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Existence of multiple solutions of critical quasilinear elliptic Neumann problems
Nonlinear Analysis 42 (2000) 613 – 629
www.elsevier.nl/locate/na
Existence of multiple solutions of critical quasilinear elliptic Neumann problems Paul A. Bindinga; ∗ , Pavel Drabekb , Yin Xi Huangc a Department
of Mathematics & Statistics, University of Calgary, Calgary, AB, Canada, T2N 1N4 of Mathematics, University of West Bohemia, P.O. Box 314, 30614 Pilsen, Czech Republic c Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA b Department
Received 21 September 1998; accepted 9 December 1998
Keywords: The p-Laplacian; Multiple solutions; Critical exponent; Neumann problem
1. Introduction We are concerned with the existence of multiple solutions of −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; x ∈ @ ; @n
∗
−2
u + h(x);
x ∈ ;
(1:1 )
where is a bounded smooth domain in RN ; p u=div(|∇u|p−2 ∇u) is the p-Laplacian with 1 ¡ p ¡ N; p∗ = Np=(N − p) is the critical Sobolev exponent, ∈ R; a ∈ L∞ ( ); b ∈ C( ); a(x) and b(x) both change sign, and h ∈ Lq ( ); q = p=(p − 1). Here we say a function f(x) changes sign if the measures of the sets {x ∈ : f(x) ¿ 0} and {x ∈ : f(x) ¡ 0} are both positive.
Research of the authors was supported by NSERC of Canada and the I.W. Killam Foundation, the Grant # 201=97=0395 of the Grant Agency of the Czech Republic, a University of Memphis Faculty Research Grant, and NATO Collaborative Research Grant OUTR.CRG 961190. ∗ Corresponding author. Tel.: +403-220-6328 E-mail addresses: [email protected] (P.A. Binding), [email protected] (Y.X. Huang)
0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 1 1 8 - 2
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In a recent paper [10], Lazzo studied the existence of multiple positive solutions to −du + u = ur−1 ; @u = 0; x ∈ @ ; @n
x ∈ ;
where d ¿ 0; r ¿ 2 if N =2 and 2 ¡ r ¡ 2N=(N −2) if N ≥ 3. For equations involving the critical exponent, the situation is trickier. Earlier, Tarantello [15] considered the existence of multiple positive solutions to a nonhomogeneous Neumann problem with critical exponent. We discuss her results later and we refer to [5,6,15,8] for more references and related problems. In this paper we take a di erent approach. Although our method is variational in nature, our emphasis is di erent. Instead, we investigate the in uence of the inde nite weight function a(x) on the existence of multiple solutions. A subcritical problem −p u = a(x)|u|p−2 u + b(x)|u| −2 u; @u = 0; x ∈ @ ; @n
x ∈ ;
(1:2 )
with 1 ¡ ¡ p∗ and 6= p has been studied in [3]. Our approach in this paper is a continuation of that of Binding et al. [3]. Owing to the presence of the critical exponent, nontrivial modi cations are needed. The diculties caused by the lack of compact embedding are overcome by a procedure used earlier in [7]. Existence results are obtained by minimizing the functional induced by (1.1 ) on some disjoint subsets of the solution manifold (to be de ned below), a procedure also used by Tarantello in [14,15], and by minimizing the functional on some cones. To state our results, we introduce some notation. First, consider the related eigencurve problem −p u = a(x)|u|p−2 u + |u|p−2 u; @u = 0; x ∈ @ ; @n
x ∈ ;
where we treat the eigenvalue as a function of and a. Take R R |∇u|p − a|u|p
R : (; a) := inf |u|p u∈W 1;p ( )\{0}
(1.3)
(1.4)
Then (; a) corresponds to a unique (up to constant multiple) positive eigenfunction of Eq. (1.3). The following is known, see, e.g., [1,2,9]. Proposition 1.1. Fix a ∈ L∞ ( ): Then () is continuous and concave and (0) = 0: If a(x) ¿ 0; then () is decreasing in ; andR if a(x) ¡ 0; then () is increasing. Assume; now; that a changes sign in . (i) If a 6= 0; then the graph of intersects R the line = 0 in the (; ) plane at 0 and at a unique point 1 where 1 · a ¡ 0: R Moreover; () ¿ 0 for between 0 and 1 : (ii) If a = 0; then 0 is the only zero of () = 0 and () ¡ 0 for 6= 0.
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Suppose from now on that (A1) a ∈ L∞ ( ) and a changes sign. (H1) h ∈ Lq ( ), where q = p=(p − 1). Let 1 be given in Proposition 1.1. De ne ( R 1 if a 6= 0; R 1 (a) = 0 if a = 0: Let u1 ¿ 0 denote the normalized positive eigenfunction associated with 1 (a). In case 1 (a) = 0, we can take u1 = 1=| |. Now, we can state our main results. Let k · kq denote the Lq norm. 1.1. Main results Suppose bothR a and b change R sign, ∗with b ∈ C( ). R (I) Assume a 6= 0, with b|u1 |p ¡ 0 if 6= 0 and b ¡ 0 if = 0. Then there R exists ˜ with (˜ − 1 (a)) · a ¡ 0, with the following properties: ˜ (1.1 ) has at least one (i) Suppose h ≡ 0. Then for any strictly between 0 and , positive solution. If, in addition, p ≥ 2, then (1.1 ) has at least two positive solutions for strictly between 1 (a) and ˜ (Theorem 3.7, Lemma 4.4 and Theorem 4.6). (ii) There exists ¿ 0, such that for any h with 0 ¡ khkq ¡ , equation (1.1 ) has at least two nontrivial solutions if either p ≥ 2 and is strictly between 0 and 1 (a), or 1 (a) ≤ ¡ R˜ (Theorems R 5.1 and 5.4). (II) Assume a = 0; b ¡ 0 and 6= 0. Then equation (1.1 ) has at least two nontrivial solutions for small enough and h 6≡ 0 small enough, and two positive solutions if h ≡ 0; small and p ≥ 2R (Remarks 4.8 and 5.6). ∗ We note that the integral condition b|u1 |p ¡ 0 is not needed for the case strictly between 0 and 1 (a). We speci cally remark that Theorem 5.4 establishes the existence multiple solutions of (1.1 ) for h 6≡ 0 when the principal functional Rof nontrivial p p (|∇u| −a|u| ) is no longer positive de nite. To the best of our knowledge, results
of this kind are new even for the case p = 2. Let us brie y compare and contrast our results with those of Tarantello [15]. Consider ∗
−u + u = |u|2 −2 u + f(x); x ∈ ; @u = 0; x ∈ @ ; @n where 2∗ = 2N=(N − 2) and ¿ 0. Let kukr denote the Lr norm and Z cN (k∇uk22 + kuk22 )(N +2)=4 − fu ; f = inf kuk2∗ =1
(1:5 )
(1.6)
where cN = 4=(N − 2)[(N − 2)=(N + 2)](N +2)=4 . It is proved in [15] that for N ≥ 5 and f 6≡ 0, Eq. (1.5 ) has at least three solutions for any ¿ 0 if f ¿ 0. k∇uk22 + kuk22 is coercive, We point out that in Eq. (1.5 ), the principal functional R while for our problem the corresponding functional (|∇u|p − a|u|p ) may not be coercive. Thus we work with a b(x) which changes sign. The condition f ¿ 0 means
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f is “small” in some R sense. For the case = 0, [15] shows that (1:5)0 has at least two solutions if f 6= 0; f = 0 and another “smallness” condition f∗ ¿ 0 (see Eq. (1.3) of [15]) is satis ed. This paper is organized as follows. We rst study the case h ≡ 0. Speci cally, Section 2 analyses the geometric structure of the solution manifolds. In Section 3 we establish the existence of one solution for h ≡ 0 by adapting Ekeland’s variational principle to accommodate a constraint. In Section 4 we establish existence of two solutions under appropriate conditions. The case h 6= 0 is treated in Section 5 and we obtain the existence of two nontrivial solutions. 2. Geometric structure of the solution manifold for h ≡ 0 In Sections 2– 4, we study the case h ≡ 0. Throughout this paper the function b is always assumed to satisfy (B1) b+ 6≡ 0 and b ∈ C( ) where b+ (x) = max{0; b(x)}. To Rbe speci c, the arguments will be detailed for the case (A2) a ¡ 0, i.e., 1 (a) ¿ 0. Modi cations for the other cases will be indicated in due course. We introduce the following functional: Z Z ∗ 1 1 (|∇u|p − a|u|p ) − ∗ (2.1) b|u|p ; I (u) = p p R and write, for simplicity, J (u) = (|∇u|p − a|u|p ). Here and below the integrals are taken over unless otherwise stated. We will use k · k to denote the norm in W 1;p ( ). Clearly the functional I is well de ned on W 1;p ( ) and a critical point of I in W 1;p ( ) is a (weak) solution of (1.1 ). We de ne the solution manifold = {u ∈ W 1;p ( ) : (u) := hI0 (u); ui = 0} Z 1;p p∗ = u ∈ W ( ) : J (u) = b|u| :
(2.2)
It is clear that solutions of (1.1 ) belong to . We now de ne a subset of by 0 − = {u ∈ : h (u); ui ¡ 0}:
(2.3)
There are several equivalent expressions for this set. We list, for example, the following: Z ∗ p∗ = u ∈ : pJ (u) ¡ p b|u| − = u ∈ : J (u) =
Z
p∗
b|u|
¿0 :
ForRthe remainder of this section R we assume the following condition on b: ∗ (B2) b|u1 |p ¡ 0 if 6= 0 and b ¡ 0 if = 0. R We note that when 1 (a) = 0, the rst integral in (B2) also becomes b ¡ 0.
(2.4)
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Lemma 2.1. Assumption (B1) implies that − 6= ∅. Proof. Indeed, suppose b(x) ¿ 0 on a small ball B ⊂ and let ’ ∈ L∞ ( ) be such that supp ’ ⊂ B. Then by continuous dependence of the principal eigenvalue on the domain and the fact that as |B| → 0, the principal eigenvalue on B tends to +∞, so we can choose ’ such that J (’) ¿ 0. Now, for t ¿ 0, let Z ∗ ∗ b|’|p : (t) := (t’) = t p J (’) − t p R ∗ ∗ That is, (t) = mt p − ct p , with m = J (’) ¿ 0; c = b|’|p ¿ 0. Choosing t0 to satisfy ∗ ct0p −p = m we see that (t0 ) = 0 and 0 (t0 ) ¡ 0, and so t0 ’ ∈ − . Remark 2.2. It is clear that ∈ [0; 1 (a)] implies J (u) ≥ 0. Also, is closed in W 1;p ( ). Observe that for any u ∈ , Z ∗ 1 b|u|p : (2.5) I (u) = N It follows from Lemma 2.1 and Eq. (2.4) that inf − I ≥ 0, so I is always bounded
below on − .
