C O L L O Q U I U M M A T H E M A T I C U M VOL. 81 1999 NO. 1

VOL. 81 1999 NO. 1

MULTIPLE SOLUTIONS FOR NONLINEAR DISCONTINUOUS ELLIPTIC PROBLEMS NEAR RESONANCE

BY

NIKOLAOS C. K O U R O G E N I S AND

NIKOLAOS S. P A P A G E O R G I O U (ATHENS)

Abstract. We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when λ → λ1 from the left, λ1 being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.

1. Introduction. In a recent paper (see Kourogenis–Papageorgiou [10]), we examined quasilinear elliptic problems at resonance with discontinuous right hand side and we proved the existence of a nontrivial solution. In the present paper we examine quasilinear elliptic problems near resonance with discontinuities. Semilinear problems near resonance with a continuous right hand side were studied by Mawhin–Schmitt [12], [13], Chiappinelli–De Figueiredo [4] and Chiappinelli–Mawhin–Nugari [5]. In [13] the equation under consideration is an ordinary differential equation (i.e. N = 1) and the authors employ a sign condition to establish the existence of three non- trivial solutions. In [12] an analogous abstract result with the sign condition replaced by a Landesman–Lazer type hypothesis can be found. The authors obtain two solutions, one negative and the other positive. In [4], a similar multiplicity result is shown under the hypothesis of linear growth as x →-

∞. In all three papers the approximation of the first eigenvalue is from the left. In [5] the parameter λ is to the right of the first eigenvalue. Again the authors prove the existence of two solutions, one of them positive. All the aforementioned papers use bifurcation theory.

1991 Mathematics Subject Classification: Primary 35J20.

Key words and phrases: multiple solutions, discontinuous function, elliptic inclusion, first eigenvalue, p-Laplacian, Rayleigh quotient, nonsmooth Palais–Smale condition, coer- cive functional, Clarke subdifferential, critical point, generalized directional derivative.

The first named author supported by the General Secretariat of Research and Tech- nology of Greece.

[89]

Recently there appeared the interesting works of Ambrosetti–Garcia Azorero–Peral [1] and Ramos–Sanchez [15]. The authors of [1] study (Sec- tion 4) the existence of positive solutions for the eigenvalue problem −∆

x = λf (x), x|

= 0, where f (x) ≃ x

near 0 and infinity, and they prove a bifurcation result, both from zero and from infinity. Their approach is based on degree-theoretic arguments. The work of Ramos–Sanchez [15] is closer to ours. They study the semilinear version (i.e. p = 2) of our problem with the right hand side function f (z, x) continuous in both variables. In Section 2 of [15], they examine the case when λ approaches λ

(the first eigenvalue of (−∆, H

(Z))) from the left. In Theorem 2.6, they prove the existence of three nontrivial solutions (as we do here in Theorem 7). As we already said, in their problem f (z, x) is jointly continuous, they assume that F is bounded below on Z × R

, they have a hypothesis similar to our hypothesis H(f )(iii), but they impose the asymptotic condition on f instead of F (the potential function corresponding to f ) as we do, and they also have hypoth- esis H(f )(iv). Their approach is different from ours and uses the theory of elliptic variational inequalities. It should be mentioned that in Section 4 of [15] they also study the case when λ approaches λ

from the right.

Our approach here is variational and is based on the critical point theory for nonsmooth locally Lipschitz functionals, as developed by Chang [3]. For the convenience of the reader, in the next section we outline the basic aspects of this theory.

2. Preliminaries. Chang’s critical point theory for locally Lipschitz functionals is based on the subdifferential theory of Clarke [6] for such func- tionals. In the previous paper [10], in Section 2, we presented the basic definitions and facts from these theories that are needed in our analysis.

