EXISTENCE AND MULTIPLICITY OF NONTRIVIAL ... - Project Euclid

Advances in Di↵erential Equations

Volume 1, Number 2, March 1996, pp. 219 – 240

EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS IN SEMILINEAR CRITICAL PROBLEMS OF FOURTH ORDER* F. Bernis, J. Garcia-Azorero, and I. Peral Departamento de Matem´ aticas, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain

(Submitted by: James Serrin) ⇤

Abstract. In this paper we consider the equation 2 u = |u|q 2 u + |u|2 2 u ⌘ f (u) in @u a smooth bounded domain ⌦ ⇢ RN with boundary conditions either u|@⌦ = @n |@⌦ = 0 or ⇤ u|@⌦ = u|@⌦ = 0, where N > 4, 1 < q < 2, > 0 and 2 = 2N/(N 4). We prove the existence of 0 such that for 0 < < 0 the above problems have infinitely many solutions. For the problem with the second boundary conditions, we prove the existence of a positive solution also in the supercritical case, i.e., when we have an exponent larger than 2⇤ . Moreover, in the critical case, we show the existence of at least two positive solutions.

1. Introduction. In this paper we study the following fourth-order problems, 2

u = |u|q

u|@⌦ = 0, and

2

@u @n

u = |u|q

u|@⌦ = 0,

2

⇤

u + |u|2

2

u

in ⌦, (P1)

= 0, @⌦

2

⇤

u + |u|2

2

u

in ⌦,

u|@⌦ = 0,

(P2)

where ⌦ ⇢ RN is a smooth bounded domain, N > 4, 1 < q < 2, > 0 and 2⇤ = 2N/(N 4), the critical Sobolev exponent for fourth-order problems. Hereafter we ⇤ write f (u) = |u|q 2 u + |u|2 2 u. Critical growth in semilinear and quasilinear problems of second order has been extensively studied in the last years, starting with the seminal paper [6]. See [10] and [11] for an extensive list of references. For fourth-order equations there are some results for the case q = 2 and 2⇤ > q > 2, see the references [9] and [16] for existence results and [18] for nonexistence theorems. Other results about existence and nonexistence for (P 1) and (P 2) and also for related systems can be seen in [17] and in [12]. In this work one of the main points is to prove the existence of infinitely many solutions for problems (P 1) and (P 2), independently of the dimension. The proof of Received for publication May 1995. *Work partially supported by D.G.I.C.Y.T. (M.E.C., Spain) Project PB94-0187. AMS Subject Classifications: 35J35, 35J40, 35J70, 46E35, 58E05, 58E30.

219

220

F. BERNIS, J. GARCIA-AZORERO, I. PERAL

the existence of at least two positive solutions for problem (P 2) is another main point of our work. We study the existence of solutions understood as critical points of the energy functional Z Z Z ⇤ 1 1 2 q J(u) = | u| dx |u| dx |u|2 dx. (1.1) ⇤ 2 ⌦ q ⌦ 2 ⌦ For the first problem (P 1), J is defined in W02,2 (⌦); for the problem (P 2), J is defined in W 2,2 (⌦) \ W01,2 (⌦). Then a critical point must be understood in the following way: (1) u 2 W02,2 (⌦) is a critical point associated to problem (P 1) if 0=

Z

u

dx

⌦

Z

f (u) dx for all

⌦

2 W02,2 (⌦),

(2) u 2 W 2,2 (⌦) \ W01,2 (⌦) is a critical point associated to problem (P 2) if 0=

Z

⌦

u

dx

Z

f (u) dx for all

⌦

2 W 2,2 (⌦) \ W01,2 (⌦).

In (1) integration by parts shows that the critical points of J are weak solutions of problem (P 1). In (2) it is not directly clear why the second boundary condition must be satisfied by a critical point. We need some information about the regularity of such critical points. Then the organization of the paper is the following. In Section 2 we prove the regularity results that we need. Section 3 will be devoted to the proof of a Local PalaisSmale condition. The main tool here is the P.L. Lions concentration-compactness result. See [13], [14]. We will use at this point the result about the best constant of the Sobolev inclusion in [22]. The application of Ljusternik-Schnirelmann methods allow us to establish the existence of infinitely many solutions for small enough. This is the content of Section 4. In Section 5 we obtain a positive solution for (P 2), also for supercritical problems, by classical methods and for 0 < < ⇤. Section 6 contains the extension of a well-known result by Brezis and Nirenberg ([7]) that we apply in Section 7 to show the existence of a second positive solution for problem (P 2) on the whole interval (0, ⇤). In the last section we obtain further results, for instance, some extensions to the quasilinear contexts. 2. About regularity. The regularity for problem (P 1) can be seen in the paper by S. Luckhaus ([15]). The problem (P 2) must be considered in a di↵erent way because the second boundary condition is not included in the natural space W 2,2 (⌦) \ W01,2 (⌦). Consider the linear problem 2

u = g(x) in ⌦,

u|@⌦ = 0,

u|@⌦ = 0,

(PL2)

CRITICAL PROBLEMS OF FOURTH ORDER

221

where we assume g 2 C 1 (⌦). It is well known by the standard L2 theory that (P L2) has a unique C 1 solution. Then we can use, for instance, the classical Agmon-DouglisNirenberg a priori estimates, ||u||W 4,p  c||g||p ,

1 < p < 1;

(2.1)

see [2]. Moreover, given g 2 Lr , 1r + 21⇤ = 1, the problem (P L2) has a unique weak solution satisfying Z Z 0= u dx g dx for all 2 W 2,2 (⌦) \ W01,2 (⌦). ⌦

⌦

Approximate g in Lr by C01 (⌦) functions and get ||u||W 4,r  c||g||r ; then in the sense of traces u 2 W 4 (1/r),r on @⌦. Hence, integration by parts shows that u satisfies the second boundary condition. Therefore a critical point u of J in W 2,2 (⌦) \ W01,2 (⌦) is a weak solution of (P L2). Let u be a critical point of J in W 2,2 (⌦) \ W01,2 (⌦); i.e., Z Z 0= u dx f (u) dx for all 2 W 2,2 (⌦) \ W01,2 (⌦). ⌦

⌦

Then g(x) = f (u(x)) 2 Lr (⌦), shows that

1 r

=1

1 2⇤

⇤

because u 2 L2 (⌦), and the argument above

||u||W 4,r  c||g||r .