Lemma 2.3. Assume (A1); (A2); (B1) and (B2) hold. Then there exist 0 ¿ 1 (a) and ¿ 0; such that for any ∈ [0; 0 ); J (u) ¿ kukp for u ∈ − . Proof. Suppose the contrary. Then there exist n ; un ∈ − n such that n → ˆ ∈ [0; 1 (a)];
1 Jn (un ) ¡ kun kp : n
(2.6)
Let vn = un =kun k. We may assume that vn → v0 weakly in W 1;p ( ) and strongly in R R ˆ ≥0 Lp ( ), for some v0 ∈ W 1;p ( ). Hence we have a|vn |p → a|v0 |p . Since ( ) by Proposition 1.1, it follows from weak lower semicontinuity of the W 1;p ( ) norm that 1 = 0: 0 ≤ J ˆ (v0 ) ≤ lim inf Jn (vn ) ≤ lim n→∞ n n→∞ Passing to a subsequence if necessary, we can therefore assume that Jn (vn ) → 0 = J ˆ (v0 ):
(2.7)
There are three possibilities: (1) v0 = 0; (2) v0 = k u1 , where 0 ¡ |k| ≤ 1 and 0 ¡ ˆ ≤ 1 (a); and (3) ˆ = 0. The rst case is impossible since it implies that vn → 0 in Lp ( ) and so by Eq. (2.7) we have vn → 0 in W 1;p ( ), contradicting kvn k = 1. For (2), we have Z Jn (vn ) = (|∇vn |p − n a|vn |p ) → 0:
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Moreover, Z Z Z n a|vn |p → ˆ a|k u1 |p = |∇(k u1 )|p ; R R by Eq. (2.7). This implies that |∇vn |p → |∇(k u1 )|p , so kvn k → kk u1 k. Thus weak convergence of vn to k u1 implies that vn → k u1 strongly in W 1;p ( ), so vn → k u1 ∗ strongly in Lp ( ). Now, we derive from un ∈ − n , Eq. (2.7) and (B2) that Z Z ∗ ∗ ∗ p∗ p∗ b|vn |p → b|k u1 |p ¡ 0: kun kp−p · Jn (vn ) ¡ p p This contradicts Eq. (2.7) R since Jn (un ) ¿ 0 by Eq. (2.4). Thus (2) cannot occur. For (3), we let vn = vn =| |. It then follows from Z Z p |vn − vn | ≤ |∇vn |p that v0 must be a constant and vn → v0 strongly in W 1;p ( ). Now, v0 6= 0 since kvn k=1, so we obtain Z Z ∗ p∗ − 1 p−p∗ p∗ b|vn | → Jn (vn ) ¡ b|v0 |p ¡ 0; 0 ≤ |un k p−1 by (B2), a contradiction. This completes the proof. Remark 2.4. An analogous result (replacing by −) holds if 1 (a) ¡ 0. If 1 (a) = 0, then there exist 0 ¿ 0 and ¿ 0 such that for with || ¡ 0 ; J (u) ≥ kukp for u ∈ − . The proof is similar and will be omitted. Similarly for Lemma 2.5 below, etc. From now on we will assume that (A1), (A2), (B1) and (B2) all hold. Lemma 2.5. There exists ¿ 0 such that −h 0 (u); ui ≥ for ∈ [0; 0 ); u ∈ − . Proof. We rst claim that there exists ¿ 0, such that kuk ¿ for ∈ [0; 0 ) and − p u ∈ − . Indeed, suppose for some un ∈ n ; kun k → 0. Dividing Eq. (2.4) by kun k and using Lemma 2.3, one obtains the contradiction Z ∗ ∗ p∗ b|vn |p · kun kp −p → 0; 0 ¡ ¡ Jn (vn ) ¡ p where vn = un =kun k. By Lemma 2.3 and the above claim, we have −h 0 (u); ui = (p∗ − p)J (u) ≥ (p∗ − p)kukp ≥ (p∗ − p)p : The lemma then follows. Corollary 2.6. For ∈ [0; 0 ); − is a closed set in . Proof. This is an immediate consequence of Lemma 2.5 and the continuity of h 0 (u); ui with respect to the W 1;p ( ) norm.
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3. Existence of one positive solution − Since I is bounded below on − , minimizing sequences exist for I on . Next, we state a generalized Ekeland variational principle, which plays a critical role at various places in this paper.
Lemma 3.1. For ∈ [0; 0 ); for any bounded minimizing sequence {un } of I on − ; there exists a modiÿed bounded minimizing sequence {zn } such that I0 (zn ) → 0. Proof. Since − is a closed set by Corollary 2.6, [11, Theorem 4:1 and Remark 4:1] imply that we can replace {un } by another minimizing sequence {zn } ⊂ − such that , kun − zn k ¡ 1=n, and for any y ∈ − 1 I (y) ¿ I (zn ) − ky − zn k: n
(3.1)
Next, we show that I0 (zn ) → 0. Choose wn of unit norm so that hI0 (zn ); wn i ≥ kI0 (zn )k − o(1) as n → ∞. It will suce to show that hI0 (zn ); wn i → 0:
(3.2)
For each n, let gn (t; s) = (t zn − s wn ). Then gn (1; 0) = 0 and @gn = h 0 (zn ); zn i = 6 0 @t
at t = 1; s = 0:
It follows from the C 1 Implicit Function Theorem that for each n, for small enough s, there exists tn ∈ C 1 so that (tn (s)zn − swn ) = 0 and h 0 (zn ); zn itn0 (0) − h 0 (zn ); wn i = 0:
(3.3)
Since zn is a bounded sequence, so is k 0 (zn )k, and we then conclude from Eq. (3.3) and Lemma 2.5 that tn0 (0) is uniformly bounded in n:
(3.4)
Until further notice we x n, and we consider vn (s) = tn (s)zn − swn − zn . Since wn is of unit norm, we have kvn (s)k ≤ |s|(1 + (|tn0 (0) + o(1)|)kzn k)
(3.5)
as s → 0. Moreover, zn ∈ gives hI0 (zn ); zn i = 0, so I (zn ) − I (tn (s)zn − swn ) = hI0 (zn ); −vn (s)i + o(vn (s)) = hI0 (zn ); swn i + o(s) (3.6) follows from Eq. (3.5). By continuity of
h 0 (u); ui,
we have
h 0 (tn (s)zn − swn ); tn (s)zn − swn i − h 0 (zn ); zn i → 0
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as s → 0. We then conclude from this and Lemma 2.5 that h 0 (tn (s)zn − swn ); tn (s)zn − swn i ¡ 0 for s small enough, so tn (s)zn − swn ∈ − . Dividing Eq. (3.6) by s and using Eqs. (3.1) and (3.5), we obtain |hI0 (zn ); wn i| ≤ n−1 (1 + (|tn0 (0)|)kzn k): Now, we let n → ∞ to show that hI0 (zn ); wn i tends to zero by boundedness of zn and Eq. (3.4). This establishes Eq. (3.2) and proves the lemma. Let S be the best Sobolev embedding constant, i.e., ) ( k∇ukpp 1;p N : u ∈ W (R )\{0} : S = inf kukpp∗ To show that a minimizing sequence as in Lemma 3.1 gives rise to a weak solution of Eq. (1:1 ), we need the concentration-compactness principle of Lions [10]: ∗
Proposition 3.2. Let {un } converge weakly to u in W 1;p ( ) such that |un |p and |∇un |p converge weakly to nonnegative measures and on ; respectively. Then; for some at mostP countable set ; we have ∗ (i) = |u|p + j∈ j xj ; P (ii) ≥ |∇u|p + j∈ j xj ; ∗
(iii) Sp=p ≤ j , j where xj ∈ ; xj is the Dirac measure at xj ; j and j are nonnegative constants.
The next result can be proved in the same way as Proposition 2:4 of [7]. Proposition 3.3. Let {un } be a bounded sequence in W 1;p ( ) such that I (un ) → c and I0 (un ) → 0. Assume also that un → u weakly in W 1;p ( ). Then the set given in Proposition 3:2 is ÿnite; b(xj ) ¿ 0 for j ∈ ; j = b(xj )j and j ≥ (S=b(xj ))N=p . Moreover; there exists a subsequence; denoted again by {un }; such that |∇un |p−2 ∇un → |∇u|p−2 ∇u weakly in ([Lp ( )]N )0 . At this stage, it is possible to obtain a solution to Eq. (1:1)0 . To obtain one that is positive, however, more preparation is needed. We make the following additional assumptions. (B3) Let b(x0 ) = kbk∞ and for some r ¿ 0, b(x) ¿ 0 for x ∈ B(x0 ; 2r) ⊂ . Without loss of generality, we assume below that x0 = 0. (A3) a(x) ≥ a0 ¿ 0 in B(0; 2r), and (B4) for x ∈ B(0; 2r), b(x) = b(0) + o(|x| )
as x → 0;
=
N −p : p−1
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De ne, for ¿ 0, u (x) = where
(x) ; ( + |x|p=(p−1) )(N −p)=p
v (x) =
u (x) ; ku (x)kp∗
∈ C0∞ (B(0; 2r)) is such that 0 ≤ (x) ≤ 1 and
(x) ≡ 1 on B(0; r).
