- Email: [email protected]
the existence of solutions of quasilinear elliptic equations with change of sign
2001,21B(4):469-482
THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN
1
Li Gongbao ( ~..I:.1.: ) Young Scientist Labomtory of Mathematical Sciences, Wuhan Institute of Physics and Mathematics, Chinese Academsj of Sciences, P.D.Box 71010, Wuhan 430071, China Yu ChUrL ( ~t
Abstract
This paper considers the following quasilinear elliptic problem -div(l\7ulp-z\7u) = a(x)g(u.) { u=O
in 0 on
where 0 is a bounded regular domain in R N (N~3), N suitable conditions and g(u)u - (3
Iou g(s)ds
>
ao p
>
1. When g(u) satisfies
is unbounded, a(x) is a Holder continuous
In- la(x)ldx is suitably small.
function which changes sign on 0 and
the existence of a nonnegative nontrivial solution for N
> p > 1, in
The authors prove
particular, the existence
of a positive solution to the problem for N > p~2. Our main theorem generalizes it recent result of Samia Khanfir and Leila Lassoued (see [1]) concerning the case where p = 2. They prove also that if g(u)
= lulq-zu.
with 1) < q
< 1)*
and 0+
=
{xEOI(£(~;)
>
O} is a
non empty open set, then the above problem possesses infinitely many solutions. Key words Qasilinear elliptic equation, (PS) condition, mountain-pass Lemma, infinite solution 1991 MR Subject Classification
1
35J60, 35B05
Introduction In this paper, we consider the following elliptic problem -div(IV'lLlp-2VU) {
= a(x)g(u)
u=O
m 0
where 0 is a bounded regular domain in R N (N 2: 3), N > p > 1, and a : continuous function which changes sign on 0 so that the open sets 0+ I
(1.1)
on 80
= {x
n
--+
R is a Holder
E Ola(x) > O}
Received July 5,2000; revised May 14, 2001. The first author is partially supported by NSFC and Academy
of Finland.
470
ACTA MATHEMATIC A SCIENTIA
and
Vo1.21 Ser.B
= {x E Ola(x) < O}
0-
are both nonempty. We use the standard notations. Let 11,11, denote the norm of L'(O) for s2:1. Denote W~'P(O) the completion of Cgo(O) under the norm Ilull = IIV'ullp, the norm of the dual space of W~'P(O) will be denoted by 11,11., (".) will denote the pairing between W~'P(O) and its dual space. "-+" represents the strong convergence and "~,, the weak convergence in related function spaces. 60 p will be the p- Laplacian operator, i.e., 60pu = div(lV'ulp-2V'u) and 602 is the usual Laplacian. We shall write a(x)
= a+(x) -a-(x) where a+(x) = a(x)Xn+ and a-(x) = -a(x)xn-
and
XE is the characteristic function of the set E in R N. Throughout this paper, we assume that
Ila(x)lloo:SC o, where Co is a FIXED constant. The conditions imposed on g(x) will be the following (g1) g(x) is a locally Holder continuous function on R+; (g2) g(u) = o(uP- 1 ) as u -+ 0+; u (g3) There exists a real ,BE]p,p·[ such that g(u)u2:,B fo g(s)ds
> 0 for u > 0 where
p. = Npj(N - p); (g4) There is a real 'Y E [,B,p.[ and a constant C 1 > 0 such that
We say that u E W~'P(O) is a weak solution to (1.1) if
L
plV'ul 2V'u.V'vdx
= La(x)g(u)VdX,
"Iv E
W~'P(O).
In the following, when we speak of solutions, we mean the weak solutions unless specified. It is clear that if g(O)
= 0 then u == 0 is the trivial solution to
(1.1). We are interested in u the existence of a positive solution and multiple solutions of (1.1). Setting G(u) = fo g(s)ds for u 2: 0 and G(u) = 0 for u < 0, it is well known that the nontrivial solutions to (1.1) corresponding to the nonzero critical points of the functional
I(u)
=~
r lV'ulPdx - inr a(x)G(u)dx
(1.2)
pin
defined on W~,P(O), which is in C 1 under the assumptions on a(x) and (g1)-(g4) and
(I'(u), v)
=
L
plV'ul 2V'v'V'v -
L
a(x)g(u)vdx,
Vu, v E
W~'P(O).
We say that I(u) satisfies the Palais-Smale condition (PS) in W5'P(0) iffor any sequence {un} in W~'P(O) such that {I(u n)} is bounded and I'(u n) -+ 0 in the dual space of W~'P(O) where I'(u) is the Frechet derivative of I(u), there exists a strongly convergent subsequence of {un}. If a(x)2:0 and (g1)-(g4) hold, it is easy to see that I(u) satisfies the (PS) condition and the geometric conditions of the Mountain-Pass Lemma (See [2]), hence the existence of a nonnegative nontrivial solution to (1.1) is obtained (See [2] for the case p = 2 and [3] for p # 2). Of course, for p = 2, one sees that the solution obtained is actually positive by the strong
471
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
No.4
maximal principle. When a( c) changes sign as in the case we are interested in, if g(u)u - f3G( u)
is bounded (e.g. g(u) = lulq- 2u,p < q < p' and f3 = q), then I(u) still satisfies the (PS) condition and the geometric assumptions of the Mountain-Pass Lemma.When g(u)u - f3G( u) is unbounded, a recent paper[l] written by Samia Khanfir and Leila Lassoued discussed the case p = 2 and obtained the existence of a positive solution to (1.1) under the assumptions (gl )-(g4) with p = 2 and f3 = 'Y in (g4), together with the smallness of In a- (x)dx (See Theorem 2 in [1]). However, the (g4) in [1] with 'Y = f3 where f3 is given in (g3) seems unnatural. This can be seen from the following observation. Let g(u) then g(u)u = hand,
m
I: luIQi,G(u)
;=1
=
m
I:
;=1
m
= I: IU!qi- 2u(2 < q1 ;=1
< q2 < ... < qm <
~~2)'
..!.Iul qi. From (g4) in [1], we must have f3?qm' Onthe other q,
t
;=1
lujqi?
t ~Iulqi, ;=1 q.
\fu > 0
(1.3)
only if f3~Qm, hence f3 =qm' If m > 1, then (1.3) implies qm ~qm-1, a contradiction. So m = 1, and g(u) = lulq,-2 u, hence g(u) -f3G(u) == 0 excluding the possibility of the unboundedness of
g(u)u - f3G(u). In this paper, we obtain the following main result Theorem 1.4 Suppose g(u) satisfies (gl)-(g4) and a(x) E C"'(IT) for given 0
<
a
<
1
with Ilalloo~Co, then there is a e = £(n,p,N,f3,'Y,Co) > 0, such that if Ina-(x)dx < e, then (A) problem (1.1) possesses at least a positive solution for 2~p < N; (B) problem (1.1) possesses at least a nontrivial nonnegative solution for 1 < P < 2.
Remark 1.5 Theorem 1.4 extends one of the main results of [1] (See Theorem 2 of [1]) in two aspects: firstly, it gives the existence of positive solution to (1.1) for p :f. 2; secondly, we do not require 'Y = f3 in (g4). However, our theorem 1.4 when p = 2 is in a slightly different setting from [1] but the difference is not essential. To prove our main result, we use similar idea employed in [1]. Using suitable cut-off
gR(U),GR(u) of g(u) and G(u), such that gR == g,GR(u) == G(u) on [O,R] and the fact that gR(u)u - f3G R(u) is bounded, we can obtain a sequence of solutions {u,,} to the modified problems (See (2.13) below), it remains to show that for R large, we actually have
o < uR~R. If p = 2, the classical maximum principle easily gives that UR > 0, while in the case where p:f. 2, we have to use a more delicate maximal principle in [4]. To show uR~R when p = 2, [1] uses the classical LP-estimate for second order uniformly elliptic equations, but for p :f. 2,
. the method seems not applicable. Also, a result in [5] (See Proposition 1.3 in [5]) gives the i
Loo-estimate of the solution u to -6 pu = f as Ilulloo~llfll:-', by assuming f E LS(n) for some s ;» N'[p, but this result seems not lead to the uniform L oo _ estimate (see (3.9) below) we want for {UR}, we thus use the variational structure of the problem and the Nash-Moser argument used in [6] (see also [7]) to overcome this difficulty. By [8], the solution u to (1.1) when p :f. 2, is in C,1~:(n). Our main result in the case 1 < P < 2 shows only the existence of a nontrivial nonnegative solution due to that the maximal principle in [4] requires that 6 p u E Lfoc(n) which can be guaranteed by Proposition 1 of [8J only if p?2.