Taking in particular as test functions 2 C01 (⌦) and taking into account the regularity of @⌦ we see that the equation is satisfied in ⌦ and the boundary conditions are satisfied in the sense of traces as above. This remark is crucial because now we can prove the following regularity lemma, essentially contained in [22]. Theorem 2.1. Assume that @⌦ is a C 4,↵ manifold. Let u be a critical point for J in W 2,2 (⌦) \ W01,2 (⌦). Then u 2 C 4,↵ (⌦) \ C 3 (⌦).

Proof. Let u be a critical point of J in W 2,2 (⌦) \ W01,2 (⌦). Then by the previous remark u is a solution of the problem 2

u = a(x)u + g(x)

u|@⌦ = 0,

u|@⌦ = 0,

in ⌦,

(PL)

where a(x) = 0 if u < 1, a(x) = f (u)/u if u 1 and g(x) = f (u) if u < 1, g(x) = 0 otherwise. It is obvious that a 2 LN/4 (⌦) and g 2 L1 (⌦). Moreover in [22] the following is proved. Claim: For all ✏ > 0 there exist q✏ 2 LN/4 (⌦) and g✏ 2 L1 (⌦) such that i) ||q✏ ||N/4  ✏ ii) q✏ (x)u(x) + g✏ (x) = a(x)u(x) + g(x), x 2 ⌦.

222


Let q✏ and g✏ be as in the claim, so that 2 u = q✏ u + g✏ . Consider the operator ( 2 ) 1 defined in the space of functions satisfying our boundary conditions. ⇤ Put A✏ u = ( 2 ) 1 (q✏ u) and h✏ = ( 2 ) 1 g✏ ; then u A✏ u = h✏ , and u 2 L2 , or in an equivalent way u = (I A✏ ) 1 h✏ . Now, for ✏ small enough, A✏ : Lp ! Lp

for all p

2⇤ ,

with ||A✏ ||p,p < 1/2. Indeed, by the Hardy-Littlewood-Sobolev inequality we get ||A✏ v||p  c(p)||q✏ v||r

if

1 1 4 = + , r p N

and H¨ older’s inequality provides the final estimate ||A✏ v||p  c(p)||q✏ ||N/4 ||v||p 

1 ||v||p , 2

if c(p)✏ < 1/2. In conclusion, for such an ✏, we obtain ||(I

A✏ )

1

||p,p  2,

and in turn, ||u||p  ||(I

A✏ )

1

h✏ ||p  2||h✏ ||p  2C(p)||g✏ ||1 ;

in others words u 2 Lp for all p 2 [1, 1). This is sufficient to deduce the desired regularity, by the Sobolev inclusion and the Agmon-Douglis-Nirenberg estimates. The local Palais-Smale condition. We denote by E either W02,2 (⌦) or W 2,2 (⌦) \ W01,2 (⌦) and define ||u||E = || u||2 . A sequence {uj } ⇢ E is said to be a Palais-Smale sequence for J, defined by (1.1), if 3.

(

J(uj ) ! c

J 0 (uj ) ! 0 in E0 .

(3.1)

If (3.1) implies the existence of a subsequence {ujk } ⇢ {uj } which converges strongly in E, we say that J satisfies the Palais-Smale condition. If this strongly convergent subsequence exists only for some values of c, we say that J satisfies a local Palais-Smale condition. ⇤ In our case, the main difficulty is the lack of compactness in the inclusion of E in L2 . Here, we shall prove a local Palais-Smale condition, which is sufficient for the problem. The technical results used are based on a measure representation lemma, given by P. L. Lions in the proof of the concentration-compactness principle (see [13] and [14]).


223

Let {uj } be a bounded sequence in E. Then there is a subsequence such that uj * u weakly in E,

and

⇤

| uj |2 * dµ,

|uj |2 * d⌫

weakly-* in the sense of measures. If we take 2 C 1 , the Sobolev inequality applied to uj gives: ✓Z ◆ 21⇤ ✓Z ◆ 12 ✓Z ◆ 12 ✓Z ⇤ 1 | |2 d⌫ S2  | |2 dµ +2 hr , rui2 dx + | ⌦

⌦

⌦

⌦

◆ 12 |2 |u|2 dx ,

(3.2) where S = inf{||u||2E : u 2 E, ||u||2⇤ = 1} is the best constant in the Sobolev inclusion. The main idea is that if u ⌘ 0 in (3.2), then we have a reverse H¨ older inequality for two di↵erent measures, for which we have the following representation result (See P. L. Lions, [13] and [14]): Lemma 3.1. Let µ, ⌫ be two nonnegative and bounded measures on ⌦, such that for 1  p < r < 1 there exists some constant C > 0 for which ✓Z ◆ r1 ✓Z ◆ p1 | |r d⌫ C | |p dµ 8 2 C 1 (⌦), with supp( ) bounded. ⌦

⌦

Then there exist {xj }j2I ⇢ ⌦ and {⌫j }j2I ⇢ (0, 1), where I is at most countable, such that: X X p ⌫= ⌫j xj , µ C p ⌫jr xj , j2I

where

xj

j2I

is the Dirac mass supported at xj .

Remark. If ⌦ is bounded then I is finite. By application of this result to vj = uj u, P.L. Lions obtains (see [13] and [14]): Lemma 3.2. Let {uj } be a weakly convergent sequence in E with weak limit u. Assume also i) | uj |2 converges in the weak-* sense of measures to a measure µ ii)

⇤

|uj |2

converges in the weak-* sense of measures to a measure ⌫.

Then there exist {xj } ⇢ ⌦, 1)

2) 3)

j = 1, 2, . . . , l, such that ⇤

⌫ = |u|2 +

l X

| u|2 +

µ 2 2⇤

⌫j

µj  . S

⌫j

xj ,

⌫j > 0,

j=1

N X

µj

xj ,

µj > 0,

j=1

Lemma 3.2 allows us to prove the following important result.

