Explosive solutions of quasilinear elliptic equations: existence and uniqueness

Nonlinear Anolysrs, Theory. Methods & Applicarrons. Vol. 20, No 2, pp. 97-125, 1993. 0362-546X/93 $6.00+ .OO

Prmted in Great Britain. 0 1993 Pergamon Press Ltd

EXPLOSIVE SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS: EXISTENCE AND UNIQUENESS

G. D~Az~-$ and R. LETELIER~

1‘Departamento de Matematica Aplicada, Fact&ad de C.C. Matematicas, Universidad Complutense de Madrid, 28040-Madrid, Spain; and 0 Departamento de Matematicas, Facultad de Ciencias, Universidad de Conception,

Casilla 3-C, Conception, Chile

(Received 1 August 1990; received for publication 26 February 1992)

Key words and phrases: Nonlinear elliptic equations, explosive solutions, interior bounds, existence, uniqueness.

1. INTRODUCTION

THIS PAPER deals with the quasilinear elliptic equation

-div(Q(IVul) VU) + A/~(U) = f infi c RN, N> 1;

more precisely, existence, uniqueness of local solutions satisfying

U(X) + aJ as dist(x, X2) + 0

and other properties are the main goals here. In the bibliography, these kinds of functions are called explosive solutions.

First of all, we point out that no behaviour at boundary to be prescribed is a priori imposed. However, we are going to show that, under an adequate strong interior structure condition on the equation, explosive behaviour near 80 cannot be arbitrary. In fact, there exists a unique such singular character governed by a uniform rate of explosion depending only on the terms

Q, 1, Bandf. On the other hand, since explosive solutions are not defined on the boundary, certain

classical consequences of the maximum principle must be carefully used. In this sense, we are going to prove the comparison u I u in ~2, for any local solutions u and u of (&) verifying

limsupz I 1 as dist(x, %2) --t 0.

In order to simplify the presentation we are concerned with the illustrative model

-div(IVulP-2 Vu) + Au” = f @Xl

posed, in local sense, in a bounded open set L2 of IRN, N > 1, with 82 E C2, providedf E C(sZ), f 2 0, and the strong interior structure condition

m>p--I. ws)p,

As in [l], the study of the behaviour near &2 uses an argument on the distance function dist(x, X2). Essentially, it consists of working on internal and external artificial boundaries.

$ Partially supported by DGICYT Grant No. 86/0405.

97

98 G. D~AZ and R. LETELIER

Arguing on dist(x, %Je4, we develop two schemes

div(lVUlp-2 VU) = f - Aurn, where eventually f Q Lu” can be satisfied

and

div(lVulp-2 VU) Q AU” = f, whenever f(x) goes to 00 as dist(x, %J) -+ 0.

Thus, we obtain the maximal growth at boundary of all functions satisfying

-div(lVuIP-’ VU) + Aum of in Sz.

We also obtain that the minimal growth at boundary of any explosive function satisfying

-div(lVuIP-2 VU) + )3zP 2 f in Q

which coincides exactly with the maximal growth at boundary. Consequently, a unique explosion rate at boundary is valid for all explosive local solutions of

-div(IVulP-2VU) + Au” = f in L2.

In fact, by the local comparison criterion, this property implies that there exists, at most, a unique explosive local solution of (&>“, . We note, also, that the explosive rate is independent on the dimension N.

Next, a natural question is to know whether some explosive forcing term f can transfer their singular character to the local solutions. In this sense, we prove that if

f(x) = c(dist(x, an))-*, q 2 p and c > 0 near afi

all functions satisfying

-div((VulP-2 VU) + 2~” L f in Sz,

verifies

U(X) --f 00 as dist(x, aQ) -+ 0.

Then, in view of the above comments, the equation

-div(IVuIP-2 VU) + Au” = f in Q,

admits, at most, a unique local solution, that, a fortiori, is explosive. In order to obtain existence results, some interior properties are required. So, we prove that

for all functions u satisfying

-div(lVulP-2 Vu) + Lum 5 f in Q,

in a bound like

U(X) cc CRP’@-m-l) + (21-i supf)i’m, if Ix - 21 5 2(1-P)‘PR B

holds, for any B = B&) CC L2. Since m > p - 1, an estimate on the decay at infinity and a version of Liouville theorem could be derived whenever Q is the whole space [RN.

Relative to the gradient, we obtain an interior bound by using the Bernstein method, provided aQ has positive mean curvature and f E w;,‘;” (Q). Thus, existence of explosive solution is obtained by classical arguments.

Quasilinear elliptic equations 99

The paper is organized into another five sections. 2: comparing local solutions; 3: behaviour near the boundary. Uniqueness of explosive solutions; 4: making explosive local solutions by external forcing terms; 5: some interior properties; 6: existence of explosive solution.

As it was pointed out, some results use arguments already introduced in [l]. More precisely, in this reference, one studies the problem

-(1/2)Au + (l/q)]V@ + Au = f in C2, (q > 1)

u(x) 4 co as dist(x, an) + 0,

as the Dynamic Programming approach of an optimization problem involving constraints on the state of the system. Here, at least in an heuristic way, a close device can be tried for the semilinear problems

-Au+Lu”=f in CJ, (m > 1)

u(x) + 00 as dist(x, XJ) + 0.

In both cases, the nonlinear terms

x I-# Ivu(x)]q-2 Vu(x) and x u (u(x))“_’

denote optimal feedback controls for the respective problems. Adequate choices of (&) also appear in several applications. So, the equilibrium of a charged

gas in a container [2], PDEs invariant under conformal or projective transformations [3], or related questions to classical Thomas-Fermi equations [4] are governed by similar equations. See [5] and the bibliography therein for other models appearing in: reaction-diffusion systems, nonNewtonian fluids, flows through porous media, plasma physical, etc.

In [6] is shown an overview on unbounded local solutions of

-(Q(lu’I)u’)’ + Afi(u) = 0 inSZ C R.

2. COMPARING LOCAL SOLUTIONS

As pointed out, our aim is concerned with the quasilinear equation

-div(Q(IVu]) Vu) + nfi(u) = f in Sz (@

where Q: R, + R, and /I: R, + IR, are continuous functions, n is a domain of RN, N 2 1, ;1 is a positive constant andfe Lii,,(sZ), f 2 0.

More precisely, we deal with the following notion.

Definition. Let 1 < s < 43 and f E L&(C2) ((l/s) + (l/s’) = 1). We say that u E I+$“@) is a local solution of (&) if

i n, QdVdPuVy,~ + A

s *,PWY,du = fvk

s v a, E W,‘q2’)

0’

for every S’Y CC Sz, provided Q(]Vul) Vu E (JC&(CJ))~, p(u) E &(~).

100 0. DiAZ and R. LETELIER

As usual, some general structural conditions on the equation are required. So, denoting

44 = QW, r>O

we will assume

a E C(lR+) n C’(IR+), a(O+) = 0 and u’(r) > 0 for r > 0.

Involved with the function a we construct

(2.1)

‘r

A(r) = ! a’(.s)s ds, r 2 0, 0

with A E C(f?+) n C’(M?+).

Remark 2.1. Among the functions Q satisfying (2.1), we pick up the choice

QJr) = rpd2, p > 1, for which a,(r) = rpP1 and A,(r) = (p - l)p-‘rP,

and the unhomogeneous one

Q(r) = (1 + r2)-1’2, for which a(r) = r(l + r2)-1’2 and A(r) = 1 - (1 + r2)-1’2.

Obviously, the contributions apply to a more wide class of Q. In order to obtain a comparison, we start with a technical result. For simplicity, we omit the

proof of the following lemma.

