- Email: [email protected]
On a superlinear elliptic p-Laplacian equation
ARTICLE IN PRESS
J. Differential Equations 198 (2004) 149–175
On a superlinear elliptic p-Laplacian equation Thomas Bartscha, and Zhaoli Liub,1 b
a Mathematisches Institut, Universita¨t Giessen, Arndtstr. 2, 35392 Giessen, Germany Department of Mathematics, Shandong University, Jinan, Shandong 250100, PR China
Received February 26, 2003
Abstract We prove the existence of four solutions for the p-Laplacian equation Dp u ¼ f ðx; uÞ;
uAW01;p ðOÞ;
on a bounded domain OCRN with smooth boundary @O provided that the nonlinearity f is superlinear and subcritical and that the equation has a pair of subsolution and supersolution. In addition to the existence of a positive and a negative solution, the existence of a sign changing solution is proved. We present a new approach to obtain existence and nodal properties of solutions of p-Laplacian equations. r 2003 Elsevier Inc. All rights reserved. Keywords: p-Laplace operator; Quasilinear Dirichlet problem; Sign changing solution
1. Introduction We consider the quasilinear elliptic equation Dp u ¼ f ðx; uÞ;
uAW01;p ðOÞ;
ð1:1Þ
on a bounded domain OCRN with smooth boundary @O; here Dp u ¼ divðjrujp2 ruÞ % R-R is continuous. The p-Laplacian is the p-Laplacian, p41; and f : O equation (1.1) arises naturally in various contexts of physics, for instance, in the
Corresponding author. Fax: +49-641-9932179. E-mail address: [email protected] (T. Bartsch). 1 Supported by the Alexander von Humboldt foundation in Germany and FUKTME in China. 0022-0396/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2003.08.001
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study of non-Newtonian fluids (the case of a Newtonian fluid corresponding to p ¼ 2), and in the study of nonlinear elasticity problems. It also appears in the search for solitons of certain Lorentz-invariant nonlinear field equations. We refer to [5,14,19] for details and further references. The goal of this paper is to find solutions of (1.1) inside or outside of certain order cones. In particular, we shall prove the existence of a sign changing solution when the nonlinearity f is superlinear and subcritical as juj-N: Results of this type are quite recent even in the semilinear case p ¼ 2: In this case, the mountain pass theorem yields a positive and a negative solution. The existence of a third nontrivial solution is due to Wang [28]. The question of the nodal properties of the third solution was left open in [28]. It was settled by several authors, see e.g. [1,2,4,6,8,16,17,21]. In these papers, one can also find various generalizations of Wang’s result and different methods of proof. A direct extension of these methods to the quasilinear case pa2 is faced with serious difficulties. First of all, the energy functional associated to (1.1) is defined on W01;p ðOÞ which is not a Hilbert space for pa2: In addition, for po2 the energy is only of class C1 so one has to construct an appropriate pseudo-gradient vector field. This needs more care as usual when one is interested in nodal properties of the solutions. Another important ingredient is the generalized Morse lemma which is based on the linearization of (1.1), or the computation of the Hessian of the energy at a critical point. This is needed in order to construct a linking and to obtain the third solution via Lusternik–Schnirelmann methods. It is also needed when one wants to compute critical groups and apply Morse-type arguments. A generalized Morse lemma is not available for pa2: Since it requires the functional to be of class C2 it cannot possibly be extended to the case po2: A third difficulty is the lack of a powerful regularity theory if pa2: For the Laplace operator there exists a sequence % and such of Banach spaces E0 +E1 +?+En with W01;2 ðOÞ+En and E0 +C1 ðOÞ; 1 that the nonlinear operators Ek -Ek1 ; u/ðD þ mÞ ðf ð:; uÞ þ muÞ; where m40; are continuous and map bounded sets to bounded sets. There is no similar regularity theory for the corresponding operator in the case pa2: Finally, an essential ingredient in the earlier approaches is the strongly order preserving property of the operator ðD þ mÞ1 ; i.e. u1 ou2 in O implies v1 :¼ ðD þ mÞ1 ðu1 Þov2 :¼ ðD þ mÞ1 ðu2 Þ in O; and @n v1 4@n v2 on @O; here n is the exterior normal to O: This property is a consequence of the strong comparison principle which holds for D; but which does not hold for Dp : In [3], we developed some critical point theory for C1 -functionals on partially ordered Banach spaces which allows to localize critical points. This theory will now be adapted in order to obtain existence results for solutions of (1.1) together with nodal information on the solutions. In particular, we generalize the above-mentioned results from [1,2,4,6,8,16,17,21] on the existence of a sign changing solution to the case pa2: In fact, technically several cases have to be distinguished, namely the cases poN; ¼ N; 4N: In addition, the critical point theorems from [3] distinguish between po2 and 42: We would like to mention that
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already the existence statements in our theorems are new for pa2: Moreover, the methods developed here and in [3] are different from earlier methods even in the case p ¼ 2:
2. Statement of results In order to keep the paper within reasonable bounds we shall not present the most general version of our results, but indicate possible extensions without proofs (cf. Remark 2.5). % : uj@O ¼ 0g: For u; vAY we say u5v if and Let X :¼ W01;p ðOÞ and Y :¼ fuAC1 ðOÞ only if uðxÞovðxÞ for xAO and @uðxÞ=@n4@vðxÞ=@n for xA@O; here n is the outer unit normal to @O: Define hp ðtÞ :¼ jtjp2 t: For m40 we need the operator Am : X -X ;
Am ðuÞ ¼ ðDp þ mhp ð ÞÞ1 ððx; uÞ þ mhp ðuÞÞ:
Finally, we set p :¼ Np=ðN pÞ for poN; and p :¼ N for pXN: Now we assume: ðH1 Þ If 1opoN then f ðx; tÞ
lim
jtj-N
jtjp
1
¼0
uniformly in x;
if p ¼ N then lim
jtj-N
lnj f ðx; tÞj jtjN=ðN1Þ
¼0
uniformly in x:
ðH2 Þ There exist m4p and M40 such that 0oF ðx; tÞ :¼
Z
t 0
1 f ðx; sÞ dsp tf ðx; tÞ m
for jtjXM:
ðH3 Þ There exist m40 and j; cAY with jpc such that uALN ðOÞ; uXj ) Am ðuÞbj and uALN ðOÞ; upc ) Am ðuÞ5c: In dimensions NX6; we also require the following technical condition on p: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðÞ If NX6 then either 1opoðN þ 1 N 2 6N þ 1Þ=4 or p4ðN þ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 2 6N þ 1Þ=4:
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The main result of this paper is Theorem 2.1. Assume ðH1 Þ–ðH3 Þ and ðÞ: Then (1.1) has four solutions u1 ; y; u4 with the properties u1 Xj;
u1 4 / c;
/ j; u2 5
u2 pc;
/ j; u3 5
u3 4 / c;
u4 Xj;
u4 pc:
There are two important cases in which ðH3 Þ holds. ðH03 Þ f ðx; 0Þ ¼ 0; and there exists m40 such that f ðx; tÞ þ mhp ðtÞ is increasing in t; and lim sup t-0
j f ðx; tÞj ol1 jtjp1
uniformly in x;
here l1 is the first eigenvalue of: Dp u ¼ lhp ðuÞ; uAX ¼ W01;p ðOÞ: ðH003 Þ There exist t1 o0ot2 such that f ðx; t1 Þ404f ðx; t2 Þ and there exists m40 such that f ðx; tÞ þ mhp ðtÞ is increasing in t: In the situation of ðH03 Þ; 0 is a solution of (1.1). In this case, we can get nodal properties of the three nontrivial solutions. Theorem 2.2. Assume ðH1 Þ; ðH2 Þ; ðH03 Þ and ðÞ: Then (1.1) has a positive solution, a negative solution and a sign changing solution. We have a similar result in the case of ðH003 Þ: Theorem 2.3. Assume ðH1 Þ; ðH2 Þ; ðH003 Þ and ðÞ: Then (1.1) has a sign changing solution. Moreover, if f ðx; 0Þ ¼ 0 then (1.1) also has a positive solution and a negative solution. Remark 2.4. (a) Condition ðH1 Þ means that the nonlinearity f is subcritical and ðH2 Þ means that f is ‘‘superlinear’’. These two conditions enable us to use a variational approach for the study of (1.1); they also provide compactness in some sense. As in the case p ¼ 2 without ðH1 Þ the above theorems may not be true. This can be seen from Pohozaev’s identity for p-Laplacian equations: If u is a solution of Dp u ¼ f ðuÞ;
uAW01;p ðOÞ;
ð2:1Þ
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then we obtain the Pohozaev identity Z Z Z N p p1 F ðuÞ dx f ðuÞu dx ¼ ðx nÞjrujp ds: N p p O O @O This identity can be proved by integrating (2.1) multiplied with x ru; and using the formulas 1 N p p2 p jrujp div jruj ðx ruÞru jruj x ¼ ðx ruÞDp u p p and divðF ðuÞxÞ ¼ ðx ruÞf ðuÞ þ NF ðuÞ: From the Pohozaev identity one sees that if O is star shaped, 1opoN; and f ðuÞ ¼ jujq2 u with qXp then (2.1) has only the trivial solution u ¼ 0: Condition ðH2 Þ corresponds to the standard superlinearity condition of Ambrosetti–Rabinowitz in the case p ¼ 2: The existence of positive solutions in this case has been proved in [9,20]. Condition ðH3 Þ provides a pair of subsolution and supersolution and is essential in order to obtain the solutions u1 and u2 in Theorem 2.1, or the positive and the negative solutions in Theorems 2.2, 2.3. We believe that there should be a sign changing solution without ðH3 Þ but do not know how to prove this. (b) In [3] we developed methods for the general case p41 which can be applied to asymptotically sublinear nonlinearities. Remark 2.5. (a) The assumption in ðH03 Þ and ðH003 Þ that f ðx; tÞ þ mhp ðtÞ is increasing in t for some m40 is not essential but is assumed for simplicity. If such m does not exist, then we approximate f by a sequence of functions so that m as above exists, and obtain the solutions by passing to limits. The details will not be given here. (b) Similarly, the inequality f ðx; t1 Þ404f ðx; t2 Þ in ðH003 Þ may be replaced with the weaker one: f ðx; t1 ÞX0Xf ðx; t2 Þ: This can be proved via an approximation argument starting from Theorem 2.3. The details will again be left to the interested reader. (c) Condition ðÞ is a technical condition which can probably be avoided. We will indicate where this inequality is used in the proof; cf. Remark 5.7. Observe that in the case NX6 our results do not contain the case p ¼ 2: Throughout this paper, we will use ci and di to represent various positive constants. jj jjq denotes the standard norm in Lq ðOÞ for 1pqp þ N and jj jj1;p the standard norm in X :¼ W01;p ðOÞ: We shall also use the norm jjujjX :¼
Z
ðjrujp þ mjujp Þ O
1=p ;
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where m is from hypothesis ðH3 Þ: This is, of course, equivalent to the usual norm in X : 3. Properties of the operator Am In this section, we study properties of Am defined in Section 2. We shall need the following stronger version of ðH1 Þ: ðH01 Þ There exist c40 and p 1oqop 1 such that j f ðx; tÞjpcð1 þ jtjq Þ: Throughout this section we assume that ðH01 Þ and ðH3 Þ hold. ðH2 Þ is not required here. Solutions of (1.1) correspond to critical points of the functional J : X -R defined by Z Z 1 jrujp F ðx; uÞ JðuÞ :¼ p O O and also to fixed points of the operator Am : X -X : Note that J is of class C1 and Am is compact, i.e. it is continuous and maps bounded subsets of X into relatively compact subsets of X ; cf. [11]. If pXN and qX1 then X +Lq ðOÞ and Am : X -Lq ðOÞ are compact operators. For 1opoN; we have P Lemma 3.1. Let 1opoN and define s0 :¼ p q 1 and qn :¼ p þ s0 ni¼1 ðpp Þi ; nAN0 : Then Am : Lqn ðOÞ-Lqnþ1 ðOÞ is bounded, i.e. it maps bounded subsets of Lqn ðOÞ into bounded subsets of Lqnþ1 ðOÞ; nAN0 : Proof. Consider uALq0 ðOÞ and set v ¼ Am ðuÞAX so that Dp v þ mhp ðvÞ ¼ f ðx; uÞ þ mhp ðuÞ: Using jvjs0 v as a testing function, (3.1), ðH01 Þ; and the Ho¨lder inequality imply Z Z ðs0 þ 1Þpp s0 =p p jrðjvj vÞj þ m jvjpþs0 ðs0 þ pÞp O O Z Z ¼ ðs0 þ 1Þ jvjs0 jrvjp þ m jvjpþs0 O O Z ¼ ð f ðx; uÞjvjs0 v þ mjujp2 ujvjs0 vÞ O Z p ðcð1 þ jujq Þ þ mjujp1 Þjvjs0 þ1 O Z pc1 ðjujq þ 1Þjvjs0 þ1 O s þ1
pc2 ðjjujjqp þ 1Þjjvjjp0 :
ð3:1Þ
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This together with the Sobolev inequality gives s þp q pc3 ðjjujjp 0 þpÞp =p
jjvjjðs0
s þ1
þ 1Þjjvjjp0
and therefore jjvjjq1 pc4 ðjjujjq=ðp1Þ þ 1Þ: q0 Thus the result is true for n ¼ 0: Now suppose that Am : Lqn1 ðOÞ-Lqn ðOÞ is bounded and consider uALqn ðOÞ; v ¼ Am ðuÞ: Denoting n i X p sn :¼ qn ðq þ 1Þ ¼ s0 p i¼0 and using jvjsn v as a testing function in (3.1), we obtain Z Z ðsn þ 1Þpp sn =p p jrðjvj vÞj p c1 ðjujq þ 1Þjvjsn þ1 ðsn þ pÞp O O p c5 ðjjujjqqn þ 1Þjjvjjsqnnþ1 : Therefore, sn þp q sn þ1 jjvjjðs ; pc6 ðjjujjqn þ 1Þjjvjjqn n þpÞp =p
which implies jjvjjqnþ1 pc7 ðjjujjq=ðp1Þ þ 1Þ: qn By induction, Am : Lqn ðOÞ-Lqnþ1 ðOÞ is bounded for every nAN0 :
&
As a consequence of Lemma 3.1, any solution u of (1.1) lies in Lr ðOÞ; for any r41: We will see that u has much stronger regularity. For this purpose, we need the next lemma which was proved in [15] by a similar argument as in the proof of [13, Theorem 8.15]. Lemma 3.2. If 1oppN and r4Np then ðDp þ mhp ð ÞÞ1 maps Lr ðOÞ into LN ðOÞ and there exists a constant c40 such that jjðDp þ mhp ð ÞÞ1 ujjN pcjjujj1=ðp1Þ r
for every uALr ðOÞ:
Lemma 3.2 implies that Am : Lr ðOÞ-LN ðOÞ is bounded if 1oppN and r4qN=p: Observe that if p4N then X +LN ðOÞ: Consequently, any solution of (1.1) is in
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LN ðOÞ: A stronger regularity of solutions of (1.1) is implied by the next lemma of Liebermann [18]. Lemma 3.3. There exist 0oao1 and c40 such that jjAm ðuÞjjC 1;a pcðjjujjq=ðp1Þ þ 1Þ N
for uALN ðOÞ;
hence Am : LN ðOÞ-C01;a ðOÞ is bounded. Proof. According to [18], there exist 0oao1 and c1 40 such that jjðDp þ mhp ð ÞÞ1 ujjC1;a pc1 jjujj1=ðp1Þ N
for uALN ðOÞ:
This yields pc2 ðjjujjq=ðp1Þ þ 1Þ jjAm ðuÞjjC1;a pc1 jj f ðx; uÞ þ mhp ðuÞjj1=ðp1Þ N N for uALN ðOÞ; as claimed.