(3.3)

224


Lemma 3.3. Let {vj } ⇢ E be a Palais-Smale sequence for J, defined by (1.1); that is, J(vj ) ! c J 0 (vj ) ! 0

(3.4) E0 .

in

(3.5)

2 N 2⇤ S 4 K , where = ⇤ and K depends on q, N and ⌦, then there exists N 2 q a subsequence {vjk } ⇢ {vj }, which converges strongly in E. If c
0 we have 1 0 J(vj ) hJ (vj ), vj i h✏j , vj i 2◆⇤ Z ✓ ✓ 1 1 1 2 = | v | dx j 2 2⇤ q ⌦

1 2⇤

◆Z

⌦

|vj |q dx

h✏j , vj i

and the boundedness of {vj } in E follows easily from the Sobolev inequality. By Lemma 3.2 there exists a subsequence, which we continue to denote by {vj }, such that 8 vj * v weakly in E, > > > > > vj ! v in Lr , 1 < r < 2⇤ , and a.e. in ⌦, > > > X < | vj |2 * dµ | v|2 + µk xk , (3.6) > > k2J > > X > > 2⇤ 2⇤ > |v | * d⌫ = |v| + ⌫k xk . j > : k2J

Take xk 2 ⌦ in the support of the singular part of dµ, d⌫, and ⌘ 1 on B(xk , ✏),

⌘ 0 on B(xk , 2✏)c ,

|r | 

Consider the sequence { vj }, where (x) = (x) is bounded in E. Then, by the hypothesis (2.5),

⌦ (x);

2 ✏

2 C 1 (RN ) , such that

and |

⌦

⌦

j!1

⌦

2 . ✏2

(3.7)

it is obvious that this sequence

limhF 0 (vj ), vj i = 0, where h, i is the duality pairing. Moreover Z Z Z d⌫ + |v|q dx = lim

|

vj (vj ) dx.


225

By (3.6), (3.7), the weak convergence and the H¨ older inequality, we can estimate Z Z Z lim vj (vj ) dx = dµ + lim vj [2hrvj , r i + vj ] dx . j!1

⌦

j!1

⌦

⌦

Now Z

0  lim

j!1

C C C

vj hrvj , r i dx  lim

j!1

⌦

✓Z

✓Z

B(xk ,✏)\⌦

✓Z

B(xk ,✏)\⌦

✓Z

B(xk ,✏)\⌦

◆1/2 |r |2 |rv|2 dx ◆1/N ✓Z |r |N dx

⌦

◆1/2 ✓Z ◆1/2 2 2 | vj | dx |r | |rvj | dx 2

⌦

B(xk ,✏)\⌦

◆(N 2) dx

|rv|2N/(N

|rv|2N/(N

2)

2)/2N

◆(N dx

2)/2N

! 0 as ✏ ! 0,

and 0  lim j!1 ⇣Z C

Z

vj vj

⌦

B(xk ,✏)\⌦

C

⇣Z

|

dx  lim

j!1

⇣Z

⌦

⌘ 12 ⇣Z |2 |v|2 dx  C

⌘1/2 ⇣ Z | vj |2 dx |

B(xk ,✏)\⌦

|

⌘1/2⇤ |v|2 dx ! 0 as ✏ ! 0. ⇤

B(xk ,✏)\⌦

Then 0 = lim

⇢Z

✏!0

Z

d⌫ +

⌦

By Lemma 3.2, we have µk

⌦

|v|q dx

2

S⌫k2⇤ ; that is, ⌫k

⌘1/2 |2 |vj |2 dx ⌦ ⌘2/N ⇣Z ⌘1/2⇤ ⇤ |N/2 dx |v|2 dx B(xk ,✏)\⌦

Z

dµ

= ⌫k

µk .

⌦ 2

S⌫k2⇤ . Hence, either ⌫k = 0, or

N

⌫k

S4.

(3.8)

We shall prove that (3.8) does not occur. Assume for the sake of contradiction that N there exists a k0 with ⌫k0 6= 0; i.e., ⌫k0 S 4 . From the hypotheses (3.4) and (3.5), Z Z ⇤ 2 2 N 1 1 c = lim J(vj ) = lim {J(vj ) 12 hJ 0 (vj ), vj i} |v|2 + S 4 + |v|q . j!1 j!1 N ⌦ N 2 q ⌦ (3.9) Because 1 < q < 2 , by applying the H¨ older inequality in (3.9), we have c

2 N 2 S4 + N N

Z

⌦

2⇤

|v|

✓

1 q

◆ ✓Z ◆ 2q⇤ 2⇤ q ⇤ 1 2 ⇤ |⌦| 2 |v| . 2 ⌦

226

F. BERNIS, J. GARCIA-AZORERO, I. PERAL ⇤

Let f (x) = c1 x2 c2 xq . This function attains its absolute minimum (for x > 0) at ✓ ◆ 1 c2 q 2⇤ q the point x0 = . Hence, 2⇤ c1 f (x)

f (x0 ) =

K

2 4 which contradicts the hypothesis that c < S N N proof is completed.

2⇤ 2⇤ q

,

K

. Hence ⌫k = 0 8k, and the

4. Existence P of infinitely many solutions. Let E be the Hilbert space defined in Section 3. Let be the class of subsets of E {0} which are closed and symmetric P with respect to the origin. For A 2 , we define the genus (A) by (A) = min{k 2 N : 9 2 C(A; Rk

{0}), (x) =

( x)}.

If such a minimum is not defined then we consider (A) = +1. The main properties of the genus are the following (see [20] for the details): P Proposition 4.1. Let A, B 2 . Then 1) 2) 3) 4) 5) 6) 7)

If there exists an odd function f 2 C(A, B), then (A)  (B). If A ⇢ B , then (A)  (B). If there exists an odd homeomorphism between A and B, then (A) = (B). If S N 1 is the sphere in RN , then (S N 1 ) = N . (A [ B)  (A) + (B). If (B) < +1, then (A B) (A) (B). If A is compact, then (A) < +1, and there exists > 0 such that (A) = (N (A)), where N (A) = {x 2 E : d(x, A)  }. 8) If X0 is a subspace of E with codimension K, and (A) > K, then A \ X0 6= ;.