LEMMA 2.1. Assume (2.1). Then

(Q(kl>t - QdtkK - 0 1 0, r, i E RN. (2.2)

Moreover, for every ball 63 of IRN, N 2 1, there exists a positive constant c(a) such that

(QdCk - Q(lCk>(t - 0 2 c(@dt - Cl27 5, c E 65. (2.3)

Remark 2.2. By means of scaling arguments, it is not difficult to show inequalities

<Q,(kbt - Q,<lCk><t - t-1 2 C,k - rip, if25p

2 C,(l<l + lrl)p-21r - cl27 if 1 <p 5 2, ItI + 151 f 0,

for a suitable positive constant C, (see, for instance, [5, lemma 4.101). The main contribution in this section is the criterion of local comparison given by the

following theorem.

THEOREM 2.2. Assume (2.1) and

fi is increasing and /3(0’) = 0.

Let U, v E W$$“(fi) n C(n) be two nonnegative functions satisfying

-div(Q(]Vu() VU) + J&u) % -div(Q((Vu]) Vu) + J&(u)

(2.4)

in Q (2.5)


and

Then

limsupz 5 1, as dist(x, %I) -+ 0. (2.6)

n(x) 5 W, x E &2. (2.7)

Proof. By continuity, if (2.7) fails, there exists a ball B CC i2, small enough, such that (U - V) > 0 in B and, consequently,

0 < K(B) = A (/3(u) - jT(u))(u - u)(x) dx. B

Furthermore, there exists E(B) > 0 such that

0 < u - (1 + E)U in B, for all E E IO, c(B)[.

On the other hand, assumption (2.6) implies that, for every E E IO, e(B)[, there exists 6 > 0, small enough, such that

(24 - (1 + E)V)(X) I 0, if 0 i dist(x, an) 5 6.

Fix E E IO, &(B)[. Then, again by continuity, the existence of a smooth open Q’ C (x E i2: dist(x, &2) > 6) follows such that B CC a and

(U - (1 f &)U)+(X) = 0, Vx E K?’ (r+ = max(r, 0)).

So that, from (2.9, the divergence theorem and the Holder inequality yield

.r n,tQ(iVui)vu - QdU +GVd)U + E)VW’(U - (1 + dW+(X)~

+A .i

(P(u) - P(U + Eb4)@ - (1 + 4@+(X) dx n’

5 dIiQ(IVd) Vu - QdVd V&#‘&s + ~I&4 - PWbh&l

+ [ikQ(b’ub vu - Q(I(l + E) Vd) Vu)O’(u - (1 + ~~))~IL&‘(~ - (1 f Wi,s

+ JIIP(4 - P(U + d~)ll,~~Il~ - (1 + ~)4l,~l.

Therefore, from lemma 2.1, it follows

2 !

(P(u) - P((1 + c)@)(n - (1 + e)@+(x) dx B

5 dllQ<b’d> Vu - Q(b’d V&dlVdb + ~hN4 - P(~)I~r~~Il~ll~~l

+ [IIQdVuI) Vu - QdU + E) Vu/) V(u - (1 + Wll,+IIV(u - (1 + Wll,s

+ nllp(u) - P(U + EMllL.5’ll~ - (1 + ~)4l,~l.

Since B is independent of E, one derives the contradiction 0 < K(B) I 0 by sending E L 0. n

102 G. Dial and R. LETELIER

Remark 2.3. We note that M and v are not required to be defined on the boundary an. Uniqueness follows whenever equalities hold in (2.5) and (2.6).

3. BEHAVIOUR NEAR THE BOUNDARY. UNIQUENESS OF EXPLOSIVE SOLUTIONS

Due to the interior character of the local solutions of

-div(Q(lVul) Vu) + AD(u) = f in n, @)

auxiliary boundaries are required whenever we want to argue near a&J (see [l] for general details of this kind of reasoning). Then, the distance function

d(x) = dist(x, aa), x E mN

plays an important role. First of all, we recall the following lemma.

LEMMA 3.1 [7-91. Let IR be a bounded domain of iI?, N > 1, with aa nonempty. Then, d(e) is a Lipschitz continuous function in [RN. Furthermore, if 8J E Ck, k L 2, there exists p = ,uun ,

depending only on 51, such that

d(*) E Ck(K@)) and

IVd(*)l = 1 in B&(%2)

where B&(&2) = (x E 0: 0 < d(x) < 2~). In order to simplify the presentation, we only detail the illustrative case

-div(lVUI”-’ Vu) + M” = f in Q. (G

A first result is obtained by straightforward computations.

LEMMA 3.2. Assume

tTl>p-1.

Then, for every v > 0 the function

o(r; v) = c(m, p, v)rP’@-lmm), r>O

satisfies

((-o’(*; v))P_‘)’ + v(w(*; v))” = 0 in lF?+,

where c(m,p, v) = [(p/(m + 1 - ~))~(((rn + l)(p - l))/~p)]“(~+‘-~). Upper estimates near the boundary of local solutions can be derived from the following

theorem.

THEOREM 3.3. Assume (SIS)p, . Let Sz be a bounded open of IRN, N > 1, ZJ E C2 and f E C(a), f 1 0, verifying

lim sup f (x)(d(x))mp”“+‘-p’ 5 c(m, p, A - fo)“fo as d(x) + 0 (3.2)

with f. E [0, A[.


Then, for every nonnegative function u E H$A;.“(Q) fl C(Q) satisfying

-div()VuIP-’ VU) + AU” I f in a,

one has

lim sup ~4(x)(d(x))~““+‘-~’ 5 c(m, p, A - Jo) as d(x) --t 0. (3.3)

Proof. Let 6 E ]O,fiCln[, c E R, and v E IO, A - fO[. We consider in d_, 3 {x E M: 6 < d(x) < pccn) the function

W(x) = o@(x) - 6; v) + c.

Clearly, we have

VW(x) = o’@(x) - 6; v) Vd(x) and IVd(x)l = 1 ifxE&.

Then, denoting 11 AdI1 = max{)Ad(x)l: x E 8&(afi)), one has

-div(lVWlp-2 VW) + h(W)”

= ((-o’(d - 6; v))~-I)’ + (-o’(d - 6; v))~-’ Ad + v(co(d - 6; v))”

> A(co(d - 6; v))“(l - VA-’ - A-I(-o’(d - 6; v))P-lIIAd]l) in d_,.

On the other hand, for every E > 0, assumption (3.2) implies the inequality

f (x)(d(x)) mp’(m+‘-p) I c(rn,p, A - fo)“A(A-tfo + E) if 0 < d(x) < 6,

for some 6, = &I, E) E IO, ,u~[. So, c(m,p, A - fJ I c(m,p, v) leads to

-f(x) 2 -(A-‘f, + &)A(co(d(x) - 6; v))” if 0 < d(x) < 6,.

So that, if 6 < d(x) < 6, < p,, we obtain

-div(]VW(x)IP-2 VW(x)) + A(W(x))” -f(x)

> A(co(d(x) - 6; v))“(l-‘(f. - A - v) - E - A-‘C(m,p, v)&/[Adll]

where C(m,p, v) = (p(m + 1 - p)-l)P-l(c(m,p, v))~-~-~. Therefore, since v < A - fO, choosing E < X1@ - f0 - v) and 6r small enough, one concludes

-div( 1 V w(x)\~-~ V W(x)) + A( W(x))” 2 f(x) if 6 < d(x) < 6,.

If we take

c = ~(6,) = max{u(y): d(y) 2 S,],

property

o(d(x) - 6; v) = 00, if d(x) = 6

implies, by comparing in the region (x E Q: 6 < d(x) < 6,), the inequality

U(X) I w(d(x) - 6; v) + ~(6,) if 6 < d(x) < 6,;

thus

u(x)(d(x) - B)p’(m+l-p) I c(m,p, v) + c(d,)(d(x) - 8)p’(m+1-p) if 6 < d(x) < 6,.