Lemma 3.4. Let (A1)–(A3) and (B1)–(B4)hold and ¿ 0: (i) For ¿ 0 small enough; sup I (tv ) ¡ S0 := t≥0
1 N=p )=p : S kbk(p−N ∞ N
(3.7)
(ii) Assume moreover p ≥ 2. Then for any w ∈ W 1;p ( )\{0}; for ¿ 0 small enough; sup I (w + Rv ) ¡ I (w) + S0 :
(3.8)
R≥0
Proof. These are essentially local properties of the functional I so the proofs presented in [7, Lemmas 4:1 and 5:6] remain valid. Speci cally, we note that p−1 if p2 ¡ N; S0 − K (3.9) sup I (tv ) ≤ S0 − Kp−1 |ln | if p2 = N; t≥0 (N −p)=p 2 if p ¿ N; S0 − K and I (w + Rv ) ≤ I (w) + S0 − K () + C1 (N −p)(p−1)=p 2
2
−C(N −p)=p + C 0 (N −p)=p ;
2
(3.10)
where () is one of the functions of given in Eq. (3.9), depending on the situation, K; C1 ; C; C 0 are positive constants and C 0 = C=2. The lemma then follows. De ne c1 = inf I (u). Lemma 3.5. Under the assumptions of Lemma 3:4; for ∈ [0; 0 ); any minimizing sequence {un } of I on − satisfying I (un ) ¡ c1 + S0 has a subsequence converging to a solution u ∈ − of (1:1 ) and I (u) ¿ 0. Proof. Suppose {un } is an unbounded sequence that minimizes I on − . Dividing R ∗ I (un ) by kun k, we conclude that, since I (un ) is bounded, b|un |p ·kun k−1 is bounded by Eq. (2.5). Thus J (un ) · kun k−1 is also bounded by Eq. (2.2). Let vn = un =kun k. Then 0 ≤ J (vn ) → 0. This contradicts Lemma 2.3. Thus any minimizing sequence {un } in − is bounded. ∗ Assume that un → u weakly in W 1;p ( ) and in Lp ( ). Since un is a minimizing sequence, we conclude from Lemma 3.1 that I0 (un ) → 0. Then, by Proposition 3.3, |∇un |p−2 ∇un → |∇u|p−2 ∇u weakly in ([Lp ( )]N )0 . We then conclude, as in the proof of Lemma 3.1 of [7], −p u = a|u|p−2 u + b|u|p
∗
−2
u
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in W 1;p ( ), that is, u is a solution of Eq. (1:1); u ∈ and I0 (u) = 0. In particular, I (u) ≥ c1 . Suppose that j given by Proposition 3.2 is not zero for some j. We have Z 1 1 0 1 − (|∇un |p − a|un |p ) S0 + c1 ¿ I (un ) − ∗ hI (un ); un i = p p p∗ Z 1 X 1 (|∇u|p − a|u|p ) + j : (3.11) ≥ N N j∈ Using Proposition 3.3 and the fact that I0 (u) = 0 we get 1 X j S0 + c1 ¿ I (un ) ≥ I (u) + N j∈ ≥ I (u) +
S N=p 1 X ≥ c1 + S0 ; N j∈ b(xj )(N −p)=p
R ∗ aR contradiction. Thus j =0 for all j. We conclude from Proposition 3.2 that |un |p → ∗ ∗ |u|p . This together with the weak convergence of un to u in Lp ( ) implies that ∗ un → u strongly in Lp ( ). We can then show as in the proofs of [9, Lemmas 3:1 and 1;p ( ), u ∈ − 5:5(i)] that un → u strongly in W 1;p ( ). Since − is closed in W . It is easy to see from Eqs. (2.4) and (2.5) that I (u) ¿ 0. We note that the functional I is even, so we can always assume that the critical points of I are nonnegative functions. Proposition 3.6. Let u 6≡ 0 be a nonnegative nontrivial solution of Eq. (1:1 ). Then u ¿ 0 in . Proof. We note that the proof of Lemma 4:3 of [13] remains valid in our situation. It shows that the solution u ∈ Lt ( ) for all t ≥ p∗ . Then for F(x) = a|u|p−2 u + b(x)|u|p
∗
−2
u;
F ∈ L ( ) for any ¿ N=p. Then a Harnack-type inequality by Serrin [12] implies that u ¿ 0. This ends the proof. Theorem 3.7. Assume that (A1)–(A3) and (B1) – (B4) hold. Then for any ∈ (0; 1 (a)]; −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; @n
∗
−2
u;
x ∈ ;
x ∈ @ ;
has at least one positive solution in W 1;p ( ). Proof. Choose ¿ 0 small enough so that v = v in Lemma 3.4 is smooth and has support where b ¿ 0. Let g(t) = (tv) so g(0) = 0; g(t) ¿ 0 for t ¿ 0 small, and
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g(t) → − ∞ as t → ∞. Then the argument of Lemma 2.1 shows that, for some t0 ¿ 0, g(t0 )=0 and g0 (t0 ) ¡ 0. That is, t0 v ∈ − . Also, c1 =inf u∈ I (u)=0 in this case. Thus Lemmas 3:5 and 3:4(i) imply that the problem has a nonnegative nontrivial solution. It then follows from Proposition 3.6 that the solution is positive. This completes the proof. Remark 3.8. For the case ∈ (0; 1 (a)), condition (B2) is not needed. In fact one can use the Mountain Pass Theorem and a variant of Lemma 3.5 to conclude the existence of a positive solution. Remark 3.9. We speci cally note that we do not allow = 0, since it is not clear whether Lemma 3.4(i) still holds or not. Consequently, any minimizing sequence may converge only weakly to a solution u of Eq. (1.1 ), and u could be zero. 4. Existence of two positive solutions We now discuss the situation ¿ 1 (a), assuming again (A1) – (A3) and (B1) – (B4). Recall that we normalize u1 ¿ 0 such that ku1 k = 1. Lemma 4.1. Assume (B2) holds.RThere exist 1 ¿ 0 and ¿ 0 such that; for all v ∗ with kvk = 1 and kv − u1 k ¡ ; b|v|p ¡ − ¡ 0. This is an immediate consequence of (B2). Lemma 4.2. Assume (B2) holds. Then for any ˆ ∈ (0; 1); there exist ˆ ¿ 1 (a) and ˆ and any v satisfying kvk = 1 and kv − u1 k = ˆ ¿ 0; such that for any ∈ (1 (a); ) ˆ J (v) ¿ . ˆ ; Proof. Suppose the lemma is false. Then for some ˆ satisfying 1 ¿ ˆ ¿ 0; n ¿ 1 (a) ˆ we have and vn with kvn k = 1; kvn − u1 k = , 1 n → 1 (a); Jn (vn ) ¡ : n Assuming without loss of generality that vn → v0 weakly in W 1;p ( ), we conclude that, as in the proof of Lemma 2.3, either v0 = 0 or v0 = k u1 for some 0 ¡ |k| ≤ 1. The same arguments as in the proof of Lemma 2.3 show that, (i) v0 = 0 is impossible, and (ii) vn → k u1 strongly in W 1;p ( ). But kvn k = 1 and kvn − u1 k = ˆ imply that (ii) is impossible for any 0 ¡ |k| ≤ 1 since 0 ¡ ˆ ¡ 1. This concludes the proof. Corollary 4.3. Fix 0 ¡ ∗ ¡ min{1; 1 }; where 1 is given by Lemma 4:1. There exist ∗ ∗ ¿ 0; ∗ ¿ 1 (a) and ∗ ¿ 0 such R that;p∗for any∗ ∈ (1 (a); ); (i) for any v satis∗ fying kvk = 1 and kv − u1 k ≤ ; b|v| ¡ − ; (ii) for any v satisfying kvk = 1 and kv − u1 k = ∗ ; J (v) ¿ ∗ . Lemma 4.4. For ∈ (1 (a); ∗ ); there exists a solution u0 of Eq. (1:1 ) with I (u0 ) ¡ 0.
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Proof. We form a cone around u1 as follows: (4.1) U + = {tv: t ≥ 0; v ∈ W 1;p ( ); kvk = 1; kv − u1 k ≤ ∗ }: R ∗ Note that, by our construction, b|v|p ¡ 0 for v ∈ U + . For v 6= 0, v ∈ @U + , the boundary of U + , we have, by Corollary 4.3, ∗ Z ∗ tp tp tp ∗ tp ∗ p∗ b|v| ≥ + ∗ ¿ 0: (4.2) I (tv) = J (v) − ∗ p p p p On the other hand, for any v ∈ U + with kvk = 1; |J (v)| ¡ M for some M ¿ 0. We derive that ∗ Z ∗ ∗ tp tp tp tp ∗ tp tp ∗ p∗ (4.3) b|v| ≥ J (v) + ∗ ≥ − M + ∗ : I (tv) = J (v) − ∗ p p p p p p It then follows that there exists R ¿ 1 such that for any v ∈ U + with kvk=R; I (v) ¿ 0. Now, we form a bounded cone UR+ = {v ∈ U + : kvk ≤ R}. It is clear from the above that I (v) ¿ 0 for v 6= 0; v ∈ @UR+ , the boundary of UR+ . Since UR+ is bounded, I is bounded below on UR+ . On the other hand, ∗ Z ∗ tp tp b|u1 |p ¡ 0 (4.4) I (tu1 ) = J (u1 ) − ∗ p p for t ¿ 0 small enough, so inf UR+ I (u) ¡ 0. Consequently, we can take a minimizing sequence {un } of I in the interior of UR+ , and we conclude that, by Ekeland’s variational principle, (I )0 (un ) → 0. We can assume un → u0 weakly in W 1;p ( ), and hence, as in the proof of Lemma 3.5, that u0 is a weak solution of −p u0 = a(x)|u0 |p−2 u0 + b(x)|u0 |p @u0 = 0; x ∈ @ ; @n
∗
−2
u0 ;
x ∈ ;
(4.5)
R ∗ + and u0 ∈ int UR+ by Eq. (4.2) and the convexity of UR+ . Since b|v|p ¡ 0 for R v ∈ pU , we J (un ) ¡ 0 for large n. Thus by the weak convergence of un → u0 , a|un | → R have a|u0 |p , and, since Eq. (4.5) leads to the analogue of Eq. (2.5), 1 1 J (u0 ) ≤ lim inf J (un ) ¡ 0: N N n→∞ Thus u0 6= 0 is in the interior of UR+ , and I (u0 ) ¡ 0. The lemma is proved. I (u0 ) =
In order to establish the existence of a second solution, we need a local property of the functional I . Lemma 4.5. (Drabek and Huang [7, Lemma 5:7]). Given ¿ 1 (a); there are ¿ 0 small enough and R ¿ 0 large enough so that u0 + Rv ∈ − . Now, by taking w = u0 , we can apply Lemmas 3:4 (ii), 3:5 and 4:5 to conclude − that any minimizing sequence in − does converge to a nontrivial solution u ∈ of Eq. (1:1). Since I (u0 ) ¡ 0 and I (u) ¿ 0, we obtain two di erent solutions. Proposition 3.6 implies that both u0 and u are positive. We summarize as follows.
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Theorem 4.6. Let p ≥ 2. Assume that (A1) – (A3) and (B1) – (B4) hold. Then there ˜ exists ˜ ¿ 1 (a); such that for any ∈ (1 (a); ); −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; @n
∗
−2
u;
x ∈ ;
x ∈ @ ;
has at least two positive solutions in W 1;p ( ). Remark 4.7. A related inde nite Neumann problem has been studied by Tehrani [16]. Consider −u + m(x)u = b(x)up ;
u ¿ 0; x ∈
(4.6) @u = 0; x ∈ @ ; @n where p ¡ (N + 2)=(N − 2) = 2∗ − 1. Tehrani proved the following result. Suppose R b ¡ 0, then for any R ¿ 0, there exists = (R) ¡ 0 such R that for all
- Holder continuous functions m (0 ¡ ¡ 1) with |m|0; ¡ R, if (i) m ≤ 0, (ii) R ¡ 0, where (R) ¡ 1 := (−1; m) ¡ 0 (using our notation (1.4)), and (iii) b’p+1 m ’m is a positive eigenfunction associated with (−1; m), then Eq. (4.6) has a solution. Note that since −’m +m’m = R 1 ’m , we have −’m =a’m if a=1 −m, so 1 (a)=1. Proposition 1.1 then implies a ¡ 0 and a changes sign. Applying Theorem 3.7 to this situation we nd that there exists ∗ ¿ 0 such that for any ∈ (0; ∗ ), the problem −u + mu = 1 u + bu2 @u = 0; x ∈ @
@n has a solution.