Assume that 1 < q < 2 in (1.1). Then, by Sobolev’s inequality we obtain: 1 2

J(u)

Z

⌦

2

| u| dx

1 2⇤ S

2⇤ 2

✓Z

⌦

◆ 22⇤ | u| dx 2

q

Cq

✓Z

⌦

◆ q2 | u|2 dx .

Consequently J(u)

h(k uk2 ), where h(x) =

1 2 x 2

1 2⇤ S

2⇤ 2

⇤

x2

q

Cq xq .

(4.1)

There exists o > 0 such that, if 0 < < o , h attains a local minimum and a local maximum. Let R0 , R1 be such that r < R0 < R < R1 , where R is the value for which h attains its maximum and r is the value for which h attains its minimum, and h(R1 ) > h(r).


227

We make the following truncation of the functional J. Take ⌧ : R+ ! [0, 1], nonincreasing and C1 , such that ⌧ (x) = 1 if x  R0 and ⌧ (x) = 0 if x R1 . Let '(u) = ⌧ (krukp ). We consider the truncated functional Z Z Z 1 1 2 2⇤ ˜ J(u) = | u| dx |u| '(u) dx |u|q dx. (4.2) 2 ⌦ 2⇤ ⌦ q ⌦ ˜ As in (4.1), J(u)

h(k uk2 ), with h(x) =

1 2 x 2

1 ⇤

2 2⇤ S 2

⇤

x2 ⌧ (x)

q

Cq xq .

(4.3)

Observe that h = h for x  R0 , and h(x) = 12 x2 q Cq xq for x R1 . ˜ defined by (4.2), are the following: The main properties of J, Lemma 4.2. 1) J˜ 2 C1 (E, R). ˜ ˜ 2) If J(u)  0, then k uk2 < R0 , and J(v) = J(v) for all v in a small enough neighborhood of u. 3) There exists A > 0, such that, if 0 < < A, then J˜ satisfies a local Palais-Smale condition for c  0. Proof. 1) and 2) are immediate. To prove 3), observe that all the Palais-Smale sequences for J˜ with c  0 must be bounded; then, by Lemma 3.1, if verifies 1 Np S K 0 there exists a convergent subsequence. N ˜ then by 2) we have a negative Note that if we find some negative critical value for J, critical value of J. Now, we will construct an appropriate mini-max sequence of negative ˜ critical values for the functional J. The next lemma uses the same idea as in [11], and we include here the proof for the sake of completeness. Lemma 4.3. Given n 2 N, there is ✏ = ✏(n) > 0, such that ˜ ({u 2 E : J(u) 

✏})

n.

Proof. Fix n and let En be an n-dimensional subspace of E. Take un 2 En , with || un ||2 = 1. For 0 < ⇢ < R0 , we have Z Z 1 2⇤ 2⇤ q ˜ n ) = J(⇢un ) = 1 ⇢2 J(⇢u ⇢ |u| ⇢ |u|q . 2 2⇤ q ⌦ ⌦ All the norms are equivalent in En . Define ⇢Z ⇤ ↵n = inf |u|2 : u 2 En , || un ||2 = 1 > 0, ⇢Z⌦ |u|q : u 2 En , || un ||2 = 1 > 0. n = inf ⌦

228


n q ˜ n )  1 ⇢2 ↵n ⇢2⇤ Hence J(⇢u ⇢ , and we can choose ✏ (which depends on n) and 2 2⇤ q ˜ ⌘ < R0 such that J(⌘u)  ✏ if || u||2 = 1. ˜ Let S⌘ = {u 2 E : || u||2 = ⌘} so that S⌘ \ En ⇢ {u 2 E : J(u)  ✏}; therefore, by Proposition 4.1,

˜ ({u 2 E : J(u) 

✏})

(S⌘ \ En ) = n.

This lemma allows us to prove the existence of critical points. Lemma 4.4. Let ⌃k = {C ⇢ E {0} : C is closed, C = C, (C) k}. Let ck = ˜ ˜ inf C2⌃k supu2C J(u), Kc = {u 2 E : J˜0 (u) = 0, J(u) = c}, and suppose 0 < < A, where A is the constant in Lemma 4.2. If c = ck = ck+1 = · · · = ck+r , then (Kc ) r + 1. (In particular, the numbers ck are critical values of J.) Proof. In the proof we will use Lemma 4.3 and a classical deformation lemma (see [20]). ˜ For simplicity, put J˜ ✏ = {u 2 E : J(u)  ✏}. By Lemma 4.3, 8k 2 N, 9✏(k) > 0 ✏ ˜ such that (J ) k. Because J˜ is continuous and even, J˜ ✏ 2 ⌃k ; then, ck  ✏(k) < 0, 8k. But J˜ is bounded from below; hence, ck > 1 8k. Let us assume that c = ck = · · · = ck+r and observe that c < 0; therefore, J˜ satisfies the Palais-Smale condition in Kc . It is easy to see that Kc is compact. Assume for the sake of contradiction that (Kc )  r. Thus there exists a closed and symmetric set U , with Kc ⇢ U , such that (U )  r. (We can choose U ⇢ J˜0 , because c < 0.) By the deformation lemma, we have an odd homeomorphism ⌘ : E ! E, such that ⌘(J˜c+ U ) ⇢ J˜c , for some > 0. (Again, we must choose 0 < < c, because J˜ satisfies the Palais-Smale condition on J˜0 , and we need J˜c+ ⇢ J˜0 .) By definition, c = ck+r =

inf

˜ sup J(u).

C2⌃k+r u2C

˜ Then, there exists A 2 ⌃k+r , such that supu2A J(u) 0, such that for 0 < < A, problems (P 1) and (P 2) admit infinitely many solutions. Proof. Integration by parts shows that any critical point is a solution of (P 1). In fact, the boundary conditions are included in the choice E = W02,2 (⌦). For problem (P 2), the result is a consequence of the regularity result given in Section 2.