Finally, (3.3) follows by sending 6 IO and then v /* ,I - fO. H


Remark 3.1. Theorem 3.2 can be extended to the general case

-div(Q(lVul) Vu) + A/3(u) = f in 0,

under the strong interior structural condition

‘m A(a) = co and I (A-‘@(s)))-’ d.s < c0

03

(SW

where ‘r

A(r) = I

a’(s)s ds, b(r) = ,0 i

‘rP(s)ds, rz0 0

provided that a(r) = Q(r)r and p(r) satisfy (2.1) and (2.4) respectively. Then, the assumptions

l im a@-‘(r/W)) = o

r-m P(r) and

(3.4)

(3.5)

imply

u(x) lim sup ____ d(x) IO %(d(X))

11

for r = I - f. and o, given implicitly by

‘co

\ (A-‘(S(s)))-’ ds = r.

CJ .w

We note that, for Q(r) = rpm2, condition (3.4) becomes

lim r”-‘(P(r))-’ = 0, r+rn

moreover, for p(r) = rm, assumptions (SIS) and (3.4) hold if

m>p-1. (SW

Remark 3.2. Assumption (3.2) [and (3.5)] is very general. It contains, in particular, the case f E L”(Q). See remark 3.3 for other comments.

In terms of dist(*, an)-y, the arguments of theorem 3.3 have used the scheme

div(Q(]Vu]> Vu) = LB(u) - f,

where eventually f Q p(u) can be satisfied. Next, an analogous device is tried for

div(Q(lVul) Vu) e Ap(u) z f.

Obviously, it requires that f(x) goes to infinity as d(x) -+ 0.


THEOREM 3.4. Assume (SIS)p,. Let Sz be a bounded open of RN, N > 1, an E C2 and f E C(a), f 1 0, verifying

lim supf(x)(d(~))~ 5 c as d(x) -+ 0 (3.6)

for

q > mp(m + 1 - p)-‘. (3.7)

Then, for every nonnegative function u E lI$‘(fi) n C(Q) satisfying

-div(IVulP-2VU) + AU” 5 f in Q,

one has

lim sup u(x)(~(x))~‘” 5 (J/c)“” as d(x) + 0. (3.8)

Proof. For every p > (c/h)““, 6 E IO, pun[ and c E R, we construct the function

W(x) = W-(x; p, S) = &d(x) - &(q’m) + c, x E h-6.

Clearly, we have in d_,

-div(lVWlp-2 VW) 2 -(pq/m)P-l[((q + m)/m)(p - 1) + (d - S)(lAdll]

x (d _ ~)-Kq+m)/m)W-l

As in the proof of theorem 3.3, we use IVES = 1 if x E 6_, and the notation

IlAdll = max(jAd(x)): x E B&(X2)].

On the other hand, for every E E IO, pm - K’c[ the condition (3.6) implies the existence of

6, E IO, pn[ such that

-f(x) 2 -A(& + K’c)(d(x))-4 1 -A(E + Pc)(d(x) - 8)-q if 6 < d(x) < 6i.

So, if 6 < d(x) < 6,, we have

-div(lVW(x)jP-2 VW(x)> + n(W(x))” -f(x)

2 (d(x) - Wq]-(pq/m)P-‘]((q + Wm)(p - 1) + (d(x) - ~)llAdlh x ((qx) _ ~)4-K~+wmP-w + a* - & - K’c)).

Since

it follows

q - ((q + m)/m)(p - 1) - 1 = (m - p + l)(q/m) - p > 0

-div(jVW(~)l~-~ VW(x)) + h(W(x))” -f(x)

2 (d(x) - S)-q(-(pq/m)P-l[((q + m)/m)(p - 1) + (~Ad1161](61)q-((q’tm)‘m)@-1’-1

+ A(p” - & - K’c)], if 6 < d(x) < 6,;

choosing 6, small enough, one concludes

-div() V W(x) lpe2 V W(x)) + A( W(x))” 2 f(x) if 6 < d(x) < 6,.


Finally, the choice c = ~(6,) = max{u(y): d(y) 2 6,) enables us to argue as in the last part of the proof of theorem 3.3 and derive

lim sup u(x)(d(x))q’” 5 p-l 5 (UC)“” as d(x) L 0. H

Remark 3.3. Theorems 3.3 and 3.4 can be summarized as follows. Assuming m > p - 1, we consider the continuous and increasing bijective function 6’: [0, A[ + lR+ given by

6(r) = rc(m,p, A - r), OIr<A.

Let Q be a bounded open of RN, N > 1, a&2 E C2 and f E C(n), f L 0. (i) If

lim sup f (x)(d(x))q 5 c as d(x) + 0

for q I mp(m + 1 - p)-‘, the maximal growth at boundary of any nonnegative function u E w,‘;‘(Q) fl C(n) verifying

-div(lVuIP-’ VU) + hu” 5 f in a

is exactly

c(m, p, A - fO)(d(x))-P’@‘“-p’

where f0 = s-‘(c). In particular, if

for d(x) small enough

lim f (x)(d(x))q = 0 as d(x) -+ 0

for q < mp(m + 1 - p)-‘, the maximal growth at boundary depends only on A, p and m. However, if

lim supf(x)(d(x))“P”‘“‘l-P’ % c E R, as d(x) + 0,

this behaviour off near aQ affects the maximal growth. (ii) If

lim supf(x)(d(x))q I c as d(x) -+ 0

for q > mp(m + 1 - p)-‘, the maximal growth at boundary of any nonnegative function u E IQ”(Q) n C(Q) verifying

-div(lVUlp-2 VU) + lu” 5 f in 0

is exactly

(A/c)l’“(d(x))-q’” for d(x) small enough.

Now our interest is focused on local solutions satisfying

u(x) -+ m as dist(x, X2) --) 0.

More precisely, the purpose here is to find lower estimates on the growth at the boundary of these explosive solutions. In order to do it, exterior auxiliary boundaries are constructed by using the following lemma.


LEMMA 3.5 [7-91. Let fi be a bounded domain of IRN, N > 1, with X2 E Ck, k 2 2, there exists ,D = ,un , depending only on 0, for which the function

d(x) = dist(x, XJ), XEQ

verifies

-dist(x, aQ), X@sZ

d(*) E Ck(4,(W)

and

jVd(.)] = 1 in &(aa)

where B,,(KJ) = lx E K?: 0 < dist(x, aa) < 2pu) > XJ. Relative to the scheme

div(Q()Vu]) Vu) E &3(u) - f,

we have the following theorem.

THEOREM 3.6. Assume (SIS)p,. Let c;Z be a bounded open of IRN, N > 1, as2 E C2 and f E C(n), f 2 0, verifying

lim inf f (x)(d(x))mp”“+l-p) 2 c(m, p, A - fO)“fO as d(x) --t 0 (3.9)

with f0 E [0, A[. Then, for every nonnegative function u E wt;,‘;“(a) fl C(n) satisfying

and

one has

-div(JVujpe2 VU) + AU” r f in Q

u(x) -+ CD as d(x) -+ 0

lim inf ~(x)(d(x))~‘(“+ 1-P) 2 c(m, p, A - fJ as d(x) -+ 0. (3.10)

Proof. Let 6 E IO, ,D*[. Using the notation of lemmas 3.1 and 3.5, we have

Bs(X-J) = (x E lRN: -6 < d(x) < 6]

&+(lX2) = {x E sz: 0 < d(x) < 6)

and B,C@SZ) c &(&2) c B,(Kl).