&
% for some 0oao1 for any solution u of (1.1). We By Lemma 3.3, uAC 1;a ðOÞ N remark that for uAL ðOÞ; local C1;a regularity of Am ðuÞ was obtained in [10,26] and boundary C1;a regularity was proved in [18]. Having the regularity of Am as in Lemma 3.3, we now show that ðH3 Þ is satisfied if either ðH03 Þ or ðH003 Þ holds. Lemma 3.4. ðH03 Þ implies ðH3 Þ: Proof. Let e1 40 be an eigenfunction of Dp in W01;p ðOÞ corresponding to the first % eigenvalue l1 40: Then e1 AC1;a 0 ðOÞ and e1 b0 as a consequence of the strong maximum principle from [27] (see also [12,22,23]). By ðH03 Þ we may choose e40 and t0 40 such that ðl1 eÞjtjp1 pf ðx; tÞpðl1 eÞjtjp1
for jtjpt0 :
Fixing d0 40 with d0 jje1 jjN ot0 we set j :¼ d0 e1 and c :¼ d0 e1 : Now we consider uALN ðOÞ with uXj: Then we have for v ¼ Am ðuÞ Dp v þ mhp ðvÞ ¼ f ðx; uÞ þ mhp ðuÞ X f ðx; d0 e1 Þ þ mhp ðd0 e1 Þ p1 p1 mdp1 X ðl1 eÞdp1 0 e1 0 e1
X
l1 þ m e ðDp ðd0 e1 Þ þ mhp ðd0 e1 ÞÞ: l1 þ m
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As a consequence of the weak comparison principle [7,12,25] we obtain vX d1 d0 e1 ; Þ1=ðp1Þ : Since 0od1 o1; we have vb d0 e1 ¼ j: where d1 ¼ ðl1lþme 1 þm Similarly, if uALN ðOÞ satisfies upc then Am ðuÞ5c: & Lemma 3.5. ðH003 Þ implies ðH3 Þ: Proof. Let f ¼ w1 and c ¼ w2 be the unique solutions of Dp wi þ mhp ðwi Þ ¼ mhp ðti Þ;
wi AX :
The strong maximum principle [27] implies j505c; jXt1 ; and cpt2 : Consider uALN ðOÞ with uXj and set v ¼ Am ðuÞ: Since f ðx; t1 Þ40 we have Dp v þ mhp ðvÞ ¼ f ðx; uÞ þ mhp ðuÞ X f ðx; t1 Þ þ mhp ðt1 Þ X dðDp j þ mhp ðjÞÞ for some 0odo1: Then vXd1=ðp1Þ jbj by the weak comparison principle. Similarly, if uALN ðOÞ and upc then Am ðuÞ5c: & In order to set up our variational framework we now discuss the relation between J 0 : X -X and Am : X -X : For doing this we need the following result from [7]. Lemma 3.6. There exist positive constants c1 ; y; c4 such that for all x; ZARN jjxjp2 x jZjp2 Zjpc1 ðjxj þ jZjÞp2 jx Zj;
ð3:2Þ
ðjxjp2 x jZjp2 ZÞ ðx ZÞXc2 ðjxj þ jZjÞp2 jx Zj2 ;
ð3:3Þ
jjxjp2 x jZjp2 Zjpc3 jx Zjp1
if 1opp2;
ðjxjp2 x jZjp2 ZÞ ðx ZÞXc4 jx Zjp
if p42:
ð3:4Þ ð3:5Þ
Lemma 3.7. There exists a constant c5 40 such that, if 1opp2 then /J 0 ðuÞ; u Am ðuÞSX ;X Xc5 jju Am ðuÞjj2X ðjjujjX þ jjAm ðuÞjjX Þp2
ð3:6Þ
and if p42 then /J 0 ðuÞ; u Am ðuÞSX ;X Xc5 jju Am ðuÞjjpX hold for all uAX :
ð3:7Þ
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Proof. Let uAX and set v ¼ Am ðuÞ: Then Z Z p2 0 jruj ru rðu vÞ f ðx; uÞðu vÞ /J ðuÞ; u vSX ;X ¼ O O Z ¼ ðjrujp2 ru jrvjp2 rvÞðru rvÞ O Z þ mðhp ðuÞ hp ðvÞÞðu vÞ O Z X ðjrujp2 ru jrvjp2 rvÞðru rvÞ: O
If p42; it follows from (3.5) that /J 0 ðuÞ; u vSX ;X Xdjju vjjpX : Now consider the case 1opp2: Then (3.3) yields Z /J 0 ðuÞ; u vSX ;X Xc2 ðjruj þ jrvjÞp2 jru rvj2 : O
The Ho¨lder inequality implies Z p jju vjjX p d1 jru rvjp ðjruj þ jrvjÞpðp2Þ=2 ðjruj þ jrvjÞpð2pÞ=2 O
p d1
Z
2
jru rvj ðjruj þ jrvjÞ
O
p d2
Z
Z
ðjruj þ jrvjÞp
p2
1p=2
O 2
p=2
jru rvj ðjruj þ jrvjÞ
p2
O
p=2
ðjjujjX þ jjvjjX Þpð1p=2Þ :
and therefore /J 0 ðuÞ; u vSX ;X Xd3 jju vjj2X ðjjujjX þ jjvjjX Þp2 :
&
Lemma 3.8. There exists a constant c6 40 such that jjJ 0 ðuÞjjX pc6 jju Am ðuÞjjp1 X
ð3:8Þ
if 1opp2
and jjJ 0 ðuÞjjX pc6 jju Am ðuÞjjX ðjjujjX þ jjAm ðuÞjjX Þp2 hold for all uAX :
if p42
ð3:9Þ
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p Proof. Let uAX and v ¼ Am ðuÞ: As usual we denote by p0 ¼ p1 the dual exponent. For wAX we have Z /J 0 ðuÞ; wSX ;X ¼ ðjrujp2 ru jrvjp2 rvÞ rw þ mðhp ðuÞ hp ðvÞÞw: O
The Ho¨lder inequality now yields /J ðuÞ; wSX ;X p 0
Z
jjruj
p2
ru jrvj
p2
rvj
1=p0 Z
p0
O
þ m
p
1=p
jrwj
O
Z
jhp ðuÞ hp ðvÞjp
0
1=p0 Z
O
jwjp
1=p :
O
Therefore, jjJ 0 ðuÞjjX p d1
Z
0
jjrujp2 ru jrvjp2 rvjp
1=p0
O
þ d1
Z
jhp ðuÞ hp ðvÞj
p0
1=p0 :
O
If 1opp2; then jjJ 0 ðuÞjjX p d2
Z
jru rvjðp1Þp
0
1=p0 þd2
Z
O
ju vjðp1Þp
0
1=p0
O
p d3 jju vjjp1 X as a consequence of (3.4). If p42; then (3.2) and the Ho¨lder inequality imply 0
jjJ ðuÞjjX p d4
Z
ðjruj þ jrvjÞ
ðp2Þp0
p0
1=p0
jru rvj
O
þ d4
Z
0
1=p0
0
ðjuj þ jvjÞðp2Þp ju vjp
O
p d4
Z
ðjruj þ jrvjÞ
p
1=p0 1=p Z
O
þ d4
jru rvjp
O
Z
p
ðjuj þ jvjÞ
1=p0 1=p Z
O
p d5 ðjjujjX þ jjvjjX Þp2 jju vjjX :
p
ju vj
O
&
1=p
1=p
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For the last lemma of this section, we make the following assumption which is stronger than ðH01 Þ: ðH001 Þ There exist c40 and q with p 1oqominfpp =N; p 1g such that j f ðx; tÞjpcð1 þ jtjq Þ: Observe that p 1opp =N is equivalent to hypothesis ðÞ; hence ðH001 Þ can only be satisfied if ðÞ holds. Lemma 3.9. If ðH001 Þ holds then Am : X -LN ðOÞ is bounded. Proof. First we consider the case 1opoN: Observe that p =q4N=p: Lemma 3.2 implies that ðDp þ mhp ð ÞÞ1 : Lp =q ðOÞ-LN ðOÞ is bounded. On the other hand, f ðx; Þ : X +Lp ðOÞ-Lp =q ðOÞ is also bounded. Therefore, Am : X -LN ðOÞ is bounded. Now we assume p ¼ N and choose r4N=p: In this case the result is a consequence of the boundedness of f ðx; Þ : W01;N ðOÞ+Lrq ðOÞ-Lr ðOÞ; and the boundedness of ðDp þ mhp ð ÞÞ1 : Lr ðOÞ-LN ðOÞ: In the case p4N the result is obvious since we have a continuous embedding X +LN ðOÞ: &
4. Constructing a vector field Bm In this section, we replace Am by a locally Lipschitz continuous operator Bm which serves as a pseudo-gradient vector field for J: The usual conditions for a pseudogradient vector field and the construction as in the appendix of [24] need to be modified for pa2: Throughout the section we will always assume ðH001 Þ and ðH3 Þ: In addition to the % we also need the Ho¨lder space Z :¼ spaces X ¼ W01;p ðOÞ and Y ¼ X -C1 ðOÞ; 1;a % Y -C ðOÞ: With f and c from ðH3 Þ we define D1 :¼ fuAY : uXjg; D2 :¼ fuAY : upcg; and D3 :¼ D1 -D2 : Let D˜ i be the closure of Di in X and set K :¼ fuAX : J 0 ðuÞ ¼ 0g; X0 :¼ X \K: Note that KCZ by the regularity results from Section 3. Lemma 4.1. Let 1opp2: There exists a locally Lipschitz continuous operator Bm : X0 -Z with the following properties: (i) For uAX0 and c5 from Lemma 3.7 we have /J 0 ðuÞ; u Bm ðuÞSX ;X X
c5 jju Am ðuÞjj2X ðjjujjX þ jjAm ðuÞjjX Þp2 2
ð4:1Þ
and 1 2jju
Bm ðuÞjjX pjju Am ðuÞjjX p2jju Bm ðuÞjjX :
ð4:2Þ
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(ii) Bm ðD˜ i -X0 ÞCintY Di for i ¼ 1; 2; 3: (iii) Bm : X0 -LN ðOÞ is bounded. (iv) Bm : ðX0 -LN ðOÞ; jj jjN Þ-Z is bounded. Proof. Let c5 and c6 be as in (3.6) and (3.8), respectively. For any uAX0 ; setting D1 ðuÞ ¼ 12jju Am ðuÞjjX
ð4:3Þ
p2 D2 ðuÞ ¼ 2cc5 jju Am ðuÞjj3p ; X ðjjujjX þ jjAm ðuÞjjX Þ
ð4:4Þ
and 6
we choose gðuÞAð0; 1Þ such that jjAm ðvÞ Am ðwÞjjX ominfD1 ðvÞ; D1 ðwÞ; D2 ðvÞ; D2 ðwÞg
ð4:5Þ
holds for every v; wANðuÞ :¼ fvAX : jjv ujjX ogðuÞg: Let V be a locally finite open refinement of fNðuÞ: uAX0 g: We need to refine V in order to construct the required operator Bm : For any subset UCX we define IU :¼ fiAf1; 2; 3g: U-D˜ i a|g and [
W :¼
[
fV g,
V AV;IV af1;2g
fV \D˜ 1 ; V \D˜ 2 g:
V AV;IV ¼f1;2g
Clearly, W is a locally finite open refinement of V: We need to refine W once more. If IW ¼ fig with i ¼ 1 or 2 and if inf
uAW -LN ðOÞ
jjujjN o
inf
uAW -D˜ i -LN ðOÞ
jjujjN 1 ¼: bW
for some W AW; then we replace W in the covering W by the following two open subsets of X0 : W \fuALN ðOÞ: jjujjN pbW g
and
W \D˜ i ;
otherwise W is left unchanged. The new open covering U obtained in this way from W is a locally finite open refinement of W; hence of V and fNðuÞ: uAX0 g; and in addition, any UAU satisfies: U-D˜ 1 a|aU-D˜ 2
implies U-D˜ 3 a|
ð4:6Þ
and inf
uAU-LN ðOÞ
jjujjN X
inf
uAU-D˜ i -LN ðOÞ
jjujjN 1
if IU ¼ fig with i ¼ 1 or 2:
ð4:7Þ
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Now we are ready to construct the operator Bm : Let fpU : UAUg be the standard partition of unity subordinated to U which is defined by X
pU ðuÞ :¼
!1 aV ðuÞ
aU ðuÞ;
V AU
where aU ðuÞ :¼ distX ðu; X0 \UÞ: For UAU we choose aU such that aU AU-LN ðOÞ and
jjaU jjN o
inf
uAU-LN ðOÞ
jjujjN þ 1
ð4:8Þ
if IU ¼ |; and aU AU-
\
