Existence of nontrivial solutions of partial differential equations with discontinuous nonlinearities

Nonlinear Anolysa. Theory, Melhods & Applicofions, Printed in Great Britain.

Vol. 16, No. 11, pp. 1025-1034. 1991. 0

0362-546X/91 $3.@0+ .OO 1991 Pergamon Press plc

EXISTENCE OF NONTRIVIAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS NONLINEARITIES NORIKO MIZOGUCHI Department of Information Science, Tokyo Institute of Technology, Oh-okayama, (Received Key words and phrases:

I February Discontinuous

1990; received for publication nonlinearities,

multiple

Meguro-ku,

6 November

solutions,

Tokyo

152, Japan

1990)

generalized

gradients.

1. INTRODUCTION

LET SI c R” BE A bounded domain with smooth boundary aa. Suppose that g: R -+ R is piecewise continuous on any bounded closed interval and that g(0) = 0. We study the existence of nontrivial solutions of the boundary value problem of the form

- Au E [g(u), t?(u)1

in a,

ulan = 0,

(I)

where g(r) = max ,vmOg(@, li;,g(s)

1

and

when g(t)/t crosses finitely many eigenvaiues of -A as t varies from - co to + co. In the case that g is a continuous function on the whole of R, the problem (1) becomes the following form -Au = g(u) in Q ulan = 0. (2) Many authors have studied the existence of multiple nontrivial solutions or one nontrivial solution for the problem (2) under the assumption that g is of class C’ or C2 (see [l, 2, 3, 4, 61). Moreover, in [12] and [13], Hirano proved the similar results for a continuous function g not necessarily of class Cl. However, in the case that g is discontinuous, the functional f(U) = + sD

(vu]2 dx -

U(X) g(t) dt b ssD 0

(3)

may be nondifferentiable. Therefore we need the critical point theory for nondifferentiable functions. Chang [8] applied the generalized gradients for locally Lipschitz functions on Banach spaces introduced by Clarke [lo], to the problem of partial differential equations with discontinuous nonlinearities. Let X be a real Banach space with a norm 1)- (( and its dual space X* andf: X -+ R be a locally Lipschitz function, that is, for each u E X, there exist a neighborhood U of u and a constant 1025

1026

N. MIZOGUCHI

K (depending on

U)

such that If(u) -f(w)1

for all u, w E U.

5 K/u - ~11

Then Clarke [lo] defined the generalized gradient af(u) off at u E X by the subdifferential the continuous and convex function f “(u; a) at 0. Here f”(u;

II) = lim sup

f(u

+ w +hu)

- f(z.4 + w)

h

w-0,hlO

of

for all u E X.

If f is convex, then df coincides with the subdifferential off in the sense of convex analysis. Also, if f admits the Gateaux derivative Df(u) at each u in a neighborhood of u and Df: X -+ X* is continuous, then df(u) = (Df(u)). Further, if f attains a local minimum at U, then 0 E af(u). Note that af is upper semicontinuous from the strong topology of X to the w*-topology of X* in the sense of multivalued mappings, that is, for each u E X and any w*-neighborhood I/ of df(u), there exists a strong neighborhood U of u such that df(u) c V for all u E U. Refer to [S] and [lo] for various properties of the generalized gradients. In this paper, we extend Hirano’s results in [ 121and [ 131to the problem (1) using the generalized gradients. Throughout this paper, we assume that g: R 4 R is piecewise continuous on any bounded closed interval and that g(0) = 0. We write L2, H,‘, H2 and H-’ instead of L2(Cl), H,‘(Q), H2(sZ) and H-‘(Q), respectively. We denote by I\*)) and 1. I the norms of Hi and L2, respectively. The pairing between Hi and H-’ is denoted by (. , a>. Let A1 < A2 5 ... % A, I a+. be the eigenvalues of the self-adjoint realization in L2(sZ) of -A with the boundary condition in (1). Putting L = -A, we have L: H,’ -+ H-l. 2. VARIATIONAL

METHOD

In this section, we first obtain the variational method on subsets of a Banach space and apply it to an existence theorem of multiple nontrivial solutions of the problem (1). Let X be a real Banach space with a norm II- )I and its dual space X* and f: X + R be a locally Lipschitz function. Then, f is said to satisfy the Palais-Smale condition on a subset C of X if any sequence {u,) c C for which [ f(u,)) is bounded and A&,) = min IIu*[I~~converges to 0 u*Eaf(u,) has a subsequence strongly convergent to some point in C. The following theorem is the main tool in this section. 1. Let C be a closed subset of a Banach space X and f: X -P R be a locally Lipschitz function which is bounded below on C and satisfies the Palais-Smale condition on C. If the following condition (i) or (ii) holds, then there exists u. E C such that 0 E af(u,): (i) For small enough E > 0 and any u E K?, there exists u E int C such that

THEOREM

f(u)

-f(u)

2

4lU - 4.

