N},
N ≥1,p, q >1,r > Nmax(1, q−1) and

u_{0}∈W_{0}^{1,r}(Ω)^{+}:={u∈W^{1,r}(Ω) ;u≥0 in Ω andu= 0 on∂Ω}.

It is known (see e.g. [A1]) that (DGT) generates a local semiflow on W_{0}^{1,r}(Ω)^{+}
and that for any u_{0} ∈ W_{0}^{1,r}(Ω)^{+} there exists a unique maximal solution u ∈
C [0, T), W_{0}^{1,r}(Ω)^{+}

, whereT =T(u0) is the maximal existence time for u. More- over, this semiflow is order-preserving.

Byk · k∞we shall denote the norm inL^{∞}(Ω). The main result of this section is
the following

Theorem 5.1. (i) IfT <∞thenlim sup_{t→T}_{−}ku(t,·)k∞= +∞.
(ii) Ifq≥pthenT = +∞andsup_{t≥0}ku(t,·)k∞<∞.

(iii) Ifq≥pandut≥0 thensup_{t≥t}_{0}k∇u(t,·)k∞<∞for anyt0 >0.

Proof: To prove (i) it is sufficient to show that an L^{∞}-estimate for u implies
also an L^{∞}-estimate for ∇u. More precisely, let 0 < t_{0} < T_{0} < T <∞, C_{1} :=

max_{t≤T}_{0}ku(t,·)k∞<∞andC0:=k∇u(t0,·)k∞. Then we shall show thatC0<∞
and that there exists a constant C2 =C2(C0, C1, T) such thatk∇u(t,·)k∞ ≤C2

for anyt∈[t0, T0].

By [A1, Theorem 14.6] we have u∈C (0, t_{0}], W^{1,rq}(Ω)

hence|∇u|^{q} ∈ C (0, t_{0}],
L^{r}(Ω)

. Since W^{1,r}(Ω) ֒→ C(Ω), we have also u^{p} ∈ C [0, t_{0}], L^{r}(Ω)

and the
variation of constants formula for u on the interval [t0/2, t0] gives us u(t0,·) ∈
W^{2−ε,r}(Ω) for any ε >0. Since W^{2−ε,r}(Ω)֒→C^{1}(Ω) forε >0 small enough, we
haveC_{0}<∞.

Now putf(y) :=y^{q},g(y) :=λy^{p} and chooseC^{∞}-functions f_{k}, g_{k} (k= 1,2, . . .)
such that

• f_{k}=f andg_{k}=g on [1,∞),

• f_{k}≥f andg_{k}≤g on [0,1],f_{k}^{′}(0) = 0,

• f_{k}→f andg_{k}→g inC^{1} [0,∞)

as k→ ∞.
Letu_{k} be the solution of the problem

(DGT)_{k}

vt=△v−f_{k}(|∇v|) +g_{k}(v) in (t_{0},∞)×Ω,

v= 0 on (t_{0},∞)×∂Ω,

v(t_{0}, x) =u(t_{0}, x) x∈Ω.

Recall that u(t_{0},·) ∈ W^{2−ε,r}(Ω) for any ε > 0. By [A2, Theorem 7.3 and
Corollary 9.4], the problem (DGT)_{k} generates a local semiflow inW_{0}^{1+δ,r}(Ω)^{+} for
0 < δ <min(^{1}_{r},1−^{N}_{r},1−(q−1)^{N}_{r}) and, denoting byT_{k} the maximal existence
time of u_{k} in this space, we have u_{k} ∈ C^{∞} (t_{0}, T_{k})×Ω

. We shall show that
T_{k} > T_{0} and k∇u_{k}(t,·)k^{∞} ≤C_{2} for any t ∈[t_{0}, T_{0}] where C_{2} =C_{2}(C_{0}, C_{1}, T) is
independent of k. Then the variation of constants formula for z_{k} := u−u_{k}, the
Gronwall’s inequality forkz_{k}(t,·)kW^{2}^{−ε,r}(Ω)and a pass to the limit fork→ ∞gives
us|∇u| ≤C2.

First notice thatu_{k}≤uby the maximum principle and that it is sufficient to find
the estimatek∇u_{k}(t,·)k∞≤C2for anyt∈[t0,min(T_{k}, T0)) since then the variation
of constants formula gives an apriori bound also inW^{1+δ,r}(Ω), henceT_{k}> T_{0}.