472
ACTA MATHEMATICA SCIENTIA
It is well known that if a(x)
Vo1.21 Ser.B
°in Q, g(tt) satisfies (gl )-(g4) together with g(-u) = -g( u),
>
then (1.1) possesses infinitely many solutions. (see e.g.[2]). One might expect that similar result could be proved when a( x) change sign using the same idea as what we use to prove Theorem 1.4, but this seems not true. However, in a special case i.e. g(u) the following result: Theorem 1.6 p
<
q
= lul q - 2u, P < q < p"; we have =
Suppose that a(x) E C(Q) and Q+ is nonempty, g(u)
lul Q -
2u
with
< pO, then (1.1) possesses infinitely many solutions {ud with Ilukll-+ 00.
As g(u)u - qG(u)
= 0, it
is trivial that l(u) given by (1.2) with G(u)
= *Iulq
satisfies
(PS) condition. But since Q- may be nonempty, it is impossible to show that for any finite dimensional subspace V of wg,P(Q) there is an R
= R(V)
such that l(u):::;O on V
n B'R, a
condition which is required in the standard Z2-Symmetric version of Mountain-Pass Theorem (see e.g. Theorem 9.12 of [9]). However, as it was mentioned in [2), the above condition can be replaced by the following: there exist a sequence of finite dimensional subspace {V,n} of
wg'P(Q) with VmCVm+l and dim V = m, and a sequence of real numbers {R m} such that 1( u):::;O if u E V n B'R m • We mention that this idea was used also in [10] to deal with the problems in unbounded domains. In Section 2, we give some lemmas and prove the existence of positive solutions to the modified problems. In Section 3, we get the uniform LOO-estimates for the solutions to the modified problems obtained in Section 2 and prove main results.
2
The Existence of Solutions to Modified Problems In this section, we prove some preliminary results to get ready for the proof of our main
results. Since a(x) changes sign, although we have by (g3) g(u)'u?J3 J~' g(s)ds, 'v'u?O, we do not have a(:r)g(u)?:.f3a(x) J~' g(s)ds. So any (PS) sequence {un} C wg'P(Q) with lI(un)I:::;C and
1'(u n )-+ O may not be uniformly bounded in W~'P(Q) and one can not show that l(u) satisfies the (PS) condition. As we stated in Section 1, if g(u)u - f3G( u) is bounded, then it is still true that l(u) satisfies the (PS) condition. But since g(u)u - f3G(u) is in general unbounded, we study first of all the "modified problems". To set up such problems following [I), we define 7/J(u) = G(u)/u f3 for u > and 'lj!(0) = 0. Then since 1/;'(u) = 'u- f3 - 1[g(u)u - f3G(u)), (g3) implies that til' ('1£)?:. for U > 0, hence 'lj!( u) is nonnegative and nondecreasing on R+. We define for R > 1 a function 7/J R on R+ in the following way
°
°
if u E]O,R] if u E]R,R+ 1]
u E]R + 1, +00] which is C 1 , nonnegative and nondecreasing. We set GR(U) satisfies
= 7/JR(U)U f3
if u
>
0 and GR(U)
GR(U) = G(u) GR(u)
= CRuf3
=
°
if u:::;O. Then GR(U) is Clover Rand
if u:::;R,
(2.1)
if u?:.R + 1.
(2.2)
No.4
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
We also set for R
>0 MR
= .E[O,R] max [g(s)s -
,6G(s)].
473
(2.3)
First of all, we have the following Lemma 2.4
(i) gR
= G~ satisfies (g3) and (2.5)
(ii) there exists for R> 1 a constant C2 > 0 such that (2.6)
!Proof It is evident that gR(U) satisfies (g3) for u~R or u?R since g(u) satisfies (g3), we have
gR(U)U - ,6GR(u)
+ 1.
For u E]R, R
+ 1[,
= ("p~(u)u + ,6"pR(U)Uf3-1)u - ,6"pR(U)Uf3 = "p~(u)uf3+1 = g(R)~;~G(R) (R + 1- u)uf3+1 ? O.
Moreover, (2.5) holds clearly for u E]O, R[U[R + 1, +00[,
gR(U)U - ,6GR(U)
= ("p'(R)(R+ 1- u)u f3+ 1 + R)f3+ 1"p'(R) = (1 + ~)f3+l Rf3+l"p'(R)~2f3+1[g(R)R ~ uf3+l"p'(R)~(1
,6G(R)]
~ 2f3 +1 MR,
(2.5) follows. For R> 1, by (g3), we get C
= ot'(R)
R-
< - Rf3-1 By (g4), g(R)~Cl(R'"I-l
~ot,l(R)
= G(R)
Rf3 -
Rf3-1'
Rf3+1
+ 1) gives g(R)
(2.6) follows since C 2
~. g(R)R - ,6G(R)
+ 2 Of' Rf3 + 2 g(R)' + g(R) <2 g(R)
Of'
+ _1_)<2C Rf3-1 -
1
R'"I-f3
= 4C1 .
Our next lemma is Lemma 2.7
and
U
(i) For any
f
> 0, there exists a constant C, > 0 such that for all R > 1
>0 (2.8)
(2.9)
474
ACTA MATHEMATIC A SCIENTIA
Vo1.21 Ser.B
(ii) There exists a positive constant C such that for all R> 1 and U> 0 (2.10) Proof Firstly, we prove that there exists a constant Cb > 0 such that for all R
> 1 and
u>O (2.11) Indeed, (2.11) is a result of (g4) because gR(U) = g(u) for u5,R. For u2:R + 1, using (2.2) and (2.6), we have 05, gR(U)
= G~(u) =(3CRuf3-1
5, C2{3R-r-f3 Uf3- 1 5, C2{3u,-I.
For u E]R, R + 1[, we get where and 1/J~(u)uf3 = 1/J'(R)(R + 1 - u)uf3
< g(R)R - (3G(R) .uf3
+ l)uf3-1
g(R) 13-1 < C -r- 1 < _ 2u . _ 2 Rf3+ 1u
hence (2.11) holds where Cb = max{Cll ({3 + I)C2} = 4({3 + I)C1. By (g2),for any e > O,there exists a 6([) E]O, 1[ such that for 0
< u < 6([) (2.12)
Taking C.
= max{Cb(1 + 6,,~,),2Cb}, by (2.11) and (2.12),
we get
and gR(U)
1 _ 2C'0 u,-1 < _ C'0 + C'0 u-r- <
5, [U p - 1 + C.u,-I,
for
u2:1.
We concludes (2.8). (2.9) holds since (g3) and (2.8) where C. = C. (Cll (3). To show (ii), we notice that (2.10) is equivalent to
475
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEARELLIPTIC EQUATIONS
No.4
we obtain (2.10) by choosing C:5.'l/JR(l) because 'l/JR(U) is nonnegative and nondecreasing. We consider the following modified problem -div(IVuIP-2Vu) {
U
=0
= a(x)YR(U) in n on an.