Let us choose v E ]A - fO, A[, if f0 > 0 and v > 2 if f0 = 0. Thus, we define the function

W(X) = o(d(x) + 6; v) - 0(6* + 6; v), -6 < d(x) < 6*

for 0<6<6*<~,, 6* to be chosen. Notice “4 < d(x) < 6* =$ x E B&(%2) n B&X2)“. First of all, we observe that w > 0 holds in B&(aa) fl Z3,(&2). Moreover, one has

VW(X) = o’(d(x) + 6; v) Vd(x) and IVd(x)] = 1 if x E Bs.(cX2).


Straightforward computations yield

-div(]Vrvlp-2 VW) + n(w)”

I i(u(d + 6; v))“(l - v/%-l + A-‘(-o’(d + 6; v))P-l\lAdll) in Bs.(K2)

where IlAd = max{IAd(x)I: x E B&(X2)). On the other hand, involved with E > 0, assumption (3.9) implies the existence of some

6, = &A, E) E IO, po[ such that: (a) if f0 > 0, inequality

-f(x) 5 A( & - A-‘f,)(c(m,p, /J - f,)(d(x) + B)p’(m+i-p))m if 0 < d(x) < 6,)

holds, provided E E IO, L-‘(fO + v - A)[; (b) if f0 = 0, inequality

-f(x) 5 -L&(c(m,p, I - f&d(x) + B)p’@+l-p))m if 0 < d(x) c c&,

holds. Then, if f0 > 0, inequalities

& < P(& + v - A), A-f,<v<A and @,P, A - fo) 5 c(m,p, v)

lead to

I(w(d(x) + 6; ~))~(l - VA-’ + A-‘(-o’@(x) + 6; v))P-lllAdl]) -f(x)

I A(o(d(x) + 6; v))~(E - A-‘(f. + v - A) + A-‘C(m, p, v)6,]]Ad]]]

if 0 < d(x) < 6 < 6,.

Whenever f. = 0, inequalities v > ,J and c(m,p, A - f,,) I c(m,p, v) yield

L(o(d(x) + 6; ~))~(l - VA-’ + X1(-o’(d(x) + 6; v))P-lIIAdl]] -f(x)

5 A(o(d(x) + 6; v))~{A-‘(A - E - v) + A-‘C(m,p, v)S,llAdl]] if 0 < d(x) < 6 < 6,.

(Again we use the notation C(m, p, v) = (p(m + 1 - p)-l)P-‘(c(m, p, v))~-‘-“.) In both cases

L(w(d(x) + 6; v))“(l - v/%--l + I-‘(-o’@(x) + 6; v))P-l]lAd]l] s f(x) if 0 < d(x) < 6,)

for 6, small enough, and consequently

-div(jVw(x)Ip-2 VW(X)) + I(w(x))” 5 f(x) if 0 < d(x) < 6

for 0 < 6 < 6* < 6,. Since w(x) = CO if d(x) = -6, the inclusion X2 C B,(dQ) C B&(X2) fl B&(X2) shows

w(x) I u(x) if 0 < d(x) < 6.

Then, by comparison results, one obtains the inequality

c(m,p, v)((d(x) + d)-p’(m+l-p) - (6* + 8)-p’(m+1-p)) 5 u(x), x E B&(X2)


for all 6 E IO, a*[; thus

c(m,p, v)(l - [(a* + d)/@(x) + B)]p’(m+l-p)) 5 u(x)@(x) + ay’(m+l-p), x E B,+,(&2)

for all 6 E 10, a*[. Finally, (3.10) follows by sending d(x) + 6 L 0, x E Sz and then v L I -fO. n

By using the scheme

div(Q(IVul) VU) 4 A/~(U) E f

we have the following theorem.

THEOREM 3.7. Assume (SIS)p, . Let a be a bounded open of iRN, N > 1, aa E C2 and f E C(n), f r 0, verifying

lim inff(x)(d(x))4 > c as d(x) + 0 (3.11)

for

q > mp(m + 1 - p)-‘. (3.7)

Then, for every nonnegative function u E II$“(a) tl C(a) satisfying

-div(]VU]p’-2VU) + lu” 2 f in a

and

U(X) + aJ as d(x) -+ 0

one has

lim sup ~(x)(d(x))~‘” 2 (I/c)“” as d(x) --+ 0. (3.12)

Proof. Here the notations of theorem 3.4 are considered. So, for every p < (c/13)““, 6* E IO, pun[ and 6 E ]0,6*[ we define in the open

B&(Kq fl B&X2) = (x E RN: -6 < d(x) < S*j

the function

w(x) = p(d(x) + ~3-‘~‘*) - p(6* + 8)-cq’m’.

Clearly, w > 0 in B,+,(aa) n B#fi). Moreover, by (3.7), one has

-div(IVwlP-2 Vw) I -(pq/m)P-“[((q + m)/m)(p - 1) - (d - S)llAdll](d - 6)-r(q+m”m1@-1)-1

I (pq/w+-‘(d - 6))lAdll(d - 8)-‘(q+m)‘m1@-1’ in B&(*(aQ),

with I/Ad]/ = max([Ad(x)l: x E B&(%J)). From (3.11) we deduce the existence of 6, = ai E 10, pun[ such that

-f(x) I -A(& - A-‘c)@(x))-4 I --A(& - K’c)(d(x) + d)-q if 0 < d(x) < 6 < 6,

for every E E IO, I-k - p”[. So that,

-div(IVw(x)]P-2 VW(X)) + A(w(x))” -f(x)

< (d(x) + ~)-q((pq/m)P-1(281)q-((q+m)‘m)@-1) t- l(p” + & - n-k)) -

if 0 < d(x) < 6 < 6,.

110 G. Di.u and R. LETELIER

Therefore, we have

-div( 1 V w(x) 1 P* V w(x)) + d( w(x))” 5 f(x) if 0 < d(x) < 6 < 6*

for 6* < 6,) 6, small enough. Arguing now, as in the last part of the proof of theorem 3.6, we obtain (3.12). n

Theorems 3.6 and 3.7 show that fhe maximal growth at boundary of all local substitutions of

-div([VuIP-* VU) + Au”’ = f in Sz @Xl

coincides exactly with the minimal growth at boundary of all explosive local supersolutions of (&>“,, provided adequate behaviour of f at XJ. In fact, in these circumstances, a unique behaviour at boundary of local solutions of (&)P, is available, as it is derived from the following theorem.

THEOREM 3.8. Assume (SIS)p,. Let Q be a bounded open of RN, N > 1, an E C* and f E C(Q), f 2 0. Then, the equation

-div(jVuIP-* VU) + ilu” = f in Q 0%

admits, at most, a unique explosive local solution in the class W&!;“(Q) n C(n). In particular, if

f(x) = c(dist(x, aO))-q, 4 E R c E G, for dist(x, aa) small (3.13)

the eventual explosive local solution u of (&)p, verifies:

u(x)(dist(x, aSZ))p’(m+l-p) -+ c(m,p, ,I - f,,) as dist(x, an) -+ 0, forf, = g-‘(c),

provided q 5 mp(m + 1 - p)-‘, and

u(x)(dist(x, as2))q’” -+ (A/C)"" as dist(x, aa) + 0

provided q < mp(m + 1 - p)-‘.

Proof. In view of theorems 3.3, 3.4, 3.6 and 3.7, the results follow by the criterion of local comparison given in theorem 2.2. n

Remark 3.4. As in remark 3.1, some extensions are available. Let us assume the strong interior structural condition

‘cc A(m) = 0;) and \ (A - ‘(B(s)))- ’ ds < m (SIS)

where ” ,

A(r) = /

a’(s)s ds, B(r) = ,0 1

‘rp(s) ds, r 2 0, 0

provided that a(r) = Q(r)r and P(r) satisfy (2.1) and (2.4) respectively. Suppose also

(3.4)


Let Sz be a bounded open of RN, N > 1, X2 E C* and f E C(a), f 2 0, verifying

f(x) P(o,(dist(x, an)))

-fo E P, AI as dist(x, &2) --t 0 (3.14)

for r = A - f. and w, given implicitly by

Then, the equation .i

co (A-‘(rB(s)))-’ ds = r.

