Abstract. We study eigenvalue problems with discontinuous terms. In particular we consider two problems: a nonlinear problem and a semilinear problem for elliptic equations.

POLONICI MATHEMATICI LXXV.2 (2000)

Existence and multiplicity results for nonlinear eigenvalue problems with discontinuities

by Nikolaos S. Papageorgiou (Athens) and Francesca Papalini (Ancona)

Abstract. We study eigenvalue problems with discontinuous terms. In particular we consider two problems: a nonlinear problem and a semilinear problem for elliptic equations.

In order to study the existence of solutions we replace these two problems with their multivalued approximations and, for the first problem, we estabilish an existence result while for the second problem we prove the existence of multiple nontrivial solutions. The approach used is variational.

1. Introduction. The aim of this paper is to study nonlinear and semi- linear eigenvalue problems with discontinuous terms. So let Z ⊂ R

be a bounded domain with a C

boundary Γ . We start with the following eigen- value problem:

(1)

 − div(kDx(z)k

Dx(z))−λ|x(z)|

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

= 0, 2 ≤ p < ∞.

Here f : Z × R → R is a function such that for all z ∈ Z, f(z, ·) is locally bounded but not necessarily continuous and this implies that the problem (1) may not have any solutions. So in order to develop a reasonable exis- tence theory, we need to pass to a multivalued extension of problem (1) by, roughly speaking, filling in the gaps at the discontinuity points of f (z, ·).

More precisely, we introduce the following two functions:

f

(z, x) = lim inf

x^′→x

f (z, x

), f

(z, x) = lim sup

x^′→x

f (z, x

).

Note that when the one-sided limits f (z, x

) and f (z, x

) exist then f

(z, x) = min{f(z, x

), f (z, x

)} and f

(z, x) = max{f(z, x

), f (z, x

)}.

2000 Mathematics Subject Classification: 35R70, 35J20, 58E05.

Key words and phrases: locally Lipschitz functional, subdifferential, discontinuous term, multivalued problem, eigenvalues, p-Laplacian.

[125]

Instead of (1) we consider the following multivalued version:

(2)

 



− div(kDx(z)k

Dx(z)) − λ|x(z)|

x(z)

∈ [f

(z, x(z)), f

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

= 0, 2 ≤ p < ∞.

Using a variational approach (for nonsmooth, locally Lipschitz energy functionals) we show that problem (2) has a solution.

We will also study the semilinear problem (3)

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

= 0.

In this case f (z, ·) need not be continuous either. So after introducing the functions f

(z, x) and f

(z, x), we replace (3) with its multivalued coun- terpart

(4)

 −∆x(z) − λx(z) ∈ [f

(z, x(z)), f

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

= 0.

For problem (4), using a result for locally Lipschitz functionals due to D. Goeleven, D. Motreanu and P. D. Panagiotopoulos [9], we prove the existence of multiple nontrivial solutions.

Similar eigenvalue problems were studied by K. J. Brown and H. Budin [3], P. Hess [10], P. Rabinowitz [14], D. G. De Figueiredo [8], C. Lefter and D. Motreanu [12], M. Ramos [15]. All these works deal with semilin- ear equations (i.e. p = 2) and with exception of [12], assume f (·, ·) to be continuous. The approaches vary although all use some aspects of the varia- tional method. More specifically P. Rabinowitz [14], D. G. De Figueiredo [8], C. Lefter and D. Motreanu [12], M. Ramos [15] follow the variational ap- proach. P. Hess [10] has a proof which combines variational and topological degree arguments, while K. J. Brown and H. Budin [3] employ a combination of variational and monotone iteration methods.

In this paper our approach is solely variational. We use the critical point theory for nonsmooth local energy functionals (see K. C. Chang [5]).

2. Mathematical preliminaries. Let X be a reflexive Banach space and X

its topological dual. A function Φ : X → R is said to be locally Lipschitz if for every x ∈ X, there exists a neighbourhood U of x and a constant k > 0 depending on U such that |Φ(z) − Φ(y)| ≤ kkz − yk for all z, y ∈ U. For this kind of functions the generalized directional derivative Φ

(x; h) at x ∈ X in the direction h ∈ X is defined by

Φ

(x; h) = lim sup

x^′→x, λ↓0

Φ(x

+ λh) − Φ(x

)

λ .

It is known that the function h 7→ Φ

(x; h) is sublinear, continuous and it is the support function of the nonempty convex w

-compact set

∂Φ(x) = {x

∈ X : (x

, h) ≤ Φ

(x; h) for all h ∈ X}.

The set ∂Φ(x) is called the generalized or Clarke subdifferential of Φ at x.