(2.13)
Its nontrivial solution is a nonzero critical point of the following functional in W~,p(n)
By (gl)-(g3), it is clear that I R is C 1 in W~,p(n). For fixed R> 1, YR(U)U - (3G R(u) is bounded by (2.5) and we have Lemma 2.14 IR(u) satisfies the (PS) condition in W~,p(n).
Proof Suppose that {un} in W~,P (n) satisfies (i) IR(u) is bounded; (ii) Ik(u n ) -+ 0 in the dual space of W~,p(n), we shall prove that {un} is precompact in W~,p(n). Following [2] and [11] we first show that I/u n /I is bounded. Since
we have by Lemma 2.4
(~- 1) II Un IIP:5.(3IR(un) + 213+ 1 M R P
inr la(x)ldx + Ilunlllllk(un)II*.
If Ilunll is not bounded, there exists a subsequence of {un}, still denoted by {Un}, such that Ilunll-+
00
as n -+
00.
So
By (i),(ii) and 1 < P < (3, this inequality gives that Un -+ 0 as n -+ 00, which is a contradiction. Thus, since W~,p(n) is reflexive, we may assume that there is a subsequence {un} of {Un} and a Uo E W~,p(n) with
(2.15) By Sobolev's imbedding, we may assume that
Un -+ Uo in L"(n) for s E [P,p*[ Un -+ Uo a.e, inn. To prove that {un} is precompact, we need only show that
(see Lemma 2.7 of [12]).
(2.16)
476
ACTA MATHEMATICA SCIENTIA
Since I' (Un)
-+
0 and lIu n II ::; c
< +00,
Vo1.21 Ser.B
we have
(Ik(U n), un} = 0(1), (Ik(un),uo)
= 0(1).
(2.17)
So, by (2.15) and (2.17), we get
l
(IVu n!P-2VUn -IVuolp-2Vuo)-(VUn - Vuo)dx
L
= llVunlPdx -
IVunIP-2Vun·Vuodx
-llvuoIP-2vuo,V(un - uo)dx
l =l =
a(x)YR(un)undx
-l
a(x)YR(un)uodx
+(Ik(un), un) - (Ik(u n), uo} a(x)YR(un)undx -
J
+ 0(1)
a(x)YR(un)uodx + 0(1).
By (2.16) and (2.8), we see that {a(x)YR(Un)U n} and {a(x)YR(Un)UO} are uniformly integrable on n i.e. "Ie> 0, 3b(e) > 0 such that
L
la(x)YR(un)unldx < e
L
and
la(x)YR(un)uoldx < e
as long as lEI < b( e) where lEI denotes the Lebesgue measure of E in RN. Hence, (2.16) and (gl) together with Vitali's convergence theorem show that lim
n-+oo
[10a(:Z:)YRCU" )u"dx - 10f a(x)YR(un)uodX] = o.
So we have shown that and the lemma is proved. The next proposition shows that I R satisfies the geometric asanmptions of the MountainPass Theorem uniformly with respect to R.
Proposition 2.18 There exist q > 0,15 > 0 depending only on CO,Cl,{3,'Y,p,N,n, and U E W~,p(n) satisfying Ilull > q and u 2 0 in n such that for all R> 1 (i) lIuli = q :=} IR(u) 2 15; (ii) IR(u) ::; O. Proof Using (2.9), the Sobolev inequality and the Poincare inequality, we have for all R>l
No.4
where C
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
= C(E,CO,Cl,{3,1',p,N,n)
is a constant independent of R. Letting E small enough,
(i) follows since l' > p. On the other hand, we choose tp E W5,p(n) such that Iltpll (tp) C n+ hence (2.10) leads to
IR(ttp)
= ~ { 1V'(ttp)IPdx pin
::;
477
2: 0 in n and support
{ a(x)GR(ttp)dx
in
~tP - C { a(x)(t/3tp/3 p
= 1, tp
in+
l)dx
(2.19)
< ~tP - Ct/3 + C -p :::
> 0 and R> 1, where C and C are positive constants depending on C, cs.s,« N, n. Since {3 > p, so for to large enough, u = totp is such that Ilull > q and IR(u :s 0 for all R > 1
for all t
and the Proposition is proved. By the Mountain-Pass Theorem (see [2]), IR possesses a critical value dR 2: 0 given by
where r = h E C([O, 1], W5,p(n)) 11'(0) .; 0 and 1'(1) = u}. We denote by UR a critical point of IR such that IR( un) = dR. Our next lemma generalized the classical maximum principle in[7] (pI79-180) in the case p = 2. We believe that the following lemma exists somewhere in the literature although we can not trace it. For the reader's convenience we state and prove it. (see Theorem 4.2 of [13]) Lemma 2.20 Assume U E W~,p(n) is a weak solution of the following equation (2.21) then ess inf U > inf Un - an where u: ::::' min{u(x),O},bj(x) 2: 0 in n(j u
< O(j = 1,2).
= 1,2),hj(u)
2: 0 for u 2: 0 and hj(u)
= 0 for
Proof Let 1= infan u: :S' 0, the function tp = (k - u)+ for Vk < I is in W~,p(n) where w+ = max{w(x), O} for any function w(x). By equation (2.21) we have
L
lV'ulp -
2V'u.V'tpdx
+
L
b1(x)h1(u)tpdx =
L
b2(x)h 2(u)tpdx.
Let n* = {x E nlk 2: u(x)}, notice that tp = k-u for k 2: u and tp = 0 for k :s u, the integration above is actually carried out on no. hj(u(x)) = O(j = 1,2) on n* because u(x):Sk < l:S0 on n°. Thus
hence
This shows that u
> I a.e.in n.
478
ACTA MATHEMATICA SCIENTIA
Vo1.21 Ser.B
The result follows. We can rewrite the equation
as hence UR 2:: 0 by Lemma 2.20. If U is a solution to -.0.p u = f(x), it was shown in e.g.[8] that U E C,1~~(n) for p > 1 and .0. p u E L;oc(n) for p 2:: 2. To prove UR > 0, we extract the strong maximum principle for p-Laplacian type of equations from [4] as our Proposition 2.22. Proposition 2.22 Let U E C 1 (n ) be such that .0. p u E L;oc(n), U 2:: 0 a.e. in n, .0.p u ~ (3(u) a.e. in n with (3: [0,=[-> R continuous, nondecreasing, (3(0) = 0 and either (3(s) = 0 for some s > 0 or f3( s) > 0 for all s > 0 and
Then if
U
n it
does not vanish identically on
Because .0.p uR
= -a(x)YR(UR) ~
l\YR(s)s)-idS 2::
1
is positive everywhere in
IlallooYR(UR) and by (2.8)
1
(ESP
+ Ces"Y)-ids 2:: (E + Co)-i
n.
1 1
s-1ds
= oo
Proposition 2.22 shows that the solution UR of (2.13) is positive everywhere in
n for p 2:: 2.
Now we have proved the following theorem T'heorem 2.23
For R > 1,
(A)R if 2 ~ p < N, problem (2.13) possesses at least a positive solution; (B)R if 1 < p < 2, problem (2.13) possesses at least a nonnegative nontrivial solution.
3
Proof of the Main Results To complete the proof of Theorem 1.4, suppose that UR is a solution to (2.13) for suitable
R> 1, we want to show that IluR//:::;R. To this end we need several lemmata. Firstly, we have the following lemma. Lemma 3.1
There exists a constant C3
> 0 such
that (3.2)
where C3
= C3(C,Ca,(3,p,N,n).