D˜ i -LN ðOÞ and jjaU jjN o
iAIU
uAU-
T inf ˜ iAIU
Di -LN ðOÞ
jjujjN þ 1
ð4:9Þ
if IU a|: We define Bm : X0 -Z by Bm ðuÞ :¼
X
pU ðuÞAm ðaU Þ:
UAU
That Bm : X0 -Z is locally Lipschitz continuous is a consequence of the Lipschitz continuity of pU ; the locally finiteness of the covering U; and the fact that Am ðLN ðOÞÞCZ: In order to see (i) we observe that jjBm ðuÞ Am ðuÞjjX p
X
pU ðuÞjjAm ðaU Þ Am ðuÞjjX
for uAX0 :
UAU
This together with (4.3)–(4.5) implies for uAX0 ; 1 jjBm ðuÞ Am ðuÞjjX o jju Am ðuÞjjX 2 and c5 p2 jju Am ðuÞjj3p : jjBm ðuÞ Am ðuÞjjX o X ðjjujjX þ jjAm ðuÞjjX Þ 2c6
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As an immediate consequence we have (4.2), while (3.6) and (3.8) imply, for uAX0 ; /J 0 ðuÞ; u Bm ðuÞSX ;X X/J 0 ðuÞ; u Am ðuÞSX ;X jjJ 0 ðuÞjjX jjBm ðuÞ Am ðuÞjjX c5 X jju Am ðuÞjj2X ðjjujjX þ jjAm ðuÞjjX Þp2 : 2 This proves (4.1). If uAD˜ i -X0 then uAD˜ i -U for any U with pU ðuÞa0: From the construction it follows that aU AU-D˜ i -LN ðOÞ for any U with pU ðuÞa0: This implies Bm ðuÞAconv Am ðD˜ i -LN ðOÞÞ; so (ii) follows from ðH3 Þ: In order to prove (iii) we suppose uAX0 and jjujjX pd1 : If pU ðuÞa0 then uAU: Since u; aU AUCNðvÞ for some vAX0 ; and since gðvÞo1 we have jjaU ujjX pjjaU vjjX þ jju vjjX o2 and thus jjaU jjX pd1 þ 2: Then jjAm ðaU ÞjjN pd2 for any UAU with pU ðuÞa0 by Lemma 3.9. Therefore, jjBm ðuÞjjN p
X
pU ðuÞjjAm ðaU ÞjjN pd2 :
UAU
It remains to prove (iv). Suppose uAX0 -LN ðOÞ and jjujjN pd3 : If pU ðuÞa0 then uAU-LN ðOÞ: By (4.8), in the case IU ¼ |; jjaU jjN o
inf
uAU-LN ðOÞ
jjujjN þ 1pd3 þ 1:
In the case IU ¼ fig with i ¼ 1 or 2, (4.7) and (4.9) imply jjaU jjN o
inf
uAU-D˜ i -LN ðOÞ
jjujjN þ 1p
inf
uAU-LN ðOÞ
jjujjN þ 2pd3 þ 2:
If IU a| and IU af1g; f2g; then (4.6) implies IU ¼ f1; 2; 3g; hence (4.9) implies aU AU-D˜ 3 : Since D˜ 3 is bounded in LN ðOÞ; we have jjaU jjN pd4 : In any case, Lemma 3.3 implies jjAm ðaU ÞjjZ pd5 for any UAU with pU ðuÞa0 and jjBm ðuÞjjZ p
X UAU
pU ðuÞjjAm ðaU ÞjjZ pd5 :
&
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In a similar way, we have Lemma 4.2. Let p42: Then the same results as in Lemma 4.1 hold except that (4.1) is now replaced with c5 /J 0 ðuÞ; u Bm ðuÞSX ;X X jju Am ðuÞjjpX ð4:10Þ 2 for uAX0 : 5. A descending flow argument In this section, we will prove Theorem 2.1 provided that the stronger assumptions ðH001 Þ; ðH2 Þ and ðH3 Þ are satisfied. A complete proof will be given in the last section. As mentioned before Lemma 3.9, ðH001 Þ can only be satisfied if ðÞ holds. The proof proceeds via a series of lemmas using a descending flow argument. For uAY0 :¼ Y \KCX0 we consider the following initial value problem, both in X0 and Y0 : jðtÞ ¼ jðtÞ þ Bm ðjðtÞÞ; ’ ð5:1Þ jð0Þ ¼ u: Since Bm : X0 -Z is locally Lipschitz continuous, the solution of (5.1) considered in X0 and the solution of (5.1) considered in Y0 are exactly the same. We denote it by jt ðuÞ with maximal interval of existence ½0; tðuÞÞ: As a consequence of Lemmas 4.1 and 4.2, Jðjt ðuÞÞ is strictly decreasing in tA½0; tðuÞÞ: Lemma 5.1. If uADi \K; then jt ðuÞAintY Di for 0ototðuÞ: A proof can be found in [21]. Lemma 5.2. For any bAR there exists a constant c7 ¼ c7 ðbÞ40 such that jjujjX þ jjAm ðuÞjjX pc7 ð1 þ jju Am ðuÞjjX Þ holds for every uAX with JðuÞpb: Proof. For uAX we have 1 JðuÞ /J 0 ðuÞ; uSX ;X ¼ m
Z Z 1 1 1 p jruj F ðx; uÞ uf ðx; uÞ : p m O m O
If JðuÞpb then ðH2 Þ implies jjujjpX pd1 ð1 þ jjJ 0 ðuÞjjX jjujjX Þ: Now we consider the case 1opp2: Then using (3.8) we obtain jjujjpX pd2 ð1 þ jju Am ðuÞjjp1 X jjujjX Þ:
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Young’s inequality yields jjujjX pd3 ð1 þ jju Am ðuÞjjX Þ; which implies the result. Next we consider the case p42: Using (3.9) we obtain jjujjpX pd4 ð1 þ jju Am ðuÞjjX ðjjujjX þ jjAm ðuÞjjX Þp2 jjujjX Þ; hence Young’s inequality gives 0
0
jjujjpX pd5 ð1 þ jju Am ðuÞjjpX ðjjujjX þ jjAm ðuÞjjX Þpp Þ; where p0 ¼ p=ðp 1Þ: Thus, we have p0 =p
0
jjujjX þ jjAm ðuÞjjX pd6 ð1 þ jju Am ðuÞjjX ðjjujjX þ jjAm ðuÞjjX Þ1p =p Þ; which implies the result using Young’s inequality again.
&
We consider the set G :¼ fuAY0 : jt ðuÞAintY D3 for some 0ototðuÞg,intY D3 ; which is an open subset of Y : Let @G be the boundary of G in Y ; so @G is closed in Y : The following result is easy to prove (see [21]). Lemma 5.3. @G is positive invariant, i.e. if uA@G then jt ðuÞA@G for 0ptotðuÞ: In addition, we have inf JðuÞXa :¼
uA@G
inf
uAintY D3
JðuÞ4 N:
Setting e :¼ 12ðj þ cÞ we have j5e 5c and e AintY D3 by ðH3 Þ: Let e1 ; e2 AW01;p ðOÞ be linearly independent with e1 b0 and denote P ¼ e þ spanfe1 ; e2 g: Clearly ðH2 Þ implies F ðx; tÞXc8 jtjm c9
for tAR
ð5:2Þ
for some positive constants c8 and c9 : For uAX we have Z Z Z Z 1 1 p p jruj F ðx; uÞp jruj c8 jujm þ c9 jOj: JðuÞ ¼ p O p O O O So there exist constants b40 and R40 depending only on m; c8 and c9 such that max JðuÞpb; uAP
ð5:3Þ
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166
and JðuÞoa ¼
max
uAP;jjujjX XR
inf
uAintY D3
ð5:4Þ
JðuÞ:
As a consequence of (5.4), @G-P is bounded. Clearly, we have e AintY D3 CG and e þ se1 eG for s40 large enough by (5.4). Thus, there exists s0 40 such that e þ s0 e1 A@G-D1 : From Lemmas 5.1 and 5.3 we see that jt ðe þ s0 e1 ÞA@G-D1
for 0ptotðe þ s0 e1 Þ:
ð5:5Þ
apJðjt ðe þ s0 e1 ÞÞpb for 0ptotðe þ s0 e1 Þ
ð5:6Þ
Lemma 5.3, (5.3) and (5.5) imply
because e þ s0 e1 A@G-P: We set ut ¼ jt ðe þ s0 e1 Þ and t0 ¼ tðe þ s0 e1 Þ for the sake of brevity. Lemma 5.4. There exist u1 AK and an increasing sequence ðtn Þn with tn -t0 such that limn-N jjutn u1 jjX ¼ 0: Proof. By (5.1) and (4.2) we have for 0ot1 ot2 ot0 Z
jjut2 ut1 jjX p
t2
t1
jjus Bm ðus ÞjjX dsp2
Z
t2
t1
jjus Am ðus ÞjjX ds:
ð5:7Þ
We now consider four cases depending on t0 and p: At first, we assume t0 o þ N and 1opp2: The Schwarz inequality and Lemma 5.2 yield Z
t2
t1
jjus Am ðus ÞjjX ds Z
p
!1=2
t2
s
jju Am ðu
t1
Z
pd1
s
ðjju jjX þ jjAm ðu ÞjjX Þ s
jju Am ðu t2 t1
p2
ds
2p
ds !1=2
t2
Z
s
þ jjAm ðu ÞjjX Þ
s
t1
Þjj2X ðjjus jjX
!1=2
t2 t1
Z
s
s
Þjj2X ðjjus jjX
s
þ jjAm ðu ÞjjX Þ !1=2
s
s
ð1 þ jju Am ðu ÞjjX Þ
2p
ds
:
p2
ds
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Therefore, (4.1), (5.1) and the Ho¨lder inequality imply Z
t2
jjus Am ðus ÞjjX ds
t1
pd2 ðJðu Þ Jðu ÞÞ t1
t2
Z
1=2
s
2p
s
ð1 þ jju Am ðu ÞjjX Þ
t1
pd3 ðJðut1 Þ Jðut2 ÞÞ1=2 2 Z 4ðt2 t1 Þ1=2 þ
!1=2
t2
ds
3
!ð2pÞ=2
t2
ðt2 t1 Þðp1Þ=2 5:
jjus Am ðus ÞjjX ds
t1
In view of (5.6) and the finiteness of t0 we see that Z
t2
jjus Am ðus ÞjjX ds ¼ 0
lim
t1 ;t2 -t0 0
t1
hence, (5.7) yields the existence of u1 AX such that lim jjut u1 jjX ¼ 0:
ð5:8Þ
t-t0 0
Since ½0; t0 Þ is the maximal interval of existence of ut in X0 ; we have u1 AX \X0 ¼ K: Now we consider the second case: t0 o þ N and p42: As a consequence of the Ho¨lder inequality we have Z
t2 t1
jjus Am ðus ÞjjX dsp
Z
t2
t1
!1=p jjus
Am ðus ÞjjpX
ds
ðt2 t1 Þ11=p :
Thus (4.10) and (5.1) imply Z
t2
t1
jjus Am ðus ÞjjX dspd4 ðJðut1 Þ Jðut2 ÞÞ1=p ðt2 t1 Þ11=p :
Then the same discussion as above leads to (5.8) for some u1 AK: The third case is: t0 ¼ þN and 1opp2: In this case, (5.6) implies the existence of an increasing sequence ftn g with tn - þ N such that lim
n-N
d Jðut Þjt¼tn ¼ 0: dt
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By (5.1), (4.1) and Lemma 5.2 we have d Jðut Þ ¼ /J 0 ðut Þ; ut þ Bm ðut ÞSX ;X dt p d5 jjut Am ðut Þjj2X ðjjut jjX þ jjAm ðut ÞjjX Þp2 p d6 jjut Am ðut Þjj2X ð1 þ jjut Am ðut ÞjjX Þp2 ; hence lim jjutn Am ðutn ÞjjX ¼ 0:
n-N
Now futn g is bounded in X by Lemma 5.2. Since Am : X -X is a compact operator it follows that (after passing to a subsequence if necessary) lim jjutn u1 jjX ¼ lim jjAm ðutn Þ u1 jjX ¼ 0
n-N
n-N
ð5:9Þ
for some u1 AK: It remains to consider the case t0 ¼ þN and p42: By (5.1) and (4.10), there exists an increasing sequence ftn g with tn - þ N such that 0pd7 jjutn Am ðutn ÞjjpX p This gives (5.9) and finishes the proof.