+(0

-div(Q((Vuj) VU) + A/?(u) = f in Sz (@

admits, at most, a unique explosive local solution in the class W$‘(0) rl C(n). Moreover, the eventual explosive local solution 2.4 of (&) satisfies

o,(dist(x, X2)) -+I as dist(x, X2) -+ 0. (3.15)

4. MAKING EXPLOSIVE LOCAL SOLUTIONS BY EXTERNAL FORCING TERMS

This section is concerned with the analysis of external forcing termsf E C@),f 2 0, blowing up at the boundary. To simplify, we only consider the choice Q(r) = rpw2, p > 1.

First of all, we show that slow growths at the boundary of the external forcing cannot be transferred to the solutions. More precisely, we have the following lemma.

LEMMA 4.1. Let Sz = IO, R[. Then, if

f(r)rP --f 0 asr-+O

for every p E ]O,p[ the function u(r) = cry, with c > 0 and 0 < y < 1, verifies

-(jU’(+4’)’ + A/I(u) L f in IO, 6[

for 6 small and /I E C(R+), p 2 0.

(4.1)

Proof. Straightforward computations yield

-(/u’(r)lp-2u’(r))’ + @(u(r)) = (cy)p-l(l - y)(p - l)r(r-l)@-l)-l + Ap(u(r)).

On the other hand, for every p E ]O,p[, assumption (4.1) implies the existence of 6 = B(E) E IO, l[ such that

f(r) < erdp, for 0 < r < 6,

for E E IO, (c~)~-‘(l - y)(p - l)[. Then, inequality

-(IW)lp-2u’(r))’ + GWr)) - f(r)

2 r-P((cy)P-l(l - y)(p - l)r(Y-l)@-l)-l+p + W/3(cry) - E]

> Er-P((r(r-l)@-l)-l+P _ -

1) 9

112 G. DiAZ and R. LETELIER

holds by choosing p < 1 + (1 - y)(p - 1). Thus, we obtain

-(IU’(r)l~-2~‘(~))’ + M(G)) 2 f(r), O<r<6. n

On the contrary, fast growths are transferred, a fortiori, to the solutions, as the following result proves.

THEOREM 4.2. Suppose that %I satisfies the interior sphere condition at x0 E XII. Assume

limsupf(x)lx - xOlp = C E R, U (co) as x + x0. (4.2)

Then, if (2.4) holds, one has

U(X” - rfi(xg)) + co asr-+O (4.3)

for every nonnegative function u E w:;“(Q) n C(Q), verifying

-div(lVU(p-2 VU) + LB(u) 1 f in Q.

Proof. For every E E ]0,2-‘C[, the condition (4.2) implies the existence of 6 = cT(c) > 0 such that

f(x) > clx - xJP, 0 < Ix - X6( < 26, x E sz.

Denote xr = x0 - rti(x,), where r E R, and “(x0) is the unit outward normal vector to 13sZ at x0. Since the inclusion B,(x’) c B,,(x,J holds, one has

f(x) L c(2r)-P, x E &(x’), O<r<6,

whence

-div((Vu(x)(P-” Vu(x)) + &~(u(x)) 2 &(2r)-P, x E B,(x’).

Choose v, E C”(RN) satisfying

O((p_=l, q(O) = 1, A = Ijdiv(lV~lp-2 Vp)(/, # 0 and suPP p c B,(O).

Let us define the function

for K > 0 to be chosen later. We note that ZI vanishes on M,(x’). Then, one obtains the inequality

-div(lVv(x)lP-2 Vu(x)) + @(v(x)) -f(x)

5 r-“(AKP-’ + CrP - ~2~~)

5 reP(AKPel + GTp - ~2-~), x E 8(x’), O<r<d,

for C = MKlldJ. S o, if K E 10, (EA-‘~-~)“(~-‘)[ one derives

-div((Vv(x)/P-2 Vu(x)) + hj?(v(x)) 5 f(x), x E &(x7, O<r<6,

(4.4)


for 6 small. By comparison arguments, it follows

u(x) 2 u(x), x E &(x7, O<r<6;

since u(x’) = K, we have

u(xO - rw(xo)) 2 K, O<r<cY,

and consequently

lim inf u(x,, - ~ti(xJ) 2 K as r -+ 0. (4.5)

Actually, K depends only on C and p, and it is independent on the solution u. Then, we may repeat the argument as follows. From (4.2) and (4.9, for every v E IO, K[ and E E IO, C[ there exists 6 = B(E, v) > 0 such that

U(X) > K - v and f(x) r E(2r)-p, x E f&(x’), O<r<6.

In this case, the function

w(x) = u(x) + K - v, x E l&(x’)

verifies

-div(lVw(x)]p-2 VW(X)) + @(w(x)) 5 f(x), x E Mx’),

and

w(x) 5 u(x), x E d&(X’),

for 0 < r < 6, 6 small enough. [Now we choose C = Ap(K(j(c+& + 1) - v) in the relative version of (4.4).] Comparing again, we obtain

u(x) 2 w(x), x E J&(x’), O<r<6;

now w(x’) = 2K - v implies

u(xO - rti(xo)) 2 2K - v, O<r<6

and, hence

lim inf u(xO - m(x,)) 2 2K

An iterative argument concludes (4.3). n

as r ---* 0.

Remark 4.1. As it follows from lemma 4.1, the assumption (4.2) is, in some sense, optimal.

Remark 4.2, The proof of theorem 4.2 does not require the condition (SIS).

Above comments enable us to derive the following theorem from theorem 3.8.

THEOREM 4.3. Assume (SISK. Let Q be a bounded open of RN, N > 1, aQ E C2 and f E C(Q), f 2 0, verifying

f(x)(dist(x, &2))p --t C E R, U (co) as dist(x, aQ) --t 0. (4.6)

114

Then, the equation

admits, at most, a explosive solution.

In particular, if

G. D~AZ and R. LETELIER

-div(lVuIp-2 VU) + iu” = f in a (%l

unique local solution u in the class u’;&‘(s2) 17 C(n). Moreover, u is an

f(x) = c(dist(x, ~XJ))-~, 4 2 PY c E IR, for dist(x, X2) small,

the eventual local solution u of (&)“, verifies:

u(x)(dist(x, %2))P’@+‘-P’ -+ c(m, p, Iz - fo) as dist(x, X2) + 0, for f. = g-'(c),

provided q 5 mp(m + 1 - p)-I, and

u(x)(dist(x, X2))q’” + (UC)“” as dist(x, X2) -+ 0

provided q < mp(m + 1 - p)-‘.

Remark 4.3. We note that the inequality p < mp(m + 1 - p)-’ holds.

Remark 4.4. Extensions of theorem 4.3, as in remark 3.4, are also available.

5. SOME INTERIOR PROPERTIES

This section deals with interior properties of local solutions of

-div(Q(]Vu]) VU) + A/?(u) = f in Sz. 6%

They will play an important role in the proof of existence results. For the sake of simplicity, we present the contribution on the homogeneous model

-div((Vu(P-2 VU) + Au” = f in a. (@Zl

So, we have the following theorem.

THEOREM 5.1. Assume (SIS)p,. Let f E L;",,(n), f 2 0. Then, for every nonnegative function 2.4 E K$$“(sZ) n C(sZ) satisfying

-div(jVUlp-2 VU) + AU” 5 f in 0,

the inequality

U(x) 5 c(m,p, A;:(1 + E)-PB-lR”(l-P)(RP’(p-l) _ Ix _ ZIP’@-l))]P’@-m-‘)

+ ((1 + &)(&n)-‘s;pfy (5.1)

holds in every ball B = B&) CC !2 and arbitrary E > 0, where c(m, p, A) is the constant given

in lemma 3.2 and

P = (*y + A-‘N(p2/(p - l)(m + 1 - p)y-'(c(m, p, A))p-l-“.