∗
−1
;
u ¿ 0; x ∈
(4.7)
Remark 4.8. As mentioned earlier in Remark 2.4, results similar Rto Lemma 2.5, R Corollary 2.6 and Lemma 3.1 hold for the cases where a ¿ 0 and a = 0 for in intervals of the form (0 ; 0], (0 ; 1 (a)) or (−0 ; 0 ). Thus existence results similar to Theorems 3.7 and 4.6 hold for these cases. 5. The inhomogeneous case h 6= 0 In this section we study the case h 6= 0. Corresponding to Eqs. (2.1), (2.2) and (2.6) we now have Z Z Z 1 1 p∗ J (u) − ∗ b|u| − hu (5.1) I (u) = p p and for u ∈ with Z Z ∗ = u ∈ W 1; p ( ): J (u) = b|u|p + hu ;
(5.2)
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P.A. Binding et al. / Nonlinear Analysis 42 (2000) 613 – 629
we have I (u) =
1 N
Z
Z ∗ 1 b|u|p − 1 − hu: p
(5.3)
We will treat the three cases 0 ¡ ¡ 1 (a); ¿ 1 (a) and = 1 (a) separately in that order. Theorem 5.1. Let p ≥ 2. Assume that ∈ (0; 1 (a)). Assume also that (A1) – (A3); (B1) – (B4) hold. Then there existsR ¿ 0 such that for any h with 0 ¡ khkq ≤ ; where q = p=(p − 1); and khkq = ( |h|q )1=q ; the problem −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; x ∈ @ ; @n has at least two nontrivial solutions.
∗
−2
u + h;
x ∈ ;
Proof. We rst observe that, for ∈ (0; 1 (a)), Z (|∇u|p − a|u|p ) ≥ kukp
(5.4)
(5.5)
for some ¿ 0 (cf. Proposition 2 of Binding et al. [3]). Hence we have, by the Sobolev embedding theorem and Young’s inequality, for any ¿ 0, there is C ¿ 0 so that ∗
I (u) ≥ kukp − C1 kbk∞ · kukp − Ckuk · khkq ∗
≥ ( − )kukp − C1 kbk∞ · kukp − C khkqq : It follows, choosing then small enough, that there exist () ¿ 0 and c00 () ¿ 0 such that I (u) ≥ c00 ();
(5.6)
for kuk = and khkq ≤ (). Choose = 0 so that c00 := c00 (0 ) ¿ 0, and write := (0 ). Now, for kuk ≤ 1 ¡ 0 , I (u) is bounded above by ∗
f(1 ) := Ap1 + Bp1 + 1 for suitable constants A and B. We choose 1 small enough to satisfy c00 and c00 (1 ) ¿ 0: 2 R Since h 6≡ 0, there exists ’ ∈ C0∞ ( ) such that h’ ¿ 0. It then follows that Z Z ∗ Z ∗ tp tp (|∇’|p − a|’|p ) − ∗ I (t’) = b|’|p − t h’ ¡ 0; p p f(1 ) ¡
for t ¿ 0 small enough. We can assume that t is so small that kt’k ¡ 1 . Thus we have c = inf {I (u) : u ∈ S1 } ¡ 0; where S1 ={u : kuk ≤ 1 }. Now, taking a minimizing sequence {un } such that kun k ≤ 1 and I (un ) → c, we have, by Ekeland’s variational principle (cf. [11]), for some z with
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627
kzk ¡ 1 , I0 (un ) → 0 and un → z weakly in W 1;p ( ). A proof similar to the rst part of that of Lemma 3.5 (using Eq. (5.3) instead of Eq. (2.5), and using Proposition 3.3) then shows that I0 (z) = 0, i.e., z is a solution of Eq. (1:1). Convexity of S1 shows that z ∈ S1 , so I (z)R ≤ c00 =2. We have, by Lemma 3.4(ii) (with v replaced by v˜ = ±v chosen so that hv˜ ≥ 0), for p ≥ 2, I (z + Rv ) ¡ I (z) + S0 ;
(5.7)
and by Theorem 2 of [4], I (z + Rv ) = I (z) + I (Rv ) + o(1) → − ∞ as R → ∞. Let R0 be such that I (z + R0 v ) ¡ 0 and kz + R0 v k ≥ . De ne = { ∈ C([0; 1]; W 1;p ( )) : (0) = z; (1) = z + R0 v }; and let c2 = inf max I ( (t)):
∈ t∈[0;1]
Then by Eq. (5.6), 0 ¡ c00 ≤ c2 ¡ I (z) + S0 . [11, Corollary 4.3] implies that there exists un such that I (un ) → c2 and I0 (un ) → 0. We claim that such un is bounded. Indeed, since I0 (un ) → 0, hI0 (un ); un i = o(kun k), so we have Z 1 1 1 1 − ∗ J (un ) − 1− ∗ hun = I (un ) − ∗ hI0 (un ); un i ¡ c3 + o(kun k) p p p p Then for some c3 ¿ c2 and n large. With Eq. (5.5) we see that kun kp is bounded. R ∗ ) hu and (1 − Eq. (5.7) and the nal part of the proof of Lemma 3.5 (with (1 − 1=p n R 1=p∗ ) hu subtracted from the right-hand side of Eq. (3.11)) show that un converges to some u strongly in W 1; p ( ), and u is a solution of Eq. (1:1). Moreover, I (u)=c2 ¿ 0. Thus u 6= z and we have obtained two nontrivial solutions. This completes the proof. Remark 5.2. Theorem 5.1 does not permit us to obtain two nontrivial solutions under the conditions of Theorem 3.7, since the solution z obtained above degenerates to zero when h = 0. Remark 5.3. Cao et al. [5] and Chabrowski [6] also proved the existence of multiple solutions similar to Theorem 5.1, on RN and bounded domains with Dirichlet boundary conditions, respectively. Their approaches are based on Mountain Pass arguments. Next, we study the case ¿ 1 (a). The idea is based on a perturbation argument. We rst construct a truncated cone UR+ , as in the proof of Lemma 4.4, so that the unperturbed functional Z Z ∗ 1 1 0 p p (|∇u| − a|u| ) − ∗ b|u|p I (u) = p p has a Rnontrivial minimizing sequence in the interior of UR+ , and then we perturb I0 by − h· (by choosing h small) so that the corresponding sequence for I remains in int UR+ .
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We note that the proof of Lemma 4.4 holds identically for I0 , so I0 (v) ≥ 0 for v ∈ @UR+ and c0 := inf + I0 (u) ¡ 0 u∈UR
by Eq. (4.4). Observe that, for u ∈ UR+ , Z Z Z ∗ 1 1 1 1 p∗ b|u| − hu ≥ J (u) − ∗ b|u|p − khkq · kuk: I (u) = J (u) − ∗ p p p p Since UR+ is bounded, one easily sees that there exists ∗ ¿ 0 such that for 0 ¡ khkq ¡ ∗ , I (u) ¿ c0 =4 for u ∈ @UR+ , and I (u0 ) ¡ c0 =2. Again, I is bounded below on UR+ , so a minimizing sequence for I can be taken from the interior of UR+ . Thus, as in the proof of Lemma 4.4, minimizing I on UR+ yields a nontrivial solution u ∈ int UR+ of −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; @n
∗
−2
u + h;
x ∈ ;
x ∈ @ :
(5:8 )
The case = 1 (a) will be treated as a limit case by letting → 1 (a). Fix ∗ ; ∗ as in Corollary 4.3 and ∗ as above. For n ∈ (1 (a); ∗ ), let n → 1 (a). Then we have a bounded sequence of solutions {un } of Eq. (5:8)n . We can assume un → u+ weakly in W 1; p ( ) and since each un satis es I0n (un ) = 0, and n → 1 (a), we can show directly that I01 (a) (u+ ) = 0. That is, u+ is a weak solution of Eq. (5:8)1 (a) . Since h 6= 0; u+ 6= 0. Note that we can carry out the identical procedure on the opposite cone U − = {tv: t ≤ 0; v ∈ W 1; p ( ); kvk = 1; kv − u1 k ≤ ∗ } and UR− = {u ∈ U − : kuk ≤ R}, and obtain a weak solution u of Eq. (5:8 ) in the interior of UR− . Noting that UR+ ∩ UR− = {0}, we obtain our nal result. Theorem 5.4. Assume that (A1); (A2); (B1) and (B2) hold. Then there exist ∗ ¿ 1 (a); ∗ ¿ 0; such that for any ∈ [1 (a); ∗ ) and 0 ¡ khkq ¡ ∗ ; Eq. (5:8 ) has at least two nontrivial solutions. Remark 5.5. To the best of our knowledge, the existence results stated in Theorem 5.4 are new even for the case p = 2. Remark 5.6. Again, we remark without proof Rthat modi ed versions of Theorems 5.1 R and 5.4 hold for the cases where a ¿ 0 and a = 0. References [1] P.A. Binding, Y.X. Huang, Two parameter problems for the p-Laplacian, in: V. Lakshmikanthan, (Ed.), Proceedings of the First International Congress on Nonlinear Analysts, Walter de Gruyter, New York, 1996, pp. 891–900.