If Φ, Ψ : X → R are locally Lipschitz functions, then ∂(Φ + Ψ)(x) ⊆ ∂Φ(x) +

∂Ψ (x), while for any λ ∈ R we have ∂(λΦ(x)) = λ∂Φ(x). Moreover, if Φ : X → R is convex, then this subdifferential coincides with the subdifferential in the sense of convex analysis. If Φ : X → R is strictly differentiable then

∂Φ(x) = {Φ

(x)}. A point x ∈ X is a critical point of Φ if 0 ∈ ∂Φ(x), while a critical value is the value assumed by Φ at a critical point. For more details we refer to the monograph of Clarke [6].

The compactness conditions for locally Lipschitz functionals Φ : X → R that we consider are the following:

• Φ satisfies the (PS )-condition if any sequence {x

}

⊂ X such that {Φ(x

)}

is bounded and m(x

) → 0 as n → ∞ has a convergent subsequence (where m(x

) = min

kx

k

; the existence of such an element follows from the fact that ∂Φ(x

) is weakly compact and the norm functional on X

is weakly semicontinuous);

• Φ satisfies the C-(P S)-condition if any sequence {x

}

⊂ X such that {Φ(x

)}

is bounded and (1 + kx

k)m(x

) → 0 as n → ∞ has a convergent subsequence.

Remark 1. The (PS)-condition is a generalization of the well known Palais–Smale condition proposed by Chang [5] in his nonsmooth critical point theory in order to obtain various minimax principles concerning the existence and characterization of critical points for locally Lipschitz func- tionals. The C-(PS)-condition is a weaker form of the (PS)-condition and it is the nonsmooth version of the condition introduced by Cerami in [4].

In a recent paper N. C. Kourogenis and N. S. Papageorgiou [11], using the C-(PS)-condition, derive minimax principles. For the convenience of the reader, we recall the following version of the Mountain Pass Theorem (cf. [11], Theorem 6).

Theorem 1. If X is a reflexive Banach space, Φ : X → R is a locally Lipschitz functional satisfying the C-(P S)-condition and there are ̺ > 0, α, β ∈ R and x

∈ X such that kx

k > ̺ and max[Φ(0), Φ(x

)] ≤ α < β ≤ inf

Φ(x) then Φ has a critical value c ≥ β characterized by

c = inf

γ∈Γ

max

0≤t≤1

Φ(γ(t))

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

}.

Moreover we recall the abstract multiplicity result due to Goeleven–

Motreanu–Panagiotopoulos (cf. [9], Theorem 2.1) that we will use to prove the existence of multiple nontrivial solutions for the problem (4).

Theorem 2. Let X be a reflexive Banach space. Suppose that Φ : X → R is an even, locally Lipschitz functional satisfying the (P S)-condition and the conditions:

(i) Φ(0) = 0,

(ii) there exists a subspace X

of X of finite codimension and numbers β, ̺ > 0 such that Φ(x) ≥ β for all x ∈ X

such that kxk = ̺,

(iii) there is a finite-dimensional subspace X

of X, with dim X

>

codim X

, such that Φ(u) → −∞ as kuk → ∞, u ∈ X

.

Then Φ has at least dim X

−codim X

pairs of nontrivial critical points.

Finally, we recall the following definitions for an operator A : X → X

:

• A is said to be monotone if hAx

−Ax

, x

−x

i ≥ 0 for all x

, x

∈ X;

• A is said to be pseudomonotone if for any sequence {x

}

⊂ X such that x

→ x weakly in X and lim sup

hAx

, x

− xi ≤ 0 it follows that hAx, x − wi ≤ lim inf

hAx

, x

− wi for all w ∈ X;

• A is said to be demicontinuous if for any sequence {x

}

⊂ X such that x

→ x in X it follows that Ax

→ Ax weakly in X.

3. Existence result. We start with the quasilinear problem. Let λ

be the first eigenvalue of the p-Laplacian −∆

x = − div(kDxk

Dx) on Z with zero Dirichlet boundary condition, which is the least real number λ for which the problem

 − div(kDx(z)k

Dx(z)) = λ|x(z)|

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

= 0,

has a nontrivial solution. It is known (cf. [13]) that this first eigenvalue λ

is positive, isolated and simple (i.e. the associated eigenfunctions are constant multiples of each other). Moreover we have the following variational characterization of λ

via the Rayleigh quotient:

λ

= min[kDxk

/kxk

: x ∈ W

(Z), x ≥ 0], where k · k

denotes the norm in the space L

(Z).

One can prove that the corresponding eigenfunction u

is almost every- where nonzero on Z. Let

λ

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

on W

(Z) and λ > λ

}.

Note that since λ

is isolated, λ

> λ

. Moreover, if p = 2, then λ

is the

second eigenvalue of the Laplacian.