Proof We have dR~
max IR(tu)
tE[a,1]
where u was given in the proof of Proposition 2.18 and it is easy to see from (2.19) that
No.4
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
479
where C = C(C, Co,{3,p, N, 0) denotes any constant independent of R. So dn is bounded for all R > 1. We can write
f3d R
= f3IR(uR) - Ik(uR)uR = (~- 1)lluRIIP + { a(x)[gR(UR)UR p
~ (~ p
1)lIuRIIP
-
f3GR(uR)]dx In In- la(x)l[gR(uR)uR - f3GR(uR)]dx. (
So using (2.5), we get
and the lemma follows. The next lemma obtains a uniform bound for UR in L oo by using the bootstrap arguement(see e.g. [10]). Lemma 3.3 Assume UR is a nonnegative solution of (2.13). Then there exists a constant C 4 = C4(Co,Cll f3" , p, N , O) > 0 independent of R such that
(3.4) where (J = (J(N,p,,)
> O.
Proof We set URL = min{uR, L} for L
> O.
Suppose that v = URU)z~-l) for a~1, then
v E W~'P(O) and
By equation (2.13), we have
So using (2.8), we get
{ P P(o-l)d ::; IIa II00 ( e n URURL z
J
I P(o-l)d ) + Ce J(n URURL x .
(3.5)
By the convexity of the power function, we obtain '
in = in lV'uRu~Ll + IV'WLIPdx
::; 2P -
Suppose that n
1
(a -
1)uRu~L2V'uRLIP dx
in lV'uRIPu)z~-l)dx
+ 2P - 1 (a - 1)P
in u)zu)z~-2)IV'uRLIPdx.
(3.6)
= {x E OIUR(X) ::; L}, then UR = URL on n, V'UR = V'URL on nand V'URL = 0
480
ACTA MATHEMATIC A SCIENTIA
Vo1.21 Ser.B
LIVURIP-2VURVURLURU~<;-1)-ldx
on [2\0, hence
= lIVuRIP-2VuRVURLURU~<;-1)-ldX =
1,vuRIPu~uj}~-2)dx
=
1 n
" 1v
(3.7)
IP P p(a-2)d x. URL URURL
By (3.5),(3.6),(3.7) and the Poincare inequality, we have
L
Lu~u~<;-l)dx +
IVWLIPdx:S CaP(c:
:S c:CaP :S c:CaP
L L
!WLIPdx
ce·l u1-Pu~~dx)
+ Cea P
IVWLIPdx
Lu1-Pu~~dx
(3.8)
+ Cea P l u1-pu~~dx.
Taking e small enough, by the Holder's inequality and the Sobolev's inequality, we write the inequality (3.8) to
where q
= p* j(p* - , + p)
and C is a constant independent of R.
We can choose a such that qpa
Suppose that s
Let a
:S p* since
< , < p*. Letting
p
L
->
+00, we obtain
= qp and X = p* j s, then we have
= xffi(m = 1,2",,), then we get IluRllx=+l :S
~-p
Cx-= Xffi X-= (1IuRII;.")X-= IluRllx='
:S c2:::o x- X2:::o ix-; (1IuRII;! )2:::0 x- IluRlls i
-y-p
i
x
:S ClluRII;."·X=1"lluRlls :S C1luRII~~e where
8
=' -
p . _X_ p X-I , - p p*
--. p
= ,- p
p* - qp
. ~ P p* - s , - p p* - ,
-_. p
+p
p* - ,
>0.
481
Li & Yu; EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
No.4
Letting m
--+
+00, we get
and this proves the lemma. Proof of Theorem 1.4 exists a constant C
By Lemma 3.1, Lemma 3.3 and Sobolev's imbedding, there
= C(Co, C I , C,f3,'Y,P, N, n)
> 0 independent
of R such that
(3.9) Taking in particular R = 2C, we see that if In- la(x)ldx is sufficiently small, then IluRllCXl :s; R. So Theorem 2.23 implies Theorem 1.4 and the solution UR to problem (2.13) is a solution to problem (1.1). To prove Theorem 1.6, we need the following proposition: Proposition 3.10
Let E be an infinite dimensional real Banach space and I E C I (F;, R)
with I even, satisfies (PS) and 1(0) (11) there exist constants o.a
= 0, and
> 0 such that if Ilull
= p then l(u) 2 a;
(12) there is a sequence of finite dimensional subspace {Em} of E with dimEm Em C E m+l for m
= 1,2"",
=m
and a sequence of real numbers {R m} such that I
:s;
and
0 on
Em nBR~; then I possesses an unbounded sequence of critical values.
Proof This result was implicitly mentioned in [2] and the proof is hidden in [2] (see also the proof of Theorem 9.12 of [9]) where (12) was explicitlv stated as:" for each finite dimensional
subspace VeE, there is an R = R(B) such that 1:S; 0 on V n B'k(m)''' Proof of Theorem 1.6 It is dear that the l(u) given by (1.2) with g(u) = lul q I
C and satisfies (PS) (see the proof of Lemma 2.14). Also 1(0)
= 0, l(u)
2u
is
is even and (11) in
Proposition 3.10 is satisfied. On the other hand, let n* be such that n* C n+ and let em be a sequence of linearly independent elements of W~,p (n) with Suppe., C n*, Em
= span{el, e2, ... , em}, e m+I
rf:. Em,
then by Proposition 3.10, to complete the proof, we need only show that there is a sequence
{Rm} of real numbers such that 1( u) Ilull = 1» 1, we have l(u)
:s;
0 on Em
= ~llullP _ P
r a(x)lul ln q
= ~lP -ZQinfa(x) p
n BR~' To this end, for any u E Em with
n-
q
dz
r
inf vEEm,lIvll=:lln-
~dx q
Since Em is finite dimensional and norms in a finite dimensional space are equivalent, we
> O. Also, since a(x) E C(n) and n* c n+, we see that inf n- a( x) > O. Hence there is an R m > 0 such that 1( u) :s; 0 on Em n Bk m and the theorem is see that infvEEm,lIvlI=:1 In- jvlqdx proved.
482
ACTA MATHEMATIC A SCIENTIA
Vo1.21 Ser.B
References 1 Samia Khanfir, Leila Lassoued. On the existence of positive solutions of a semilinear elliptic equation with change of sign, Nonlinear Analysis, Theory, Methods, Aplications, 1994, 22(11): 1039-1314 2 Ambrosetti a A, Rabinowitz P H. Dual variational methods incritical point theory and application. J Funct Anal, 1973, 14: 349-381 3 Shen Yaotian. The nontrivial solutions of quasilinear elliptic equations in W~'p. Scientia Sinica (Ser A), 1984,27: 720-730 4 Vazquez J L. A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim, 1984,12: 191-202 5 Guedda M, Veron L. Qusilinear elliptic equations involving critical Sololev exponents. Nonlinear Analysis, Theory, Methods, Applications, 1989, 13(8): 879-902 6 Li Gongbao. Some properties of weak solutions of nonlinear scalar field equations. Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1989,14: 27-36 7 Gilbarg D, Trudinger N S. Elliptic partial differential equations of second order. Heidelberg, New York: Springer-Verlag, 1983 8 Peter Tolksdorf. Regularity for a more general class of quasilinear elliptic equations. J Diff Equs, 1984,51:121-160 9 Robinowitz P H. Minimax methods in critical point theory with applications to differential equations. Regional Conference Series in Mathematics, ISSN 0160-7642, no.65,AMS,1986 10 Li Gongbao. The existence of infinitely many solutions of quasilinear partial differential equations in unbounded domains. Act.a Math Sci, 1989,9(2):175-188 11 Li Gongbao, Zhu Xiping. Nontrivial solution of nonlinear scalar field equations with strong nonlinearity. Acta Math Sci, 1988, 8(4): 431-448 12 Li Gongbao, Olli Martio. Stability in obstacle problems. Math Scand, 1994, 75: 87-100 13 Chen Yiaze, Wu Lanchen. Second order elliptic equations and elliptic systems. Science Press, 1991 (Chinese Edition)
THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN
1
Li Gongbao ( ~..I:.1.: ) Young Scientist Labomtory of Mathematical Sciences, Wuhan Institute of Physics and Mathematics, Chinese Academsj of Sciences, P.D.Box 71010, Wuhan 430071, China Yu ChUrL ( ~t
Abstract
This paper considers the following quasilinear elliptic problem -div(l\7ulp-z\7u) = a(x)g(u.) { u=O
in 0 on
where 0 is a bounded regular domain in R N (N~3), N suitable conditions and g(u)u - (3
Iou g(s)ds
>
ao p
>
1. When g(u) satisfies
is unbounded, a(x) is a Holder continuous
In- la(x)ldx is suitably small.