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[2] P.A. Binding, Y.X. Huang, The principal eigencurve for the p-Laplacian, Di erential Integral Equations 8 (1995) 405– 414. [3] P.A. Binding, P. Drabek, Y.X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Elect. JDE 1997 (5) (1997) 1–11. [4] H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486 – 490. [5] D.M. Cao, G.B. Li, H.S. Zhou, Multiple solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent, Proc. Roy. Soc. Edin. 124 A (1994) 1177–1191. [6] J. Chabrowski, On multiple solutions for the nonhomogeneous p-Laplacian with a critical Sobolev exponent, Di erential Integral Equations 8 (1995) 705–716. [7] P. Drabek, Y.X. Huang, Multiplicity of positive solutions for some quasilinear elliptic equation in RN with critical Sobolev exponent, J. Di erential Equations 140 (1997) 106 –132. [8] F. Huang, Existence of two solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions, Proc. Roy. Soc. Edin. 126 A (1996) 47–75. [9] Y.X. Huang, On eigenvalue problems for the p-Laplacian with Neumann boundary conditions, Proc. Amer. Math. Soc. 109 (1990) 177–184. [10] M. Lazzo, Morse theory and multiple positive solutions to a Neumann problem, Ann. Mat. Pura Appl. 168 (1995) 205–217. [11] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci., vol. 74, Springer, New York, 1989. [12] J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964) 247–302. [13] C.A. Swanson, L.S. Yu, Critical p-Laplacian problems in RN , Ann. Mat. Pura Appl. 169 (1995) 233–250. [14] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. I.H.P. Anal. non Lineaire 9 (1992) 281–304. [15] G. Tarantello, Multiplicity results for an inhomogeneous Neumann Problem with critical exponent, Manuscripta Math. 81 (1993) 57–78. [16] H.T. Tehrani, On inde nite superlinear elliptic equations, Calc. Var. 4 (1996) 139–153.
www.elsevier.nl/locate/na
Existence of multiple solutions of critical quasilinear elliptic Neumann problems Paul A. Bindinga; ∗ , Pavel Drabekb , Yin Xi Huangc a Department
of Mathematics & Statistics, University of Calgary, Calgary, AB, Canada, T2N 1N4 of Mathematics, University of West Bohemia, P.O. Box 314, 30614 Pilsen, Czech Republic c Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA b Department
Received 21 September 1998; accepted 9 December 1998
Keywords: The p-Laplacian; Multiple solutions; Critical exponent; Neumann problem
1. Introduction We are concerned with the existence of multiple solutions of −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; x ∈ @ ; @n
∗
−2
u + h(x);
x ∈ ;
(1:1 )
where is a bounded smooth domain in RN ; p u=div(|∇u|p−2 ∇u) is the p-Laplacian with 1 ¡ p ¡ N; p∗ = Np=(N − p) is the critical Sobolev exponent, ∈ R; a ∈ L∞ ( ); b ∈ C( ); a(x) and b(x) both change sign, and h ∈ Lq ( ); q = p=(p − 1). Here we say a function f(x) changes sign if the measures of the sets {x ∈ : f(x) ¿ 0} and {x ∈ : f(x) ¡ 0} are both positive.
Research of the authors was supported by NSERC of Canada and the I.W. Killam Foundation, the Grant # 201=97=0395 of the Grant Agency of the Czech Republic, a University of Memphis Faculty Research Grant, and NATO Collaborative Research Grant OUTR.CRG 961190. ∗ Corresponding author. Tel.: +403-220-6328 E-mail addresses: [email protected] (P.A. Binding), [email protected] (Y.X. Huang)
0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 1 1 8 - 2
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In a recent paper [10], Lazzo studied the existence of multiple positive solutions to −du + u = ur−1 ; @u = 0; x ∈ @ ; @n
x ∈ ;
where d ¿ 0; r ¿ 2 if N =2 and 2 ¡ r ¡ 2N=(N −2) if N ≥ 3. For equations involving the critical exponent, the situation is trickier. Earlier, Tarantello [15] considered the existence of multiple positive solutions to a nonhomogeneous Neumann problem with critical exponent. We discuss her results later and we refer to [5,6,15,8] for more references and related problems. In this paper we take a di erent approach. Although our method is variational in nature, our emphasis is di erent. Instead, we investigate the in uence of the inde nite weight function a(x) on the existence of multiple solutions. A subcritical problem −p u = a(x)|u|p−2 u + b(x)|u| −2 u; @u = 0; x ∈ @ ; @n
x ∈ ;
(1:2 )
with 1 ¡ ¡ p∗ and 6= p has been studied in [3]. Our approach in this paper is a continuation of that of Binding et al. [3]. Owing to the presence of the critical exponent, nontrivial modi cations are needed. The diculties caused by the lack of compact embedding are overcome by a procedure used earlier in [7]. Existence results are obtained by minimizing the functional induced by (1.1 ) on some disjoint subsets of the solution manifold (to be de ned below), a procedure also used by Tarantello in [14,15], and by minimizing the functional on some cones. To state our results, we introduce some notation. First, consider the related eigencurve problem −p u = a(x)|u|p−2 u + |u|p−2 u; @u = 0; x ∈ @ ; @n
x ∈ ;
where we treat the eigenvalue as a function of and a. Take R R |∇u|p − a|u|p
R : (; a) := inf |u|p u∈W 1;p ( )\{0}
(1.3)
(1.4)
Then (; a) corresponds to a unique (up to constant multiple) positive eigenfunction of Eq. (1.3). The following is known, see, e.g., [1,2,9]. Proposition 1.1. Fix a ∈ L∞ ( ): Then () is continuous and concave and (0) = 0: If a(x) ¿ 0; then () is decreasing in ; andR if a(x) ¡ 0; then () is increasing. Assume; now; that a changes sign in . (i) If a 6= 0; then the graph of intersects R the line = 0 in the (; ) plane at 0 and at a unique point 1 where 1 · a ¡ 0: R Moreover; () ¿ 0 for between 0 and 1 : (ii) If a = 0; then 0 is the only zero of () = 0 and () ¡ 0 for 6= 0.
P.A. Binding et al. / Nonlinear Analysis 42 (2000) 613 – 629
615
Suppose from now on that (A1) a ∈ L∞ ( ) and a changes sign. (H1) h ∈ Lq ( ), where q = p=(p − 1). Let 1 be given in Proposition 1.1. De ne ( R 1 if a 6= 0; R 1 (a) = 0 if a = 0: Let u1 ¿ 0 denote the normalized positive eigenfunction associated with 1 (a). In case 1 (a) = 0, we can take u1 = 1=| |. Now, we can state our main results. Let k · kq denote the Lq norm. 1.1. Main results Suppose bothR a and b change R sign, ∗with b ∈ C( ). R (I) Assume a 6= 0, with b|u1 |p ¡ 0 if 6= 0 and b ¡ 0 if = 0. Then there R exists ˜ with (˜ − 1 (a)) · a ¡ 0, with the following properties: ˜ (1.1 ) has at least one (i) Suppose h ≡ 0. Then for any strictly between 0 and , positive solution. If, in addition, p ≥ 2, then (1.1 ) has at least two positive solutions for strictly between 1 (a) and ˜ (Theorem 3.7, Lemma 4.4 and Theorem 4.6). (ii) There exists ¿ 0, such that for any h with 0 ¡ khkq ¡ , equation (1.1 ) has at least two nontrivial solutions if either p ≥ 2 and is strictly between 0 and 1 (a), or 1 (a) ≤ ¡ R˜ (Theorems R 5.1 and 5.4). (II) Assume a = 0; b ¡ 0 and 6= 0. Then equation (1.1 ) has at least two nontrivial solutions for small enough and h 6≡ 0 small enough, and two positive solutions if h ≡ 0; small and p ≥ 2R (Remarks 4.8 and 5.6). ∗ We note that the integral condition b|u1 |p ¡ 0 is not needed for the case strictly between 0 and 1 (a). We speci cally remark that Theorem 5.4 establishes the existence multiple solutions of (1.1 ) for h 6≡ 0 when the principal functional Rof nontrivial p p (|∇u| −a|u| ) is no longer positive de nite. To the best of our knowledge, results
of this kind are new even for the case p = 2. Let us brie y compare and contrast our results with those of Tarantello [15]. Consider ∗
−u + u = |u|2 −2 u + f(x); x ∈ ; @u = 0; x ∈ @ ; @n where 2∗ = 2N=(N − 2) and ¿ 0. Let kukr denote the Lr norm and Z cN (k∇uk22 + kuk22 )(N +2)=4 − fu ; f = inf kuk2∗ =1
(1:5 )
(1.6)
where cN = 4=(N − 2)[(N − 2)=(N + 2)](N +2)=4 . It is proved in [15] that for N ≥ 5 and f 6≡ 0, Eq. (1.5 ) has at least three solutions for any ¿ 0 if f ¿ 0. k∇uk22 + kuk22 is coercive, We point out that in Eq. (1.5 ), the principal functional R while for our problem the corresponding functional (|∇u|p − a|u|p ) may not be coercive. Thus we work with a b(x) which changes sign. The condition f ¿ 0 means
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f is “small” in some R sense. For the case = 0, [15] shows that (1:5)0 has at least two solutions if f 6= 0; f = 0 and another “smallness” condition f∗ ¿ 0 (see Eq. (1.3) of [15]) is satis ed. This paper is organized as follows. We rst study the case h ≡ 0. Speci cally, Section 2 analyses the geometric structure of the solution manifolds. In Section 3 we establish the existence of one solution for h ≡ 0 by adapting Ekeland’s variational principle to accommodate a constraint. In Section 4 we establish existence of two solutions under appropriate conditions. The case h 6= 0 is treated in Section 5 and we obtain the existence of two nontrivial solutions. 2. Geometric structure of the solution manifold for h ≡ 0 In Sections 2– 4, we study the case h ≡ 0. Throughout this paper the function b is always assumed to satisfy (B1) b+ 6≡ 0 and b ∈ C( ) where b+ (x) = max{0; b(x)}. To Rbe speci c, the arguments will be detailed for the case (A2) a ¡ 0, i.e., 1 (a) ¿ 0. Modi cations for the other cases will be indicated in due course. We introduce the following functional: Z Z ∗ 1 1 (|∇u|p − a|u|p ) − ∗ (2.1) b|u|p ; I (u) = p p R and write, for simplicity, J (u) = (|∇u|p − a|u|p ). Here and below the integrals are taken over unless otherwise stated. We will use k · k to denote the norm in W 1;p ( ). Clearly the functional I is well de ned on W 1;p ( ) and a critical point of I in W 1;p ( ) is a (weak) solution of (1.1 ). We de ne the solution manifold = {u ∈ W 1;p ( ) : (u) := hI0 (u); ui = 0} Z 1;p p∗ = u ∈ W ( ) : J (u) = b|u| :
(2.2)
It is clear that solutions of (1.1 ) belong to . We now de ne a subset of by 0 − = {u ∈ : h (u); ui ¡ 0}:
(2.3)
There are several equivalent expressions for this set. We list, for example, the following: Z ∗ p∗ = u ∈ : pJ (u) ¡ p b|u| − = u ∈ : J (u) =
Z
p∗
b|u|
¿0 :
ForRthe remainder of this section R we assume the following condition on b: ∗ (B2) b|u1 |p ¡ 0 if 6= 0 and b ¡ 0 if = 0. R We note that when 1 (a) = 0, the rst integral in (B2) also becomes b ¡ 0.