The notation introduced there will also be used in this paper. So if X is a Banach space and f : X → R is locally Lipschitz we define the generalized directional derivative

f

(x; h) = lim

x^′→x λ↓0

f (x

+ λh) − f (x

) λ

and the generalized subdifferential

∂f (x) = {x

∈ X

: (x

, h) ≤ f

(x; h) for all h ∈ X}.

A point x ∈ X is a critical point of f if 0 ∈ ∂f (x). We say that f

satisfies the nonsmooth Palais–Smale condition (nonsmooth PS-condition)

if any sequence {x

}

along which {f (x

)}

is bounded and m(x

) =

inf{kx

k : x

∈ ∂f (x

)} → 0 as n → ∞, has a strongly convergent sub-

sequence. This notion generalizes the classical one for C

-functionals (see Rabinowitz [14]).

Consider the nonnegative p-Laplacian (2 ≤ p < ∞) differential operator

−∆

x = − div(kDxk

Dx)

with Dirichlet boundary conditions; we use the notation (−∆

, W

(Z)).

The first (principal) eigenvalue λ

of this operator is the least real number λ for which the nonlinear elliptic problem

(1)

 − div(kDx(z)k

Dx(z)) = λ|x(z)|

x(z) a.e. on Z, x|

= 0,

has a nontrivial solution. The first eigenvalue λ

is positive, isolated and sim- ple (i.e. the associated eigenfunctions are constant multiples of each other).

Furthermore, we have the following variational characterization of λ

> 0 (Rayleigh quotient):

λ

= min[kDxk

/kxk

: x ∈ W

(Z)].

This minimum is realized at the normalized eigenfunction u

. Note that if u

minimizes the Rayleigh quotient, then so does |u

| and so we infer that the first eigenfuction u

does not change sign on Z. In fact, we can show that u

6= 0 a.e. on Z and so we may assume that u

(z) > 0 a.e. on Z.

Also, using nonlinear elliptic regularity theory (see Tolksdorf [16]), we can show that u

∈ C

for some α > 0. For details we refer to Lindqvist [11]. The Lyusternik–Schnirelmann theory gives, in addition to λ

, a whole strictly increasing sequence {λ

}

of positive numbers for which there exist nontrivial solutions of the eigenvalue problem (1). In other words, the spectrum σ(−∆

) of (−∆

, W

(Z)) contains at least these points.

However, in general, nothing is known about the possible existence of other points in σ(−∆

) ⊆ [λ

, ∞) ⊆ R

. Nevertheless, we can define

µ = inf{λ > 0 : λ is an eigenvalue of (−∆

, W

(Z)), λ 6= λ

}.

Because λ

> 0 is isolated, we have µ > λ

and if V is a topological com- plement of X = hu

i = Ru

, then

µ

= inf[kDvk

/kvk

: v ∈ V ] > λ

, µ = sup

V

µ

.

The following theorem is due to Chang [6] and extends to a nonsmooth setting the well-known Mountain Pass Theorem due to Ambrosetti–Rabino- witz [2].

Theorem 1. If X is a reflexive Banach space, R : X → R is a locally Lipschitz functional which satisfies the PS-condition and x

, x

∈ X, β ∈ R and ̺ > 0 are such that

(i) kx

− x

k > ̺,

(ii) max[R(x

), R(x

)] < β ≤ inf[R(u) : ku − x

k = ̺],

then R(·) has a critical point x ∈ X such that c = R(x) ≥ β. Moreover , c can be characterized by the following min-max principle :

c = inf

γ∈Γ

sup

t∈[0,1]

R(γ(t)) where Γ = {γ ∈ C([0, 1], X) : γ(0) = x

, γ(1) = x

}.

3. Auxiliary results. Let Z ⊆ R

be a bounded domain with C

- boundary Γ . We consider the following quasilinear elliptic problem:

(2)

 



− div(kDx(z)k

Dx(z)) − λ|x(z)|

x(z)

= f (z, x(z)) a.e. on Z, x|

= 0, 2 ≤ p < ∞.