229

5. Existence of a positive solution. Consider the problem 2

u = |u|q

2

u|@⌦ = 0,

u + |u|r

2

u

in ⌦,

(P3)

u|@⌦ = 0,

where ⌦ ⇢ RN is a smooth bounded domain, N > 4, 1 < q < 2, > 0, r > 2. This means that we consider also supercritical problems. (Obviously, problem (P 2) is a particular case of (P 3).) The Laplacian case of (P 3) has been already treated in [5]. Notice that for 2 with these boundary conditions, the maximum principle holds as a consequence of the maximum principle for the Laplacian. In this section, we will show the following result. Theorem 5.1. There is a constant a positive solution.

> 0 such that for 0
0 for which

Proof. For simplicity we write |u|r 2 u + |u|q 2 u = F (u). For fixed consider the solution v of the Dirichlet problem (S). Then 0 < v < K in ⌦. Define u(x) = T v(x), where T is chosen in such a way that if a) 2 u = T ( + 1) and b) F (u)  T q 1 M q 1 + T r 1 M r 1 , M = max {1, kvk1 }, then 2

u

F (T M )

F (u).

(5.1)

Note that (5.1) is equivalent to ( + 1)

Tq

2

Mq

1

and that M depends linearly on . Define (T ) = c1 T q 2 + c2 T r 2 (c1 = M q lim

T !0+

(T ) = lim

T !1

+ Tr 1

2

Mr

1

, c2 = M r

1

,

(5.2)

). Then

(T ) = 1,

because q 2 < 0 < r 2. Thus attains a minimum in [0, 1). By elementary 1 0 q 2 1 computations we have (T ) ⌘ c3 T + c4 T r 2 1 = 0 in T0 = c5 r q , where c5 = M 1 (r q) 1 (2 q).

230

F. BERNIS, J. GARCIA-AZORERO, I. PERAL r

2

For the validity of (5.2), it suffices that (T0 )  + 1, that is c6 r q < + 1. Then, there is a constant 0 such that u(x) = T0 v is a supersolution of (P 3) for 0 < < 0 . Let 1 be the positive solution of the eigenvalue problem, 2

u=

1u

u|@⌦ = 0,

u|@⌦ = 0,

with k 1 k1 = 1, corresponding to the first eigenvalue 1 of the operator 2 with these boundary conditions. (We remark that in fact 1 coincides with the first eigenfunction of the Laplacian.) Lemma 5.3 (Construction of a subsolution). For t small enough, u = t lution of (P 3) such that u  u.

1

is a subso-

Proof. Define u(x) = t 1 (x). Then 2 u = 1 t 1 , and u is a subsolution of (P3) if q 1 q 1 + tr 1 r1 1 . In turn, for t small enough, 1t 1  t 1 1t 1

q 1 1 (t 1 )



 (t

q 1 1)

+ (t )r

1

.

(5.3)

Fix the supersolution u, i.e., T . Then, for t small enough, we get 2

u=t

1 1

t

1

 F (T M ) 

2

u

by (5.1). Then u is a subsolution and by the weak comparison principle (which holds for problem (P3)), u  u. Proof of Theorem 5.1. By iteration and comparison, we get a solution between the subsolution and the supersolution. Remarks. 1) The result is independent of the dimension for small . 2) Taking into account Remark 1), it seems an interesting question to study the behavior of solutions of (P3) when q ! 2. Fix p and such that (P3) has a solution uq > 0. Then ||uq ||1 ! 0 as q ! 2. It suffices to observe that the constant c5 in the proof of Lemma 5.2 goes to zero as q ! 2. Proposition 5.4. There is solutions. Proof. Let

1

> 0 such that for all

>

problem (P 3) has no positive

> 0 be as in Lemma 5.3. Integration by parts shows Z Z Z q 1 r 1 2 ( |u| + u ) 1 dx = u 1 dx = 1 u 1 dx. ⌦

⌦

↵

q 1

But, for some ↵ > 0, c u  u of (P3) necessarily c ↵ < 1 .

r 1

+u

⌦

8u > 0, and then if u is a positive solution

6. On a result by Brezis and Nirenberg. In the paper [7], Brezis and Nirenberg obtain a remarkable result showing that for very general functionals related with semilinear problems involving the Laplacian, the local minima in C 1 are also local minima in W01,2 .


231

We need an extension of this result of Brezis and Nirenberg to our problems. The next theorem is stated solely for problem (P 2) because later applications will be done only for (P 2). The proof follows that of Brezis and Nirenberg for second-order problems, but is included for the sake of completeness. We remark that this first result is also true for problem (P 1). Consider the functional Z Z Z ⇤ 1 1 J(u) = | u|2 dx |u|q dx |u|2 dx (6.1) ⇤ 2 ⌦ q ⌦ 2 ⌦ Ru ⇤ and put f (u) = |u|q 2 u + |u|2 2 u, F (u) = 0 f (s) ds. We recall that u 2 W 2,2 (⌦) \ W01,2 (⌦) is a critical point of J associated to problem (P 2) if Z Z 0= u dx f (u) dx for all 2 W 2,2 (⌦) \ W01,2 (⌦). ⌦

⌦

Define the class of functions E0 = {v 2 C 2 (⌦) : v(x) = 0, x 2 @⌦}. Then u0 is a local minimizer of J in E0 (respectively W 2,2 (⌦) \ W01,2 (⌦)) if there is > 0 such that J(u0 )  J(u0 + v) for all v 2 E0 such that ||v||C 2 < (respectively for all v 2 W 2,2 (⌦) \ W01,2 (⌦) such that ||v||W 2,2 < ). We denote by B✏ the ball of radius ✏ in W 2,2 (⌦) \ W01,2 (⌦), centered at the origin.

Theorem 6.1. Let u0 2 W 2,2 (⌦)\W01,2 (⌦) be a local minimizer of J in the C 2 topology, then u0 is a local minimizer in the W 2,2 (⌦) \ W01,2 (⌦) topology.