(2.4)
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Lemma 2.1. Assumption (B1) implies that − 6= ∅. Proof. Indeed, suppose b(x) ¿ 0 on a small ball B ⊂ and let ’ ∈ L∞ ( ) be such that supp ’ ⊂ B. Then by continuous dependence of the principal eigenvalue on the domain and the fact that as |B| → 0, the principal eigenvalue on B tends to +∞, so we can choose ’ such that J (’) ¿ 0. Now, for t ¿ 0, let Z ∗ ∗ b|’|p : (t) := (t’) = t p J (’) − t p R ∗ ∗ That is, (t) = mt p − ct p , with m = J (’) ¿ 0; c = b|’|p ¿ 0. Choosing t0 to satisfy ∗ ct0p −p = m we see that (t0 ) = 0 and 0 (t0 ) ¡ 0, and so t0 ’ ∈ − . Remark 2.2. It is clear that ∈ [0; 1 (a)] implies J (u) ≥ 0. Also, is closed in W 1;p ( ). Observe that for any u ∈ , Z ∗ 1 b|u|p : (2.5) I (u) = N It follows from Lemma 2.1 and Eq. (2.4) that inf − I ≥ 0, so I is always bounded
below on − .
Lemma 2.3. Assume (A1); (A2); (B1) and (B2) hold. Then there exist 0 ¿ 1 (a) and ¿ 0; such that for any ∈ [0; 0 ); J (u) ¿ kukp for u ∈ − . Proof. Suppose the contrary. Then there exist n ; un ∈ − n such that n → ˆ ∈ [0; 1 (a)];
1 Jn (un ) ¡ kun kp : n
(2.6)
Let vn = un =kun k. We may assume that vn → v0 weakly in W 1;p ( ) and strongly in R R ˆ ≥0 Lp ( ), for some v0 ∈ W 1;p ( ). Hence we have a|vn |p → a|v0 |p . Since ( ) by Proposition 1.1, it follows from weak lower semicontinuity of the W 1;p ( ) norm that 1 = 0: 0 ≤ J ˆ (v0 ) ≤ lim inf Jn (vn ) ≤ lim n→∞ n n→∞ Passing to a subsequence if necessary, we can therefore assume that Jn (vn ) → 0 = J ˆ (v0 ):
(2.7)
There are three possibilities: (1) v0 = 0; (2) v0 = k u1 , where 0 ¡ |k| ≤ 1 and 0 ¡ ˆ ≤ 1 (a); and (3) ˆ = 0. The rst case is impossible since it implies that vn → 0 in Lp ( ) and so by Eq. (2.7) we have vn → 0 in W 1;p ( ), contradicting kvn k = 1. For (2), we have Z Jn (vn ) = (|∇vn |p − n a|vn |p ) → 0:
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Moreover, Z Z Z n a|vn |p → ˆ a|k u1 |p = |∇(k u1 )|p ; R R by Eq. (2.7). This implies that |∇vn |p → |∇(k u1 )|p , so kvn k → kk u1 k. Thus weak convergence of vn to k u1 implies that vn → k u1 strongly in W 1;p ( ), so vn → k u1 ∗ strongly in Lp ( ). Now, we derive from un ∈ − n , Eq. (2.7) and (B2) that Z Z ∗ ∗ ∗ p∗ p∗ b|vn |p → b|k u1 |p ¡ 0: kun kp−p · Jn (vn ) ¡ p p This contradicts Eq. (2.7) R since Jn (un ) ¿ 0 by Eq. (2.4). Thus (2) cannot occur. For (3), we let vn = vn =| |. It then follows from Z Z p |vn − vn | ≤ |∇vn |p that v0 must be a constant and vn → v0 strongly in W 1;p ( ). Now, v0 6= 0 since kvn k=1, so we obtain Z Z ∗ p∗ − 1 p−p∗ p∗ b|vn | → Jn (vn ) ¡ b|v0 |p ¡ 0; 0 ≤ |un k p−1 by (B2), a contradiction. This completes the proof. Remark 2.4. An analogous result (replacing by −) holds if 1 (a) ¡ 0. If 1 (a) = 0, then there exist 0 ¿ 0 and ¿ 0 such that for with || ¡ 0 ; J (u) ≥ kukp for u ∈ − . The proof is similar and will be omitted. Similarly for Lemma 2.5 below, etc. From now on we will assume that (A1), (A2), (B1) and (B2) all hold. Lemma 2.5. There exists ¿ 0 such that −h 0 (u); ui ≥ for ∈ [0; 0 ); u ∈ − . Proof. We rst claim that there exists ¿ 0, such that kuk ¿ for ∈ [0; 0 ) and − p u ∈ − . Indeed, suppose for some un ∈ n ; kun k → 0. Dividing Eq. (2.4) by kun k and using Lemma 2.3, one obtains the contradiction Z ∗ ∗ p∗ b|vn |p · kun kp −p → 0; 0 ¡ ¡ Jn (vn ) ¡ p where vn = un =kun k. By Lemma 2.3 and the above claim, we have −h 0 (u); ui = (p∗ − p)J (u) ≥ (p∗ − p)kukp ≥ (p∗ − p)p : The lemma then follows. Corollary 2.6. For ∈ [0; 0 ); − is a closed set in . Proof. This is an immediate consequence of Lemma 2.5 and the continuity of h 0 (u); ui with respect to the W 1;p ( ) norm.
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3. Existence of one positive solution − Since I is bounded below on − , minimizing sequences exist for I on . Next, we state a generalized Ekeland variational principle, which plays a critical role at various places in this paper.
Lemma 3.1. For ∈ [0; 0 ); for any bounded minimizing sequence {un } of I on − ; there exists a modiÿed bounded minimizing sequence {zn } such that I0 (zn ) → 0. Proof. Since − is a closed set by Corollary 2.6, [11, Theorem 4:1 and Remark 4:1] imply that we can replace {un } by another minimizing sequence {zn } ⊂ − such that , kun − zn k ¡ 1=n, and for any y ∈ − 1 I (y) ¿ I (zn ) − ky − zn k: n
(3.1)
Next, we show that I0 (zn ) → 0. Choose wn of unit norm so that hI0 (zn ); wn i ≥ kI0 (zn )k − o(1) as n → ∞. It will suce to show that hI0 (zn ); wn i → 0:
(3.2)
For each n, let gn (t; s) = (t zn − s wn ). Then gn (1; 0) = 0 and @gn = h 0 (zn ); zn i = 6 0 @t
at t = 1; s = 0:
It follows from the C 1 Implicit Function Theorem that for each n, for small enough s, there exists tn ∈ C 1 so that (tn (s)zn − swn ) = 0 and h 0 (zn ); zn itn0 (0) − h 0 (zn ); wn i = 0:
(3.3)
Since zn is a bounded sequence, so is k 0 (zn )k, and we then conclude from Eq. (3.3) and Lemma 2.5 that tn0 (0) is uniformly bounded in n:
(3.4)
Until further notice we x n, and we consider vn (s) = tn (s)zn − swn − zn . Since wn is of unit norm, we have kvn (s)k ≤ |s|(1 + (|tn0 (0) + o(1)|)kzn k)
(3.5)
as s → 0. Moreover, zn ∈ gives hI0 (zn ); zn i = 0, so I (zn ) − I (tn (s)zn − swn ) = hI0 (zn ); −vn (s)i + o(vn (s)) = hI0 (zn ); swn i + o(s) (3.6) follows from Eq. (3.5). By continuity of
h 0 (u); ui,
we have
h 0 (tn (s)zn − swn ); tn (s)zn − swn i − h 0 (zn ); zn i → 0
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as s → 0. We then conclude from this and Lemma 2.5 that h 0 (tn (s)zn − swn ); tn (s)zn − swn i ¡ 0 for s small enough, so tn (s)zn − swn ∈ − . Dividing Eq. (3.6) by s and using Eqs. (3.1) and (3.5), we obtain |hI0 (zn ); wn i| ≤ n−1 (1 + (|tn0 (0)|)kzn k): Now, we let n → ∞ to show that hI0 (zn ); wn i tends to zero by boundedness of zn and Eq. (3.4). This establishes Eq. (3.2) and proves the lemma. Let S be the best Sobolev embedding constant, i.e., ) ( k∇ukpp 1;p N : u ∈ W (R )\{0} : S = inf kukpp∗ To show that a minimizing sequence as in Lemma 3.1 gives rise to a weak solution of Eq. (1:1 ), we need the concentration-compactness principle of Lions [10]: ∗
Proposition 3.2. Let {un } converge weakly to u in W 1;p ( ) such that |un |p and |∇un |p converge weakly to nonnegative measures and on ; respectively. Then; for some at mostP countable set ; we have ∗ (i) = |u|p + j∈ j xj ; P (ii) ≥ |∇u|p + j∈ j xj ; ∗
(iii) Sp=p ≤ j , j where xj ∈ ; xj is the Dirac measure at xj ; j and j are nonnegative constants.
The next result can be proved in the same way as Proposition 2:4 of [7]. Proposition 3.3. Let {un } be a bounded sequence in W 1;p ( ) such that I (un ) → c and I0 (un ) → 0. Assume also that un → u weakly in W 1;p ( ). Then the set given in Proposition 3:2 is ÿnite; b(xj ) ¿ 0 for j ∈ ; j = b(xj )j and j ≥ (S=b(xj ))N=p . Moreover; there exists a subsequence; denoted again by {un }; such that |∇un |p−2 ∇un → |∇u|p−2 ∇u weakly in ([Lp ( )]N )0 . At this stage, it is possible to obtain a solution to Eq. (1:1)0 . To obtain one that is positive, however, more preparation is needed. We make the following additional assumptions. (B3) Let b(x0 ) = kbk∞ and for some r ¿ 0, b(x) ¿ 0 for x ∈ B(x0 ; 2r) ⊂ . Without loss of generality, we assume below that x0 = 0. (A3) a(x) ≥ a0 ¿ 0 in B(0; 2r), and (B4) for x ∈ B(0; 2r), b(x) = b(0) + o(|x| )
as x → 0;
=
N −p : p−1
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De ne, for ¿ 0, u (x) = where
(x) ; ( + |x|p=(p−1) )(N −p)=p
v (x) =
u (x) ; ku (x)kp∗
∈ C0∞ (B(0; 2r)) is such that 0 ≤ (x) ≤ 1 and
(x) ≡ 1 on B(0; r).