function which changes sign on 0 and
the existence of a nonnegative nontrivial solution for N
> p > 1, in
The authors prove
particular, the existence
of a positive solution to the problem for N > p~2. Our main theorem generalizes it recent result of Samia Khanfir and Leila Lassoued (see [1]) concerning the case where p = 2. They prove also that if g(u)
= lulq-zu.
with 1) < q
< 1)*
and 0+
=
{xEOI(£(~;)
>
O} is a
non empty open set, then the above problem possesses infinitely many solutions. Key words Qasilinear elliptic equation, (PS) condition, mountain-pass Lemma, infinite solution 1991 MR Subject Classification
1
35J60, 35B05
Introduction In this paper, we consider the following elliptic problem -div(IV'lLlp-2VU) {
= a(x)g(u)
u=O
m 0
where 0 is a bounded regular domain in R N (N 2: 3), N > p > 1, and a : continuous function which changes sign on 0 so that the open sets 0+ I
(1.1)
on 80
= {x
n
--+
R is a Holder
E Ola(x) > O}
Received July 5,2000; revised May 14, 2001. The first author is partially supported by NSFC and Academy
of Finland.
470
ACTA MATHEMATIC A SCIENTIA
and
Vo1.21 Ser.B
= {x E Ola(x) < O}
0-
are both nonempty. We use the standard notations. Let 11,11, denote the norm of L'(O) for s2:1. Denote W~'P(O) the completion of Cgo(O) under the norm Ilull = IIV'ullp, the norm of the dual space of W~'P(O) will be denoted by 11,11., (".) will denote the pairing between W~'P(O) and its dual space. "-+" represents the strong convergence and "~,, the weak convergence in related function spaces. 60 p will be the p- Laplacian operator, i.e., 60pu = div(lV'ulp-2V'u) and 602 is the usual Laplacian. We shall write a(x)
= a+(x) -a-(x) where a+(x) = a(x)Xn+ and a-(x) = -a(x)xn-
and
XE is the characteristic function of the set E in R N. Throughout this paper, we assume that
Ila(x)lloo:SC o, where Co is a FIXED constant. The conditions imposed on g(x) will be the following (g1) g(x) is a locally Holder continuous function on R+; (g2) g(u) = o(uP- 1 ) as u -+ 0+; u (g3) There exists a real ,BE]p,p·[ such that g(u)u2:,B fo g(s)ds
> 0 for u > 0 where
p. = Npj(N - p); (g4) There is a real 'Y E [,B,p.[ and a constant C 1 > 0 such that
We say that u E W~'P(O) is a weak solution to (1.1) if
L
plV'ul 2V'u.V'vdx
= La(x)g(u)VdX,
"Iv E
W~'P(O).
In the following, when we speak of solutions, we mean the weak solutions unless specified. It is clear that if g(O)
= 0 then u == 0 is the trivial solution to
(1.1). We are interested in u the existence of a positive solution and multiple solutions of (1.1). Setting G(u) = fo g(s)ds for u 2: 0 and G(u) = 0 for u < 0, it is well known that the nontrivial solutions to (1.1) corresponding to the nonzero critical points of the functional
I(u)
=~
r lV'ulPdx - inr a(x)G(u)dx
(1.2)
pin
defined on W~,P(O), which is in C 1 under the assumptions on a(x) and (g1)-(g4) and
(I'(u), v)
=
L
plV'ul 2V'v'V'v -
L
a(x)g(u)vdx,
Vu, v E
W~'P(O).
We say that I(u) satisfies the Palais-Smale condition (PS) in W5'P(0) iffor any sequence {un} in W~'P(O) such that {I(u n)} is bounded and I'(u n) -+ 0 in the dual space of W~'P(O) where I'(u) is the Frechet derivative of I(u), there exists a strongly convergent subsequence of {un}. If a(x)2:0 and (g1)-(g4) hold, it is easy to see that I(u) satisfies the (PS) condition and the geometric conditions of the Mountain-Pass Lemma (See [2]), hence the existence of a nonnegative nontrivial solution to (1.1) is obtained (See [2] for the case p = 2 and [3] for p # 2). Of course, for p = 2, one sees that the solution obtained is actually positive by the strong
471
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
No.4
maximal principle. When a( c) changes sign as in the case we are interested in, if g(u)u - f3G( u)
is bounded (e.g. g(u) = lulq- 2u,p < q < p' and f3 = q), then I(u) still satisfies the (PS) condition and the geometric assumptions of the Mountain-Pass Lemma.When g(u)u - f3G( u) is unbounded, a recent paper[l] written by Samia Khanfir and Leila Lassoued discussed the case p = 2 and obtained the existence of a positive solution to (1.1) under the assumptions (gl )-(g4) with p = 2 and f3 = 'Y in (g4), together with the smallness of In a- (x)dx (See Theorem 2 in [1]). However, the (g4) in [1] with 'Y = f3 where f3 is given in (g3) seems unnatural. This can be seen from the following observation. Let g(u) then g(u)u = hand,
m
I: luIQi,G(u)
;=1
=
m
I:
;=1
m
= I: IU!qi- 2u(2 < q1 ;=1
< q2 < ... < qm <
~~2)'
..!.Iul qi. From (g4) in [1], we must have f3?qm' Onthe other q,
t
;=1
lujqi?
t ~Iulqi, ;=1 q.
\fu > 0
(1.3)
only if f3~Qm, hence f3 =qm' If m > 1, then (1.3) implies qm ~qm-1, a contradiction. So m = 1, and g(u) = lulq,-2 u, hence g(u) -f3G(u) == 0 excluding the possibility of the unboundedness of
g(u)u - f3G(u). In this paper, we obtain the following main result Theorem 1.4 Suppose g(u) satisfies (gl)-(g4) and a(x) E C"'(IT) for given 0
<
a
<
1
with Ilalloo~Co, then there is a e = £(n,p,N,f3,'Y,Co) > 0, such that if Ina-(x)dx < e, then (A) problem (1.1) possesses at least a positive solution for 2~p < N; (B) problem (1.1) possesses at least a nontrivial nonnegative solution for 1 < P < 2.
Remark 1.5 Theorem 1.4 extends one of the main results of [1] (See Theorem 2 of [1]) in two aspects: firstly, it gives the existence of positive solution to (1.1) for p :f. 2; secondly, we do not require 'Y = f3 in (g4). However, our theorem 1.4 when p = 2 is in a slightly different setting from [1] but the difference is not essential. To prove our main result, we use similar idea employed in [1]. Using suitable cut-off
gR(U),GR(u) of g(u) and G(u), such that gR == g,GR(u) == G(u) on [O,R] and the fact that gR(u)u - f3G R(u) is bounded, we can obtain a sequence of solutions {u,,} to the modified problems (See (2.13) below), it remains to show that for R large, we actually have
o < uR~R. If p = 2, the classical maximum principle easily gives that UR > 0, while in the case where p:f. 2, we have to use a more delicate maximal principle in [4]. To show uR~R when p = 2, [1] uses the classical LP-estimate for second order uniformly elliptic equations, but for p :f. 2,
. the method seems not applicable. Also, a result in [5] (See Proposition 1.3 in [5]) gives the i
Loo-estimate of the solution u to -6 pu = f as Ilulloo~llfll:-', by assuming f E LS(n) for some s ;» N'[p, but this result seems not lead to the uniform L oo _ estimate (see (3.9) below) we want for {UR}, we thus use the variational structure of the problem and the Nash-Moser argument used in [6] (see also [7]) to overcome this difficulty. By [8], the solution u to (1.1) when p :f. 2, is in C,1~:(n). Our main result in the case 1 < P < 2 shows only the existence of a nontrivial nonnegative solution due to that the maximal principle in [4] requires that 6 p u E Lfoc(n) which can be guaranteed by Proposition 1 of [8J only if p?2.