d Jðut Þjt¼tn -0: dt
&
Lemma 5.5. For the sequence ftn g and u1 AK as in Lemma 5.4 we have lim jjutn u1 jjY ¼ 0:
n-N
Proof. We first claim that fut : 0ptot0 g is bounded in X : Suppose that jjut jjX X2c7 for tA½t1 ; t2 C½0; t0 Þ with c7 ¼ c7 ð1Þ from Lemma 5.2. Then we have jjut Am ðut ÞjjX X1
for tA½t1 ; t2 :
If 1opp2 then (5.1), (4.2) and (5.10) imply jju u jjX p 2 t2
t1
Z
p d1
t2
t1
Z
jjus Am ðus ÞjjX ds
t2
t1
jjus Am ðus Þjj2X ð1 þ jjus Am ðus ÞjjX Þp2 ds:
ð5:10Þ
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Using Lemma 5.2 again we arrive at jjut2 ut1 jjX pd2
Z
t2
t1
jjus Am ðus Þjj2X ðjjus jjX þ jjAm ðus ÞjjX Þp2 ds:
From (5.1), (4.1) and (5.6) we deduce jjut2 ut1 jjX pd3 ðJðut1 Þ Jðut2 ÞÞpd4 : In the case p42 we have jju u jjX p2 t2
t1
Z
t2
t1
jjus Am ðus ÞjjpX dspd5 ðJðut1 Þ Jðut2 ÞÞpd6 :
Here we used (5.1), (4.2), (5.10), (4.10) and (5.6). In either case, we obtain that fut : 0ptot0 g is bounded in X : It follows from (5.1) that t
t
u ¼ e ðe þ s0 e1 Þ þ e
t
Z
t
es Bm ðus Þ ds
for 0ptot0 :
ð5:11Þ
0
Now Bm ðus Þ : ½0; t0 Þ-Z is continuous as a consequence of Lemmas 4.1 and 4.2. Since e þ s0 e1 AY CLN ðOÞ and since Bm : X0 -LN ðOÞ and Bm : ðX0 -LN ðOÞ; jj Rt jjN Þ-Z are bounded, fet 0 es Bm ðus Þ ds: 0ptot0 g is bounded in Z and hence relatively compact in Y : This fact and (5.11) imply that fut : 0ptot0 g is relatively compact in Y : Then we obtain the result of Lemma 5.5 immediately from Lemma 5.4. & By (5.5) and Lemma 5.5 we have u1 A@G-D1 : Clearly, u1 eintY D3 because u1 eG: Thus, u1 ¼ Am u1 and Am ðD3 ÞCintY D3 imply u1 eD3 : Therefore, u1 is a solution of (1.1) and u1 Xj;
u1 4 / c:
ð5:12Þ
u2 pc:
ð5:13Þ
Similarly, (1.1) has a solution u2 satisfying u2 5 / j;
According to [21], there exists u A@G-P such that fjt ðu Þ: 0ptotðu Þg-ðintY D1 ,intY D2 Þ ¼ |: Using a similar argument as before we find an increasing sequence ftn g with tn -tðu Þ and u3 AK such that lim jjjtn ðu Þ u3 jjY ¼ 0:
n-N
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Note that u3 eintY D1 ,intY D2 : This implies u3 eD1 ,D2 because u3 ¼ Am u3 and Am ðDi ÞCintY Di for i ¼ 1; 2: Consequently, u3 is a solution of (1.1) satisfying u3 5 / j;
ð5:14Þ
u3 4 / c:
Finally, considering a trajectory starting at a point in D3 -P we find a solution u4 of (1.1) such that fpu4 pc:
ð5:15Þ
Since u1 ; y; u4 are in the o-limit sets of trajectories with starting points in P; we see from (5.3) that Jðui Þp max JðuÞpb;
ð5:16Þ
uAP
where b depends only on m; c8 and c9 from (5.2). To summarize, we have proved the following. Proposition 5.6. Assume ðH001 Þ; ðH2 Þ and ðH3 Þ: Then (1.1) has four solutions u1 ; y; u4 satisfying (5.12)–(5.16), where b depends only on the numbers m; c8 and c9 from (5.2). Remark 5.7. (a) In the proof of Lemma 5.5 we used the boundedness of Bm : X0 -LN ðOÞ and of Bm : ðX0 -LN ðOÞ; jj jjN Þ-Z: This boundedness comes from the boundedness of Am : X -LN ðOÞ and of Am : ðX -LN ðOÞ; jj jjN Þ-Z as shown in the proof of Lemma 4.1. Here condition ðÞ enters; cf. Lemma 3.9. (b) The boundedness of the operators Am
Am
Am
Am
Lq0 ðOÞ ! ? ! Lqn ðOÞ ! LN ðOÞ ! Z
ð5:17Þ
with n large enough has been proved in Section 3. If the operators in (5.17) are continuous and Am : X -X and Am : Y -Y are locally Lipschitz continuous, then I þ Am can be used instead of I þ Bm to generate the descending flow jt ðuÞ and one can use a bootstrap argument as in the semilinear case p ¼ 2 and condition ðÞ can be avoided. But these continuity properties of Am are not yet available for pa2: (c) In order to locate the solutions inside or outside of order cones, we made use of the open positive invariant subset intY D3 of Y. Since intX D˜ 3 ¼ | it is impossible to place the above argument completely inside X even if D˜ 1 ; D˜ 2 and D˜ 3 are also positive invariant for the descending flow. This is why we need the Y -topology in our discussion, and why we have to prove the Y -convergence as in Lemma 5.5.
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6. Proof of Theorems 2.1, 2.2 and 2.3 In this section we assume ðH1 Þ ðH3 Þ and ðÞ: Without loss of generality, we can assume that p 1om 1ominfpp =N; p 1g: Choose q such that p 1om 1oqominfpp =N; p 1g; recall that p ¼ N if pXN: For nAN we define
fn ðx; tÞ ¼
8 < f ðx; tÞ
q jtj : f ðx; 7nÞ n
if jtjpn; if 7t4n:
The following lemma is an easy consequence of ðH1 Þ and ðH2 Þ: Lemma 6.1. (i) If 1opoN then fn ðx; tÞ
lim
jtj-N
jtjp
1
¼0
uniformly in n and x:
(ii) If p ¼ N then lim
jtj-N
ln j fn ðx; tÞj jtjN=ðN1Þ
¼0
uniformly in n and x:
(iii) For any nXM; xAO and jtjXM; 0oFn ðx; tÞ :¼
Z
t
0
1 fn ðx; sÞ dsp tfn ðx; tÞ: m
Here M40 and m42 are from ðH2 Þ: Proof of Theorem 2.1. For any n; consider the modified problem Dp u ¼ fn ðx; uÞ;
uAW01;p ðOÞ
ð6:1Þ
and its associated functional Jn ðuÞ ¼
1 p
Z
jrujp
Z
O
O
Fn ðx; uÞ for uAW01;p ðOÞ:
By Lemma 6.1(iii) there exist c8 40; c9 40 such that Fn ðx; tÞXc8 jtjm c9
for nAN; xAO; tAR:
ð6:2Þ
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Proposition 5.6 implies (6.1) has four solutions un;i ; i ¼ 1; y; 4; satisfying (5.12)– (5.15) and such that Jn ðun;i Þpb
for nAN; i ¼ 1; y; 4;
here b depends only on m; c8 and c9 from (6.2). Moreover, by Lemma 6.1(iii) there exists c10 40 such that 1 tfn ðx; tÞ Fn ðx; tÞX c10 m
for nAN; xAO; tAR:
From this we obtain bX Jn ðun;i Þ 1 ¼ Jn ðun;i Þ /Jn0 ðun;i Þ; un;i SX ;X m Z Z 1 1 1 p ¼ jrun;i j Fn ðx; un;i Þ un;i fn ðx; un;i Þ p m O m O Z 1 1 X jrun;i jp c10 jOj p m O and therefore jjun;i jjX pd1
for nAN; i ¼ 1; y; 4:
ð6:3Þ
Now we estimate the LN ðOÞ norm of un;i : First we consider the case 1opoN: By Lemma 6.1(i) for any e40 there exists d2 ðeÞ40 such that j fn ðx; tÞj jtjNþ1 oejtjp
þN
% tAR: þ d2 ðeÞ for nAN; xAO;
Using jun;i jN un;i as a testing function in (6.1), we get for nAN and i ¼ 1; y; 4: Z Z ðN þ 1Þpp jrðjun;i jN=p un;i Þjp pe jun;i jp þN þ d2 ðeÞjOj: p ðN þ pÞ O O The Sobolev inequality and Ho¨lder inequality now imply Z p=p ðN þ 1Þpp ðNþpÞp =p Sp jun;i j ðN þ pÞp O Z 1p=p Z p=p p ðNþpÞp =p pe jun;i j jun;i j þd2 ðeÞO O
peSp1p
=p
O
Z
jrun;i jp O
ðp =pÞ1 Z O
jun;i jðNþpÞp
=p
p=p þd2 ðeÞjOj;
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where Sp is the best constant of the Sobolev embedding W01;p ðOÞ+Lp ðOÞ: Since un;i ALr ðOÞ for all r41 by Lemma 3.1, using (6.3) we see that Nþp jjun;i jjNþp ðNþpÞp =p ped3 jjun;i jjðNþpÞp =p þ d4 ðeÞ
for nAN; i ¼ 1; y; 4:
Choosing e ¼ 1=2d3 ; we obtain jjun;i jjðNþpÞp =p pd5
for nAN; i ¼ 1; y; 4:
Then Lemma 6.1(i) implies jj fn ðx; un;i ÞjjðNþpÞ=p pd6
for nAN; i ¼ 1; y; 4:
As a consequence of Lemma 3.2 we have jjun;i jjN pd7
for nAN; i ¼ 1; y; 4:
Now suppose that p ¼ N: In view of [13, Theorem 7.15], we have N=ðN1Þ Z juj exp pc12 jOj for uAW01;N ðOÞ c11 jjrujjN O
ð6:4Þ
ð6:5Þ
for some positive constants c11 ; c12 : By Lemma 6.1(ii), for any e40 there exists d8 ðeÞ40 such that j fn ðx; tÞj2 pd8 ðeÞ expðejtjN=ðN1Þ Þ
% tAR: for nAN; xAO;
ð6:6Þ
Using (6.3), (6.5) and (6.6) and choosing e small enough, we obtain jj fn ðx; un;i Þjj22 pd9
for nAN; i ¼ 1; y; 4:
Again (6.4) follows from Lemma 3.2. Finally, if p4N then (6.4) is a direct consequence of (6.3). Thus, we have proved the uniform bound (6.4) for any p41: Hence, un;i ; i ¼ 1; y; 4; are solutions of (1.1) satisfying (5.12)–(5.15) provided n4d7 because then jjun;i jjon: & Proof of Theorem 2.2. According to Lemma 3.4, ðH3 Þ is satisfied with j ¼ d0 e1 and c ¼ d0 e1 for some d0 40: Let u1 ; u2 and u3 be the solutions of (1.1) obtained in Theorem 2.1 and satisfying (5.12)–(5.14). Then u3 is sign changing because u3 5 /j and u3 4 / c: Suppose that O1 :¼ fxAOju1 ðxÞo0ga|: Since u1 Xj the proof of Lemma 3.4 yields Dp u1 X ðl1 eÞju1 jp1
in O1 ;
u1 AW01;p ðO1 Þ:
This together with the Poincare inequality implies Z Z Z l1 ju1 jp p jru1 jp pðl1 eÞ O1
O1
O1
ju1 jp ;
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which is a contradiction. Thus u1 X0; and u1 a0 because u1 4 / c: It follows that u1 b0 by the strong maximum principle from [27]. Similarly, u2 50 is a negative solution. & Proof of Theorem 2.3. The existence of a sign changing solution is a direct consequence of Theorem 2.1 and Lemma 3.5. A positive solution is obtained from Theorem 2.1 and Lemma 3.5 by replacing f with f˜ such that ¼ f ðx; tÞ ˜ fðx; tÞ 40
if teðt1 ; 0Þ; if tAðt1 ; 0Þ:
A negative solution can be obtained similarly.
&
Note added in proof As conjectured in Remark 2.5 the condition ðÞ is not necessary. This will be published in a joint paper with Tobias Weth.
References [1] T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001) 117–152. [2] T. Bartsch, K.C. Chang, Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z. 233 (2000) 655–677. [3] T. Bartsch, Z.L. Liu, Location and Morse indices of critical points with an application to nonlinear eigenvalue problems, preprint. [4] T. Bartsch, Z.-Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Top. Meth. Nonlinear Anal. 7 (1996) 115–131. [5] V. Benci, P. d’Avenia, D. Fortunato, L. Pisani, Solitons in several space dimensions: Derrick’s problem and infinitely many solutions, Arch. Rational Mech. Anal. 154 (2000) 297–324. [6] A. Castro, J. Cossio, J.M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997) 1041–1053. [7] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincare´. Anal. Non Line´aire 15 (1998) 493–516. [8] E.N. Dancer, Y. Du, Multiple solutions of some semilinear elliptic equations via the generalized Conley index, J. Math. Anal. Appl. 189 (1995) 848–871. [9] D.G. de Figueiredo, P.L. Lions, R.D. Nussbaum, A priori estimates and existence results for positive solutions of semilinear elliptic equations, J. Math. Pure Appl. 61 (1981) 41–63. [10] E. Di Benedetto, C 1þa local regularity of weak solutions to degenerate elliptic equations, Nonlinear Anal. 7 (1983) 827–850. [11] G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Portugal. Math. 58 (2001) 339–378. [12] J. Garci´a-Melia´n, J. Sabina de Lis, Maximum and comparison principles for operators involving the p-Laplacian, J. Math. Anal. Appl. 218 (1998) 49–65. [13] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Springer, Berlin, 1998.