In what follows let F (z, x) =

Tx

0

f (z, r) dr, x ∈ R, be the potential func- tion corresponding to f and Ψ : W

(Z) → R be the energy functional defined as Ψ (x) =

T

Z

F (z, x(z)) dz.

We introduce the following hypotheses on the function f :

H(f )

: f : Z × R → R is a Borel measurable function such that for all z ∈ Z, f(z, ·) is locally bounded and

(i) f

, f

are both N-measurable (i.e. for every measurable function x : Z → R the functions z 7→ f

(z, x(z)), i = 0, 1, are measurable);

(ii) there are a ∈ L

(Z, R

) and c

> 0 such that

|f(z, x)| ≤ a(z) + c

|x|

a.e. on Z, ∀x ∈ R;

(iii) there are c

> 0 and µ such that 1 ≤ µ ≤ p and lim sup

|x|→∞

f

(z, x)x − pF (z, x)

|x|

≤ −c

uniformly for a.e. z ∈ Z, i = 0, 1;

(iv) there are c

∈ R and q > p such that q − p < µ if 1 ≤ N ≤ p, while N (q − p)/p < µ and q < Np/(N − p) if N > p, with the property that

lim sup

|x|→∞

F (z, x)

|x|

≤ c

uniformly for a.e. z ∈ Z;

lim inf

|x|→∞

pF (z, x)

|x|

≥ 0 uniformly for a.e. z ∈ Z;

(v)

lim sup

|x|→0

pF (z, x)

|x|

≤ −λ

for a.e. z ∈ Z.

(vi)

Remark 2. Condition H(f )

(iii) was first introduced by D. G. Costa and C. A. Magalh˜aes [7]. Note that this condition follows from the well known Ambrosetti–Rabinowitz condition (see [1], when p = 2) which says:

there are θ > 2 and ξ > 0 such that 0 < θF (z, x) < xf (z, x) for a.e. z ∈ Z and all x such that |x| ≥ ξ.

We now present an example of a function f that satisfies hypotheses H(f )

. For simplicity we drop the z-dependence and we assume p > 2. Fix M > 1 and η, ξ > 0 such that pη + λ

− 1 ≤ ξ. Let k ∈ L

(R) be the function defined by

k(x) =

 



−ξ|x|

x, |x| ≤ 1, x

, 1 < |x| ≤ M, x, |x| > M.

We define f : Z × R → R by

f (z, x) = pη|x|

x − |x|

x

|x|

+ 1 + k(x).

Then the corresponding potential function is F (z, x) = η|x|

− 1

p log(|x|

+ 1) + θ(x), where

θ(x) =

x\

0

k(r) dr =

 

 



 

 

x

/2 − ξ/p + M

/5 − M

/2 − 1/5, x < −M,

−ξ/p + x

/5 + 1/5, −M < x < −1,

−(ξ/p)|x|

, |x| ≤ 1,

−ξ/p + x

/5 − 1/5, 1 < x < M , x

/2 − ξ/p + M

/5 − M

/2 + 1/5, x > M . Evidently hypothesis H(f )

(ii) is satisfied. Also for i = 1, 2 we have

f

(z, x)x − pF (z, x)

|x|

= − |x|

|x|

+ x

+1− log(|x|

+ 1)

x

− pθ(x)

x

→ 1− p 2 < 0 as |x| → ∞. So H(f)

(iii) is satisfied with µ = 2 < p. Moreover for any q > p we have F (z, x)/|x|

→ 0 as |x| → ∞ and so H(f)

(iv) holds. Now we deduce that

pF (z, x)

|x|

= pη − log(|x|

+ 1)

|x|

+ pθ(x)

|x|

→ pη as |x| → ∞, while

pF (z, x)

|x|

= pη − log(|x|

+ 1)

|x|

+ pθ(x)

|x|

→ pη − 1 − ξ as |x| → 0.

So recalling the choice of η and ξ we see that H(f )

(v) and H(f )

(vi) are satisfied.

Now we start to study the problem (2). We introduce, for every λ > 0, the functional R

: W

(Z) → R defined by

R

(x) = 1

p kDxk

− λ

p kxk

−

\

Z

F (z, x(z)) dz = 1

p kDxk

− λ

p kxk

− Ψ(x).

It is well known (cf. [5]) that Ψ is locally Lipschitz. Hence so is R

.

Proposition 3. If hypotheses H(f )

hold, then R

satisfies the C- (P S)-condition.