472
ACTA MATHEMATICA SCIENTIA
It is well known that if a(x)
Vo1.21 Ser.B
°in Q, g(tt) satisfies (gl )-(g4) together with g(-u) = -g( u),
>
then (1.1) possesses infinitely many solutions. (see e.g.[2]). One might expect that similar result could be proved when a( x) change sign using the same idea as what we use to prove Theorem 1.4, but this seems not true. However, in a special case i.e. g(u) the following result: Theorem 1.6 p
<
q
= lul q - 2u, P < q < p"; we have =
Suppose that a(x) E C(Q) and Q+ is nonempty, g(u)
lul Q -
2u
with
< pO, then (1.1) possesses infinitely many solutions {ud with Ilukll-+ 00.
As g(u)u - qG(u)
= 0, it
is trivial that l(u) given by (1.2) with G(u)
= *Iulq
satisfies
(PS) condition. But since Q- may be nonempty, it is impossible to show that for any finite dimensional subspace V of wg,P(Q) there is an R
= R(V)
such that l(u):::;O on V
n B'R, a
condition which is required in the standard Z2-Symmetric version of Mountain-Pass Theorem (see e.g. Theorem 9.12 of [9]). However, as it was mentioned in [2), the above condition can be replaced by the following: there exist a sequence of finite dimensional subspace {V,n} of
wg'P(Q) with VmCVm+l and dim V = m, and a sequence of real numbers {R m} such that 1( u):::;O if u E V n B'R m • We mention that this idea was used also in [10] to deal with the problems in unbounded domains. In Section 2, we give some lemmas and prove the existence of positive solutions to the modified problems. In Section 3, we get the uniform LOO-estimates for the solutions to the modified problems obtained in Section 2 and prove main results.
2
The Existence of Solutions to Modified Problems In this section, we prove some preliminary results to get ready for the proof of our main
results. Since a(x) changes sign, although we have by (g3) g(u)'u?J3 J~' g(s)ds, 'v'u?O, we do not have a(:r)g(u)?:.f3a(x) J~' g(s)ds. So any (PS) sequence {un} C wg'P(Q) with lI(un)I:::;C and
1'(u n )-+ O may not be uniformly bounded in W~'P(Q) and one can not show that l(u) satisfies the (PS) condition. As we stated in Section 1, if g(u)u - f3G( u) is bounded, then it is still true that l(u) satisfies the (PS) condition. But since g(u)u - f3G(u) is in general unbounded, we study first of all the "modified problems". To set up such problems following [I), we define 7/J(u) = G(u)/u f3 for u > and 'lj!(0) = 0. Then since 1/;'(u) = 'u- f3 - 1[g(u)u - f3G(u)), (g3) implies that til' ('1£)?:. for U > 0, hence 'lj!( u) is nonnegative and nondecreasing on R+. We define for R > 1 a function 7/J R on R+ in the following way
°
°
if u E]O,R] if u E]R,R+ 1]
u E]R + 1, +00] which is C 1 , nonnegative and nondecreasing. We set GR(U) satisfies
= 7/JR(U)U f3
if u
>
0 and GR(U)
GR(U) = G(u) GR(u)
= CRuf3
=
°
if u:::;O. Then GR(U) is Clover Rand
if u:::;R,
(2.1)
if u?:.R + 1.
(2.2)
No.4
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
We also set for R
>0 MR
= .E[O,R] max [g(s)s -
,6G(s)].
473
(2.3)
First of all, we have the following Lemma 2.4
(i) gR
= G~ satisfies (g3) and (2.5)
(ii) there exists for R> 1 a constant C2 > 0 such that (2.6)
!Proof It is evident that gR(U) satisfies (g3) for u~R or u?R since g(u) satisfies (g3), we have
gR(U)U - ,6GR(u)
+ 1.
For u E]R, R
+ 1[,
= ("p~(u)u + ,6"pR(U)Uf3-1)u - ,6"pR(U)Uf3 = "p~(u)uf3+1 = g(R)~;~G(R) (R + 1- u)uf3+1 ? O.
Moreover, (2.5) holds clearly for u E]O, R[U[R + 1, +00[,
gR(U)U - ,6GR(U)
= ("p'(R)(R+ 1- u)u f3+ 1 + R)f3+ 1"p'(R) = (1 + ~)f3+l Rf3+l"p'(R)~2f3+1[g(R)R ~ uf3+l"p'(R)~(1
,6G(R)]
~ 2f3 +1 MR,
(2.5) follows. For R> 1, by (g3), we get C
= ot'(R)
R-
< - Rf3-1 By (g4), g(R)~Cl(R'"I-l
~ot,l(R)
= G(R)
Rf3 -
Rf3-1'
Rf3+1
+ 1) gives g(R)
(2.6) follows since C 2
~. g(R)R - ,6G(R)
+ 2 Of' Rf3 + 2 g(R)' + g(R) <2 g(R)
Of'
+ _1_)<2C Rf3-1 -
1
R'"I-f3
= 4C1 .
Our next lemma is Lemma 2.7
and
U
(i) For any
f
> 0, there exists a constant C, > 0 such that for all R > 1
>0 (2.8)
(2.9)
474
ACTA MATHEMATIC A SCIENTIA
Vo1.21 Ser.B
(ii) There exists a positive constant C such that for all R> 1 and U> 0 (2.10) Proof Firstly, we prove that there exists a constant Cb > 0 such that for all R
> 1 and
u>O (2.11) Indeed, (2.11) is a result of (g4) because gR(U) = g(u) for u5,R. For u2:R + 1, using (2.2) and (2.6), we have 05, gR(U)
= G~(u) =(3CRuf3-1
5, C2{3R-r-f3 Uf3- 1 5, C2{3u,-I.
For u E]R, R + 1[, we get where and 1/J~(u)uf3 = 1/J'(R)(R + 1 - u)uf3
< g(R)R - (3G(R) .uf3
+ l)uf3-1
g(R) 13-1 < C -r- 1 < _ 2u . _ 2 Rf3+ 1u
hence (2.11) holds where Cb = max{Cll ({3 + I)C2} = 4({3 + I)C1. By (g2),for any e > O,there exists a 6([) E]O, 1[ such that for 0
< u < 6([) (2.12)
Taking C.
= max{Cb(1 + 6,,~,),2Cb}, by (2.11) and (2.12),
we get
and gR(U)
1 _ 2C'0 u,-1 < _ C'0 + C'0 u-r- <
5, [U p - 1 + C.u,-I,
for
u2:1.
We concludes (2.8). (2.9) holds since (g3) and (2.8) where C. = C. (Cll (3). To show (ii), we notice that (2.10) is equivalent to
475
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEARELLIPTIC EQUATIONS
No.4
we obtain (2.10) by choosing C:5.'l/JR(l) because 'l/JR(U) is nonnegative and nondecreasing. We consider the following modified problem -div(IVuIP-2Vu) {
U
=0
= a(x)YR(U) in n on an.