Proof. The regularity Theorem 2.1 shows that the minimizer u0 2 C 3 (⌦). By linearity on the di↵erential operator, without loss of generality, we can assume that u0 = 0. Then if the conclusion does not hold, for all ✏ > 0,

there exists v✏ 2 B✏ ,

such that J(v✏ ) < J(0).

We shall prove that 0 is not a minimum in the C 2 topology. Consider the truncation 8 if s  k > < k Tk (s) = s if k 0 there is a constant k = k(✏), such that Jk(✏) (v✏ ) < J(0).

(6.5)

A fortiori, if w✏ 2 W 2,2 (⌦) \ W01,2 (⌦) satisfies Jk (w✏ ) = min Jk (v),

(6.6)

Jk (w✏ ) < J(0).

(6.7)

v2B✏

then If w✏ 2 C 3,

then for some ✏ small enough

and w✏ ! 0 in C 2 ,

(6.8)

J(w✏ ) = Jk (w✏ ) < J(0), which yields a contradiction since 0 is a local minimum in C 2 . We will prove below that (6.8) holds. As in the proof of Theorem 2.1, there exists a Lagrange multiplier µ✏ 0, such that the Euler equation for w✏ is (1 + µ✏ ) and w✏ = 0,

2

w✏ = fk (w✏ ),

(6.9)

w✏ = 0 on @⌦. Moreover we have the uniform estimate ⇤

|fk (u)|  c(1 + |u|2

1

).

(6.10) ⇤

Because {w✏ } converges to zero in W 2,2 (⌦) \ W01,2 (⌦) and therefore in L2 (⌦), for some ⇤ subsequence there exists a function h 2 L2 (⌦) such that |w✏ |  h for all ✏. Hence we can write |fk (w✏ )|  C(1 + ↵|w✏ |) where ↵ = h8/(N 4) . By following the proof of Theorem 2.1 we get uniform estimates in C 3, , and the Ascoli lemma gives us (6.8). (In fact, the convergence is in C 3, ). This completes the proof. The monotonicity of f in our problem plays a very important rˆ ole. If 1 < < 2 then the corresponding solutions of problem (P 2) constructed in Theorem 5.1 satisfy i) u 1  u 2 , by comparison. ii) u 1 (respectively u 2 ) is a subsolution (respectively a supersolution) for the problem ⇤ 2 u = |u|q 2 u + |u|2 2 u ⌘ f (u) in ⌦, (P 2 ) u|@⌦ = 0, u|@⌦ = 0, and both are not solutions of (P 2 ). For simplicity of printing, put u i ⌘ ui .


233

Theorem 6.2. Let 0 < 1 < < 2 , and ui be defined as above. Then there exists a solution u of (P 2 ) such that u1  u  u2 , and moreover u is a local minimum of J in W 2,2 (⌦) \ W01,2 (⌦). Proof. We consider the following truncation 8 > < f (u1 (x)), f (x, u) = f (u), > : f (u2 (x)), and F (x, u) =

Ru 0

of f if u < u1 (x) if u1 (x)  u  u2 (x) if u2 (x) < u,

f (x, s) ds. It is easy to show that the functional 1 J(u) = 2

Z

2

⌦

| u| dx

Z

F (x, u) dx.

(6.11)

⌦

attains its minimum at a point u 2 W 2,2 (⌦) \ W01,2 (⌦) and that u satisfies the problem 2

u = f (x, u)

u|@⌦ = 0,

in ⌦,

u|@⌦ = 0.

In particular, u is regular. By monotonicity we see that (1) 2 (u1 u)  f (u1 ) f (x, u)  0 (2) 2 (u u2 )  f (u) f (x, u2 )  0, so that 2 (u1 u)  0 and 2 (u u2 )  0. By using the Hopf Lemma twice for the Laplacian, we have for some ✏ > 0, u1 (x) + ✏d(x, @⌦)  u(x)  u2 (x)

✏d(x, @⌦).

(6.12)

Thus J(u) = J(u). Moreover, if we choose a ball in E0 of radius less than ✏, i.e., v 2 C 2 (⌦) \ C01 (⌦) such that ||v u||C 2 < ✏, we have that J(v) = J(v) J(u) = J(u); that is, u is a local minimum of J in C 2 . But then, by Theorem 6.1, u is a local minimum of J in W 2,2 (⌦) \ W01,2 (⌦). We will use Theorems 6.1 and 6.2 in the next Section. Remark. Similar methods have been used by De Figueiredo, [8]. 7. Existence of at least two positive solutions. The existence of a second positive solution for (P 2) depends on whether we can apply some version of the Mountain Pass Lemma. In fact for > 0 small enough, we can proceed as in [11] and obtain in this case a second positive solution. This result takes as starting point the minimum of the truncated functional discussed in section 4. A Palais-Smale condition is thereby obtained, which depends on this minimum value. We concentrate our attention on a global result, in the spirit of [1]. More precisely, if ⇤ = sup{ > 0 : (P 2) has a positive solution}, we obtain the following result.

234


Theorem 7.1. If

2 I = (0, ⇤), then problem (P 2) has at least two positive solutions.

Proof. The proof of the theorem will be done in several steps. Step 1. Fix 2 I and consider the solution u0 of (P 2) obtained in Theorem 6.2; that is, u0 is a local minimum of the functional Z Z 1 2 J(u) = | u| dx F (u) dx (7.1) 2 ⌦ ⌦ in W 2,2 (⌦) \ W01,2 (⌦). Define ⇢ f (u0 (x) + s) f (u0 (x)), if s > 0 g (x, s) = 0, if s  0, and consider the truncation of g (x, s), 8 if s u2 (x) u0 (x) > < f (u2 (x)) f (u0 (x)), g (x, s) = g (x, s), if u2 (x) u0 (x) > s > 0 > : 0, if s  0, Ru Ru and G (x, u) = 0 g (x, s) ds, G (x, u) = 0 g (x, s) ds, the respective primitives. Here u2 ⌘ u 2 as in Theorem 6.2. It is easy to check that the functional Z Z 1 (v) = | v|2 dx G (x, v) dx (7.2) 2 ⌦ ⌦ attains its absolute minimum in W 2,2 (⌦) \ W01,2 (⌦) at some point v0 2 W 2,2 (⌦) \ W01,2 (⌦). By the maximum principle we have u2

u0

v0

0.