P r o o f. Let {x

}

⊂ W

(Z) be a sequence such that {R

(x

)}

is bounded and (1 + kx

k)m(x

) → 0 as n → ∞. Let x

∈ W

(Z) (1/p + 1/q = 1), x

∈ ∂R

(x

), n ≥ 1, be such that kx

k = m(x

). We have

x

= A(x

) − λJ(x

) − u

, where A : W

(Z) → W

(Z) is defined by

hA(x), yi =

\

Z

kDx(z)k

(Dx(z), Dy(z)) dz, ∀x, y ∈ W

(Z)

(here h·, ·i denotes the duality brackets for the pair (W

(Z), W

(Z))), and J : W

(Z) → L

(Z) ⊂ W

(Z) is given by

J(x)(·) = |x(·)|

x(·)

and u

∈ ∂Ψ(x

). From [5] we know that f

(z, x

(z)) ≤ u

≤ f

(z, x

(z)) a.e. on Z. Moreover let M > 0 be such that |R

(x

)| ≤ M for all n ≥ 1;

we have

(5) −kDx

k

+ λkx

k

+

\

Z

pF (z, x

(z)) dz ≤ pM, ∀n ≥ 1.

Also since (1 + kx

k)kx

k → 0 as n → ∞, there exists n ∈ N such that hA(x

), x

i − λ(J(x

), x

)

−

\

Z

u

(z)x

(z) dz < 1, ∀n ≥ n (here (·, ·)

denotes the duality brackets for the pair (L

(Z), L

(Z))). So we have

(6) kDx

k

− λkx

k

−

\

Z

u

(z)x

(z) dz < 1, ∀n ≥ n.

Adding (5) and (6) above we obtain (7)

\

Z

(pF (z, x

(z)) − u

(z)x

(z)) dz ≤ pM + 1, ∀n ≥ n.

From H(f )

(iii) we find that there exist bc

> 0 and M

> 1 such that for almost all z ∈ Z, all |x| ≥ M

and all u

∈ [f

(z, x), f

(z, x)], we have

(8) u

x − pF (z, x)

|x|

≤ −bc

.

Also using the growth hypothesis H(f )

(ii), we see that there exists β

> 0 such that

u

x − pF (z, x) ≤ β

for almost all z ∈ Z and all |x| < M

, hence, from (8), setting β

= β

+ bc

|M

|

, we obtain

u

x − pF (z, x) ≤ −bc

|x|

+ β

for almost all z ∈ Z and all x ∈ R.

Using this in (7), we obtain pM + 1 ≥

\

Z

(bc

|x

(z)|

− β

) dz, ∀n ≥ n, which implies that there exists β

> 0 such that

(9) kx

k

≤ β

, ∀n ≥ 1,

that is, the sequence {x

}

is bounded in L

(Z).

Now let 0 < θ < 1 be such that 1

q = 1 − θ

µ + θ

p

where p

= N p/(N − p) if p < N, otherwise any p

such that 1 ≤ µ ≤ p <

q < p

. Using the interpolation inequality (cf. [2], p. 57) and (9) we have kx

k

≤ kx

k

kx

k

≤ β

kx

k

, ∀n ≥ 1,

and so, from the Sobolev embedding theorem, there exists β

> 0 with the property

(10) kx

k

≤ β

kx

k

, ∀n ≥ 1

(k · k

denotes the norm in W

(Z)). By H(f )

(ii), (iv), for fixed bc

> c

, there exists c

> 0 such that

(11) F (z, x) ≤ bc

|x|

+ c

for almost all z ∈ Z and all x ∈ R.

From the choice of the sequence {x

}

, we find that R

(x

) ≤ M for all n ≥ 1, and so, using (11) we obtain

1

p kDx

k

≤ λ

p kx

k

+ bc

kx

k

+ β

, ∀n ≥ 1,

where β

= c

|Z| + M; hence, as p < q it follows that there exists β

> 0 such that

1

p kDx

k

≤ λ

p β

kx

k

+ bc

kx

k

+ β

, ∀n ≥ 1.

Now Young’s inequality and (10) show that there exist two positive numbers depending on λ: β

(λ) and β

(λ), with the property

1

p kDx

k

≤ β

(λ)kx

k

+ β

(λ), ∀n ≥ 1.

From our assumptions we can assume that θ q < p, in fact if p < N , then (N/p)(q − p) < µ is equivalent to θ q < p, while if p ≥ N, then we can always choose p

large and then θ in the interpolation inequality small so that θ q < p. Therefore, recalling that from Poincar´e’s inequality (cf. [2], p. 174), kDxk

is an equivalent norm in W

(Z), it follows that {x

}

is bounded in W

(Z). Thus, by passing to a subsequence if necessary, we may assume that

x

→ x weakly in W

(Z), x

→ x in L

(Z),

and so x

(z) →x(z) a.e. on Z and there exists η ∈ L

(Z) such that |x

(z)| ≤ η(z) a.e. on Z for all n ≥ 1. Hence we have

λ(J(x

), x

− x)

→ 0 as n → ∞,

(u

, x

− x)

→ 0 as n → ∞,

hx

, x

− xi → 0 as n → ∞.