(2.13)
Its nontrivial solution is a nonzero critical point of the following functional in W~,p(n)
By (gl)-(g3), it is clear that I R is C 1 in W~,p(n). For fixed R> 1, YR(U)U - (3G R(u) is bounded by (2.5) and we have Lemma 2.14 IR(u) satisfies the (PS) condition in W~,p(n).
Proof Suppose that {un} in W~,P (n) satisfies (i) IR(u) is bounded; (ii) Ik(u n ) -+ 0 in the dual space of W~,p(n), we shall prove that {un} is precompact in W~,p(n). Following [2] and [11] we first show that I/u n /I is bounded. Since
we have by Lemma 2.4
(~- 1) II Un IIP:5.(3IR(un) + 213+ 1 M R P
inr la(x)ldx + Ilunlllllk(un)II*.
If Ilunll is not bounded, there exists a subsequence of {un}, still denoted by {Un}, such that Ilunll-+
00
as n -+
00.
So
By (i),(ii) and 1 < P < (3, this inequality gives that Un -+ 0 as n -+ 00, which is a contradiction. Thus, since W~,p(n) is reflexive, we may assume that there is a subsequence {un} of {Un} and a Uo E W~,p(n) with
(2.15) By Sobolev's imbedding, we may assume that
Un -+ Uo in L"(n) for s E [P,p*[ Un -+ Uo a.e, inn. To prove that {un} is precompact, we need only show that
(see Lemma 2.7 of [12]).
(2.16)
476
ACTA MATHEMATICA SCIENTIA
Since I' (Un)
-+
0 and lIu n II ::; c
< +00,
Vo1.21 Ser.B
we have
(Ik(U n), un} = 0(1), (Ik(un),uo)
= 0(1).
(2.17)
So, by (2.15) and (2.17), we get
l
(IVu n!P-2VUn -IVuolp-2Vuo)-(VUn - Vuo)dx
L
= llVunlPdx -
IVunIP-2Vun·Vuodx
-llvuoIP-2vuo,V(un - uo)dx
l =l =
a(x)YR(un)undx
-l
a(x)YR(un)uodx
+(Ik(un), un) - (Ik(u n), uo} a(x)YR(un)undx -
J
+ 0(1)
a(x)YR(un)uodx + 0(1).
By (2.16) and (2.8), we see that {a(x)YR(Un)U n} and {a(x)YR(Un)UO} are uniformly integrable on n i.e. "Ie> 0, 3b(e) > 0 such that
L
la(x)YR(un)unldx < e
L
and
la(x)YR(un)uoldx < e
as long as lEI < b( e) where lEI denotes the Lebesgue measure of E in RN. Hence, (2.16) and (gl) together with Vitali's convergence theorem show that lim
n-+oo
[10a(:Z:)YRCU" )u"dx - 10f a(x)YR(un)uodX] = o.
So we have shown that and the lemma is proved. The next proposition shows that I R satisfies the geometric asanmptions of the MountainPass Theorem uniformly with respect to R.
Proposition 2.18 There exist q > 0,15 > 0 depending only on CO,Cl,{3,'Y,p,N,n, and U E W~,p(n) satisfying Ilull > q and u 2 0 in n such that for all R> 1 (i) lIuli = q :=} IR(u) 2 15; (ii) IR(u) ::; O. Proof Using (2.9), the Sobolev inequality and the Poincare inequality, we have for all R>l
No.4
where C
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
= C(E,CO,Cl,{3,1',p,N,n)
is a constant independent of R. Letting E small enough,
(i) follows since l' > p. On the other hand, we choose tp E W5,p(n) such that Iltpll (tp) C n+ hence (2.10) leads to
IR(ttp)
= ~ { 1V'(ttp)IPdx pin
::;
477
2: 0 in n and support
{ a(x)GR(ttp)dx
in
~tP - C { a(x)(t/3tp/3 p
= 1, tp
in+
l)dx
(2.19)
< ~tP - Ct/3 + C -p :::
> 0 and R> 1, where C and C are positive constants depending on C, cs.s,« N, n. Since {3 > p, so for to large enough, u = totp is such that Ilull > q and IR(u :s 0 for all R > 1
for all t
and the Proposition is proved. By the Mountain-Pass Theorem (see [2]), IR possesses a critical value dR 2: 0 given by
where r = h E C([O, 1], W5,p(n)) 11'(0) .; 0 and 1'(1) = u}. We denote by UR a critical point of IR such that IR( un) = dR. Our next lemma generalized the classical maximum principle in[7] (pI79-180) in the case p = 2. We believe that the following lemma exists somewhere in the literature although we can not trace it. For the reader's convenience we state and prove it. (see Theorem 4.2 of [13]) Lemma 2.20 Assume U E W~,p(n) is a weak solution of the following equation (2.21) then ess inf U > inf Un - an where u: ::::' min{u(x),O},bj(x) 2: 0 in n(j u
< O(j = 1,2).
= 1,2),hj(u)
2: 0 for u 2: 0 and hj(u)
= 0 for
Proof Let 1= infan u: :S' 0, the function tp = (k - u)+ for Vk < I is in W~,p(n) where w+ = max{w(x), O} for any function w(x). By equation (2.21) we have
L
lV'ulp -
2V'u.V'tpdx
+
L
b1(x)h1(u)tpdx =
L
b2(x)h 2(u)tpdx.
Let n* = {x E nlk 2: u(x)}, notice that tp = k-u for k 2: u and tp = 0 for k :s u, the integration above is actually carried out on no. hj(u(x)) = O(j = 1,2) on n* because u(x):Sk < l:S0 on n°. Thus
hence
This shows that u
> I a.e.in n.
478
ACTA MATHEMATICA SCIENTIA
Vo1.21 Ser.B
The result follows. We can rewrite the equation
as hence UR 2:: 0 by Lemma 2.20. If U is a solution to -.0.p u = f(x), it was shown in e.g.[8] that U E C,1~~(n) for p > 1 and .0. p u E L;oc(n) for p 2:: 2. To prove UR > 0, we extract the strong maximum principle for p-Laplacian type of equations from [4] as our Proposition 2.22. Proposition 2.22 Let U E C 1 (n ) be such that .0. p u E L;oc(n), U 2:: 0 a.e. in n, .0.p u ~ (3(u) a.e. in n with (3: [0,=[-> R continuous, nondecreasing, (3(0) = 0 and either (3(s) = 0 for some s > 0 or f3( s) > 0 for all s > 0 and
Then if
U
n it
does not vanish identically on
Because .0.p uR
= -a(x)YR(UR) ~
l\YR(s)s)-idS 2::
1
is positive everywhere in
IlallooYR(UR) and by (2.8)
1
(ESP
+ Ces"Y)-ids 2:: (E + Co)-i
n.
1 1
s-1ds
= oo
Proposition 2.22 shows that the solution UR of (2.13) is positive everywhere in
n for p 2:: 2.
Now we have proved the following theorem T'heorem 2.23
For R > 1,
(A)R if 2 ~ p < N, problem (2.13) possesses at least a positive solution; (B)R if 1 < p < 2, problem (2.13) possesses at least a nonnegative nontrivial solution.
3
Proof of the Main Results To complete the proof of Theorem 1.4, suppose that UR is a solution to (2.13) for suitable
R> 1, we want to show that IluR//:::;R. To this end we need several lemmata. Firstly, we have the following lemma. Lemma 3.1
There exists a constant C3
> 0 such
that (3.2)
where C3
= C3(C,Ca,(3,p,N,n).