Since we do not assume that f (z, ·) is continuous, problem (2) need not have a solution. To develop a reasonable existence theory, we need to pass to a multivalued version of (2) by, roughly speaking, filling in the gaps at the discontinuity points of f (z, ·). For this purpose we introduce the following two functions:

f

(z, x) = lim

x^′→x

f (z, x

) = lim

δ↓0

ess inf

|x^′−x|<δ

f (z, x

) f

(z, x) = lim

x^′→x

f (z, x

) = lim

δ↓0

ess sup

|x^′−x|<δ

f (z, x

).

We introduce b f (z, x) = {y ∈ R : f

(z, x) ≤ y ≤ f

(z, x)} and instead of (2) we consider the following differential inclusion:

(3)

 



− div(kDx(z)k

Dx(z)) − λ|x(z)|

x(z)

∈ b f (z, x(z)) a.e. on Z, x|

= 0, 2 ≤ p < ∞.

Problem (3) is a multivalued approximation of (2), which captures the dis- continuity features of f (z, ·) and permits the development of an existence theory. Of course, if f (z, ·) is continuous for almost all z ∈ Z, then the two problems coincide.

We introduce now our hypotheses on f :

H(f ): f : Z × R → R is a measurable function such that

(i) f

, f

are N-measurable functions (i.e. for every measurable function x : Z → R, the functions z → f

(z, x(z)) and z → f

(z, x(z)) are measurable; superpositional measurability);

(ii) for every r > 0, there exists a

∈ L

(Z) such that |f (z, x)| ≤ a

(z) for almost all z ∈ Z and all |x| ≤ r;

(iii) if F (z, x) =

Tx

0

f (z, r) dr then lim

pF (z, x)/|x|

= 0 uni-

formly for almost all z ∈ Z;

(iv) lim

T

Z

F (z, tu

(z)) dz = ∞;

(v) lim

pF (z, x)/|x|

< −λ

and lim

F (z, x)/|x|

> −∞

uniformly for almost all z ∈ Z.

Remark . Hypotheses H(f ) are more general than the corresponding ones used by Kourogenis–Papageorgiou [10]. Indeed, while H(f )(i) is com- mon in both papers, H(f )(ii) is weaker than the one of [10], where it is assumed that |f (z, x)| ≤ a(z) for almost all z ∈ Z and all x ∈ R, with a ∈ L

(Z). Moreover, H(f )(iii) is weaker than the corresponding one in [10], since it does not imply that lim

f (z, x) exist and are finite for almost all z ∈ Z. Finally, H(f )(v) is common in both papers and is needed in order to apply Theorem 1.

We introduce the energy functional R

: W

(Z) → R defined by R

(x) = 1

p kDxk

− λ

p kxk

−

\

Z

F (z, x(z)) dz.

Since J : W

(Z) → R defined by J(x) =

T

Z

F (z, x(z)) dz is locally Lipschitz (see Chang [3]) and

x → 1

p kDxk

→ λ p kxk

are continuous convex functions on W

(Z), hence locally Lipschitz on W

(Z), it follows that R

(·) is locally Lipschitz.

Proposition 2. If hypotheses H(f ) hold and λ < λ

, then R

(·) is coercive.

P r o o f. Suppose not. Then we can find {x

}

⊆ W

(Z) such that kx

k

→ ∞ as n → ∞, and R

(x

) ≤ M for all n ≥ 1. We have

1

p kDx

k

− λ

p kx

k

−

\

Z

F (z, x(z)) dz ≤ M.

By H(f )(iii), given ε > 0 we can find M

= M

(ε) > 0 such that F (z, x) ≤ (ε/p)|x|

for almost all z ∈ Z and all |x| > M

. Also, for al- most all z ∈ Z and all |x| ≤ M

we have |F (z, x)| ≤ a

(z) with a

∈ L

(Z) (see hypothesis H(f )(ii)). So finally, we can say that for almost all z ∈ Z and all x ∈ R we have F (z, x) ≤ a

(z) + (ε/p)|x|

. Hence

1

p kDx

k

− λ

p kx

k

− ka

k

− ε

p kx

k

≤ M, that is,

1

p kDx

k

≤ M + ka

k

+ λ + ε

p kx

k

.