In particular,


u(x) 5 c(m,p, A)[(1 + ~)-~R/(26)]~‘@-“-” + ((1 + &)(&A)-’ spf)““,

if Ix - z( 5 2c1-P)‘pR.

(5.2)

Proof. Since m > p - 1, one proves that

u(s; A) = c(m, p, A)s~‘~-~-~), s>o

verifies

(-o’(.s; A))p-*(u(s; A))-” = (p/(m + 1 - p))P-l(c(m,p, A))p-l-ms, s>o

and

((-o’(.s; A))P_‘)’ + A(@; A))” = 0 s>o

(see lemma 3.2). Let us define

w(r) = w(6- l/(1 -P)(RPm- 1) _ rPA.P-l)); A), O<r<R.

Then

Iw’(r)Ip-*w’(r)

= (P/(~ _ l))P-l++./R)(_u~(~-lR1’(l-p)(RP’@-l) _ ,.P’@-l)); A))“-‘, O<r<R.

Next, if we denote

W(X) = W(lX - A), x E BR(Z),

and r = Ix - 21, one has in B,(z)

-div()VW(x)jP-* VW(x))

= -(Iw’(r)lp-2w’(r))’ - y Iw’(r)lpe2w’(r)

= A(p/(p _ 1))Ps-P(r/R)P/(P-1)(0(8-1R1/(1-P)(RP/@-l) _ rP/(P-l)); I))”

_ N(p/(p _ l))P-lgl-PR-I(_WI(B-lR1/(l-P)(RP/(p-l) _ rP&J-l)); A))P-’

Consequently, for all x E B&z), we obtain

-div(lVW(~)l~-~VW(x)) + A(W(x))”

= A(W(x))“[l - (p/(p - l))pd-p(r/R)p’@-l) - A-‘iV(p/(p - l))p-l

x FP(p/(m + 1 - p))P-l(c(m,p, A))p-l-“(l - (r/R)p’@-l’))

2 A( W(x))“{1 - (p/(p - l))p - A-‘N(p’/(p - l)(m + 1 - p))P-l(c(m,p, A))p

=o

I-m&P] -1

provided

+ A-‘N(p*/(p - l)(m + 1 - p))P-l(c(m,p, A))p-l-“.


Finally, we consider the function

V(x) = cW(x) + (ds;pf)“‘“, x E B&z)

where c and d are two positive constants to be chosen. Inequality

(a + b)” 1 (1 + &)-l(P + &P), a, b L 0, E > 0,

and properties of W, lead to

-div(IVV(x)jp-’ VV(x)) + A(V(x))” - f(x)

- =- -cp-’ div(lV W(x)lpm2 VW(x)) + A(1 + ~)-l(cW(x))~ + (1 + E)-’ I&ds:pf - f(x)

2 S2Pf - f(x), for x E B&),

provided c = (1 + E)“~“-~ and d = (1 + s)(,Ls)-‘. Then, comparison arguments (see theorem 2.2) conclude (5.1). n

As in Sections 3 and 4, we may consider adequate extensions to the general case

-div(Q(jVul) VU) f @3(u) = f in 0, @)

under the strong interior structural condition

I ICC

A(w) = w and (A-‘@(s)))-’ ds < 00 (SIS)

and

i

r r A(r) = a’(s)s ds, B(r) = P(s) ds, r 2 0,

.O 0

provided that a(r) = Q(r)r and P(r) satisfy (2.1) and (2.4) respectively. So, we may prove the following technicality.

LEMMA 5.2. Assume (2.1), (2.4) and (SIS). Then, for each

the function

satisfies

R < m(A-1(U3(~)))-1 ds, i’ 0

V(x) = w,(N -1’2(R - Ix - zt,)), x E @3R(Z)

-div(Q(\VV() VV) + AD(V) I 0 in %(z),

where wx is given implicitly by

i

ca (A -‘W(s)))- wx(r)

and 63,(z) = (x E RN: (x - zll < R). (Here

‘ds=r, OsrcR,

IYII = Cil.Yil*)

(5.3)

(5.4)


Remark 5.1. Analogous interior properties were also obtained in [lo-121 or [13].

In this second part, the interest is concerned with interior estimates of the gradient of the local solutions of

-div(Q(/VuI) VU) + A/~(U) = f in a.

First of all, we note that if f = 0, N = 1 and Sz = ]a, b[, equation (@ becomes

-(Q(b’b’) + UW) = 0 in la, K

then assumption (2.1) yields

(g)

where

lu’(r)l = A-W(W)) - WW,))), a<r<b,

A(r) = i’

r a’(s)s ds, B(r) = ‘B(s) h,

s r L 0,

0 0

and u(ro) = min U. In several dimensions, we will use a device close to the well-known Bernstein method.

Essentially, it consists of obtaining bounds on the gradient of any local solution u of (&) by studying the auxiliary function

a(x) = A(jVu(x)() - lB(u(x)) - Cu(x), x E 0, c > 0.

In this sense, we are going to prove, later on, the following proposition,

PROPOSITION 5.3. Assume (2.1) and (2.4). Let f E C’(a), f 2 0 and u E W$‘(0) fJ C’(a) a nonnegative local solution of

-div(Q(IVu() VU) + Ap(u) = f in a. (a)

Then, if Sz is bounded and aa E C’ has positive mean curvature H,_, , one verifies

NIVu(x)l) - Wu(x)) - u(x)(a(suplVul) - wdvf b1’2

5 Ala-‘[((N - l)H,)-‘((a(suplVul) - suplVf ))1’2 + supf)]), x E Q (5.5)

where H, = min(H,_,(x): x E XJ).

Remark 5.2. A sharp estimate for the case f = 0 was proved in [14]. More precisely, they prove the inequality

AdVu(x)~) % W(u(x)) - B(m)) - 44x) - m), XE!3

where m(x) is the unit outward normal vector to &2 at x E a&2, rn = min u and

CY = min{O, min(H,_,(x)Q(/Vu(x)]) Vu(x) * e(x): x E Xl)),

for any u E C’(a) weak solution of

-div(Q(]VuI)Vu) + AD(u) = 0 in KZ.

Since maximum principle arguments imply VU * n. 2 0 on a0, it follows CY E 0 if&Z has positive mean curvature.


In view of proposition 5.3, a coercive argument enables us to deduce the following theorem.

THEOREM 5.4. Assume (2.1) and (2.4). Let f E C’(a), fz 0 and u E II$&‘(sZ) fl C’(a) a nonnegative local solution of

-div(Q((Vu() VU) + @(u) = f in Sz. (8)

Then, if Sz is bounded and XJ E C’ has positive mean curvature HN_l, there exists a positive constant C such that

lVu(x)l 5 C, XEL=J (5.6)

where C depends only on Q, A, p, Ho and upper bounds of u, f and Vf on a.

Proof. From proposition 5.3, one has

NllVull) - Il4(4llw) * IlVf IV2 - Ak-‘KW - l)Ha)-‘((a(llvulollvf b1’2 + Iif I]111

5 Wl4>,

where Ilull = I141r-cn~~ Ilf II = Ilf ILm(~), IIWI = Ilv41L~~~, and llvf II = Il~fll~-~n,. Consider the continuous function

W) = A(r) - Il4lW9ll~f IV2 - 4a-‘[W - 1)Jh-‘~@(r)l~Vf h1’2 + Ilf ]lll], r > 0.

Properties of the functions A and B imply

WO+) = -Ala-‘NW - V-W’kflll1 and S(co) = m.