Proof We have dR~
max IR(tu)
tE[a,1]
where u was given in the proof of Proposition 2.18 and it is easy to see from (2.19) that
No.4
Li & Yu: EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
479
where C = C(C, Co,{3,p, N, 0) denotes any constant independent of R. So dn is bounded for all R > 1. We can write
f3d R
= f3IR(uR) - Ik(uR)uR = (~- 1)lluRIIP + { a(x)[gR(UR)UR p
~ (~ p
1)lIuRIIP
-
f3GR(uR)]dx In In- la(x)l[gR(uR)uR - f3GR(uR)]dx. (
So using (2.5), we get
and the lemma follows. The next lemma obtains a uniform bound for UR in L oo by using the bootstrap arguement(see e.g. [10]). Lemma 3.3 Assume UR is a nonnegative solution of (2.13). Then there exists a constant C 4 = C4(Co,Cll f3" , p, N , O) > 0 independent of R such that
(3.4) where (J = (J(N,p,,)
> O.
Proof We set URL = min{uR, L} for L
> O.
Suppose that v = URU)z~-l) for a~1, then
v E W~'P(O) and
By equation (2.13), we have
So using (2.8), we get
{ P P(o-l)d ::; IIa II00 ( e n URURL z
J
I P(o-l)d ) + Ce J(n URURL x .
(3.5)
By the convexity of the power function, we obtain '
in = in lV'uRu~Ll + IV'WLIPdx
::; 2P -
Suppose that n
1
(a -
1)uRu~L2V'uRLIP dx
in lV'uRIPu)z~-l)dx
+ 2P - 1 (a - 1)P
in u)zu)z~-2)IV'uRLIPdx.
(3.6)
= {x E OIUR(X) ::; L}, then UR = URL on n, V'UR = V'URL on nand V'URL = 0
480
ACTA MATHEMATIC A SCIENTIA
Vo1.21 Ser.B
LIVURIP-2VURVURLURU~<;-1)-ldx
on [2\0, hence
= lIVuRIP-2VuRVURLURU~<;-1)-ldX =
1,vuRIPu~uj}~-2)dx
=
1 n
" 1v
(3.7)
IP P p(a-2)d x. URL URURL
By (3.5),(3.6),(3.7) and the Poincare inequality, we have
L
Lu~u~<;-l)dx +
IVWLIPdx:S CaP(c:
:S c:CaP :S c:CaP
L L
!WLIPdx
ce·l u1-Pu~~dx)
+ Cea P
IVWLIPdx
Lu1-Pu~~dx
(3.8)
+ Cea P l u1-pu~~dx.
Taking e small enough, by the Holder's inequality and the Sobolev's inequality, we write the inequality (3.8) to
where q
= p* j(p* - , + p)
and C is a constant independent of R.
We can choose a such that qpa
Suppose that s
Let a
:S p* since
< , < p*. Letting
p
L
->
+00, we obtain
= qp and X = p* j s, then we have
= xffi(m = 1,2",,), then we get IluRllx=+l :S
~-p
Cx-= Xffi X-= (1IuRII;.")X-= IluRllx='
:S c2:::o x- X2:::o ix-; (1IuRII;! )2:::0 x- IluRlls i
-y-p
i
x
:S ClluRII;."·X=1"lluRlls :S C1luRII~~e where
8
=' -
p . _X_ p X-I , - p p*
--. p
= ,- p
p* - qp
. ~ P p* - s , - p p* - ,
-_. p
+p
p* - ,
>0.
481
Li & Yu; EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS
No.4
Letting m
--+
+00, we get
and this proves the lemma. Proof of Theorem 1.4 exists a constant C
By Lemma 3.1, Lemma 3.3 and Sobolev's imbedding, there
= C(Co, C I , C,f3,'Y,P, N, n)
> 0 independent
of R such that
(3.9) Taking in particular R = 2C, we see that if In- la(x)ldx is sufficiently small, then IluRllCXl :s; R. So Theorem 2.23 implies Theorem 1.4 and the solution UR to problem (2.13) is a solution to problem (1.1). To prove Theorem 1.6, we need the following proposition: Proposition 3.10
Let E be an infinite dimensional real Banach space and I E C I (F;, R)
with I even, satisfies (PS) and 1(0) (11) there exist constants o.a
= 0, and
> 0 such that if Ilull
= p then l(u) 2 a;
(12) there is a sequence of finite dimensional subspace {Em} of E with dimEm Em C E m+l for m
= 1,2"",
=m
and a sequence of real numbers {R m} such that I
:s;
and
0 on
Em nBR~; then I possesses an unbounded sequence of critical values.
Proof This result was implicitly mentioned in [2] and the proof is hidden in [2] (see also the proof of Theorem 9.12 of [9]) where (12) was explicitlv stated as:" for each finite dimensional
subspace VeE, there is an R = R(B) such that 1:S; 0 on V n B'k(m)''' Proof of Theorem 1.6 It is dear that the l(u) given by (1.2) with g(u) = lul q I
C and satisfies (PS) (see the proof of Lemma 2.14). Also 1(0)
= 0, l(u)
2u
is
is even and (11) in
Proposition 3.10 is satisfied. On the other hand, let n* be such that n* C n+ and let em be a sequence of linearly independent elements of W~,p (n) with Suppe., C n*, Em
= span{el, e2, ... , em}, e m+I
rf:. Em,
then by Proposition 3.10, to complete the proof, we need only show that there is a sequence
{Rm} of real numbers such that 1( u) Ilull = 1» 1, we have l(u)
:s;
0 on Em
= ~llullP _ P
r a(x)lul ln q
= ~lP -ZQinfa(x) p
n BR~' To this end, for any u E Em with
n-
q
dz
r
inf vEEm,lIvll=:lln-
~dx q
Since Em is finite dimensional and norms in a finite dimensional space are equivalent, we
> O. Also, since a(x) E C(n) and n* c n+, we see that inf n- a( x) > O. Hence there is an R m > 0 such that 1( u) :s; 0 on Em n Bk m and the theorem is see that infvEEm,lIvlI=:1 In- jvlqdx proved.
482
ACTA MATHEMATIC A SCIENTIA
Vo1.21 Ser.B
References 1 Samia Khanfir, Leila Lassoued. On the existence of positive solutions of a semilinear elliptic equation with change of sign, Nonlinear Analysis, Theory, Methods, Aplications, 1994, 22(11): 1039-1314 2 Ambrosetti a A, Rabinowitz P H. Dual variational methods incritical point theory and application. J Funct Anal, 1973, 14: 349-381 3 Shen Yaotian. The nontrivial solutions of quasilinear elliptic equations in W~'p. Scientia Sinica (Ser A), 1984,27: 720-730 4 Vazquez J L. A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim, 1984,12: 191-202 5 Guedda M, Veron L. Qusilinear elliptic equations involving critical Sololev exponents. Nonlinear Analysis, Theory, Methods, Applications, 1989, 13(8): 879-902 6 Li Gongbao. Some properties of weak solutions of nonlinear scalar field equations. Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1989,14: 27-36 7 Gilbarg D, Trudinger N S. Elliptic partial differential equations of second order. Heidelberg, New York: Springer-Verlag, 1983 8 Peter Tolksdorf. Regularity for a more general class of quasilinear elliptic equations. J Diff Equs, 1984,51:121-160 9 Robinowitz P H. Minimax methods in critical point theory with applications to differential equations. Regional Conference Series in Mathematics, ISSN 0160-7642, no.65,AMS,1986 10 Li Gongbao. The existence of infinitely many solutions of quasilinear partial differential equations in unbounded domains. Act.a Math Sci, 1989,9(2):175-188 11 Li Gongbao, Zhu Xiping. Nontrivial solution of nonlinear scalar field equations with strong nonlinearity. Acta Math Sci, 1988, 8(4): 431-448 12 Li Gongbao, Olli Martio. Stability in obstacle problems. Math Scand, 1994, 75: 87-100 13 Chen Yiaze, Wu Lanchen. Second order elliptic equations and elliptic systems. Science Press, 1991 (Chinese Edition)