Since hx

, x

− xi = hA(x

), x

− xi − λ(J(x

), x

− x)

− (u

, x

− x)

for all n ≥ 1, we infer that

(12) lim sup

n→∞

hA(x

), x

− xi = lim

hA(x

), x

− xi = 0.

It is simple to see that A : W

(Z) → W

(Z) is monotone and demicontinuous and so (cf. [16], p. 596) it is pseudomonotone; therefore from (12) we obtain

n→∞

lim hA(x

), x

i = hA(x), xi, which implies that

kDx

k

→ kDxk

as n → ∞.

But we also have Dx

→ Dx weakly in L

(Z), thus, since L

(Z) is a uni- formly convex space, we deduce that x

→ x in W

(Z) as n → ∞. So R

satisfies the C-(PS)-condition.

Using the previous proposition and the above-mentioned mountain pass theorem, we can prove the existence of nontrivial solutions for the prob- lem (2).

Theorem 4. If hypotheses H(f )

hold , then for every λ ∈ (λ

, λ

) the problem (2) has a nontrivial solution.

P r o o f. From H(f )

(v), given ε > 0 we can find M

> 0 such that F (x, z) ≥ − ε

p |x|

for a.e. z ∈ Z and all x such that |x| ≥ M

; also from H(f )

(ii), there exists γ > 0 with

F (x, z) ≥ −γ for a.e. z ∈ Z and all x such that 0 ≤ |x| < M

. Thus we deduce that

F (z, x) ≥ − ε

p |x|

− γ for a.e. z ∈ Z and all x ∈ R.

Hence, for all u

∈ W

(Z), u

6= 0, and all t ∈ R, it follows that R

(tu

) = |t|

p kDu

k

− λ |t|

p ku

k

−

\

Z

F (z, tu

(z)) dz

≤ |t|

p

 1 − λ

λ



kDu

k

+ ε|t|

λ

p kDu

k

+ γ

= |t|

p



1 − λ − ε λ



kDu

k

+ γ

, where γ

= γ|Z|. Take ε > 0 so that λ

< λ − ε. Then

(13) R

(tu

) → −∞ as |t| → ∞.

Next using H(f )

(vi), given ε > 0 we can find 0 < δ < 1 such that F (z, x) ≤ 1

p (−λ

+ ε)|x|

for a.e. z ∈ Z and all x such that |x| < δ.

From (11) we know that

F (z, x) ≤ bc

|x|

+ c

for a.e. z ∈ Z and all x ∈ R, so we have

F (z, x) ≤



bc

+ c

δ



|x|

for a.e. z ∈ Z and all x such that δ ≤ |x| ≤ 1, while

F (z, x) ≤ (bc

+ c

)|x|

for a.e. z ∈ Z and all x such that |x| ≥ 1.

Thus finally we can find η > 0 such that F (z, x) ≤ 1

p (−λ

+ ε)|x|

+ η|x|

for a.e. z ∈ Z and all x ∈ R.

Therefore, taking ε > 0 such that λ < λ

− ε, for all x ∈ W

(Z) we obtain R

(x) = 1

p kDxk

− λ

p kxk

−

\

Z

F (z, x(z)) dz

≥ 1

p kDxk

− λ

p kxk

+ 1

p (λ

− ε)kxk

− ηkxk

≥ 1

p kDxk

− η

kDxk

=

 1

p − η

kDxk



kDxk

,

where η

is a positive constant found by using the Rayleigh quotient and the Sobolev embedding theorem because p < q < p

. Since kDxk

is an equivalent norm in W

(Z), from this last inequality we deduce that there are ̺ > 0 and α, β ∈ R such that

0 ≤ α < β ≤ inf

kxk=̺

R

(x).

Now we observe that R

(0) = 0 and from (13) it is possible to find x

∈ W

(Z) with kx

k > ̺ and R

(x

) ≤ 0, therefore

max[R

(0), R

(x

)] ≤ α < β ≤ inf

kxk=̺

R

(x).

So, from Proposition 3, we can apply Theorem 1 to find x ∈ W

(Z) such

that 0 ∈ ∂R

(x) and R

(x) ≥ β > 0. Hence x 6= 0.

Also, since 0 ∈ ∂R

(x), there exists u

∈ ∂Ψ(x) such that 0 = A(x) − λJ(x) − u

, hence

hA(x), ϕi = λ(J(x), ϕ)

+ (u

, ϕ)

for all ϕ ∈ C

(Z).

Thus

\

Z

kDx(z)k

(Dx(z), Dϕ(z))

dz

=

\

Z

(λ|x(z)|

x(z)ϕ(z) + u

(z)ϕ(z)) dz for all ϕ ∈ C

(Z).