(ii) There exists u E int C such that f(u) < .‘,“,f, f(u). Proof.

From Ekeland’s variational principle, for any E > 0, there exists u, E C such that f(q)

and f(u)

Zf(%)

-

5 inff C

&lb- &II

+ .s for all u E C.

(4) (5)

PDEs with discontinuous

nonlinearities

1027

Assuming the condition (i), if U, E XT, there exists u, E int C such that

This inequality and the inequality (14) imply that f(u) zf(n,)

- allu - &II

for all 24E C,

namely, u, E int C is a minimizer of the function f + E/I* - u,[[ on C. It follows that 0 E af(u,) +

mll. - ~,llx~,)c afw + EB*,

where B* is the closed unit ball in X*. Therefore, we obtain some u: E af(u,) with ]]u,*]]~*I E. Sincefsatisfies the Palais-Smale condition on C, there exists (u,) in C such that (u,) converges strongly to some u0 E C. From the upper semicontinuity of af from the strong topology of X to the w*-topology of X*, we have 0 E af(u,). In the case of (ii), let 0 < E < inff - inf f. Then ac intC the point U, satisfying the inequality (4) is contained in int C. Thus, we can prove the case (ii) by the same argument as the case of (i). Let Hi and H2 be the subspaces of L2 spanned by the eigenspaces corresponding to the eigenvalues {A,) and (d,, A,, . ..I. respectively. Then H, and H2 are orthogonal in L2. Also we have H, = (k$: k E R), where 4 is the normalized eigenfunction corresponding to Ai. Let Pi and P2 be the projections from L2 onto HI and Hz, respectively. The purpose of this section is to prove the following theorem by using theorem 1. THEOREM2. If g: R + R satisfies the following condition -A, < b, I b* < A, -c a, 5 a* < A2,

(6)

where g(t)

a* = supt,

a, = liminf?

tfo b* = limsupy

g(t)

I++0 and

ItI--(a

b * = liminfm t IfI--

’

then the problem (1) has at least two nontrivial solutions in Hi n H2. Proof.

Define p: Hi + R by V(U) =

u(x) g(t) dt dx D 0 .i’I

for each u E H,‘.

(7)

Then 9 is a locally Lipschitz function with respect to I]- II by the condition (6). Further, it was shown in [8] that

ap(f4) c (w E L2: w(x) E [&d(x)), g@(x))] a.e. on a] for all u E Hi. To prove the theorem, we need two lemmas.

1028

N. MIZOGUCHI

LEMMA 1. Let f: Hi -+ R be defined by (3). Then, u E aKi, there is u E int Ki (i = 1,2) satisfying

f(u) rf(u) where K, = (&:

there exist s > 0 and 6 > 0 such that for

+ au - 4,

k 1 s) x Hz and K2 = (k+: k 5 -s)

x H,.

Proof. The method of the proof is based on [12] and [13]. From the condition (6), we obtain p > i, and p > 0 such that g(t)/t 2 j3 for all t E R with ItI < p. Also, there exists s > 0 such that sup Iw(x)l d p/2 for all w E H, with /wll 5 s. Put w = Plu and z = Pzu for u E Hi. xeo For each U* E df(u), we have

(u*,

w>= (L(z

2 -

=

+ w), z - w) -

llzl12 - A,lw12-

1

u*(x)(z - w)(x) dx

s

12

u*(x)(z - w)(x)dx

12

for some U* E a&u).