Lemma 3.4. Let (A1)–(A3) and (B1)–(B4)hold and ¿ 0: (i) For ¿ 0 small enough; sup I (tv ) ¡ S0 := t≥0
1 N=p )=p : S kbk(p−N ∞ N
(3.7)
(ii) Assume moreover p ≥ 2. Then for any w ∈ W 1;p ( )\{0}; for ¿ 0 small enough; sup I (w + Rv ) ¡ I (w) + S0 :
(3.8)
R≥0
Proof. These are essentially local properties of the functional I so the proofs presented in [7, Lemmas 4:1 and 5:6] remain valid. Speci cally, we note that p−1 if p2 ¡ N; S0 − K (3.9) sup I (tv ) ≤ S0 − Kp−1 |ln | if p2 = N; t≥0 (N −p)=p 2 if p ¿ N; S0 − K and I (w + Rv ) ≤ I (w) + S0 − K () + C1 (N −p)(p−1)=p 2
2
−C(N −p)=p + C 0 (N −p)=p ;
2
(3.10)
where () is one of the functions of given in Eq. (3.9), depending on the situation, K; C1 ; C; C 0 are positive constants and C 0 = C=2. The lemma then follows. De ne c1 = inf I (u). Lemma 3.5. Under the assumptions of Lemma 3:4; for ∈ [0; 0 ); any minimizing sequence {un } of I on − satisfying I (un ) ¡ c1 + S0 has a subsequence converging to a solution u ∈ − of (1:1 ) and I (u) ¿ 0. Proof. Suppose {un } is an unbounded sequence that minimizes I on − . Dividing R ∗ I (un ) by kun k, we conclude that, since I (un ) is bounded, b|un |p ·kun k−1 is bounded by Eq. (2.5). Thus J (un ) · kun k−1 is also bounded by Eq. (2.2). Let vn = un =kun k. Then 0 ≤ J (vn ) → 0. This contradicts Lemma 2.3. Thus any minimizing sequence {un } in − is bounded. ∗ Assume that un → u weakly in W 1;p ( ) and in Lp ( ). Since un is a minimizing sequence, we conclude from Lemma 3.1 that I0 (un ) → 0. Then, by Proposition 3.3, |∇un |p−2 ∇un → |∇u|p−2 ∇u weakly in ([Lp ( )]N )0 . We then conclude, as in the proof of Lemma 3.1 of [7], −p u = a|u|p−2 u + b|u|p
∗
−2
u
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in W 1;p ( ), that is, u is a solution of Eq. (1:1); u ∈ and I0 (u) = 0. In particular, I (u) ≥ c1 . Suppose that j given by Proposition 3.2 is not zero for some j. We have Z 1 1 0 1 − (|∇un |p − a|un |p ) S0 + c1 ¿ I (un ) − ∗ hI (un ); un i = p p p∗ Z 1 X 1 (|∇u|p − a|u|p ) + j : (3.11) ≥ N N j∈ Using Proposition 3.3 and the fact that I0 (u) = 0 we get 1 X j S0 + c1 ¿ I (un ) ≥ I (u) + N j∈ ≥ I (u) +
S N=p 1 X ≥ c1 + S0 ; N j∈ b(xj )(N −p)=p
R ∗ aR contradiction. Thus j =0 for all j. We conclude from Proposition 3.2 that |un |p → ∗ ∗ |u|p . This together with the weak convergence of un to u in Lp ( ) implies that ∗ un → u strongly in Lp ( ). We can then show as in the proofs of [9, Lemmas 3:1 and 1;p ( ), u ∈ − 5:5(i)] that un → u strongly in W 1;p ( ). Since − is closed in W . It is easy to see from Eqs. (2.4) and (2.5) that I (u) ¿ 0. We note that the functional I is even, so we can always assume that the critical points of I are nonnegative functions. Proposition 3.6. Let u 6≡ 0 be a nonnegative nontrivial solution of Eq. (1:1 ). Then u ¿ 0 in . Proof. We note that the proof of Lemma 4:3 of [13] remains valid in our situation. It shows that the solution u ∈ Lt ( ) for all t ≥ p∗ . Then for F(x) = a|u|p−2 u + b(x)|u|p
∗
−2
u;
F ∈ L ( ) for any ¿ N=p. Then a Harnack-type inequality by Serrin [12] implies that u ¿ 0. This ends the proof. Theorem 3.7. Assume that (A1)–(A3) and (B1) – (B4) hold. Then for any ∈ (0; 1 (a)]; −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; @n
∗
−2
u;
x ∈ ;
x ∈ @ ;
has at least one positive solution in W 1;p ( ). Proof. Choose ¿ 0 small enough so that v = v in Lemma 3.4 is smooth and has support where b ¿ 0. Let g(t) = (tv) so g(0) = 0; g(t) ¿ 0 for t ¿ 0 small, and
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g(t) → − ∞ as t → ∞. Then the argument of Lemma 2.1 shows that, for some t0 ¿ 0, g(t0 )=0 and g0 (t0 ) ¡ 0. That is, t0 v ∈ − . Also, c1 =inf u∈ I (u)=0 in this case. Thus Lemmas 3:5 and 3:4(i) imply that the problem has a nonnegative nontrivial solution. It then follows from Proposition 3.6 that the solution is positive. This completes the proof. Remark 3.8. For the case ∈ (0; 1 (a)), condition (B2) is not needed. In fact one can use the Mountain Pass Theorem and a variant of Lemma 3.5 to conclude the existence of a positive solution. Remark 3.9. We speci cally note that we do not allow = 0, since it is not clear whether Lemma 3.4(i) still holds or not. Consequently, any minimizing sequence may converge only weakly to a solution u of Eq. (1.1 ), and u could be zero. 4. Existence of two positive solutions We now discuss the situation ¿ 1 (a), assuming again (A1) – (A3) and (B1) – (B4). Recall that we normalize u1 ¿ 0 such that ku1 k = 1. Lemma 4.1. Assume (B2) holds.RThere exist 1 ¿ 0 and ¿ 0 such that; for all v ∗ with kvk = 1 and kv − u1 k ¡ ; b|v|p ¡ − ¡ 0. This is an immediate consequence of (B2). Lemma 4.2. Assume (B2) holds. Then for any ˆ ∈ (0; 1); there exist ˆ ¿ 1 (a) and ˆ and any v satisfying kvk = 1 and kv − u1 k = ˆ ¿ 0; such that for any ∈ (1 (a); ) ˆ J (v) ¿ . ˆ ; Proof. Suppose the lemma is false. Then for some ˆ satisfying 1 ¿ ˆ ¿ 0; n ¿ 1 (a) ˆ we have and vn with kvn k = 1; kvn − u1 k = , 1 n → 1 (a); Jn (vn ) ¡ : n Assuming without loss of generality that vn → v0 weakly in W 1;p ( ), we conclude that, as in the proof of Lemma 2.3, either v0 = 0 or v0 = k u1 for some 0 ¡ |k| ≤ 1. The same arguments as in the proof of Lemma 2.3 show that, (i) v0 = 0 is impossible, and (ii) vn → k u1 strongly in W 1;p ( ). But kvn k = 1 and kvn − u1 k = ˆ imply that (ii) is impossible for any 0 ¡ |k| ≤ 1 since 0 ¡ ˆ ¡ 1. This concludes the proof. Corollary 4.3. Fix 0 ¡ ∗ ¡ min{1; 1 }; where 1 is given by Lemma 4:1. There exist ∗ ∗ ¿ 0; ∗ ¿ 1 (a) and ∗ ¿ 0 such R that;p∗for any∗ ∈ (1 (a); ); (i) for any v satis∗ fying kvk = 1 and kv − u1 k ≤ ; b|v| ¡ − ; (ii) for any v satisfying kvk = 1 and kv − u1 k = ∗ ; J (v) ¿ ∗ . Lemma 4.4. For ∈ (1 (a); ∗ ); there exists a solution u0 of Eq. (1:1 ) with I (u0 ) ¡ 0.
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Proof. We form a cone around u1 as follows: (4.1) U + = {tv: t ≥ 0; v ∈ W 1;p ( ); kvk = 1; kv − u1 k ≤ ∗ }: R ∗ Note that, by our construction, b|v|p ¡ 0 for v ∈ U + . For v 6= 0, v ∈ @U + , the boundary of U + , we have, by Corollary 4.3, ∗ Z ∗ tp tp tp ∗ tp ∗ p∗ b|v| ≥ + ∗ ¿ 0: (4.2) I (tv) = J (v) − ∗ p p p p On the other hand, for any v ∈ U + with kvk = 1; |J (v)| ¡ M for some M ¿ 0. We derive that ∗ Z ∗ ∗ tp tp tp tp ∗ tp tp ∗ p∗ (4.3) b|v| ≥ J (v) + ∗ ≥ − M + ∗ : I (tv) = J (v) − ∗ p p p p p p It then follows that there exists R ¿ 1 such that for any v ∈ U + with kvk=R; I (v) ¿ 0. Now, we form a bounded cone UR+ = {v ∈ U + : kvk ≤ R}. It is clear from the above that I (v) ¿ 0 for v 6= 0; v ∈ @UR+ , the boundary of UR+ . Since UR+ is bounded, I is bounded below on UR+ . On the other hand, ∗ Z ∗ tp tp b|u1 |p ¡ 0 (4.4) I (tu1 ) = J (u1 ) − ∗ p p for t ¿ 0 small enough, so inf UR+ I (u) ¡ 0. Consequently, we can take a minimizing sequence {un } of I in the interior of UR+ , and we conclude that, by Ekeland’s variational principle, (I )0 (un ) → 0. We can assume un → u0 weakly in W 1;p ( ), and hence, as in the proof of Lemma 3.5, that u0 is a weak solution of −p u0 = a(x)|u0 |p−2 u0 + b(x)|u0 |p @u0 = 0; x ∈ @ ; @n
∗
−2
u0 ;
x ∈ ;
(4.5)
R ∗ + and u0 ∈ int UR+ by Eq. (4.2) and the convexity of UR+ . Since b|v|p ¡ 0 for R v ∈ pU , we J (un ) ¡ 0 for large n. Thus by the weak convergence of un → u0 , a|un | → R have a|u0 |p , and, since Eq. (4.5) leads to the analogue of Eq. (2.5), 1 1 J (u0 ) ≤ lim inf J (un ) ¡ 0: N N n→∞ Thus u0 6= 0 is in the interior of UR+ , and I (u0 ) ¡ 0. The lemma is proved. I (u0 ) =
In order to establish the existence of a second solution, we need a local property of the functional I . Lemma 4.5. (Drabek and Huang [7, Lemma 5:7]). Given ¿ 1 (a); there are ¿ 0 small enough and R ¿ 0 large enough so that u0 + Rv ∈ − . Now, by taking w = u0 , we can apply Lemmas 3:4 (ii), 3:5 and 4:5 to conclude − that any minimizing sequence in − does converge to a nontrivial solution u ∈ of Eq. (1:1). Since I (u0 ) ¡ 0 and I (u) ¿ 0, we obtain two di erent solutions. Proposition 3.6 implies that both u0 and u are positive. We summarize as follows.