From the definition of the distributional derivative, we infer that

− div(kDx(z)k

Dx(z)) = λ|x(z)|

x(z) + u

(z) a.e. on Z.

Since u

(z) ∈ ∂Ψ(x)(z) ⊂ [f

(z, x(z)), f

(z, x(z))] a.e. on Z (cf. [5]), we deduce that x ∈ W

(Z) is a nontrivial solution of problem (2).

4. Semilinear problem. Now we pass to the semilinear eigenvalue prob- lem with discontinuous right hand side

(3)

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

= 0.

Again the discontinuity of f (z, ·) in (3) forces us to pass to the multival- ued variant of (3). Namely we pass to the elliptic eigenvalue inclusion (4)

 −∆x(z) − λx(z) ∈ [f

(z, x(z)), f

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

= 0.

Also in this case we denote by F (z, x) the function F (z, x) =

Tx

0

f (z, r) dr, x ∈ R, which is the potential function corresponding to f.

In order to prove the existence of multiple nontrivial solutions we make the following hypotheses on f :

H(f )

: f : Z × R → R is a Borel measurable function such that for all z ∈ Z, f(z, ·) is locally bounded and

(i) f

, f

are both N-measurable;

(ii) there are a

∈ L

(Z, R

) and c > 0 such that

|f(z, x)| ≤ a

(z) + c|x|

a.e. on Z, ∀x ∈ R, where 0 ≤ s ≤ (N + 2)/(N − 2) and N > 2;

(iii) there are θ > 2 and ξ > 0 such that

0 < θF (z, x) < xf (z, x) for a.e. z ∈ Z and all x such that |x| ≥ ξ;

(iv) lim inf

|x|→∞

2F (z, x)

x

≥ 0 uniformly for a.e. z ∈ Z.

Remark 3. Condition H(f )

(iii) forces the s in H(f )

(ii) to satisfy 1 < s ≤ (N + 2)/(N − 2) (see [14], p. 9).

Again we introduce, for every λ > 0, the locally Lipschitz functional R

: H

(Z) → R defined by

R

(x) = 1

2 kDxk

− λ

2 kxk

− Ψ(x), where Ψ : H

(Z) → R is defined by Ψ(x) =

T

Z

F (z, x(z)) dz.

Proposition 5. If hypotheses H(f )

hold , then R

satisfies the (PS )- condition.

P r o o f. Let {x

}

⊂ H

(Z) be a sequence with the property that there exists M

> 0 such that |R

(x

)| ≤ M

for all n ≥ 1 and m(x

) → 0 as n → ∞. As before let x

∈ H

(Z), x

∈ ∂R

(x

), n ≥ 1, be such that kx

k = m(x

). We know that

x

= A(x

) − λx

− u

, n ≥ 1, where A : H

(Z) → H

(Z) is defined by hA(x), yi =

T

Z

(Dx(z), Dy(z)) dz for x, y ∈ H

(Z) (here h·, ·i denotes the duality brackets for the pair (H

(Z), H

(Z))) and u

∈ ∂Ψ(x

). From [5] we know that f

(z, x

(z)) ≤ u

(z) ≤ f

(z, x

(z)) a.e. on Z. It is simple to see that A is linear and monotone and so (cf. [16], p. 596) it is pseudomonotone.

Let η ∈ (1/θ, 1/2) and observe that there exists n ≥ 1 such that

−ηhx

, x

i ≤ ηkx

k

, ∀n ≥ n, which implies that

−ηhA(x

), x

i + ηλkx

k

+ η(u

, x

)

≤ ηkx

k

, ∀n ≥ n, where (·, ·)

denotes the inner product in the Hilbert space L

(Z). So from the definition of R

and since 2 < θ, we deduce that there exists b β

such that

M

+ ηkx

k

≥ (1/2 − η)kDx

k

− λ(1/2 − η)b β

kx

k

(14)

−

\

Z

(F (z, x

(z)) − ηu

(z)x

(z)) dz, ∀n ≥ n.

Using H(f )

(iii), we have

0 < θF (z, x) ≤ xf

(z, x) for a.e. z ∈ Z and all x with |x| > ξ, i = 0, 1, and so, for all n ≥ 1, we obtain

(15) 0 < θF (z, x

(z)) ≤ u

(z)x

(z) for a.e. z ∈ Z with |x

(z)| > ξ.