Since kx

k

→ ∞, we have kDx

k

→ ∞ and so from the last inequality it follows that kx

k

→ ∞. Let y

= x

/kx

k

, n ≥ 1. Dividing by kx

k

we obtain

(4) 1

p kDy

k

≤ M kx

k

+ ka

k

kx

k

+ 1

p (λ + ε),

hence {y

}

⊆ W

(Z) is bounded (by Poincar´e’s inequality). Thus by passing to a subsequence if necessary, we may assume that y

→ y in W

(Z), y

→ y in L

(Z), y

(z) → y(z) a.e. on Z as n → ∞ and

|y

(z)| ≤ h

(z) a.e. on Z with h

∈ L

(Z).

Passing to the limit in (4) and using the fact that kDyk

≤ lim kDy

k

(weak lower semicontinuity of the norm functional), we obtain 1

p kDyk

≤ λ + ε p .

Let ε > 0 be such that λ + ε < λ

(recall that by hypothesis λ < λ

).

Also, since ky

k

= 1, n ≥ 1, we have kyk

= 1, i.e. y 6= 0. So we can write kDyk

< λ

kyk

,

which contradicts the variational characterization of λ

(Rayleigh quotient, see Section 2). This proves the coercivity of R

(·).

Let X = hu

i = Ru

and V a topological complement. Then W

(Z) = X ⊕ V .

Proposition 3. If hypotheses H(f ) hold, then there exists β < 0 such that R

(v) ≥ β for all v ∈ V and all 0 < λ < λ

.

P r o o f. From Section 2 we know that there exists µ > λ

such that for all v ∈ V we have

kDvk

≥ µkvk

. Also, since 0 < λ < λ

, we have

R

(v) ≥ 1

p kDvk

− λ

p kvk

−

\

Z

F (z, v(z)) dz.

From the proof of Proposition 2 we know that given ε > 0 we can find a

∈ L

(Z) (depending on ε > 0) such that for almost all z ∈ Z and all x ∈ R we have

F (z, x) ≤ a

(z) + ε p |x|

. So we can write

R

(v) ≥ 1

p kDvk

− λ

p kvk

−ka

k

− ε

p kvk

≥ 1 p



1− λ

+ ε µ



kDvk

−ka

k

.

Choose ε > 0 so that λ

+ ε < µ. From the last inequality it follows that R

(·) is coercive on V , uniformly in 0 < λ < λ

. Thus we can find β < 0 such that R

(v) ≥ β for all v ∈ V and all 0 < λ < λ

.

Proposition 4. If hypotheses H(f ) hold, then there exists b t > 0 such that for every |t| > b t there exists δ

> 0 such that R

(±tu

) < β for all λ ∈ (λ

− δ, λ

).

P r o o f. We have R

(tu

) = |t|

p kDu

k

− λ|t|

p ku

k

−

\

Z

F (z, tu

(z)) dz

= |t|

p kDu

k

− λ|t|

λ

p kDu

k

−

\

Z

F (z, tu

(z)) dz.

By H(f )(iv), we can find b t > 0 so large that for every |t| > b t,

−(β + 1) <

\

Z

F (z, ±tu

(z)) dz.

Let δ = δ

= λ

p/(t

kDu

k

) > 0. Then for λ

− δ < λ < λ

we have R

(±tu

) < t

δ

λ

p kDu

k

+ β − 1 = β.

The next proposition shows that R

(·) satisfies a sort of nonsmooth Palais–Smale condition over closed and convex subsets of W

(Z).