Fixkand let ˜T < T_{k}, ˜T ≤T0. The functionw:=^{1}_{2}|∇u_{k}|^{2} fulfils the equation
(5.1) wt=△w−X

i,j

(u_{k})^{2}_{x}_{i}_{x}_{j}−X

j

f_{k}^{′}(∇u_{k})

|∇u_{k}| (u_{k})xjwxj+ 2g^{′}_{k}(u_{k})w
Since sup_{t∈[t}

0,T˜]2g^{′}_{k}(u_{k})≤2λpmax(2, C_{1}^{p−1}) =: ˆC if kis large enough, the maxi-
mum principle implies that the functionz:=we^{−}^{C(t−t}^{ˆ} ^{0}^{)}attains its maximumZin
Q:= [t_{0},T˜]×Ω on the parabolic boundary ({t_{0}} ×Ω)∪([t_{0},T˜]×∂Ω).

IfZ≤C_{0}^{2}/2 then ^{1}_{2}|∇u_{k}|^{2}=w≤ ^{1}2C_{0}^{2}e^{CT}^{ˆ} in Qand we are done.

IfZ > C_{0}^{2}/2 then Z=z(t, x0) for somet∈(t0,T] and˜ x0 ∈∂Ω. Consequently,

∂u_{k}

∂n(t, x_{0})

=|∇u_{k}(t, x_{0})|= max

x∈Ω|∇u_{k}(t, x)|=√

2Ze^{C(t−t}^{ˆ} ^{0}^{)/2}.
Sinceu_{k}is smooth at (t, x0), we have

(5.2) 0 = (u_{k})t(t, x0) =△u_{k}(t, x0)− |∇u_{k}(t, x0)|^{q}.
Ifν is any unit tangential vector to∂Ω atx0 then, obviously,

∂u_{k}

∂ν (t, x) ≤C˜

∂u_{k}

∂n(t, x)

|x−x0| forx∈∂Ω, x→x0,

where ˜C is some constant depending only on the curvature of ∂Ω at x_{0}. Conse-
quently,

(5.3)

△u_{k}−∂^{2}u_{k}

∂n^{2}

(t, x_{0})
≤C˜

∂u_{k}

∂n (t, x_{0})
.
Since|∇u_{k}(t,·)|attains its maximum atx_{0}, we have ∂^{2}u_{k}

∂n^{2} (t, x_{0})≤0. This inequal-
ity together with (5.2) and (5.3) imply|∇u_{k}(t, x_{0})|^{q−1}≤C, which gives the desired˜
estimate.

(ii) Ifq≥pthen it follows from [F] that the functionψ(x) :=α^{2/(p−1)}e^{α(}

P

ixi+C)

is a supersolution foruifαand C are large enough. Hence,u(t, x)≤max_{Ω}ψ for
anyt < T andx∈Ω. Now the assertion (ii) follows from (i).

Note that choosing ϕ(x) := min{ψ(x), Kdist (x, ∂Ω)} with K sufficiently large
we obtain a supersolution ϕ for u(t,·), t ≥ t_{0}, which gives us an apriori bound

|∇u|=

∂u

∂n

≤K on the boundary∂Ω.

(iii) Letq≥pandut≥0. Then

△u=ut+|∇u|^{q}−λu^{p} ≥ |∇u|^{q}−C1 for some C1 >0,
and, consequently,

X

i,j

u^{2}_{x}_{i}_{x}_{j} ≥C_{2}(△u)^{2}≥C_{3}|∇u|^{2q}−C_{4} for someC_{2}, C_{3}, C_{4}>0.

By the note in the proof of (ii), the functionw:= ^{1}_{2}|∇u|^{2}is bounded on∂Ω so that
the last inequality together with (5.1), the boundedness of u and the maximum
principle imply the boundedness ofwin [0,∞)×Ω.

6. Stationary solutions for (DGT).

Throughout this section we suppose that Ω is a smoothly bounded domain inR^{N},
N ≥1. By a (stationary) solution we mean always a classical positive stationary
solution.

Lemma 6.1. Letq≥p, λ_{1} >0. Then there exists K=K(λ_{1})>0such that any
positive stationary solutionuof(DGT) withλ≤λ_{1} fulfils kuk_{C}^{1}_{(Ω)}≤K.

Proof: We shall use similar arguments as in the proof of Theorem 5.1 (ii), (iii).

One can easily find a function

ϕ(x) =ϕ_{α}(x) = min{ψ_{α}(x), Kdist (x, ∂Ω)},
whereψα(x) =α^{2/(p−1)}e^{α(}

P

ixi+C) and K=K(α) is a continuous nondecreasing function ofα, lim

α→∞K(α) = +∞, such that forα≥α_{0},ϕis a strict supersolution
for (DGT) with anyλ≤λ_{1}. Now suppose that uis a positive stationary solution
of (DGT) with λ ≤ λ_{1} which does not lie below ϕα0. Choosing α_{1} := inf{α;
ϕα ≥ u} we have ϕα1 ≥ u and either ∂ϕα1

∂n (x_{1}) = u(x_{1}) for some x_{1} ∈ ∂Ω or
ϕα1(x_{2}) =u(x_{2}) for somex_{2}∈Ω. Since both possibilities lead to the contradiction
with the maximum principle, we haveu≤ϕα0, i.e. we have an apriori bound (say
C_{1}) foruinL^{∞}(Ω) and an apriori bound for

∂u

∂n

=|∇u|on∂Ω.