Now, passing to integrals, we have

\

Z

(ηu

(z)x

(z) − F (z, x

(z))) dz

=

\

A

(ηu

(z)x

(z) − F (z, x

(z))) dz +

\

B

ηu

(z)x

(z) dz

−

\

B

F (z, x

(z)) dz, ∀n ≥ 1,

where A = {z ∈ Z : |x

(z)| > ξ} and B = {z ∈ Z : |x

(z)| ≤ ξ}. Note that, from H(f )

(ii), |ηu

(z)x

(z)| ≤ η(a

(z) + c|x

(z)|

)|x

(z)| ≤ b β

a.e. on B, where b β

= η(ka

k

+ cξ

)ξ. Hence, using (15) we obtain

(16)

\

Z

(ηu

(z)x

(z) − F (z, x

(z))) dz

≥

\

A

(ηθ − 1)F (z, x

(z)) dz −

\

B

F (z, x

(z)) dz − b β

, ∀n ≥ 1, where b β

= b β

|Z|.

Hypothesis H(f )

(iii) implies (cf. [14], p. 9) that there exist ec

, ec

> 0 such that F (z, x) ≥ ec

|x|

− ec

a.e. on Z and for all x ∈ R. So we have (17)

\

A

(ηθ − 1)F (z, x

(z)) dz

≥ (ηθ − 1)

\

Z

(ec

|x

(z)|

− ec

) dz − (ηθ − 1)

\

B

(ec

|x

(z)|

− ec

) dz

≥ (ηθ − 1)(ec

kx

k

− ec

|Z|) − (ηθ − 1)ec

ξ

|Z|

+ (ηθ − 1)ec

|B|, ∀n ≥ 1.

Also, by H(f )

(ii) there exists b β

> 0 such that (18)

\

B

F (z, x

(z)) dz ≤ b β

, ∀n ≥ 1.

Using (17) and (18) in (16) we find that there exists b β

> 0 such that (19)

\

Z

(ηu

(z)x

(z) − F (z, x

(z))) dz ≥ (ηθ − 1)ec

kx

k

− b β

, ∀n ≥ 1.

Now, using (19) in (14), we have

M

+ ηkx

k

≥ (1/2 − η)kDx

k

− λ(1/2 − η)b β

kx

k

+ (ηθ − 1)ec

kx

k

− b β

, ∀n ≥ n.

Since by hypothesis 2 < θ, from Young’s inequality, for fixed ε > 0, there

exists δ(ε) > 0 such that kx

k

≤ δ(ε) + εkx

k

for all n ≥ 1; so we deduce

that

M

+ ηkx

k

≥ (1/2 − η)kDx

k

− λ(1/2 − η)b β

εkx

k

+ (ηθ − 1)ec

kx

k

− b β

(ε), ∀n ≥ n,

where b β

(ε) = b β

+ λ(1/2 − η)b β

δ(ε). Choose ε > 0 so that (ηθ − 1)ec

− λ(1/2 − η)b β

ε > 0. Then

M

+ ηkx

k

≥ (1/2 − η)kDx

k

− b β

(ε), ∀n ≥ n.

Since η < 1/2 and using Poincar´e’s inequality, we infer that {x

}

is bounded in H

(Z); thus we may assume that x

→ x weakly in H

(Z) and x

→ x in L

(Z) as n → ∞. Since by hypothesis s < (N + 2)/(N − 2), we have s + 1 < p

= 2N/(N − 2) and so H

(Z) is compactly embedded in L

(Z). Hence x

→ x in L

(Z) as n → ∞.

Moreover from Theorem 2.1 of [5] we find that u

∈ ∂Ψ(x

) ⊂ L

(Z) for all n ≥ 1 where 1/ν + 1/(s + 1) = 1, and since f

(z, x

(z)) ≤ u

(z) ≤ f

(z, x

(z)) a.e. on Z and for all n ≥ 1, from H(f)

(ii) we deduce that {u

}

is bounded in L

(Z).

Now, from our assumptions, we know that

x

= A(x

) − λx

− u

, n ≥ 1, and so

hA(x

), x

− xi = hx

, x

− xi + λ(x

, x

− x)

+ (u

, x

− x)

, n ≥ 1.

Therefore

lim sup

n→∞

hA(x

), x

− xi = 0,

and so from the pseudomonotonicity of A : H

(Z) → H

(Z) we deduce kDx

k

→ kDxk

as n → ∞.

But we also have Dx

→ Dx weakly in L

(Z), thus, since L

(Z) is a uni- formly convex space, we deduce that x

→ x in H

(Z) as n → ∞. So R

satisfies the (PS)-condition.

In what follows we denote by (λ

)

the increasing sequence of positive numbers which are distinct eigenvalues of −∆ with the Dirichlet boundary condition (cf. [2], p. 192). Using the previous proposition we can prove our multiplicity result:

Theorem 6. If hypotheses H(f )

hold , if for a.e. z ∈ Z, f(z, ·) is odd and if there exists k ≥ 1 such that

lim sup

x→0

2F (z, x)

x

≤ −λ

uniformly for a.e. z ∈ Z,

then for every λ ∈ (λ

, λ

) the problem (4) has k pairs of nontrivial

solutions.