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[14] Z.M. Guo, J.R.L. Webb, Large and small solutions of a class of quasilinear elliptic eigenvalue problems, J. Differential Equations 180 (2002) 1–50. [15] S.J. Li, Z.L. Liu, Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents, preprint. [16] S.J. Li, Z.-Q. Wang, Mountain pass theorems in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math. 81 (2000) 373–396. [17] S.J. Li, Z.-Q. Wang, Ljusternik–Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc. 354 (2002) 3207–3227. [18] G.M. Liebermann, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988) 1203–1219. [19] P. Lindqvist, On nonlinear Rayleigh quotients, Potential Anal. 2 (1993) 199–218. [20] P.L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Review 24 (1982) 441–467. [21] Z.L. Liu, J.X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations 172 (2001) 257–299. [22] P. Pucci, J. Serrin, A note on the strong maximum principle for elliptic differential equations, J. Math. Pures Appl. 79 (2000) 57–71. [23] P. Pucci, J. Serrin, H. Zou, A strong maximum principle and a compact support principle for singular elliptic inequalities, J. Math. Pures Appl. 78 (1999) 769–789. [24] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Vol. 65, American Mathematical Society, Providence, RI 1986. [25] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8 (1983) 773–817. [26] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984) 126–150. [27] J.L. Va´zquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984) 191–202. [28] Z.-Q. Wang, On a semilinear elliptic equation, Ann. Inst. H. Poincare´. Anal. Non Line´aire 8 (1991) 43–57.
J. Differential Equations 198 (2004) 149–175
On a superlinear elliptic p-Laplacian equation Thomas Bartscha, and Zhaoli Liub,1 b
a Mathematisches Institut, Universita¨t Giessen, Arndtstr. 2, 35392 Giessen, Germany Department of Mathematics, Shandong University, Jinan, Shandong 250100, PR China
Received February 26, 2003
Abstract We prove the existence of four solutions for the p-Laplacian equation Dp u ¼ f ðx; uÞ;
uAW01;p ðOÞ;
on a bounded domain OCRN with smooth boundary @O provided that the nonlinearity f is superlinear and subcritical and that the equation has a pair of subsolution and supersolution. In addition to the existence of a positive and a negative solution, the existence of a sign changing solution is proved. We present a new approach to obtain existence and nodal properties of solutions of p-Laplacian equations. r 2003 Elsevier Inc. All rights reserved. Keywords: p-Laplace operator; Quasilinear Dirichlet problem; Sign changing solution
1. Introduction We consider the quasilinear elliptic equation Dp u ¼ f ðx; uÞ;
uAW01;p ðOÞ;
ð1:1Þ
on a bounded domain OCRN with smooth boundary @O; here Dp u ¼ divðjrujp2 ruÞ % R-R is continuous. The p-Laplacian is the p-Laplacian, p41; and f : O equation (1.1) arises naturally in various contexts of physics, for instance, in the
Corresponding author. Fax: +49-641-9932179. E-mail address: [email protected] (T. Bartsch). 1 Supported by the Alexander von Humboldt foundation in Germany and FUKTME in China. 0022-0396/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2003.08.001
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study of non-Newtonian fluids (the case of a Newtonian fluid corresponding to p ¼ 2), and in the study of nonlinear elasticity problems. It also appears in the search for solitons of certain Lorentz-invariant nonlinear field equations. We refer to [5,14,19] for details and further references. The goal of this paper is to find solutions of (1.1) inside or outside of certain order cones. In particular, we shall prove the existence of a sign changing solution when the nonlinearity f is superlinear and subcritical as juj-N: Results of this type are quite recent even in the semilinear case p ¼ 2: In this case, the mountain pass theorem yields a positive and a negative solution. The existence of a third nontrivial solution is due to Wang [28]. The question of the nodal properties of the third solution was left open in [28]. It was settled by several authors, see e.g. [1,2,4,6,8,16,17,21]. In these papers, one can also find various generalizations of Wang’s result and different methods of proof. A direct extension of these methods to the quasilinear case pa2 is faced with serious difficulties. First of all, the energy functional associated to (1.1) is defined on W01;p ðOÞ which is not a Hilbert space for pa2: In addition, for po2 the energy is only of class C1 so one has to construct an appropriate pseudo-gradient vector field. This needs more care as usual when one is interested in nodal properties of the solutions. Another important ingredient is the generalized Morse lemma which is based on the linearization of (1.1), or the computation of the Hessian of the energy at a critical point. This is needed in order to construct a linking and to obtain the third solution via Lusternik–Schnirelmann methods. It is also needed when one wants to compute critical groups and apply Morse-type arguments. A generalized Morse lemma is not available for pa2: Since it requires the functional to be of class C2 it cannot possibly be extended to the case po2: A third difficulty is the lack of a powerful regularity theory if pa2: For the Laplace operator there exists a sequence % and such of Banach spaces E0 +E1 +?+En with W01;2 ðOÞ+En and E0 +C1 ðOÞ; 1 that the nonlinear operators Ek -Ek1 ; u/ðD þ mÞ ðf ð:; uÞ þ muÞ; where m40; are continuous and map bounded sets to bounded sets. There is no similar regularity theory for the corresponding operator in the case pa2: Finally, an essential ingredient in the earlier approaches is the strongly order preserving property of the operator ðD þ mÞ1 ; i.e. u1 ou2 in O implies v1 :¼ ðD þ mÞ1 ðu1 Þov2 :¼ ðD þ mÞ1 ðu2 Þ in O; and @n v1 4@n v2 on @O; here n is the exterior normal to O: This property is a consequence of the strong comparison principle which holds for D; but which does not hold for Dp : In [3], we developed some critical point theory for C1 -functionals on partially ordered Banach spaces which allows to localize critical points. This theory will now be adapted in order to obtain existence results for solutions of (1.1) together with nodal information on the solutions. In particular, we generalize the above-mentioned results from [1,2,4,6,8,16,17,21] on the existence of a sign changing solution to the case pa2: In fact, technically several cases have to be distinguished, namely the cases poN; ¼ N; 4N: In addition, the critical point theorems from [3] distinguish between po2 and 42: We would like to mention that
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already the existence statements in our theorems are new for pa2: Moreover, the methods developed here and in [3] are different from earlier methods even in the case p ¼ 2:
2. Statement of results In order to keep the paper within reasonable bounds we shall not present the most general version of our results, but indicate possible extensions without proofs (cf. Remark 2.5). % : uj@O ¼ 0g: For u; vAY we say u5v if and Let X :¼ W01;p ðOÞ and Y :¼ fuAC1 ðOÞ only if uðxÞovðxÞ for xAO and @uðxÞ=@n4@vðxÞ=@n for xA@O; here n is the outer unit normal to @O: Define hp ðtÞ :¼ jtjp2 t: For m40 we need the operator Am : X -X ;
Am ðuÞ ¼ ðDp þ mhp ð ÞÞ1 ððx; uÞ þ mhp ðuÞÞ:
Finally, we set p :¼ Np=ðN pÞ for poN; and p :¼ N for pXN: Now we assume: ðH1 Þ If 1opoN then f ðx; tÞ
lim
jtj-N
jtjp
1
¼0
uniformly in x;
if p ¼ N then lim
jtj-N
lnj f ðx; tÞj jtjN=ðN1Þ
¼0
uniformly in x:
ðH2 Þ There exist m4p and M40 such that 0oF ðx; tÞ :¼
Z
t 0
1 f ðx; sÞ dsp tf ðx; tÞ m
for jtjXM:
ðH3 Þ There exist m40 and j; cAY with jpc such that uALN ðOÞ; uXj ) Am ðuÞbj and uALN ðOÞ; upc ) Am ðuÞ5c: In dimensions NX6; we also require the following technical condition on p: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðÞ If NX6 then either 1opoðN þ 1 N 2 6N þ 1Þ=4 or p4ðN þ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 2 6N þ 1Þ=4:
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The main result of this paper is Theorem 2.1. Assume ðH1 Þ–ðH3 Þ and ðÞ: Then (1.1) has four solutions u1 ; y; u4 with the properties u1 Xj;
u1 4 / c;
/ j; u2 5
u2 pc;
/ j; u3 5
u3 4 / c;
u4 Xj;
u4 pc:
There are two important cases in which ðH3 Þ holds. ðH03 Þ f ðx; 0Þ ¼ 0; and there exists m40 such that f ðx; tÞ þ mhp ðtÞ is increasing in t; and lim sup t-0
j f ðx; tÞj ol1 jtjp1
uniformly in x;
here l1 is the first eigenvalue of: Dp u ¼ lhp ðuÞ; uAX ¼ W01;p ðOÞ: ðH003 Þ There exist t1 o0ot2 such that f ðx; t1 Þ404f ðx; t2 Þ and there exists m40 such that f ðx; tÞ þ mhp ðtÞ is increasing in t: In the situation of ðH03 Þ; 0 is a solution of (1.1). In this case, we can get nodal properties of the three nontrivial solutions. Theorem 2.2. Assume ðH1 Þ; ðH2 Þ; ðH03 Þ and ðÞ: Then (1.1) has a positive solution, a negative solution and a sign changing solution. We have a similar result in the case of ðH003 Þ: Theorem 2.3. Assume ðH1 Þ; ðH2 Þ; ðH003 Þ and ðÞ: Then (1.1) has a sign changing solution. Moreover, if f ðx; 0Þ ¼ 0 then (1.1) also has a positive solution and a negative solution. Remark 2.4. (a) Condition ðH1 Þ means that the nonlinearity f is subcritical and ðH2 Þ means that f is ‘‘superlinear’’. These two conditions enable us to use a variational approach for the study of (1.1); they also provide compactness in some sense. As in the case p ¼ 2 without ðH1 Þ the above theorems may not be true. This can be seen from Pohozaev’s identity for p-Laplacian equations: If u is a solution of Dp u ¼ f ðuÞ;
uAW01;p ðOÞ;
ð2:1Þ
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then we obtain the Pohozaev identity Z Z Z N p p1 F ðuÞ dx f ðuÞu dx ¼ ðx nÞjrujp ds: N p p O O @O This identity can be proved by integrating (2.1) multiplied with x ru; and using the formulas 1 N p p2 p jrujp div jruj ðx ruÞru jruj x ¼ ðx ruÞDp u p p and divðF ðuÞxÞ ¼ ðx ruÞf ðuÞ þ NF ðuÞ: From the Pohozaev identity one sees that if O is star shaped, 1opoN; and f ðuÞ ¼ jujq2 u with qXp then (2.1) has only the trivial solution u ¼ 0: Condition ðH2 Þ corresponds to the standard superlinearity condition of Ambrosetti–Rabinowitz in the case p ¼ 2: The existence of positive solutions in this case has been proved in [9,20]. Condition ðH3 Þ provides a pair of subsolution and supersolution and is essential in order to obtain the solutions u1 and u2 in Theorem 2.1, or the positive and the negative solutions in Theorems 2.2, 2.3. We believe that there should be a sign changing solution without ðH3 Þ but do not know how to prove this. (b) In [3] we developed methods for the general case p41 which can be applied to asymptotically sublinear nonlinearities. Remark 2.5. (a) The assumption in ðH03 Þ and ðH003 Þ that f ðx; tÞ þ mhp ðtÞ is increasing in t for some m40 is not essential but is assumed for simplicity. If such m does not exist, then we approximate f by a sequence of functions so that m as above exists, and obtain the solutions by passing to limits. The details will not be given here. (b) Similarly, the inequality f ðx; t1 Þ404f ðx; t2 Þ in ðH003 Þ may be replaced with the weaker one: f ðx; t1 ÞX0Xf ðx; t2 Þ: This can be proved via an approximation argument starting from Theorem 2.3. The details will again be left to the interested reader. (c) Condition ðÞ is a technical condition which can probably be avoided. We will indicate where this inequality is used in the proof; cf. Remark 5.7. Observe that in the case NX6 our results do not contain the case p ¼ 2: Throughout this paper, we will use ci and di to represent various positive constants. jj jjq denotes the standard norm in Lq ðOÞ for 1pqp þ N and jj jj1;p the standard norm in X :¼ W01;p ðOÞ: We shall also use the norm jjujjX :¼
Z
ðjrujp þ mjujp Þ O
1=p ;
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where m is from hypothesis ðH3 Þ: This is, of course, equivalent to the usual norm in X : 3. Properties of the operator Am In this section, we study properties of Am defined in Section 2. We shall need the following stronger version of ðH1 Þ: ðH01 Þ There exist c40 and p 1oqop 1 such that j f ðx; tÞjpcð1 þ jtjq Þ: Throughout this section we assume that ðH01 Þ and ðH3 Þ hold. ðH2 Þ is not required here. Solutions of (1.1) correspond to critical points of the functional J : X -R defined by Z Z 1 jrujp F ðx; uÞ JðuÞ :¼ p O O and also to fixed points of the operator Am : X -X : Note that J is of class C1 and Am is compact, i.e. it is continuous and maps bounded subsets of X into relatively compact subsets of X ; cf. [11]. If pXN and qX1 then X +Lq ðOÞ and Am : X -Lq ðOÞ are compact operators. For 1opoN; we have P Lemma 3.1. Let 1opoN and define s0 :¼ p q 1 and qn :¼ p þ s0 ni¼1 ðpp Þi ; nAN0 : Then Am : Lqn ðOÞ-Lqnþ1 ðOÞ is bounded, i.e. it maps bounded subsets of Lqn ðOÞ into bounded subsets of Lqnþ1 ðOÞ; nAN0 : Proof. Consider uALq0 ðOÞ and set v ¼ Am ðuÞAX so that Dp v þ mhp ðvÞ ¼ f ðx; uÞ þ mhp ðuÞ: Using jvjs0 v as a testing function, (3.1), ðH01 Þ; and the Ho¨lder inequality imply Z Z ðs0 þ 1Þpp s0 =p p jrðjvj vÞj þ m jvjpþs0 ðs0 þ pÞp O O Z Z ¼ ðs0 þ 1Þ jvjs0 jrvjp þ m jvjpþs0 O O Z ¼ ð f ðx; uÞjvjs0 v þ mjujp2 ujvjs0 vÞ O Z p ðcð1 þ jujq Þ þ mjujp1 Þjvjs0 þ1 O Z pc1 ðjujq þ 1Þjvjs0 þ1 O s þ1
pc2 ðjjujjqp þ 1Þjjvjjp0 :
ð3:1Þ
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This together with the Sobolev inequality gives s þp q pc3 ðjjujjp 0 þpÞp =p
jjvjjðs0
s þ1
þ 1Þjjvjjp0
and therefore jjvjjq1 pc4 ðjjujjq=ðp1Þ þ 1Þ: q0 Thus the result is true for n ¼ 0: Now suppose that Am : Lqn1 ðOÞ-Lqn ðOÞ is bounded and consider uALqn ðOÞ; v ¼ Am ðuÞ: Denoting n i X p sn :¼ qn ðq þ 1Þ ¼ s0 p i¼0 and using jvjsn v as a testing function in (3.1), we obtain Z Z ðsn þ 1Þpp sn =p p jrðjvj vÞj p c1 ðjujq þ 1Þjvjsn þ1 ðsn þ pÞp O O p c5 ðjjujjqqn þ 1Þjjvjjsqnnþ1 : Therefore, sn þp q sn þ1 jjvjjðs ; pc6 ðjjujjqn þ 1Þjjvjjqn n þpÞp =p
which implies jjvjjqnþ1 pc7 ðjjujjq=ðp1Þ þ 1Þ: qn By induction, Am : Lqn ðOÞ-Lqnþ1 ðOÞ is bounded for every nAN0 :
&
As a consequence of Lemma 3.1, any solution u of (1.1) lies in Lr ðOÞ; for any r41: We will see that u has much stronger regularity. For this purpose, we need the next lemma which was proved in [15] by a similar argument as in the proof of [13, Theorem 8.15]. Lemma 3.2. If 1oppN and r4Np then ðDp þ mhp ð ÞÞ1 maps Lr ðOÞ into LN ðOÞ and there exists a constant c40 such that jjðDp þ mhp ð ÞÞ1 ujjN pcjjujj1=ðp1Þ r
for every uALr ðOÞ:
Lemma 3.2 implies that Am : Lr ðOÞ-LN ðOÞ is bounded if 1oppN and r4qN=p: Observe that if p4N then X +LN ðOÞ: Consequently, any solution of (1.1) is in
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LN ðOÞ: A stronger regularity of solutions of (1.1) is implied by the next lemma of Liebermann [18]. Lemma 3.3. There exist 0oao1 and c40 such that jjAm ðuÞjjC 1;a pcðjjujjq=ðp1Þ þ 1Þ N
for uALN ðOÞ;
hence Am : LN ðOÞ-C01;a ðOÞ is bounded. Proof. According to [18], there exist 0oao1 and c1 40 such that jjðDp þ mhp ð ÞÞ1 ujjC1;a pc1 jjujj1=ðp1Þ N
for uALN ðOÞ:
This yields pc2 ðjjujjq=ðp1Þ þ 1Þ jjAm ðuÞjjC1;a pc1 jj f ðx; uÞ þ mhp ðuÞjj1=ðp1Þ N N for uALN ðOÞ; as claimed.