Puttingw:= ^{1}_{2}|∇u|^{2}and assuming thatwattains its maximum at somex_{0} ∈Ω,
we get by (5.1) (withwt= 0,△w(x_{0})≤0,wxj(x_{0}) = 0) and (DGT)

2λ_{1}pC_{1}^{p−1}w(x_{0}) ≥2λpu^{p−1}(x_{0})w(x_{0})≥X

i,j

u^{2}_{x}_{i}_{x}_{j}(x_{0})

≥C_{2} △u(x_{0})2

=C_{2} |∇u(x_{0})|^{q}−λu^{p}(x_{0})2

≥C_{3}w^{q}(x_{0})−C_{4},

which gives an apriori bound forw(x_{0}).

Remark 6.1. The apriori bound inC^{1}(Ω) and standard regularity results for the
stationary problem related to (DGT) imply also an apriori bound in W^{2,r}(Ω) for
any r >1 so that the set of positive stationary solutions for λ≤ λ_{1} is relatively
compact inC^{1}(Ω).

Theorem 6.1. Let q ≥ p. Then there exists λ_{0} > 0 such that the stationary
problem corresponding to(DGT)

(i) does not have positive solutions for λ < λ_{0},

(ii) has at least one positive solution forλ=λ0 and at least two positive solu- tions forλ > λ0.

Proof: To prove (i) suppose the contrary, i.e. there exist solutionsun with λ=
λn↓0. By Lemma 6.1, these solutions are uniformly bounded inC(Ω) by some con-
stantC_{1}. Denoting byν_{n}the norm ofu_{n}inW^{1,2}(Ω), multiplying the (stationary)

equation in (DGT) byu_{n} and integrating by parts we get

(6.1)

cν_{n}^{2} ≤
Z

Ω|∇u_{n}|^{2}dx=−
Z

Ω|∇u_{n}|^{q}u_{n}dx+λ_{n}
Z

Ω

u^{p+1}_{n} dx

≤λnC_{1}^{p−1}
Z

Ω

u^{2}_{n}dx≤λnC_{1}^{p−1}ν_{n}^{2}

for suitablec >0, which gives us a contradiction.

(ii) Suppose that (DGT) has a positive stationary solutionu0 for some λ0 >0
and letλ > λ_{0}. Then u_{0} ∈ W^{2,r}(Ω) by Remark 6.1 and u_{0} is a (strict) subsolu-
tion for (DGT). By the maximum principle, ut ≥ 0 for the solution u of (DGT)
starting atu_{0}. Consequently, the function u(t,·) is bounded inW^{1,∞}(Ω) by The-
orem 5.1 (iii). Standard parabolic regularity results imply now the boundedness
of u(t,·) in W^{2−ε,r}(Ω) for any r > 1 and ε > 0 so that the orbit {u(t,·)}t≥0 is
relatively compact inC^{1}(Ω). Sinceu_{t}≥0, we haveu(t,·)→u˜ ast→+∞, where

˜

uis a positive stationary solution of (DGT).

To see that (DGT) has a positive stationary solution at least for some λ, let
u0 be a nonnegative C^{2}(Ω)-function such that u0 = 0 on ∂Ω, △u0 ≥ |∇u|^{q} in
a neighbourhoodU of∂Ω andu_{0}≥ε >0 in Ω\U (It is sufficient to chooseu_{0}(x) :=

w(dist (x, ∂Ω)) for x close to ∂Ω, where w is the solution of O.D.E. w(0) = 0,
w^{′}(0) =C ≫1,w^{′′}(y) = 2w^{′}^{q}(y) fory ∈(0, δ],u_{0}(x) :=w(δ) for dist (x, ∂Ω)> δ,
and then regularizeu_{0} in the δ/2-neighbourhood of {x; dist (x, ∂Ω) = δ}.). Then
u_{0} is a subsolution for (DGT) if λis sufficiently large, hence (similarly as above)
we get the existence of a positive stationary solution.

Until now, we have shown the existence of a λ_{0} > 0 such that the stationary
problem corresponding to (DGT) has

(j) no solutions forλ < λ0,

(jj) at least one solution forλ > λ0.

To prove the existence of a solution forλ=λ_{0}, let u_{n} be solutions corresponding
toλ_{n}↓λ_{0}. Due to the apriori bounds (Lemma 6.1 and Remark 6.1) we know that
u_{n} converge to some nonnegative stationary solution of (DGT) with λ =λ_{0}. To
showu6≡0, suppose the contrary. Then similarly as in (6.1) we get

cν_{n}^{2} ≤
Z

Ω|∇un|^{2}dx≤λn

Z

Ω

u^{p+1}_{n} dx≤λn

Z

Ω

u^{2}_{n}dx
max

Ω

u^{p−1}_{n}

≤λnν_{n}^{2}max

Ω u^{p−1}_{n}

and sinceun→0 (even inC^{1}(Ω)), we get a contradiction.