Proposition 5. If hypotheses H(f ) hold, K ⊆ W

(Z) is a nonempty, closed and convex set and {x

}

⊆ K and ε

> 0, ε

↓ 0 satisfy

|R

(x

)| ≤ M, 0 ≤ R

(x

; y − x

) + ε

ky − x

k for all y ∈ K, then {x

}

⊆ W

(Z) has a strongly convergent subsequence.

P r o o f. Since by hypothesis {R

(x

)}

is bounded and because R

(·) is coercive (see Proposition 2) we infer that {x

}

⊆ W

(Z) is bounded.

So by passing to a subsequence if necessary, we may assume that x

→ x in W

(Z) and x

→ x in L

(Z) as n → ∞. By h·, ·i we denote the duality brackets of the pair (W

(Z), W

(Z)). We know that R

(x

; x − x

) = sup{hx

, x − x

i : x

∈ ∂R

(x

)}, n ≥ 1. Because ∂R

(x

) ⊆ W

(Z) is weakly compact, we can find x

∈ ∂R

(x

) such that R

(x

; x − x

) = hx

, x − x

i, n ≥ 1. We have

x

= A(x

) − λk(x

) − v

, n ≥ 1, k(x) = |x|

x, where A : W

(Z) → W

(Z) is defined by

hA(x), yi =

\

Z

kDx(z)k

(Dx(z), Dy(z))

dz

for all y ∈ W

(Z) and v

∈ ∂J(x

) with J(x

) =

T

Z

F (z, x

(z)) dz. It is easy to see that A is monotone, demicontinuous, hence maximal monotone (see Hu–Papageorgiou [8]) and f

(z, x

(z)) ≤ v

(z) ≤ f

(z, x

(z)) a.e. on Z for all n ≥ 1 (see [10]). Then {v

}

⊆ L

(Z) is bounded and

0 ≤ hx

, x − x

i + ε

kx − x

k

= hA(x

), x − x

i −

\

Z

v

(z)(x − x

)(z) dz

− λ(k(x

), x − x

)

+ ε

kx − x

k, hence limhA(x

), x

− xi ≤ 0 since

\

Z

v

(z)(x − x

)(z) dz

−−→ 0, (k(x

), x − x

)

−−→ 0, ε

kx − x

k −−→ 0.

But A, being maximal monotone, is generalized pseudomonotone (see for example [8], Remark III.6.3, p. 365). So we have

hA(x

), x

i −−→ hA(x), xi,

hence kDx

k

−−→ kDxk

. Recall that Dx

→ Dx in L

(Z, R

) and that L

(Z, R

) is uniformly con- vex. So it has the Kadec–Klee property (see [8], Definition I.1.72, p. 28).

Thus Dx

→ Dx in L

(Z, R

), from which we conclude that x

→ x in W

(Z) as n → ∞.

Because ∂R

(x) ⊆ W

(Z) is weakly compact, we can find x

∈

∂R

(x) such that kx

k = m(x) = inf{ky

k : y

∈ ∂R

(x)}. Then the same proof as for Proposition 5 gives us the following result:

Proposition 6. If hypotheses H(f ) hold, then R

(·) satisfies the non- smooth (PS )-condition.

4. Multiplicity theorem. In this section we state and prove the main result of this paper, which shows that problem (3) has at least three non- trivial solutions for all λ ∈ (λ

− δ, λ

) with δ > 0 small enough (i.e. for equations near resonance).

Theorem 7. If hypotheses H(f ) hold, then there exists δ > 0 such that for all λ ∈ (λ

− δ, λ

) problem (3) has at least three nontrivial solutions.

P r o o f. From Proposition 3 of [10], we know that we can find β

, β

> 0 such that for all 0 < λ ≤ λ

and all x ∈ W

(Z), we have R

(x) ≥ β

kxk

− β

kxk

with θ > p. So we can find ̺ > 0 small enough so that inf[R

(x) : kxk

= ̺] > 0 for all λ ∈ (0, λ

]. Let β ∈ R be as in Proposition 3 and let b t > 0 be as in Proposition 4. Choose t

> max{b t, ̺} and let δ = δ

be as in Proposition 4. Then t

u

∈ B

(0) and R

(t

u

) < β < 0.