Then the inequality

%ll~u]]) 5 W]]ull)

[see (5.5)] concludes the result. n

In existence results, we will use the consequence of the following corollary.

COROLLARY 5.5. Assume (2.1) and (2.4). Let fi be a bounded open set whose boundary has positive mean curvature HN_l and an E C’.

Let f E %;;“(&2), f 2 0 and u E II$;‘(a) fl C(sZ) a nonnegative local solution of

-div(Q(IVuI) VU) + J.&u) = f in a. G)

Then, for every 6 > 0, one has

IWx)l 5 c,, x E CL6 = {x E i2: dist(x, aa) > 6) (5.7)

for a constant C8 depending only on Q, A, fi, H, and upper bounds of u, f and Vf on the set a_&.

Proof. Clearly, k6 is a bounded open subset of Sz satisfying the assumptions of theorem 5.4. Then, (5.7) follows from (5.6) applied to a_,. H


The section ends with the following proof.

Proof of proposition 5.3. In order to simplify the presentation, we will assume smooth data. Then, the regularity of CD implies the existence of y E 0, such that

for

CD(x) = A(lVu(x)J) - U(u(x)) - CU(X), XEi=J

and C* = adDull) - ll~fll + E. Here IIv4 = IIV~IP~Q~~ llvfll = IIvI~-~~~ and E > 0. (i) Assume y E X2. By means of straightforward computations, equation (g) on the boundary

a&2 becomes

= w44) - f(x), x E asz,

where a(x) denotes the unit outward normal vector to 22 at x E &2 (see, e.g. [15, p. 611). Since @ attains a maximum at the boundary point y, inequality

0 5 E(Y)

holds. Then the above representation leads to

at4 a,(Y) + f(Y) +

whence

Monotonicity of a(r) = Q(r)r implies

I I 2 (Y) 5 A@-‘[(W - l)H,)-‘(C + f(~))ll,

consequently, inequality (5.5) follows by sending E to 0 in

A(IVu(x)l) - nNu(x)) - CU(X) 5 WY) 5 Na-‘KW - V-W’(C + .f(~))lL XE CL

(ii) Let us suppose, now, that y E Sz. We claim Vu(y) = 0. Indeed, assume Vu(y) # 0. Then, by using the summation convention, we have on !2+ = (x E Sz: Vu(x) # 0)

Oj~ = (Q’(IVU/)IVU/ + Q(lVu()]DiUDijU - AP(U)DjU - CDjU (5.8)

and

Djk@ = (Q”(Jvul) + 2Q’(IVuJ)IVuI-‘1DiuD~uD,UDlkU + (Q’(lVul)lVul + Q(IVUl))DikDijU

+ (Q’(lvul)lvul + Q(JVu()]DiuD,,ju - AP’(U)D,UDjU - lP(U)D,U - CDkjU.


Next, if we write (&) as

Q’((Vul)lVul-‘DjuD,uDj, u + Q(IVuI)Djju = AP(u) - _L @)

their derivative with respect to Xi is

Q”(/Vu()lVu1-*D,UD,iUDjUDkUDjk u - Q’(IVul)(VuJ-3D,uD,iuDjuD,uDj,u

+ Q’(JVul)(Vul-‘(DjiuD,uDj,u + DjUD,,UDj,U + DjUD,UDjklU + D,UD,iUDjjU)

+ Q(IVuI)Djjiu

= ~p’(U)OiU - Oif.

Thus, @ satisfies

Djj~ + Q’(IVul)(Q(IVul)(Vul)-‘DjuD,uDj,~

2 ((Q’(b’u~))3(Q(b’~l)~~~~)-1 + 4(Q’(IVub)2(Q(IVul))-1b’ui-2

- Q”(IvuI)(vuI-~ + Q’(~Vul)lVul-3)(DjUD~UDj~U)2

+ (Q’(IVuI)IVuI + Q(IvuI>l(Diju12 + {Q”(IVuI) - (Q(Iv~I))-‘(Q’(IVUI))~ + Q’(IVuI)IVuI-‘)DiuD,uDijuD,,u

- ((Q’(lv~l))2(Q(Iv~(>)-2 + Q’(IVuI)(Q(IVuI))-‘IVuI-‘)~P(u)DjuD,uDj,u

+ ((Q’(IVU())~(Q(IVU())-~ + Q’(IVuI)(Q(IVuI))-‘IVuI-‘lfDjuD,uDj,u

- Q’(IvuI)(Q((vuI>>-lIV~I-l~~~(~) + ClDjUD,UDj,U

- (Q’(IVuI)IVuI(Q(IVuI))-’ + lPiuDif_ tAP(u) + ClDjjU. Suitable manipulations at y are now required. So, V@(y) = 0, (5.8) and the Cauchy-Schwarz inequality yield

Djj@(Y) + Q’(Ivu(Y)I)(Q(IVu(Y)/)IVu(Y)I)-‘Dju(Y)Dku(Y)Dj,~(Y)

L (Q”(IVU(Y)~))-~@IVU(Y)~)IV~(Y)~ + Q(IWY)\)W~ - Q(IV~Y)I)~V~Y)~/V~(Y)II.

Denoting by d: the operator

we have

C@ = Djj~ + Q’(IVuI)(Q(IVul)lVul)-‘DjuD,uDj,Q, - ~kDk@,

C@(Y) 2 (Q(IV~(Y)I))-~~Q’(IVU(Y)I)~VU(Y)~ + Q(IV~(Y)~)W~ - Q(~VY(Y)~)~~Y)~~Y)IL

Since a’(r) > 0 implies

-Q’(lvul)(Q(b’ul)iVuI)-’ < IVUI-~,

the ellipticity strict of 6: on L2+ follows from

I<I2 + Q’(IV~I)(Q(IvuI)IVuI)-‘,Dj~D~~~j~~ > 0, < E RN*

Again using a’(r) > 0, we have

(Q’(~WY)I)IV~(Y)~ + Q(/Vu(y)l)I > 0,


consequently, uk(y)Dk@(y) = 0 implies

g@(y) 2 (Q(tVu(y)I))-21Q’(IVu(~)i)lVu(~)l + Q(]Vu(y)l)]lC’ - Q(llVd)lb'd * IlVflil.

The choice of C and the ellipticity strict of 6: derive the contradiction

0 > &X(y) > 0.

Thus, the claim Vu(y) = 0 is proved, and inequality

A(lVu(x)l) - AB(u(x)) - CU(X) 5 Q(y) I 0, XESZ

holds. Finally, (5.5) follows, sending E to 0. n

6. EXISTENCE OF LOCAL SOLUTIONS

General assumptions (2.1), (2.4), (SIS), f E l+$iirn(fi) and f 2 0 are assumed through this section on the quasilinear equation

-div(Q(lVuj) VU) + AD(u) = f in fi (@

where G is a bounded domain of RN, N > 1, whose boundary has positive mean curvature and an E c’.

First of all, for each A4 > 0, we consider the truncation

A&) = minlf(x), ML x E 0.

Remark 6.1. Clearly, (f,)M is an increasing family satisfying

fM E W’*“(Q) and (fM1 - f in B$“(fi) asM7co.

Thus, there exists a unique weak solution u,+_, E W’~“(~) of

-div(Q(IVvl) VU) + Ap(v) = fM in a (6.1)

v=M on aa, (6.2)

provided (2.1) and (2.4) hold.

Remark 6.2. For existence of uM, we refer, for instance, to [5] or [14]. We observe that the estimations of the gradient (see theorem 5.4) are now required.

Comparison results prove the following theorem.

LEMMA 6.1. (u,), is an increasing family.

Furthermore, the interior bound obtained in theorem 5.1 and the consequences of lemma 5.2 enable us to derive the following proposition.