&
% for some 0oao1 for any solution u of (1.1). We By Lemma 3.3, uAC 1;a ðOÞ N remark that for uAL ðOÞ; local C1;a regularity of Am ðuÞ was obtained in [10,26] and boundary C1;a regularity was proved in [18]. Having the regularity of Am as in Lemma 3.3, we now show that ðH3 Þ is satisfied if either ðH03 Þ or ðH003 Þ holds. Lemma 3.4. ðH03 Þ implies ðH3 Þ: Proof. Let e1 40 be an eigenfunction of Dp in W01;p ðOÞ corresponding to the first % eigenvalue l1 40: Then e1 AC1;a 0 ðOÞ and e1 b0 as a consequence of the strong maximum principle from [27] (see also [12,22,23]). By ðH03 Þ we may choose e40 and t0 40 such that ðl1 eÞjtjp1 pf ðx; tÞpðl1 eÞjtjp1
for jtjpt0 :
Fixing d0 40 with d0 jje1 jjN ot0 we set j :¼ d0 e1 and c :¼ d0 e1 : Now we consider uALN ðOÞ with uXj: Then we have for v ¼ Am ðuÞ Dp v þ mhp ðvÞ ¼ f ðx; uÞ þ mhp ðuÞ X f ðx; d0 e1 Þ þ mhp ðd0 e1 Þ p1 p1 mdp1 X ðl1 eÞdp1 0 e1 0 e1
X
l1 þ m e ðDp ðd0 e1 Þ þ mhp ðd0 e1 ÞÞ: l1 þ m
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As a consequence of the weak comparison principle [7,12,25] we obtain vX d1 d0 e1 ; Þ1=ðp1Þ : Since 0od1 o1; we have vb d0 e1 ¼ j: where d1 ¼ ðl1lþme 1 þm Similarly, if uALN ðOÞ satisfies upc then Am ðuÞ5c: & Lemma 3.5. ðH003 Þ implies ðH3 Þ: Proof. Let f ¼ w1 and c ¼ w2 be the unique solutions of Dp wi þ mhp ðwi Þ ¼ mhp ðti Þ;
wi AX :
The strong maximum principle [27] implies j505c; jXt1 ; and cpt2 : Consider uALN ðOÞ with uXj and set v ¼ Am ðuÞ: Since f ðx; t1 Þ40 we have Dp v þ mhp ðvÞ ¼ f ðx; uÞ þ mhp ðuÞ X f ðx; t1 Þ þ mhp ðt1 Þ X dðDp j þ mhp ðjÞÞ for some 0odo1: Then vXd1=ðp1Þ jbj by the weak comparison principle. Similarly, if uALN ðOÞ and upc then Am ðuÞ5c: & In order to set up our variational framework we now discuss the relation between J 0 : X -X and Am : X -X : For doing this we need the following result from [7]. Lemma 3.6. There exist positive constants c1 ; y; c4 such that for all x; ZARN jjxjp2 x jZjp2 Zjpc1 ðjxj þ jZjÞp2 jx Zj;
ð3:2Þ
ðjxjp2 x jZjp2 ZÞ ðx ZÞXc2 ðjxj þ jZjÞp2 jx Zj2 ;
ð3:3Þ
jjxjp2 x jZjp2 Zjpc3 jx Zjp1
if 1opp2;
ðjxjp2 x jZjp2 ZÞ ðx ZÞXc4 jx Zjp
if p42:
ð3:4Þ ð3:5Þ
Lemma 3.7. There exists a constant c5 40 such that, if 1opp2 then /J 0 ðuÞ; u Am ðuÞSX ;X Xc5 jju Am ðuÞjj2X ðjjujjX þ jjAm ðuÞjjX Þp2
ð3:6Þ
and if p42 then /J 0 ðuÞ; u Am ðuÞSX ;X Xc5 jju Am ðuÞjjpX hold for all uAX :
ð3:7Þ
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158
Proof. Let uAX and set v ¼ Am ðuÞ: Then Z Z p2 0 jruj ru rðu vÞ f ðx; uÞðu vÞ /J ðuÞ; u vSX ;X ¼ O O Z ¼ ðjrujp2 ru jrvjp2 rvÞðru rvÞ O Z þ mðhp ðuÞ hp ðvÞÞðu vÞ O Z X ðjrujp2 ru jrvjp2 rvÞðru rvÞ: O
If p42; it follows from (3.5) that /J 0 ðuÞ; u vSX ;X Xdjju vjjpX : Now consider the case 1opp2: Then (3.3) yields Z /J 0 ðuÞ; u vSX ;X Xc2 ðjruj þ jrvjÞp2 jru rvj2 : O
The Ho¨lder inequality implies Z p jju vjjX p d1 jru rvjp ðjruj þ jrvjÞpðp2Þ=2 ðjruj þ jrvjÞpð2pÞ=2 O
p d1
Z
2
jru rvj ðjruj þ jrvjÞ
O
p d2
Z
Z
ðjruj þ jrvjÞp
p2
1p=2
O 2
p=2
jru rvj ðjruj þ jrvjÞ
p2
O
p=2
ðjjujjX þ jjvjjX Þpð1p=2Þ :
and therefore /J 0 ðuÞ; u vSX ;X Xd3 jju vjj2X ðjjujjX þ jjvjjX Þp2 :
&
Lemma 3.8. There exists a constant c6 40 such that jjJ 0 ðuÞjjX pc6 jju Am ðuÞjjp1 X
ð3:8Þ
if 1opp2
and jjJ 0 ðuÞjjX pc6 jju Am ðuÞjjX ðjjujjX þ jjAm ðuÞjjX Þp2 hold for all uAX :
if p42
ð3:9Þ
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159
p Proof. Let uAX and v ¼ Am ðuÞ: As usual we denote by p0 ¼ p1 the dual exponent. For wAX we have Z /J 0 ðuÞ; wSX ;X ¼ ðjrujp2 ru jrvjp2 rvÞ rw þ mðhp ðuÞ hp ðvÞÞw: O
The Ho¨lder inequality now yields /J ðuÞ; wSX ;X p 0
Z
jjruj
p2
ru jrvj
p2
rvj
1=p0 Z
p0
O
þ m
p
1=p
jrwj
O
Z
jhp ðuÞ hp ðvÞjp
0
1=p0 Z
O
jwjp
1=p :
O
Therefore, jjJ 0 ðuÞjjX p d1
Z
0
jjrujp2 ru jrvjp2 rvjp
1=p0
O
þ d1
Z
jhp ðuÞ hp ðvÞj
p0
1=p0 :
O
If 1opp2; then jjJ 0 ðuÞjjX p d2
Z
jru rvjðp1Þp
0
1=p0 þd2
Z
O
ju vjðp1Þp
0
1=p0
O
p d3 jju vjjp1 X as a consequence of (3.4). If p42; then (3.2) and the Ho¨lder inequality imply 0
jjJ ðuÞjjX p d4
Z
ðjruj þ jrvjÞ
ðp2Þp0
p0
1=p0
jru rvj
O
þ d4
Z
0
1=p0
0
ðjuj þ jvjÞðp2Þp ju vjp
O
p d4
Z
ðjruj þ jrvjÞ
p
1=p0 1=p Z
O
þ d4
jru rvjp
O
Z
p
ðjuj þ jvjÞ
1=p0 1=p Z
O
p d5 ðjjujjX þ jjvjjX Þp2 jju vjjX :
p
ju vj
O
&
1=p
1=p
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160
For the last lemma of this section, we make the following assumption which is stronger than ðH01 Þ: ðH001 Þ There exist c40 and q with p 1oqominfpp =N; p 1g such that j f ðx; tÞjpcð1 þ jtjq Þ: Observe that p 1opp =N is equivalent to hypothesis ðÞ; hence ðH001 Þ can only be satisfied if ðÞ holds. Lemma 3.9. If ðH001 Þ holds then Am : X -LN ðOÞ is bounded. Proof. First we consider the case 1opoN: Observe that p =q4N=p: Lemma 3.2 implies that ðDp þ mhp ð ÞÞ1 : Lp =q ðOÞ-LN ðOÞ is bounded. On the other hand, f ðx; Þ : X +Lp ðOÞ-Lp =q ðOÞ is also bounded. Therefore, Am : X -LN ðOÞ is bounded. Now we assume p ¼ N and choose r4N=p: In this case the result is a consequence of the boundedness of f ðx; Þ : W01;N ðOÞ+Lrq ðOÞ-Lr ðOÞ; and the boundedness of ðDp þ mhp ð ÞÞ1 : Lr ðOÞ-LN ðOÞ: In the case p4N the result is obvious since we have a continuous embedding X +LN ðOÞ: &
4. Constructing a vector field Bm In this section, we replace Am by a locally Lipschitz continuous operator Bm which serves as a pseudo-gradient vector field for J: The usual conditions for a pseudogradient vector field and the construction as in the appendix of [24] need to be modified for pa2: Throughout the section we will always assume ðH001 Þ and ðH3 Þ: In addition to the % we also need the Ho¨lder space Z :¼ spaces X ¼ W01;p ðOÞ and Y ¼ X -C1 ðOÞ; 1;a % Y -C ðOÞ: With f and c from ðH3 Þ we define D1 :¼ fuAY : uXjg; D2 :¼ fuAY : upcg; and D3 :¼ D1 -D2 : Let D˜ i be the closure of Di in X and set K :¼ fuAX : J 0 ðuÞ ¼ 0g; X0 :¼ X \K: Note that KCZ by the regularity results from Section 3. Lemma 4.1. Let 1opp2: There exists a locally Lipschitz continuous operator Bm : X0 -Z with the following properties: (i) For uAX0 and c5 from Lemma 3.7 we have /J 0 ðuÞ; u Bm ðuÞSX ;X X
c5 jju Am ðuÞjj2X ðjjujjX þ jjAm ðuÞjjX Þp2 2
ð4:1Þ
and 1 2jju
Bm ðuÞjjX pjju Am ðuÞjjX p2jju Bm ðuÞjjX :
ð4:2Þ
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161
(ii) Bm ðD˜ i -X0 ÞCintY Di for i ¼ 1; 2; 3: (iii) Bm : X0 -LN ðOÞ is bounded. (iv) Bm : ðX0 -LN ðOÞ; jj jjN Þ-Z is bounded. Proof. Let c5 and c6 be as in (3.6) and (3.8), respectively. For any uAX0 ; setting D1 ðuÞ ¼ 12jju Am ðuÞjjX
ð4:3Þ
p2 D2 ðuÞ ¼ 2cc5 jju Am ðuÞjj3p ; X ðjjujjX þ jjAm ðuÞjjX Þ
ð4:4Þ
and 6
we choose gðuÞAð0; 1Þ such that jjAm ðvÞ Am ðwÞjjX ominfD1 ðvÞ; D1 ðwÞ; D2 ðvÞ; D2 ðwÞg
ð4:5Þ
holds for every v; wANðuÞ :¼ fvAX : jjv ujjX ogðuÞg: Let V be a locally finite open refinement of fNðuÞ: uAX0 g: We need to refine V in order to construct the required operator Bm : For any subset UCX we define IU :¼ fiAf1; 2; 3g: U-D˜ i a|g and [
W :¼
[
fV g,
V AV;IV af1;2g
fV \D˜ 1 ; V \D˜ 2 g:
V AV;IV ¼f1;2g
Clearly, W is a locally finite open refinement of V: We need to refine W once more. If IW ¼ fig with i ¼ 1 or 2 and if inf
uAW -LN ðOÞ
jjujjN o
inf
uAW -D˜ i -LN ðOÞ
jjujjN 1 ¼: bW
for some W AW; then we replace W in the covering W by the following two open subsets of X0 : W \fuALN ðOÞ: jjujjN pbW g
and
W \D˜ i ;
otherwise W is left unchanged. The new open covering U obtained in this way from W is a locally finite open refinement of W; hence of V and fNðuÞ: uAX0 g; and in addition, any UAU satisfies: U-D˜ 1 a|aU-D˜ 2
implies U-D˜ 3 a|
ð4:6Þ
and inf
uAU-LN ðOÞ
jjujjN X
inf
uAU-D˜ i -LN ðOÞ
jjujjN 1
if IU ¼ fig with i ¼ 1 or 2:
ð4:7Þ
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162
Now we are ready to construct the operator Bm : Let fpU : UAUg be the standard partition of unity subordinated to U which is defined by X
pU ðuÞ :¼
!1 aV ðuÞ
aU ðuÞ;
V AU
where aU ðuÞ :¼ distX ðu; X0 \UÞ: For UAU we choose aU such that aU AU-LN ðOÞ and
jjaU jjN o
inf
uAU-LN ðOÞ
jjujjN þ 1
ð4:8Þ
if IU ¼ |; and aU AU-