Now let λ > λ_{0} and let u_{λ} be the positive stationary solution which we have
got as the limit of the solution ˆu of (DGT) starting atu_{0} (= positive stationary
solution corresponding toλ0). ChooseK >0,K >sup_{t≥0}ku(t,ˆ ·)k_{C}1(Ω)and letf_{K}

be a smooth cut-off function for the functiony 7→y^{q}; more precisely, f_{K}(y) =y^{q}
fory ∈[0, K],f_{K}(y) =K^{q}+ 1 fory ≥K+ 1,f_{K}^{′} >0 on [K, K+ 1),f_{K}(y)≤y^{q}
for anyy≥0. Consider the problem

(DGT)_{K}

ut=△u−f_{K}(|∇u|) +λu^{p} in (0,∞)×Ω,

u= 0 on (0,∞)×∂Ω,

u(0, x) =u_{1}(x) x∈Ω,

where 0≤ u1 ≤u_{λ}, u1 ∈ W^{2−ε,r}(Ω)∩W_{0}^{1,r}(Ω) for somer > N/(1−2ε). Since
f_{K}(y) =y^{q} fory ≤ ku(t,ˆ ·)k_{C}^{1}_{(Ω)}, the functionu_{λ} is a positive stationary solution
of (DGT)_{K} and ˆu(t,·) is a nondecreasing solution of (DGT)_{K} connecting u_{0} to
u_{λ}. Moreover, 0 is a stable stationary solution of (DGT)_{K} and one can easily find
a positive function ˜u0such that∂˜u_{0}

∂n >0 on∂Ω and the solution of (DGT)_{K}starting
at ˜u0 tends (in a monotone way) to 0 ast→ ∞. DenotingS^{τ}u1:=u(τ,·) whereu
is the solution of (DGT)_{K}starting atu_{1}, we get thatS^{τ}is (for anyτ >0) an order-
preserving discrete semigroup which maps the order interval [0, u_{λ}]⊂ W^{2−ε,r}(Ω)
into a relatively compact subset of [0, u_{λ}]. Moreover, 0 or u_{λ} is an equilibrium of
S^{τ} which is stable from above or from below, respectively. Due to [AH, Lemma 5],
there exists another equilibriumu^{τ} of S^{τ} which lies between 0 andu_{λ}. Since u^{τ}
lies neither aboveu0 nor below ˜u0, we have

(6.2) min{ku^{τ}−u_{λ}k_{C}^{1}_{(Ω)},ku^{τ}k_{C}^{1}_{(Ω)}} ≥c_{0}>0

for some c_{0} which is independent of τ. The variation of constants formula and
a straightforward estimate imply that the set{u^{τ}}τ∈(0,τ0)is bounded inW^{2−ε,r}(Ω)
and hence we may find a sequenceτ_{k}↓0 such thatu^{τ}^{k} →u_{K} inW^{2−2ε,r}(Ω). Due
to (6.2),u_{K}is a positive stationary solution of (DGT)_{K}which lies in [0, u_{λ}]\{0, u_{λ}}.
Putw_{K} := ^{1}_{2}|∇u_{K}|^{2}. We show thatw_{K}≤ ^{1}2K^{2} forKsufficiently large so thatu_{K}
is also a stationary solution of (DGT).

Since w_{K} ≤ ^{1}_{2}|∇u_{λ}|^{2} ≤ ^{1}_{2}K^{2} on the boundary ∂Ω, we may assume that w_{K}
attains its maximum at somex_{0} ∈Ω. Supposew_{K}(x_{0})> ^{1}_{2}K^{2}. Using an analogue
to (5.1) we get, similarly as in the proof of Lemma 6.1,

(6.3) C1w_{K}(x0) ≥2λpu^{p−1}_{K} (x0)w_{K}(x0)≥X

(u_{K})^{2}_{x}_{i}_{x}_{j}(x0)≥C2 △u_{K}(x0)2

=C2 f_{K}(|∇u_{K}(x0)|)−λu^{p}_{K}(x0)2

≥C3K^{2q}−C4

On the other hand, due to theL^{r}-estimates for the stationary problem correspond-
ing to (DGT)_{K} we have

ku_{K}kW^{2,r}(Ω) ≤C_{5}+C_{6}kf(|∇u_{K}|)kL^{r}(Ω)

≤C5+C6(K^{q}+ 1)^{(r−1)/r}Z

Ωf(|∇u_{K}|)dx1/r

≤C_{7}+C_{8}K^{q(r−1)/r},