If Iz(x) + w(x)1 2 p, we have Iz(x)l 2 Iw(x)l and hence

-A,IW(X)12 - u*(x)(z - w)(x) L --a*(z(x)l* by the condition have

(6). Next suppose -&lw(x)l2

In the case that

Iz(x) + w(x)1 < p. In the case that Iz(x)l 1 Iw(x)l, we

- v*(x)(z - w)(x) 1 --a*Jz(x)12

+ (a* - qlw(x)j2.

Iz(x)l < Iw(x)1, we have

-A,Iw(x)12It follows

that

+ (a* - &)Iw(x))*

u*(x)(z - w)(x) 1

-a*lz(x)12+

(a, - A1)Iww1*.

that (u*, z - w> z 11211* - a*Izl* + (a, - /%,)(w(2 2 asllz -

for some CY> 0. Putting

Since f’(u;

WII

6, = CYS,we have

*) is the support

function

fO(u; u) =

of df(u), i.e. max (u*, u) U* E L?f(U)

for all u E Ht’,

we obtain

From

the definition

off’,

it follows

lim sup

that

f(u + h((w -

f(u)

h

h10

Therefore,

z)/llw - ZIIN -

< _6 0.

for each u E dKi (i = 1,2) we can take 0 < 6 < 6, and h > 0 so small that u

E

int Ki

and

f(u)

where u = u + h((w - z>/II w - ~11). This completes

2 f(u)

+

au - 41,

the proof.

1029

PDEs with discontinuous nonlinearities

LEMMA 2. The function f: H,’ + R defined Palais-Smale condition on Ki (i = 1, 2).

by

(3) is bounded

below

and

satisfies

the

Proof. From the assumption (6), we easily see that f(u) 2 C,llz# - C, for all u E H,-j and some C, , C, > 0. This implies that f is bounded below on Hi. To prove that f satisfies the Palais-Smale condition, take (u,] C Ki for which (f(u,)] is bounded and A(u,) converges to 0 as n + 03. Then, by the above inequality, lu,) is a bounded sequence in H,’ and hence has a subsequence (u,J of (u,] convergent weakly to some u in H,‘. Since Hd is compactly embedded in L2, (u,] converges strongly to u in L2. On the other hand, there exists a sequence (w,) such that w, E a&u,) for each n and (Lu, - w,] converges strongly to 0 in H-’ since A(u,) converges to 0. Then lim sup (Lu, - w,, 24, - U> 5 0. n+m From the boundedness

of (w,] c L2, it follows

that

lim (w,, , u,, - 24) = 0, j - cc and therefore lim sup (Lunj, u,, - 24) 5 0. j -9co Since (Lu,,~] converges

weakly to Lu in H-l,

lim Uu,,.,

j + cc

so (unj] converges

strongly

we obtain

un,>= (IA,

to u in H,‘. This completes

u),

the proof.

Now lemmas 1 and 2 enable us to apply theorem 1 to the function Ki c Hi (i = 1, 2). Thus there exist at least two nontrivial solutions 3. PSEUDOMONOTONE

The purpose

of this section

the following

by (3) and each

METHOD

is to prove the following

THEOREM 3. If g: R + R satisfies

f defined of (1).

theorem.

condition

where a* = supgO

tzo

t

a

’

b* = limsupy Itl-ro then the problem

(1) has at least two nontrivial

*

and

=

liminfg(t) Id+-

b

solutions

*

t

’

= infg(l) ttro t

’

in HJ fl H2.

Before proving the theorem, we obtain four lemmas. Let C be a closed convex subset of a Banach space X and T be a mapping Then T is said to be pseudomonotone if the following condition holds.

from C into 2x”.

N . MIZOGUCHI

1030

Condition. If (u,) C Cconverging satisfy that

weakly to u E C and {w,) c X* with w, E T(u,) for all n 1 1 lim sup (w, , u, - U) 5 0, n+m

then for each u E C there exists w E T(U) such that (w, 24 - v) I lim inf (w,, 2.4,- u). “-CO We call T bounded when T maps bounded subsets of C into bounded subsets The following lemma is a multivalued version of theorem A of [12].

of X*.

LEMMA 3. Let C be a closed convex subset of a reflexive Banach space X and T: C + p* be a bounded pseudomonotone mapping which is upper semicontinuous from the strong topology of X to the weak topology of X*. Suppose that K is a bounded closed convex subset of C. If for each z E aK and each w E Tz, there exists x E int K such that (w, z - x> 2 0, then there exist x0 E K and w. E TX,, such that ( wo, y - x0>2 0 for all y E C.