So we can apply Theorem 1 and obtain y

6= 0 such that R

(y

) > 0 > β

and 0 ∈ ∂R

(y

).

Let U

= {x ∈ W

(Z) : x = ±tu

+ v, t > 0, v ∈ V }. We show that R

(·) attains its infimum on both subsets U

and U

. To this end let m

= inf[R

(x) : x ∈ U

] = inf[R

(x) : x ∈ U

] (since R

(·) is locally Lipschitz). Let

R

(x) =

 R

(x) if x ∈ U

, +∞ otherwise.

Evidently, R

(·) is a lower semicontinuous function on the Banach space W

(Z) which is bounded below (see Proposition 2). By Ekeland’s vari- ational principle (strong form, see for example De Figueiredo [7] or Hu–

Papageorgiou [8]), we can find {x

}

⊆ U

such that R

(x

) ↓ m

as n → ∞ and

R

(x

) ≤ R

(y) + ε

ky − x

k for all y ∈ W

(Z), hence

R

(x

) ≤ R

(y) + ε

ky − x

k for all y ∈ U

.

Because U

is convex, for every t ∈ (0, 1) and every w ∈ U

, we have y

= (1 − t)x

+ tw ∈ U

for all n ≥ 1. So we have

−ε

kw − x

k ≤ R

(x

+ t(w − x

)) − R

(x

) t

and therefore

0 ≤ R

(x

; w − x

) + ε

kw − x

k for all w ∈ U

.

Proposition 5 says that by passing to a subsequence if necessary, we may assume that x

→ y

as n → ∞ in W

(Z). If y

∈ ∂U

= V , then because of Proposition 3, we have lim R

(x

) = R

(y

) = m

> β. On the other hand, Proposition 4 implies that there exists δ > 0 such that m

< β for λ ∈ (λ

− δ, λ

). So we have a contradiction from which it follows that y

∈ U

, y

6= 0. Thus y

is a local minimum of R

(·) and so 0 ∈ ∂R

(y

) (see Section 2). Working similarly on the other open set U

, we obtain y

∈ U

, y

6= y

6= y

such that 0 ∈ ∂R

(y

) for all λ ∈ (λ

− δ, λ

).

Finally, let y = y

, k ∈ {1, 2, 0}. Since 0 ∈ ∂R

(y) (where λ

− δ

<

λ < λ

, δ

= δ

), we have

A(y) − λ|y|

y = v in W

(Z)

with f

(z, y(z)) ≤ v(z) ≤ f

(z, y(z)) a.e. on Z. Thus for θ ∈ C

(Z) we have

hA(y), θi − λ

\

Z

|y(z)|

y(z)θ(z) dz =

\

Z

v(z)θ(z) dz, hence

\

Z

kDy(z)k

(Dy(z), Dθ(z))

dz =

\

Z

(v(z) + λ|y(z)|

y(z))θ(z) dz

and consequently

h− div(kDyk

Dy), θi =

\

Z

(v(z) + λ|y(z)|

y(z))θ(z) dz

(by Green’s formula; see for example Kenmochi [9]). Since C

(Z) is dense in W

(Z), we conclude that y ∈ W

(Z) solves problem (3).

This proves that y

, y

, y

are three dinstinct nontrivial solutions of (3).

Remark . It will be very interesting to know if a similar multiplicity theorem is also valid for the resonant problem.

Acknowledgments. The authors wish to express their gratitude to a very knowledgeable referee for his corrections, remarks, constructive criti- cism and for furnishing additional relevant references.

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Department of Mathematics National Technical University Zografou Campus

Athens 15780, Greece E-mail: npapg@math.ntua.gr

Received 16 September 1998;

revised 28 January 1999