P r o o f. From H(f )

(iv), given ε > 0, we can find M

> 0 such that F (z, x) ≥ − ε

2 x

for a.e. z ∈ Z and all x such that |x| > M

. Also from the growth hypothesis H(f )

(ii) we know that there exists b β

such that

|F (z, x)| ≤ b β

for a.e. z ∈ Z and all x such that |x| ≤ M

. Hence we infer that

(20) F (z, x) ≥ − ε

2 x

− b β

for a.e. z ∈ Z and all x ∈ R.

Let V = span{u

}

, where u

, . . . , u

are the eigenfunctions corre- sponding to the first k eigenvalues of the Laplacian. Recall that for all x ∈ V we have kDxk

≤ λ

kxk

, so, using (20), we obtain

R

(x) = 1

2 kDxk

− λ

2 kxk

−

\

Z

F (z, x(z)) dz

≤ 1

2 kDxk

− λ

2 kxk

+ ε

2 kxk

+ b β

≤ 1 2



1 − λ − ε λ



kDxk

+ b β

,

where b β

= b β

|Z|. Take ε > 0 so that λ

< λ − ε. Then we see that

(21) R

(x) → −∞ as kxk

→ ∞.

Next using our assumption, given ε > 0 we can find 0 < δ < 1 such that F (z, x) ≤ (−λ

+ ε) x

2 for a.e. z ∈ Z and all x such that |x| < δ.

Also from H(f )

(ii) we have

F (z, x) ≤ γ

|x|

for a.e. z ∈ Z and all x such that δ ≤ |x| ≤ 1, where γ

= (ka

k

/δ

+ c). Moreover, there exists eγ > 0 such that for almost all z ∈ Z and all x such that |x| ≥ 1 we have

F (z, x) ≤ eγ|x|

≤ eγ|x|

,

with 2 < r ≤ 2N/(N − 2) and r ≥ s + 1. Therefore we can find bγ > 0 large enough such that

F (z, x) ≤ (−λ

+ ε) x

2 + bγ|x|

for a.e. z ∈ Z and all x ∈ R.

Hence, for all x ∈ H

(Z), taking ε > 0 such that λ + ε < λ

, it follows

that

R

(x) = 1

2 kDxk

− λ

2 kxk

−

\

Z

F (z, x(z)) dz

≥ 1

2 kDxk

− λ

2 kxk

+ λ

− ε

2 kxk

− bγkxk

≥ 1

2 kDxk

− γ

kDxk

=

 1

2 − γ

kDxk



kDxk

,

where γ

is a positive constant found by using Poincar´e’s inequality.

Since kDxk

is an equivalent norm in H

(Z), from this last inequality we deduce that there are ̺, β > 0 such that

R

(x) ≥ β for all x ∈ H

(Z) such that kxk

= ̺.

Now, since R

is even and R

(0) = 0, by Proposition 5 and (21), we can apply Theorem 2 with X

= H

(Z) and X

= V to deduce the existence of k pairs of nontrivial critical points for R

. As in the proof of Theorem 4 we can verify that these pairs of nontrivial critical points are nontrivial solutions of problem (4).

We conclude by presenting an example of a function f that satisfies hypotheses H(f )

and the conditions of our Theorem 6. Again we drop the z-dependence and we suppose for simplicity that N = 3. Let f : Z × R → R be defined by

f (z, x) =

 − √

x, |x| ≤ 1, 6x

, |x| > 1.

Evidently f (z, ·) is odd and F (z, x) =

 −(3/4) √

x

, |x| ≤ 1, x

− 7/4, |x| > 1.

Note that if s = 5 = (N + 2)/(N − 2), then f satisfies the growth hypothesis H(f )

(ii). Also, assuming θ = 6 and ξ = 2, for |x| > ξ we obtain

0 < θF (z, x) = 6(x

− 7/4) ≤ 6x

= xf (z, x).

So H(f )

(iii) is satisfied . Moreover if |x| > 1 then 2F (z, x)

x

= 2x

− 7

2x

→ +∞ as |x| → ∞;

while if |x| < 1 then 2F (z, x)

x

= − 3

2 x

→ −∞ as |x| → 0.

Therefore the function f satisfies hypotheses H(f )

and the conditions of

Theorem 6.

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Department of Mathematics National Technical University Zografou Campus

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

Department of Mathematics University of Ancona Via Brecce Bianche Ancona 60131, Italy E-mail: papalini@dipmat.unian.it Re¸ cu par la R´ edaction le 18.10.1999

R´ evis´ e le 17.4.2000