\
D˜ i -LN ðOÞ and jjaU jjN o
iAIU
uAU-
T inf ˜ iAIU
Di -LN ðOÞ
jjujjN þ 1
ð4:9Þ
if IU a|: We define Bm : X0 -Z by Bm ðuÞ :¼
X
pU ðuÞAm ðaU Þ:
UAU
That Bm : X0 -Z is locally Lipschitz continuous is a consequence of the Lipschitz continuity of pU ; the locally finiteness of the covering U; and the fact that Am ðLN ðOÞÞCZ: In order to see (i) we observe that jjBm ðuÞ Am ðuÞjjX p
X
pU ðuÞjjAm ðaU Þ Am ðuÞjjX
for uAX0 :
UAU
This together with (4.3)–(4.5) implies for uAX0 ; 1 jjBm ðuÞ Am ðuÞjjX o jju Am ðuÞjjX 2 and c5 p2 jju Am ðuÞjj3p : jjBm ðuÞ Am ðuÞjjX o X ðjjujjX þ jjAm ðuÞjjX Þ 2c6
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163
As an immediate consequence we have (4.2), while (3.6) and (3.8) imply, for uAX0 ; /J 0 ðuÞ; u Bm ðuÞSX ;X X/J 0 ðuÞ; u Am ðuÞSX ;X jjJ 0 ðuÞjjX jjBm ðuÞ Am ðuÞjjX c5 X jju Am ðuÞjj2X ðjjujjX þ jjAm ðuÞjjX Þp2 : 2 This proves (4.1). If uAD˜ i -X0 then uAD˜ i -U for any U with pU ðuÞa0: From the construction it follows that aU AU-D˜ i -LN ðOÞ for any U with pU ðuÞa0: This implies Bm ðuÞAconv Am ðD˜ i -LN ðOÞÞ; so (ii) follows from ðH3 Þ: In order to prove (iii) we suppose uAX0 and jjujjX pd1 : If pU ðuÞa0 then uAU: Since u; aU AUCNðvÞ for some vAX0 ; and since gðvÞo1 we have jjaU ujjX pjjaU vjjX þ jju vjjX o2 and thus jjaU jjX pd1 þ 2: Then jjAm ðaU ÞjjN pd2 for any UAU with pU ðuÞa0 by Lemma 3.9. Therefore, jjBm ðuÞjjN p
X
pU ðuÞjjAm ðaU ÞjjN pd2 :
UAU
It remains to prove (iv). Suppose uAX0 -LN ðOÞ and jjujjN pd3 : If pU ðuÞa0 then uAU-LN ðOÞ: By (4.8), in the case IU ¼ |; jjaU jjN o
inf
uAU-LN ðOÞ
jjujjN þ 1pd3 þ 1:
In the case IU ¼ fig with i ¼ 1 or 2, (4.7) and (4.9) imply jjaU jjN o
inf
uAU-D˜ i -LN ðOÞ
jjujjN þ 1p
inf
uAU-LN ðOÞ
jjujjN þ 2pd3 þ 2:
If IU a| and IU af1g; f2g; then (4.6) implies IU ¼ f1; 2; 3g; hence (4.9) implies aU AU-D˜ 3 : Since D˜ 3 is bounded in LN ðOÞ; we have jjaU jjN pd4 : In any case, Lemma 3.3 implies jjAm ðaU ÞjjZ pd5 for any UAU with pU ðuÞa0 and jjBm ðuÞjjZ p
X UAU
pU ðuÞjjAm ðaU ÞjjZ pd5 :
&
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In a similar way, we have Lemma 4.2. Let p42: Then the same results as in Lemma 4.1 hold except that (4.1) is now replaced with c5 /J 0 ðuÞ; u Bm ðuÞSX ;X X jju Am ðuÞjjpX ð4:10Þ 2 for uAX0 : 5. A descending flow argument In this section, we will prove Theorem 2.1 provided that the stronger assumptions ðH001 Þ; ðH2 Þ and ðH3 Þ are satisfied. A complete proof will be given in the last section. As mentioned before Lemma 3.9, ðH001 Þ can only be satisfied if ðÞ holds. The proof proceeds via a series of lemmas using a descending flow argument. For uAY0 :¼ Y \KCX0 we consider the following initial value problem, both in X0 and Y0 : jðtÞ ¼ jðtÞ þ Bm ðjðtÞÞ; ’ ð5:1Þ jð0Þ ¼ u: Since Bm : X0 -Z is locally Lipschitz continuous, the solution of (5.1) considered in X0 and the solution of (5.1) considered in Y0 are exactly the same. We denote it by jt ðuÞ with maximal interval of existence ½0; tðuÞÞ: As a consequence of Lemmas 4.1 and 4.2, Jðjt ðuÞÞ is strictly decreasing in tA½0; tðuÞÞ: Lemma 5.1. If uADi \K; then jt ðuÞAintY Di for 0ototðuÞ: A proof can be found in [21]. Lemma 5.2. For any bAR there exists a constant c7 ¼ c7 ðbÞ40 such that jjujjX þ jjAm ðuÞjjX pc7 ð1 þ jju Am ðuÞjjX Þ holds for every uAX with JðuÞpb: Proof. For uAX we have 1 JðuÞ /J 0 ðuÞ; uSX ;X ¼ m
Z Z 1 1 1 p jruj F ðx; uÞ uf ðx; uÞ : p m O m O
If JðuÞpb then ðH2 Þ implies jjujjpX pd1 ð1 þ jjJ 0 ðuÞjjX jjujjX Þ: Now we consider the case 1opp2: Then using (3.8) we obtain jjujjpX pd2 ð1 þ jju Am ðuÞjjp1 X jjujjX Þ:
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165
Young’s inequality yields jjujjX pd3 ð1 þ jju Am ðuÞjjX Þ; which implies the result. Next we consider the case p42: Using (3.9) we obtain jjujjpX pd4 ð1 þ jju Am ðuÞjjX ðjjujjX þ jjAm ðuÞjjX Þp2 jjujjX Þ; hence Young’s inequality gives 0
0
jjujjpX pd5 ð1 þ jju Am ðuÞjjpX ðjjujjX þ jjAm ðuÞjjX Þpp Þ; where p0 ¼ p=ðp 1Þ: Thus, we have p0 =p
0
jjujjX þ jjAm ðuÞjjX pd6 ð1 þ jju Am ðuÞjjX ðjjujjX þ jjAm ðuÞjjX Þ1p =p Þ; which implies the result using Young’s inequality again.
&
We consider the set G :¼ fuAY0 : jt ðuÞAintY D3 for some 0ototðuÞg,intY D3 ; which is an open subset of Y : Let @G be the boundary of G in Y ; so @G is closed in Y : The following result is easy to prove (see [21]). Lemma 5.3. @G is positive invariant, i.e. if uA@G then jt ðuÞA@G for 0ptotðuÞ: In addition, we have inf JðuÞXa :¼
uA@G
inf
uAintY D3
JðuÞ4 N:
Setting e :¼ 12ðj þ cÞ we have j5e 5c and e AintY D3 by ðH3 Þ: Let e1 ; e2 AW01;p ðOÞ be linearly independent with e1 b0 and denote P ¼ e þ spanfe1 ; e2 g: Clearly ðH2 Þ implies F ðx; tÞXc8 jtjm c9
for tAR
ð5:2Þ
for some positive constants c8 and c9 : For uAX we have Z Z Z Z 1 1 p p jruj F ðx; uÞp jruj c8 jujm þ c9 jOj: JðuÞ ¼ p O p O O O So there exist constants b40 and R40 depending only on m; c8 and c9 such that max JðuÞpb; uAP
ð5:3Þ
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166
and JðuÞoa ¼
max
uAP;jjujjX XR
inf
uAintY D3
ð5:4Þ
JðuÞ:
As a consequence of (5.4), @G-P is bounded. Clearly, we have e AintY D3 CG and e þ se1 eG for s40 large enough by (5.4). Thus, there exists s0 40 such that e þ s0 e1 A@G-D1 : From Lemmas 5.1 and 5.3 we see that jt ðe þ s0 e1 ÞA@G-D1
for 0ptotðe þ s0 e1 Þ:
ð5:5Þ
apJðjt ðe þ s0 e1 ÞÞpb for 0ptotðe þ s0 e1 Þ
ð5:6Þ
Lemma 5.3, (5.3) and (5.5) imply
because e þ s0 e1 A@G-P: We set ut ¼ jt ðe þ s0 e1 Þ and t0 ¼ tðe þ s0 e1 Þ for the sake of brevity. Lemma 5.4. There exist u1 AK and an increasing sequence ðtn Þn with tn -t0 such that limn-N jjutn u1 jjX ¼ 0: Proof. By (5.1) and (4.2) we have for 0ot1 ot2 ot0 Z
jjut2 ut1 jjX p
t2
t1
jjus Bm ðus ÞjjX dsp2
Z
t2
t1
jjus Am ðus ÞjjX ds:
ð5:7Þ
We now consider four cases depending on t0 and p: At first, we assume t0 o þ N and 1opp2: The Schwarz inequality and Lemma 5.2 yield Z
t2
t1
jjus Am ðus ÞjjX ds Z
p
!1=2
t2
s
jju Am ðu
t1
Z
pd1
s
ðjju jjX þ jjAm ðu ÞjjX Þ s
jju Am ðu t2 t1
p2
ds
2p
ds !1=2
t2
Z
s
þ jjAm ðu ÞjjX Þ
s
t1
Þjj2X ðjjus jjX
!1=2
t2 t1
Z
s
s
Þjj2X ðjjus jjX
s
þ jjAm ðu ÞjjX Þ !1=2
s
s
ð1 þ jju Am ðu ÞjjX Þ
2p
ds
:
p2
ds
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167
Therefore, (4.1), (5.1) and the Ho¨lder inequality imply Z
t2
jjus Am ðus ÞjjX ds
t1
pd2 ðJðu Þ Jðu ÞÞ t1
t2
Z
1=2
s
2p
s
ð1 þ jju Am ðu ÞjjX Þ
t1
pd3 ðJðut1 Þ Jðut2 ÞÞ1=2 2 Z 4ðt2 t1 Þ1=2 þ
!1=2
t2
ds
3
!ð2pÞ=2
t2
ðt2 t1 Þðp1Þ=2 5:
jjus Am ðus ÞjjX ds
t1
In view of (5.6) and the finiteness of t0 we see that Z
t2
jjus Am ðus ÞjjX ds ¼ 0
lim
t1 ;t2 -t0 0
t1
hence, (5.7) yields the existence of u1 AX such that lim jjut u1 jjX ¼ 0:
ð5:8Þ
t-t0 0
Since ½0; t0 Þ is the maximal interval of existence of ut in X0 ; we have u1 AX \X0 ¼ K: Now we consider the second case: t0 o þ N and p42: As a consequence of the Ho¨lder inequality we have Z
t2 t1
jjus Am ðus ÞjjX dsp
Z
t2
t1
!1=p jjus
Am ðus ÞjjpX
ds
ðt2 t1 Þ11=p :
Thus (4.10) and (5.1) imply Z
t2
t1
jjus Am ðus ÞjjX dspd4 ðJðut1 Þ Jðut2 ÞÞ1=p ðt2 t1 Þ11=p :
Then the same discussion as above leads to (5.8) for some u1 AK: The third case is: t0 ¼ þN and 1opp2: In this case, (5.6) implies the existence of an increasing sequence ftn g with tn - þ N such that lim
n-N
d Jðut Þjt¼tn ¼ 0: dt
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By (5.1), (4.1) and Lemma 5.2 we have d Jðut Þ ¼ /J 0 ðut Þ; ut þ Bm ðut ÞSX ;X dt p d5 jjut Am ðut Þjj2X ðjjut jjX þ jjAm ðut ÞjjX Þp2 p d6 jjut Am ðut Þjj2X ð1 þ jjut Am ðut ÞjjX Þp2 ; hence lim jjutn Am ðutn ÞjjX ¼ 0:
n-N
Now futn g is bounded in X by Lemma 5.2. Since Am : X -X is a compact operator it follows that (after passing to a subsequence if necessary) lim jjutn u1 jjX ¼ lim jjAm ðutn Þ u1 jjX ¼ 0
n-N
n-N
ð5:9Þ
for some u1 AK: It remains to consider the case t0 ¼ þN and p42: By (5.1) and (4.10), there exists an increasing sequence ftn g with tn - þ N such that 0pd7 jjutn Am ðutn ÞjjpX p This gives (5.9) and finishes the proof.