Since the proof of this lemma proceeds by connecting [5, theorem 7.81 with [12, theorem A], we omit it here. Let Hi, H2 and H3 be the subspaces of L2 spanned by the eigenspaces corresponding to the eigenvalues (Ak+i, Ak+2 ,... 1, (A,) and 12,) AZ, . . . . A,_,), respectively. Then Hi, Hz and H3 are orthogonal in L’. Also we have H, = {I+: k E R),where #J is the normalized eigenfunction corresponding to 2,. Let P,, P2 and P3 be the projections from L2 onto H,, H2 and H3, respectively. We set Tu = L(u - 2(P2 + P&) - &y(u - 2(P2 + P3)u) for each u E Hi. We can see that u - 2(P, + P& if TM 3 0.

is a solution

of the problem

LEMMA 4. The

tinuous

mapping T: H,’ 4 2H-’ is bounded, pseudomonotone from the strong topology of Hi to the weak topology of H- ‘.

and

(1) if and only

upper

semicon-

Proof. From the assumption (8), acp maps bounded subsets of H,j into bounded subsets of H-’ and hence T does so. Since ay, is upper semicontinuous from the strong topology of Hi to the weak topology of H-‘, so is T. To prove the pseudomonotonicity of T, suppose that (u,] c Hi converging weakly to u E Hi and {w,] c H-’ with w, E a&u,, - (Pz + P&J for all n 1 1 satisfy that lim sup (L(z.4, - (P2 + P&4,) - w,, 24, - 24) 5 0. (9) n-00 Then we have that (Lu,) converges weakly to Lu in H-l. Since H,’ is compactly embedded in L2, (u,) converges strongly to f.4in L2, so ((P2 + P&J is strongly convergent to (P2 + P&.4 in L2. From the boundedness of ap, it follows that lim (L(P, n*m Therefore

the inequality

(9) implies

+ P&4, + w,, u, - u) = 0. that

lim sup (Lu,,

n+m

24, - u) 5 0,

PDEs with discontinuous

1031

nonlinearities

that is, lim (Lu,, 24,) = (Lu, u). n+m Thus we obtain the strong convergence of (u,) to u in H,‘. By the weak compactness of ap(u) in H-l, for each v E II,’ there exists w(v) E 13&u) such that (w(v), u - v> = z ZrrU, (z, u - u>. Since ap is bounded and upper semicontinuous from the strong topology of Ho to the weak topology of H-l, we can take a subsequence (wnj) of (w,) which converges weakly to some w0 E a&u). Then lim ( w,, , u,, - v) = (w,, 24- u> 1 (w(v), 24- v>

j-m

for each u E Hi. It follows that (w(u), u - u) 5 lim inf (w,, u, - tj>. n*m

This completes the proof. The following two lemmas can be shown by the similar methods to the proofs of [12, lemma 2 and lemma 31, which are the results corresponding to the case that g is a continuous function on the whole of R. 1 0 for each u E Hi with llPzull I s and

LEMMA 5. There exists s > 0 such that (w, u - 2P,u) each w E Tu.

LEMMA 6. There exists r > 0 with r > 2s such that (w, U) 2 0 for each u E H,’ with IIu(] L r and

each w E Tu. Proof of theorem 3. We put s = (u E Hi + H,: JIUJ]I r), S1 = {&:s

I k I r),

Sz = (k4: -rs

k 5 --sJ

Ki = S X Si

(i= 1,2).

and Considering that X = C = H,’ and K = Ki (i = 1,2) in lemma 3, lemmas 4-6 imply that all the assumptions in lemma 3 are satisfied. Therefore we obtain some Ui E Ki such that Tui 3 0 for i = 1, 2 (see [12]). This completes the proof of theorem 3. 4. APPROXIMATION

METHOD

In this section, we treat the existence of one nontrivial solution of the problem (1). THEOREM 4. If g: R + R satisfies the following condition

&,

< b, I b* -c Ak 5 A, < a, I a* < A,,,

for some k, m 2 2,

(10)

1032

N. MIZOOUCHI

where a* = limsupt, g(t) lrl-m

a, = liminf--, ltl -m

6* = limsupy ItI-0 then the problem Proof.