(7.3)

2

If v0 6= 0 in ⌦ we are done, since v0 = g (x, v0 ) = g (x, v0 ), and u1 = u0 + v0 is the second positive solution of (P 2), namely, 2

u1 =

2

v0 +

2

u0 = f (u0 + v0 )

f (u0 ) + f (u0 ) = f (u1 ).

Our problem is now reduced to the case when v0 ⌘ 0. Step 2. Assume that the minimum of is attained only when v0 ⌘ 0. First, we want to show that v0 = 0 is a local minimum in W 2,2 (⌦) \ W01,2 (⌦) of the functional Z Z 1 2 (v) = | v| dx G (x, v) dx. (7.4) 2 ⌦ ⌦ Now we have 2 (u2 u0 ) f (u2 ) f (u0 ) 0 and u2 u0 = 0, (u2 u0 ) = 0 on @⌦. Thus, by the Hopf Lemma for the Laplacian we can conclude that for ✏ > 0 small enough, if h 2 C 2 \ C01 and ||h||C 2  ✏, then (h) = (h) (v0 ) ⌘ (0) = (0). This means that v0 = 0 is a local minimum of in C 2 \ C01 . Then by Theorem 6.1 it is a local minimum in W 2,2 (⌦) \ W01,2 (⌦). In Step 2 above, we reduced the problem to the case where the minimum of is zero. All the critical points of are nonnegative. To finish the proof, assume that v0 = 0 is the unique critical point of . We shall prove that in this case the Mountain Pass Theorem applies, giving a nontrivial solution, and hence a contradiction. This argument is developed in two further steps. Step 3. We will prove the following lemma.


Lemma 7.2. If v0 = 0 is the only critical point of condition below the level c0 = N2 S N/4 .

, then

235

satisfies the Palais-Smale

Proof. Assume that {vj }j2N ⇢ W 2,2 (⌦) \ W01,2 (⌦) is a Palais-Smale sequence below the level c0 for the functional defined by (7.4); i.e., i) limj!1 (vj ) = c < c0 = N2 S N/4 , ii) limj!1 0 (vj ) = 0 in the dual space of W 2,2 (⌦) \ W01,2 (⌦). In a way similar to Lemma 3.3 we get the boundedness of the sequence in W 2,2 (⌦) \ 1,2 W0 (⌦), and as a consequence we get a convenient subsequence which converges weakly in W 2,2 (⌦)\W01,2 (⌦) and satisfies the inequalities (3.3). We also obtain that the masses for the singular part are as in (3.8); i.e., either ⌫k = 0, or ⌫k S N/4 . From i) and weak convergence we have, 0 = lim

j!1

Z

vj

dx

⌦

Z

g (x, vj ) dx =

⌦

Z

v

dx

⌦

Z

g (x, v) dx

⌦

for all 2 W 2,2 (⌦) \ W01,2 (⌦); hence v is a critical point of in W 2,2 (⌦) \ W01,2 (⌦), and so necessarily v = 0 by hypothesis. Then, if for some k, ⌫k 6= 0, we get 2 N/4 S > c = lim (vj ) = lim { (vj ) j!1 j!1 N

1 2h

0

(vj ), vj i}

2 N/4 S , N ⇤

and this is a contradiction. Thus the subsequence converges strongly in L2 (⌦), and in turn in W 2,2 (⌦) \ W01,2 (⌦). The last result which we have to show is that there exists a Palais-Smale sequence below the critical level N2 S N/4 . More precisely Lemma 7.3. If v0 = 0 is the unique critical point of , then there exists a Palais-Smale sequence such that 2 lim (vj ) = c < c0 = S N/4 . j!1 N Proof. We assume for simplicity in printing that 0 2 ⌦. Consider the best constant of the Sobolev inclusion defined in (3.2). It is known ([16]) that for ⌦ = RN the best constant is attained by the following minimizers, V✏ (x) = K1 (

✏2

N 4 ✏ ) 2 , 2 + |x|

and V✏ satisfies the problem S

N 4

✏ > 0,

K1 = [N (N

4)(N 2

4)](N

4)/8

,

N +4

2

u = u N 4 in RN with N > 4 and Z Z ⇤ = | V1 |2 dx = |V1 |2 dx. RN

RN

The best constant is the same for equivalent norms. (See [13].)

(7.5)

236


The idea is to perform a truncation with a cuto↵ function ⇢(x) 0, smooth, such that, ⇢(x) = 1 if |x| < R, ⇢(x) = 0 if |x| > 2R; where we take R > 0 in such way that all x satisfying |x|  2R belong to ⌦. More precisely, define v✏ (x) = ⇢(x)V✏ (x).

(7.6)

For ✏ small enough, the concentration of V✏ will give us that sup (tv✏ ) = c✏ < t 0

2 N/4 S , N

(7.7)

that is sufficient to have the result. We proceed to prove (7.7). We have the estimates, Z Z | v✏ |2 dx = | V1 |2 dx + O(✏N

4

)

(7.8)

RN

⌦

Z

2⇤

⌦

|v✏ |

dx =

Z

⇤

RN

|V1 |2 dx + O(✏N ),

(7.9)

and for some positive k, 8 (N 4)r (N 4)r N > > k✏ 2 + o(✏ 2 ) if r < > > N 4 > > Z < (N 4)r (N 4)r N 2 2 |v✏ |r dx = k✏N | log ✏| + o(✏N | log ✏|) if r = > N 4 ⌦ > > > (N 4)r (N 4)r > N > N N 2 2 : k✏ + o(✏ ) if r > . N 4

(7.10)

The key for the estimate (7.7) is G (x, s)

⇤ 1 2⇤ s + u0 (x)s2 ⇤ 2

1

⇤

+ Cu0 (x)2

s ,

2(

N

N +4 ), 4 N 4 ,

N

which is a consequence of the following calculus inequality: If r > 2 then given 2 (1, r 1) there exists a constant C > inf

⇢

(1 + t)r

(1 + tr + rt + rtr t

t>0

1

)

1 such that

C.