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Theorem 4.6. Let p ≥ 2. Assume that (A1) – (A3) and (B1) – (B4) hold. Then there ˜ exists ˜ ¿ 1 (a); such that for any ∈ (1 (a); ); −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; @n
∗
−2
u;
x ∈ ;
x ∈ @ ;
has at least two positive solutions in W 1;p ( ). Remark 4.7. A related inde nite Neumann problem has been studied by Tehrani [16]. Consider −u + m(x)u = b(x)up ;
u ¿ 0; x ∈
(4.6) @u = 0; x ∈ @ ; @n where p ¡ (N + 2)=(N − 2) = 2∗ − 1. Tehrani proved the following result. Suppose R b ¡ 0, then for any R ¿ 0, there exists = (R) ¡ 0 such R that for all
- Holder continuous functions m (0 ¡ ¡ 1) with |m|0; ¡ R, if (i) m ≤ 0, (ii) R ¡ 0, where (R) ¡ 1 := (−1; m) ¡ 0 (using our notation (1.4)), and (iii) b’p+1 m ’m is a positive eigenfunction associated with (−1; m), then Eq. (4.6) has a solution. Note that since −’m +m’m = R 1 ’m , we have −’m =a’m if a=1 −m, so 1 (a)=1. Proposition 1.1 then implies a ¡ 0 and a changes sign. Applying Theorem 3.7 to this situation we nd that there exists ∗ ¿ 0 such that for any ∈ (0; ∗ ), the problem −u + mu = 1 u + bu2 @u = 0; x ∈ @
@n has a solution.
∗
−1
;
u ¿ 0; x ∈
(4.7)
Remark 4.8. As mentioned earlier in Remark 2.4, results similar Rto Lemma 2.5, R Corollary 2.6 and Lemma 3.1 hold for the cases where a ¿ 0 and a = 0 for in intervals of the form (0 ; 0], (0 ; 1 (a)) or (−0 ; 0 ). Thus existence results similar to Theorems 3.7 and 4.6 hold for these cases. 5. The inhomogeneous case h 6= 0 In this section we study the case h 6= 0. Corresponding to Eqs. (2.1), (2.2) and (2.6) we now have Z Z Z 1 1 p∗ J (u) − ∗ b|u| − hu (5.1) I (u) = p p and for u ∈ with Z Z ∗ = u ∈ W 1; p ( ): J (u) = b|u|p + hu ;
(5.2)
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we have I (u) =
1 N
Z
Z ∗ 1 b|u|p − 1 − hu: p
(5.3)
We will treat the three cases 0 ¡ ¡ 1 (a); ¿ 1 (a) and = 1 (a) separately in that order. Theorem 5.1. Let p ≥ 2. Assume that ∈ (0; 1 (a)). Assume also that (A1) – (A3); (B1) – (B4) hold. Then there existsR ¿ 0 such that for any h with 0 ¡ khkq ≤ ; where q = p=(p − 1); and khkq = ( |h|q )1=q ; the problem −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; x ∈ @ ; @n has at least two nontrivial solutions.
∗
−2
u + h;
x ∈ ;
Proof. We rst observe that, for ∈ (0; 1 (a)), Z (|∇u|p − a|u|p ) ≥ kukp
(5.4)
(5.5)
for some ¿ 0 (cf. Proposition 2 of Binding et al. [3]). Hence we have, by the Sobolev embedding theorem and Young’s inequality, for any ¿ 0, there is C ¿ 0 so that ∗
I (u) ≥ kukp − C1 kbk∞ · kukp − Ckuk · khkq ∗
≥ ( − )kukp − C1 kbk∞ · kukp − C khkqq : It follows, choosing then small enough, that there exist () ¿ 0 and c00 () ¿ 0 such that I (u) ≥ c00 ();
(5.6)
for kuk = and khkq ≤ (). Choose = 0 so that c00 := c00 (0 ) ¿ 0, and write := (0 ). Now, for kuk ≤ 1 ¡ 0 , I (u) is bounded above by ∗
f(1 ) := Ap1 + Bp1 + 1 for suitable constants A and B. We choose 1 small enough to satisfy c00 and c00 (1 ) ¿ 0: 2 R Since h 6≡ 0, there exists ’ ∈ C0∞ ( ) such that h’ ¿ 0. It then follows that Z Z ∗ Z ∗ tp tp (|∇’|p − a|’|p ) − ∗ I (t’) = b|’|p − t h’ ¡ 0; p p f(1 ) ¡
for t ¿ 0 small enough. We can assume that t is so small that kt’k ¡ 1 . Thus we have c = inf {I (u) : u ∈ S1 } ¡ 0; where S1 ={u : kuk ≤ 1 }. Now, taking a minimizing sequence {un } such that kun k ≤ 1 and I (un ) → c, we have, by Ekeland’s variational principle (cf. [11]), for some z with
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kzk ¡ 1 , I0 (un ) → 0 and un → z weakly in W 1;p ( ). A proof similar to the rst part of that of Lemma 3.5 (using Eq. (5.3) instead of Eq. (2.5), and using Proposition 3.3) then shows that I0 (z) = 0, i.e., z is a solution of Eq. (1:1). Convexity of S1 shows that z ∈ S1 , so I (z)R ≤ c00 =2. We have, by Lemma 3.4(ii) (with v replaced by v˜ = ±v chosen so that hv˜ ≥ 0), for p ≥ 2, I (z + Rv ) ¡ I (z) + S0 ;
(5.7)
and by Theorem 2 of [4], I (z + Rv ) = I (z) + I (Rv ) + o(1) → − ∞ as R → ∞. Let R0 be such that I (z + R0 v ) ¡ 0 and kz + R0 v k ≥ . De ne = { ∈ C([0; 1]; W 1;p ( )) : (0) = z; (1) = z + R0 v }; and let c2 = inf max I ( (t)):
∈ t∈[0;1]
Then by Eq. (5.6), 0 ¡ c00 ≤ c2 ¡ I (z) + S0 . [11, Corollary 4.3] implies that there exists un such that I (un ) → c2 and I0 (un ) → 0. We claim that such un is bounded. Indeed, since I0 (un ) → 0, hI0 (un ); un i = o(kun k), so we have Z 1 1 1 1 − ∗ J (un ) − 1− ∗ hun = I (un ) − ∗ hI0 (un ); un i ¡ c3 + o(kun k) p p p p Then for some c3 ¿ c2 and n large. With Eq. (5.5) we see that kun kp is bounded. R ∗ ) hu and (1 − Eq. (5.7) and the nal part of the proof of Lemma 3.5 (with (1 − 1=p n R 1=p∗ ) hu subtracted from the right-hand side of Eq. (3.11)) show that un converges to some u strongly in W 1; p ( ), and u is a solution of Eq. (1:1). Moreover, I (u)=c2 ¿ 0. Thus u 6= z and we have obtained two nontrivial solutions. This completes the proof. Remark 5.2. Theorem 5.1 does not permit us to obtain two nontrivial solutions under the conditions of Theorem 3.7, since the solution z obtained above degenerates to zero when h = 0. Remark 5.3. Cao et al. [5] and Chabrowski [6] also proved the existence of multiple solutions similar to Theorem 5.1, on RN and bounded domains with Dirichlet boundary conditions, respectively. Their approaches are based on Mountain Pass arguments. Next, we study the case ¿ 1 (a). The idea is based on a perturbation argument. We rst construct a truncated cone UR+ , as in the proof of Lemma 4.4, so that the unperturbed functional Z Z ∗ 1 1 0 p p (|∇u| − a|u| ) − ∗ b|u|p I (u) = p p has a Rnontrivial minimizing sequence in the interior of UR+ , and then we perturb I0 by − h· (by choosing h small) so that the corresponding sequence for I remains in int UR+ .
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We note that the proof of Lemma 4.4 holds identically for I0 , so I0 (v) ≥ 0 for v ∈ @UR+ and c0 := inf + I0 (u) ¡ 0 u∈UR
by Eq. (4.4). Observe that, for u ∈ UR+ , Z Z Z ∗ 1 1 1 1 p∗ b|u| − hu ≥ J (u) − ∗ b|u|p − khkq · kuk: I (u) = J (u) − ∗ p p p p Since UR+ is bounded, one easily sees that there exists ∗ ¿ 0 such that for 0 ¡ khkq ¡ ∗ , I (u) ¿ c0 =4 for u ∈ @UR+ , and I (u0 ) ¡ c0 =2. Again, I is bounded below on UR+ , so a minimizing sequence for I can be taken from the interior of UR+ . Thus, as in the proof of Lemma 4.4, minimizing I on UR+ yields a nontrivial solution u ∈ int UR+ of −p u = a(x)|u|p−2 u + b(x)|u|p @u = 0; @n
∗
−2
u + h;
x ∈ ;
x ∈ @ :
(5:8 )
The case = 1 (a) will be treated as a limit case by letting → 1 (a). Fix ∗ ; ∗ as in Corollary 4.3 and ∗ as above. For n ∈ (1 (a); ∗ ), let n → 1 (a). Then we have a bounded sequence of solutions {un } of Eq. (5:8)n . We can assume un → u+ weakly in W 1; p ( ) and since each un satis es I0n (un ) = 0, and n → 1 (a), we can show directly that I01 (a) (u+ ) = 0. That is, u+ is a weak solution of Eq. (5:8)1 (a) . Since h 6= 0; u+ 6= 0. Note that we can carry out the identical procedure on the opposite cone U − = {tv: t ≤ 0; v ∈ W 1; p ( ); kvk = 1; kv − u1 k ≤ ∗ } and UR− = {u ∈ U − : kuk ≤ R}, and obtain a weak solution u of Eq. (5:8 ) in the interior of UR− . Noting that UR+ ∩ UR− = {0}, we obtain our nal result. Theorem 5.4. Assume that (A1); (A2); (B1) and (B2) hold. Then there exist ∗ ¿ 1 (a); ∗ ¿ 0; such that for any ∈ [1 (a); ∗ ) and 0 ¡ khkq ¡ ∗ ; Eq. (5:8 ) has at least two nontrivial solutions. Remark 5.5. To the best of our knowledge, the existence results stated in Theorem 5.4 are new even for the case p = 2. Remark 5.6. Again, we remark without proof Rthat modi ed versions of Theorems 5.1 R and 5.4 hold for the cases where a ¿ 0 and a = 0. References [1] P.A. Binding, Y.X. Huang, Two parameter problems for the p-Laplacian, in: V. Lakshmikanthan, (Ed.), Proceedings of the First International Congress on Nonlinear Analysts, Walter de Gruyter, New York, 1996, pp. 891–900.
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