PROPOSITION 6.2. Suppose (SIS). Then, for every 6 > 0, there exists a positive constant cg such that

UM 5 cs in Q-6 (6.3)

here Q-8 = lx E fi: dist(x, aa) > 6).


Remark 6.3. The inequality

f&x) I sup(&): x E fi

shows that cg is independent on M. Then, we construct

6 < dist@‘, &2). Since L”(a’), 1 < s < co, is a reflexive space, it follows from

u(x) = lim uM(x) = sup u&x), M-m M>O

A first property of u is the following corollary.

6)

x E a.

COROLLARY 6.3. u E WrA;.“(n).

Proof. The definition and (6.3) imply u E L;*,,(n). On the other hand, let a’ CC Q and

(6.4)

u,,,f 5 cg in a_,

the convergence

UMM’ -+ u in a(P(G!‘), L”‘(a’)) asM’/co

for some subsequence (uM,JM’ . Then, by lemma 6.1 and the dominated convergence theorem, we obtain

Llw --f u in L”(a’) as M’/*co,

whence

uw --t u in G,(W asM’7m.

Finally, classical arguments on Sobolev space conclude the result from uM, E W1,m(O’) and

\vuMr(x)i 5 c&v x E Q-6 (see corollary 5.5). n

Remark 6.4. In fact, the above reasoning shows

UM’ -+ u weakly in W1,‘(,‘) asM’/Ico.

Now, we may face up the existence of explosive solutions of (g).

(6.5)

THEOREM 6.4. Assume (2. l), (2.4) and (SIS). Let &2 be a bounded domain of IRN, N L 1, whose boundary has positive mean curvature and aa E C’. Then, for every f E Ul;A;m(Cl), f 1 0, the function u, constructed in (6.4), is the minimal local solution of

-div(Q((Vu\) Vu) + AD(u) = f in Sz G) satisfying

u(x) -+ 00 as dist(x, a&2) + 0.

Proof. Regularity u E w,‘;“(sZ) and (6.2) show

u(x) -+ co as dist(x, KZ) + 0.


On the other hand, fix a’ CC 62 and s E 11, a~[. The continuity of ,8 and the dominated convergence theorem imply

W(%M’) - fW1.W -+ UW) - f in L’(n’) asM’7co. (6.6)

Then, by (6.1),

div(Q(IV+h+) -+ 434 - f in L’(O’) asM’roo

holds. Next, in the domain

63 = (v E W’*“(W): Q(lVvj) Vu E (W’,“(W))N),

one defines the operator

d:: 03 -+ (W’J(sz’))’

by

A?(v) = div(Q([Vvj) Vu), v E 63.

Condition (2.1) implies that 6: is a monotone and hemicontinuous operator (see [16] for definition and related properties). Moreover, the embedding W’*“(L2’) C 63, and the convergence

U&f’ + u weakly in W1*‘(GI’) asM’roo

lead to

.qU~) --f I weakly in ( W1,‘(L’J’))’ asM’Pw,

for some I E (W’s”@‘))‘. Therefore, from (6.6), we obtain the representation

v a, E wlqwr),

for the duality bracket (W’*“(L’I’))’ x W1,“(s2’). Finally, from Q(IVu() Vu E (L”(L2’))N, one obtains

thus

I = g(u) = div(Q(lVu1) Vu),

(div QdVul) Vu, ~0 = QdVul) Vu * VP k, v (0 E wl*s(a’), R’

whence one concludes

1 Q(IVul) Vu VP dx + A /3(u)p dw = fo, d-G v v, E W,‘q2’).

0’ nc ns

Since comparison arguments imply

u,(x) 5 v(x), XESZ

for any explosive local solution v of (&), it follows that

u(x) 5 v(x), XEM. n


Remark 6.5. Complementary regularity CIPYsZ) can be derived by arguing, for instance, as in [17-191.

The results of Sections 3 and 4 can be applied to function u. In particular, we have the following corollary.

COROLLARY 6.5. Assume (SIS)p, . Let Q be a bounded domain of IRN, N 2 1, whose boundary has positive mean curvature and aQ E C2. Then, for every f E K$~;“(Cl), f 2 0, satisfying

f(x)(dist(x, aO))p + C E R+ U (CO] as dist(x, aa) --t 0.

Then, the function u, constructed in (6.4), is the unique nonnegative local solution of

-div(lVU(p-2VU) + Au” = f in 0,

in the class F,‘;“(Q) fl C(Q). Moreover, if

f(x) = c(dist(x, aQ))-q, 4rP7 c E iR+ for dist(x, aQ) small,

the function u verifies:

u(x)(dist(x, aM))p’(m+l-p) --t c(m, p, A - fo) as dist(x, X2) -+ 0, for f. = s-'(c),

provided q I mp(m + 1 - p)-‘, and

u(x)(dist(x, aQ))q’m --t (,I/c)“~ as dist(x, aQ) + 0

provided q < mp(m + 1 - p)- ‘.

1.

2.

3.

4.

5.

6. I. 8.

9.

10.

11. 12. 13.

14.

LASRY J. M. & LIONS P. L., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state contraints, Math. Annln. 283, 583-630 (1989). KELLER J. B., Electrohydrodynamics I. The equilibrium of a charged gas in a container, J. ration. Mech. Analysis 4, 715-724 (1956). LOEWNER CH. & NIRENBERG L., Partial differential equations invariant under conformal or projective transformations, Contributions to Analysis, pp. 245-272. Academic Press, New York (1974). ROBINSON P. D., Complementary variational principles. In Nonlinear Functional Analysis and Applications. Academic Press, New York (1971). D~AZ .I. I., Nonlinear Partial Differential Equations and Free Boundaries, Vol. I. Elliptic Equations. Res. Notes Math. 106. Pitman (1985). DiAz G. Sr LETELIER R., Unbounded solutions of one dimensional quasilinear elliptic equations, Applic. Analysis. SERRIN J., Local behavior of solutions of quasilinear equations, Acta Math. 111, 247-302 (1964). SERRIN .I., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. R. Sot. London 264, 413-496 (1969). GILBARC D. & TRUDINGER N. S., Elliptic Partial Differential Equations of Second Order, 2nd edition. Springer, Berlin (1983). DiAz G. & LETELIER R., Uniqueness for viscosity solutions of quasilinear elliptic equations in RN without conditions at infinity, Diff. Integral Eqns 5, 999-1016 (1992). KELLER J. B., On solutions of Au = f(u), Communspure uppl. Math. X, 503-510 (1957). OSSERMAN R., On the inequality Au 2 f(u), Pacific J. math. 7, 1641-1647 (1957). VAZQUEZ J. L., An apriori interior estimate for the solutions of nonlinear problems representing weak diffusion, Nonlinear Analysis 5, 95-103 (1981). D~AZ J. I., SAA .I. E. & THIEL U., Sobre la ecuacibn de curvatura media prescrita y otras ecuaciones cuasilineales elipticas con soluciones anulPndose localmente. Aparecera en las Actas de1 Homenaje a Julio Rey Pastor, Buenos Aires (1988).

15. SPERB R., Maximum Principles and Their Applications. Academic Press, New York (1981).

REFERENCES


16. LIONS J. L., Quelques Methodes de Resolutions de Problemes aux Limited non Lineares. Dunod (1969). 17. EVANS L. C., A new proof of local Cl+” regularity for certain degenerate elliptic PDE, J. difJ Eqns 45, 356-373

(1982). 18. DI BENEDETTO E., Cl+” local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis 7,

827-850 (1983). 19. TOLKSDORFF P., Regularity for a more general class of quasilinear elliptic equations, J. difJ Eqns 51, 126-150

(1984).

Explosive solutions of quasilinear elliptic equations: existence and uniqueness

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