d Jðut Þjt¼tn -0: dt
&
Lemma 5.5. For the sequence ftn g and u1 AK as in Lemma 5.4 we have lim jjutn u1 jjY ¼ 0:
n-N
Proof. We first claim that fut : 0ptot0 g is bounded in X : Suppose that jjut jjX X2c7 for tA½t1 ; t2 C½0; t0 Þ with c7 ¼ c7 ð1Þ from Lemma 5.2. Then we have jjut Am ðut ÞjjX X1
for tA½t1 ; t2 :
If 1opp2 then (5.1), (4.2) and (5.10) imply jju u jjX p 2 t2
t1
Z
p d1
t2
t1
Z
jjus Am ðus ÞjjX ds
t2
t1
jjus Am ðus Þjj2X ð1 þ jjus Am ðus ÞjjX Þp2 ds:
ð5:10Þ
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Using Lemma 5.2 again we arrive at jjut2 ut1 jjX pd2
Z
t2
t1
jjus Am ðus Þjj2X ðjjus jjX þ jjAm ðus ÞjjX Þp2 ds:
From (5.1), (4.1) and (5.6) we deduce jjut2 ut1 jjX pd3 ðJðut1 Þ Jðut2 ÞÞpd4 : In the case p42 we have jju u jjX p2 t2
t1
Z
t2
t1
jjus Am ðus ÞjjpX dspd5 ðJðut1 Þ Jðut2 ÞÞpd6 :
Here we used (5.1), (4.2), (5.10), (4.10) and (5.6). In either case, we obtain that fut : 0ptot0 g is bounded in X : It follows from (5.1) that t
t
u ¼ e ðe þ s0 e1 Þ þ e
t
Z
t
es Bm ðus Þ ds
for 0ptot0 :
ð5:11Þ
0
Now Bm ðus Þ : ½0; t0 Þ-Z is continuous as a consequence of Lemmas 4.1 and 4.2. Since e þ s0 e1 AY CLN ðOÞ and since Bm : X0 -LN ðOÞ and Bm : ðX0 -LN ðOÞ; jj Rt jjN Þ-Z are bounded, fet 0 es Bm ðus Þ ds: 0ptot0 g is bounded in Z and hence relatively compact in Y : This fact and (5.11) imply that fut : 0ptot0 g is relatively compact in Y : Then we obtain the result of Lemma 5.5 immediately from Lemma 5.4. & By (5.5) and Lemma 5.5 we have u1 A@G-D1 : Clearly, u1 eintY D3 because u1 eG: Thus, u1 ¼ Am u1 and Am ðD3 ÞCintY D3 imply u1 eD3 : Therefore, u1 is a solution of (1.1) and u1 Xj;
u1 4 / c:
ð5:12Þ
u2 pc:
ð5:13Þ
Similarly, (1.1) has a solution u2 satisfying u2 5 / j;
According to [21], there exists u A@G-P such that fjt ðu Þ: 0ptotðu Þg-ðintY D1 ,intY D2 Þ ¼ |: Using a similar argument as before we find an increasing sequence ftn g with tn -tðu Þ and u3 AK such that lim jjjtn ðu Þ u3 jjY ¼ 0:
n-N
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Note that u3 eintY D1 ,intY D2 : This implies u3 eD1 ,D2 because u3 ¼ Am u3 and Am ðDi ÞCintY Di for i ¼ 1; 2: Consequently, u3 is a solution of (1.1) satisfying u3 5 / j;
ð5:14Þ
u3 4 / c:
Finally, considering a trajectory starting at a point in D3 -P we find a solution u4 of (1.1) such that fpu4 pc:
ð5:15Þ
Since u1 ; y; u4 are in the o-limit sets of trajectories with starting points in P; we see from (5.3) that Jðui Þp max JðuÞpb;
ð5:16Þ
uAP
where b depends only on m; c8 and c9 from (5.2). To summarize, we have proved the following. Proposition 5.6. Assume ðH001 Þ; ðH2 Þ and ðH3 Þ: Then (1.1) has four solutions u1 ; y; u4 satisfying (5.12)–(5.16), where b depends only on the numbers m; c8 and c9 from (5.2). Remark 5.7. (a) In the proof of Lemma 5.5 we used the boundedness of Bm : X0 -LN ðOÞ and of Bm : ðX0 -LN ðOÞ; jj jjN Þ-Z: This boundedness comes from the boundedness of Am : X -LN ðOÞ and of Am : ðX -LN ðOÞ; jj jjN Þ-Z as shown in the proof of Lemma 4.1. Here condition ðÞ enters; cf. Lemma 3.9. (b) The boundedness of the operators Am
Am
Am
Am
Lq0 ðOÞ ! ? ! Lqn ðOÞ ! LN ðOÞ ! Z
ð5:17Þ
with n large enough has been proved in Section 3. If the operators in (5.17) are continuous and Am : X -X and Am : Y -Y are locally Lipschitz continuous, then I þ Am can be used instead of I þ Bm to generate the descending flow jt ðuÞ and one can use a bootstrap argument as in the semilinear case p ¼ 2 and condition ðÞ can be avoided. But these continuity properties of Am are not yet available for pa2: (c) In order to locate the solutions inside or outside of order cones, we made use of the open positive invariant subset intY D3 of Y. Since intX D˜ 3 ¼ | it is impossible to place the above argument completely inside X even if D˜ 1 ; D˜ 2 and D˜ 3 are also positive invariant for the descending flow. This is why we need the Y -topology in our discussion, and why we have to prove the Y -convergence as in Lemma 5.5.
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6. Proof of Theorems 2.1, 2.2 and 2.3 In this section we assume ðH1 Þ ðH3 Þ and ðÞ: Without loss of generality, we can assume that p 1om 1ominfpp =N; p 1g: Choose q such that p 1om 1oqominfpp =N; p 1g; recall that p ¼ N if pXN: For nAN we define
fn ðx; tÞ ¼
8 < f ðx; tÞ
q jtj : f ðx; 7nÞ n
if jtjpn; if 7t4n:
The following lemma is an easy consequence of ðH1 Þ and ðH2 Þ: Lemma 6.1. (i) If 1opoN then fn ðx; tÞ
lim
jtj-N
jtjp
1
¼0
uniformly in n and x:
(ii) If p ¼ N then lim
jtj-N
ln j fn ðx; tÞj jtjN=ðN1Þ
¼0
uniformly in n and x:
(iii) For any nXM; xAO and jtjXM; 0oFn ðx; tÞ :¼
Z
t
0
1 fn ðx; sÞ dsp tfn ðx; tÞ: m
Here M40 and m42 are from ðH2 Þ: Proof of Theorem 2.1. For any n; consider the modified problem Dp u ¼ fn ðx; uÞ;
uAW01;p ðOÞ
ð6:1Þ
and its associated functional Jn ðuÞ ¼
1 p
Z
jrujp
Z
O
O
Fn ðx; uÞ for uAW01;p ðOÞ:
By Lemma 6.1(iii) there exist c8 40; c9 40 such that Fn ðx; tÞXc8 jtjm c9
for nAN; xAO; tAR:
ð6:2Þ
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Proposition 5.6 implies (6.1) has four solutions un;i ; i ¼ 1; y; 4; satisfying (5.12)– (5.15) and such that Jn ðun;i Þpb
for nAN; i ¼ 1; y; 4;
here b depends only on m; c8 and c9 from (6.2). Moreover, by Lemma 6.1(iii) there exists c10 40 such that 1 tfn ðx; tÞ Fn ðx; tÞX c10 m
for nAN; xAO; tAR:
From this we obtain bX Jn ðun;i Þ 1 ¼ Jn ðun;i Þ /Jn0 ðun;i Þ; un;i SX ;X m Z Z 1 1 1 p ¼ jrun;i j Fn ðx; un;i Þ un;i fn ðx; un;i Þ p m O m O Z 1 1 X jrun;i jp c10 jOj p m O and therefore jjun;i jjX pd1
for nAN; i ¼ 1; y; 4:
ð6:3Þ
Now we estimate the LN ðOÞ norm of un;i : First we consider the case 1opoN: By Lemma 6.1(i) for any e40 there exists d2 ðeÞ40 such that j fn ðx; tÞj jtjNþ1 oejtjp
þN
% tAR: þ d2 ðeÞ for nAN; xAO;
Using jun;i jN un;i as a testing function in (6.1), we get for nAN and i ¼ 1; y; 4: Z Z ðN þ 1Þpp jrðjun;i jN=p un;i Þjp pe jun;i jp þN þ d2 ðeÞjOj: p ðN þ pÞ O O The Sobolev inequality and Ho¨lder inequality now imply Z p=p ðN þ 1Þpp ðNþpÞp =p Sp jun;i j ðN þ pÞp O Z 1p=p Z p=p p ðNþpÞp =p pe jun;i j jun;i j þd2 ðeÞO O
peSp1p
=p
O
Z
jrun;i jp O
ðp =pÞ1 Z O
jun;i jðNþpÞp
=p
p=p þd2 ðeÞjOj;
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where Sp is the best constant of the Sobolev embedding W01;p ðOÞ+Lp ðOÞ: Since un;i ALr ðOÞ for all r41 by Lemma 3.1, using (6.3) we see that Nþp jjun;i jjNþp ðNþpÞp =p ped3 jjun;i jjðNþpÞp =p þ d4 ðeÞ
for nAN; i ¼ 1; y; 4:
Choosing e ¼ 1=2d3 ; we obtain jjun;i jjðNþpÞp =p pd5
for nAN; i ¼ 1; y; 4:
Then Lemma 6.1(i) implies jj fn ðx; un;i ÞjjðNþpÞ=p pd6
for nAN; i ¼ 1; y; 4:
As a consequence of Lemma 3.2 we have jjun;i jjN pd7
for nAN; i ¼ 1; y; 4:
Now suppose that p ¼ N: In view of [13, Theorem 7.15], we have N=ðN1Þ Z juj exp pc12 jOj for uAW01;N ðOÞ c11 jjrujjN O
ð6:4Þ
ð6:5Þ
for some positive constants c11 ; c12 : By Lemma 6.1(ii), for any e40 there exists d8 ðeÞ40 such that j fn ðx; tÞj2 pd8 ðeÞ expðejtjN=ðN1Þ Þ
% tAR: for nAN; xAO;
ð6:6Þ
Using (6.3), (6.5) and (6.6) and choosing e small enough, we obtain jj fn ðx; un;i Þjj22 pd9
for nAN; i ¼ 1; y; 4:
Again (6.4) follows from Lemma 3.2. Finally, if p4N then (6.4) is a direct consequence of (6.3). Thus, we have proved the uniform bound (6.4) for any p41: Hence, un;i ; i ¼ 1; y; 4; are solutions of (1.1) satisfying (5.12)–(5.15) provided n4d7 because then jjun;i jjon: & Proof of Theorem 2.2. According to Lemma 3.4, ðH3 Þ is satisfied with j ¼ d0 e1 and c ¼ d0 e1 for some d0 40: Let u1 ; u2 and u3 be the solutions of (1.1) obtained in Theorem 2.1 and satisfying (5.12)–(5.14). Then u3 is sign changing because u3 5 /j and u3 4 / c: Suppose that O1 :¼ fxAOju1 ðxÞo0ga|: Since u1 Xj the proof of Lemma 3.4 yields Dp u1 X ðl1 eÞju1 jp1
in O1 ;
u1 AW01;p ðO1 Þ:
This together with the Poincare inequality implies Z Z Z l1 ju1 jp p jru1 jp pðl1 eÞ O1
O1
O1
ju1 jp ;
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which is a contradiction. Thus u1 X0; and u1 a0 because u1 4 / c: It follows that u1 b0 by the strong maximum principle from [27]. Similarly, u2 50 is a negative solution. & Proof of Theorem 2.3. The existence of a sign changing solution is a direct consequence of Theorem 2.1 and Lemma 3.5. A positive solution is obtained from Theorem 2.1 and Lemma 3.5 by replacing f with f˜ such that ¼ f ðx; tÞ ˜ fðx; tÞ 40
if teðt1 ; 0Þ; if tAðt1 ; 0Þ:
A negative solution can be obtained similarly.
&
Note added in proof As conjectured in Remark 2.5 the condition ðÞ is not necessary. This will be published in a joint paper with Tobias Weth.
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