(1) has at least one nontrivial

For each E > 0, we define g,(s) =

b

and

*

= liminfm ItI-

solution

a continuous

g(t)

t

in Hi n Hz.

function

g,: R -+R by

g(s) i g(t - E) + (s - t + &)(g(t + E) - g(t - &))/(2&)

where t E R is an arbitrary point where g (10) remains valid with g replaced by g,. Let by the eigenfunctions corresponding to the Denote by P, I4 9?l2, . .., A2,_,), respectively. and Hj, respectively. We put Ki = (U E Hi: llull I rj

’

if Is - tl 1 .c if 1s - t( < c,

has a jump discontinuity. Then, the condition H, , Hz and Hj be the subspaces of L2 spanned eigenvalues [Am+l, Am+2,. . .], (Ak, . . . , A,) and , Pz and Pj the projections from L2 onto H,, Hz

(i = 1,2, 3),

S,=K,xK,xK,, S2 =

u E S,: IP,u( < $,

IP2uJ <&and

IP3ul < 5

i

1

and K = S,\S, for large enough r > 0 and small enough c > 0. Then, using [13, theorem there exists at least one solution U, in K of the following problem -Au

in Q,

= g,(u)

ulan = 0.

11, for each E > 0,

(11)

Indeed, we can take positive numbers c and r so that the solutions (u,) obtained for each g, in [ 131 are contained in the same set K. Denote by g, the corresponding function to E = 1/n and let u, E K be the solution of (11) corresponding to g, . By the boundedness of (u,], we may assume that (u,) converges weakly to some u in Hi. Since Hi is compactly embedded in L2, (u,) converges strongly to u in L2. Thus we have u E K. From the assumption (lo), it is easily seen that (g,(u,)) is bounded in L2, so we can suppose that (g,(u,)) converges weakly to some w in L3. Since (u,) converges to u a.e. on a, for each q > 0, there exists a subset A of Q with meas < q such that (u,) converges to u uniformly on Q\A. That is, for each p > 0, we obtain NE N with N > 2/p such that

14z(4- m


for all n L N and all x E Q\A.

Putting &(I)

= supjg(s):

1 - p 5 s 5 t + p)

g,(t)

= inf(g(s):

t - p 5 s 5 t + p],

and

PDEs with discontinuous

1033

nonlinearities

we have & (N-9) 5 g, (%I (-9) 5 &J(u(x)) for all n 2 N and all x E QW. a.e. on L2U. Then we have gp (Nx))w(x)

Take an arbitrary

function

I,YE L’(QW)

that I,UL 0

s, (u(x))rv(x) d-x

g, (%I (x))w(x) dx 5

dJC 5 i R\A

fl\A

satisfying

O\A

and hence

gp(wMx)

.R\A i

It follows

dx 5

wcw(x) dx 5

s

sp(4eMx) dx.

fl\A

O\A

that g(@))y/(x)

W(X) w(x)dx 5

d-x 5 i R\A

s WATherefore

I

E(u(x))ll/(x) dx. i fi\A

we obtain g@(x)) 5 w(x) 5 g(U))

a.e. on s2W.

Since q > 0 is arbitrary, g(u(x)) 5 w(x) 5 E(M)

a.e. on M,

that is, w E @P(U). Since {LA,) converges weakly to Lu in H-l, {Lu, - g,(u,)) converges to Lu - w E df(u) in H-l, so 0 = Lu - w E df(u). This completes the proof.

Remark.

We can also prove theorem

Acknowledgement-The during the preparation

author wishes to thank of this paper.

3 by the same argument

Professor

N. Hirano

as in the proof

for many

helpful

weakly

of theorem

suggestions

4.

and kind advice

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N. MIZOGUCHI

10. CLARKE F. H., A new approach to Lagrange multipliers, Math. Oper. Res. 1, 165-174 (1976). 11. FLEISHMANB. A. & MAHAR T. J., Analytic methods for approximate solution of elliptic free boundary problems, Nonlinear Analysis 1, 561-569 (1979). 12. HIRANO N., Multiple nontrivial solutions of semilinear elliptic equations, Proc. Am. math. Sot. 103, 468-472 (1988). 13. HIRANO N., Existence of nontrivial solutions of semilinear elliptic equations, Nonlinear Analysis 13, 695-705 (1989).