Now, from (7.11) we have t2 (tv✏ )  2

Z

⌦

⇤

2

| v✏ | dx ⇤

+ |C|m21

t

Z

t2 2⇤

⌦

Z

⌦

2⇤

|v✏ |

|v✏ (x)| dx

dx

2⇤ 1

m1 t

Z

⌦

⇤

|v✏ |2

1

dx

(7.11)


237

(here we use that 0 < m1 = inf x2B2R u0 (x)). Then t2 (tv✏ )  2

Z

RN

Z

⇤

| V1 | dx

t2 2⇤

| V1 |2 dx

t2 2⇤

2

⇤

⇤

|V1 |2 dx

RN

m1 t2

1

k✏

N

4 2

+ o(✏

N

4 2

).

Consider the function h✏ (t) =

t2 2

Z

RN

⇤

Z

⇤

RN

⇤

|V1 |2 dx

Ct2

1

✏

N

4 2

+ o(✏

N

4 2

).

2 N/4 S (because of the N relationship between V1 and the best Sobolev constant S). It is clear that h✏ (t) < ho (t); hence we conclude that 2 max h✏ (t) < ho (to ) = S N/4 . t>0 N When ✏ = 0, ho attains its maximum in [0, 1) at to and ho (to ) =

To finish the proof, we need to analyze the influence of the error term. If we denote by t✏ the point where h✏ attains its maximum, it is easily seen that 0 < t✏ < to and t✏ ! to as ✏ ! 0. Therefore, we can write t✏ = to x✏ , where x✏ ! 1 as ✏ ! 0. Taking into account that h0✏ (t✏ ) = 0, we get to x✏

Z

RN

⇤

| V1 |2 dx

t2o

1 2⇤ 1 x✏

Z

⇤

RN

⇤

|V1 |2 dx = C(2⇤

2 2⇤ 2 N 2 x✏ ✏

1)t2o

Using the precise value of to , after some computations we arrive at ⇤

x2✏

1 where

By Taylor’s expansion: x✏ )(2⇤

Therefore, 1

x✏ = M ✏

N

4 2

⇤

2)x2✏

+ o(✏

⇤

= Ax2✏

N

4 2

3

N

4 4

,

⇤

), for M =

2 N/4 S N

and the conclusion follows at once.

✏

x✏ ) = Ax2✏

+ o(1

Finally, this identity allows us to prove that h✏ (t✏ ) =

3

R 1 ( N | V1 |2 dx) 2⇤ 2 1) R R . 1 ( RN |V1 |2⇤ dx)1 2⇤ 2

A = C(2⇤

(1

2

⇤

Ct2o

1

A 2⇤

✏

N

2 4

2

3

✏

N

.

+ o(✏

N

4 2

),

4 2

.

4

.

238


Step 4. Assume that v0 is the unique critical point of . Consider the function w✏ = r✏ v✏ , with r✏ large enough, such that (w✏ ) < 0 and the mini-max value c✏ = inf max

2P t2[0,1]

( (t)),

where P = { : [0, 1] ! W 2,2 (⌦) \ W01,2 (⌦) : continuous, (0) = 0, (1) = w✏ }. N

Because v0 = 0 is the local minimum, then 0  c✏ < N2 S 4 . If c✏ > 0 the Mountain Pass Lemma by Ambrosetti and Rabinowitz, [4], gives us a second positive critical point, in contradiction with the hypothesis. In the case c✏ = 0, we get the same contradiction by using a result by Pucci-Serrin, [19]. This contradiction finishes the proof. Remark. We can say that the solutions constructed in Sections 4 and 5 correspond to the sublinear term, because, for instance, when ! 0 they converge to the trivial solution. The same behavior is obtained for q ! 2 in the case of the minimal positive solution obtained in Section 5. The second positive solution obtained in Theorem 7.1, however, tends to a Dirac mass as ! 0. This behavior was obtained for solutions of the p-Laplacian in [11]. 8. Further results. The results given above can be generalized without difficulty to second members of the form f (x, u), where f is increasing in u, satisfies some regularity, growth and oddness properties. We prefer to avoid more technicalities. We have, on the other hand, several remarks about possible applications of the methods above to similar cases. A) Subcritical Problems. Obviously all the results obtained above also hold for the problems 2 u = |u|q 2 u + |u|r 2 u in ⌦, (S1) @u u|@⌦ = 0, = 0, @n @⌦ and

2

u = |u|q

u|@⌦ = 0,

2

u + |u|r

2

u

in ⌦,

(S2)

u|@⌦ = 0,

where 1 < q < 2 < r < 2⇤ . In these cases there is compactness and all arguments are much easier. B) Quasilinear Problems. The results in Section 4 can be extended to the problems ⇤ (| u|p 2 u) = |u|q 2 u + |u|p 2 u in ⌦, (Q1) @u u|@⌦ = 0, = 0, @n @⌦ and

(| u|p

2

u) = |u|q

u|@⌦ = 0,

2

⇤

u + |u|p

u|@⌦ = 0,

2

u

in ⌦,

(Q2)


239

where N > 2p, 1 < q < p and p⇤ = NN p2p . The mini-max theorem for quasilinear equations of second order is given in [10], and here only a minor change gives the result. The existence result in Section 5 can be obtained also for the problem (| u|p

2

u) = |u|q

u|@⌦ = 0,

2

u + |u|r

u|@⌦ = 0,

2

u

in ⌦,

(P)

where 1 < q < p < r, since a maximum principle can be found by iteration of the maximum principle for the Laplacian and the monotonicity of the function h(s) = |s|p 2 s. Acknowledgments. We are grateful to Professors P. Pucci and J. Serrin for helpful suggestions about the paper. REFERENCES [1] [2]

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