Leszek Gasi´ nski

ON THE EXISTENCE OF FIVE NONTRIVIAL SOLUTIONS FOR RESONANT PROBLEMS

WITH p-LAPLACIAN

Leszek Gasi´ nski

Jagiellonian University Institute of Computer Science, Poland

e-mail: gasinski@softlab.ii.uj.edu.pl

and

Nikolaos S. Papageorgiou National Technical University Department of Mathematics, Greece

e-mail: npapgamath.ntua.gr

Abstract

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential.

The hypotheses on the nonsmooth potential allow resonance with re- spect to the principal eigenvalue λ

> 0 of (−∆

, W

(Z)). We prove the existence of five nontrivial smooth solutions, two positive, two nega- tive and the fifth nodal.

Keywords: p-Laplacian, Clarke subdifferential, linking sets, upper- lower solutions, second eigenvalue, nodal and constant sign solutions, second deformation theorem.

2000 Mathematics Subject Classification: 35J20, 35J60, 58J70.

∗

This research has been partially supported by the Ministry of Science and Higher

Education of Poland under Grant no. N201 027 32/1449.

1. Introduction

Let Z ⊆ R

be a bounded domain with a C

-boundary ∂Z. In this paper we study the following nonlinear elliptic problem with a nonsmooth potential:

(1.1)

 −div k∇x(z)k

∇x(z)

∈ ∂j(z, x(z)) for a.a. z ∈ Z, x|

= 0.

with p ∈ (1, +∞). Here (z, ζ) 7−→ j(z, ζ) is a measurable potential which for almost all z ∈ Z, as a function of ζ ∈ R, is locally Lipschitz and in general nonsmooth. By ∂j(z, ζ) we denote the Clarke subdifferential of ζ −→ j(z, ζ) (see Section 2.). Our goal is to prove a multiplicity result for problem (1.1), when the problem at infinity is resonant with respect to λ

> 0, the principal eigenvalue of (−∆

, W

(Z)). This implies that the corresponding Euler functional of the problem is indefinite. Moreover, we will provide full information about the sign of the solutions we obtain.

Let us mention some recent papers containing multiplicity results for the p-Laplacian equation. In Carl-Perera [3], Jiu-Su [12], Zhang-Chen-Li [20], Liu-Liu [15], Liu [16], Papageorgiou-Papageorgiou [18] and Carl-Motreanu [2], the Euler functionals of the problem is coercive and the authors produce at most three nontrivial solutions. Precise information about the sign of their solutions, can be found in Carl-Perera [3], Zhang-Chen-Li [20] and Carl-Motreanu [2]. Papers of Ambrosetti-Garcia Azorero-Peral Alonso [1]

and Garcia Azorero-Manfredi-Peral Alonso [9], deal with an indefinite Euler functional.

2. Mathematical Background

Let X be a Banach space and X

its topological dual. By k · k we denote the norm in X, by k · k

the norm in X

, and by h·, ·i the duality brackets for the pair (X, X

). If ϕ : X 7−→ R is a locally Lipschitz function, then the generalized directional derivative of ϕ at x ∈ X in the direction h ∈ X is defined by

ϕ

(x; h) = lim sup

x⁰→ x t& 0

ϕ(x

+ th) − ϕ(x

)

t .

It is easy to check that the function X 3 h 7−→ ϕ

(x; h) ∈ R is sublinear,

continuous, hence it is the support function of a nonempty, convex and

w

-compact set, defined by

∂ϕ(x) = {x

∈ X

: hx

, hi 6 ϕ

(x; h) for all h ∈ X}.

The multifunction x 7−→ ∂ϕ(x) is called Clarke subdifferential of ϕ. For a given locally Lipschitz functional ϕ : X −→ R, we say that x is a critical point of ϕ, if 0 ∈ ∂ϕ(x). It is easy to see that, if x ∈ X is a local extremum of ϕ (i.e., a local minimum or a local minimum of ϕ), then x ∈ X is a critical point of ϕ.

We say that a locally Lipschitz functional ϕ : X −→ R satisfies the Ce- rami condition at level c ∈ R (the C

-condition for short), if every sequence {x

}

⊆ X, such that ϕ(x

) −→ c and (1 + kx

k)m

(x

) −→ 0, with m

(x

) = inf 

kx

k : x

∈ ∂ϕ(x

)

, has a strongly convergent subsequence.

We say that ϕ satisfies the C-condition, if it satisfies the C

-condition for every c ∈ R.

Let Y be a Hausdorff topological space, E

, E, D are nonempty closed subsets of Y and E

⊆ E. We say that the pair {E

, E } is linking with D in Y if and only if E

∩ D = ∅, and for any γ ∈ C(E; Y ) such that γ|

= id|

we have γ(E) ∩ D 6= ∅. Using this general topological notion, we can prove the following minimax principle for the critical values of a locally Lipschitz function.

Theorem 2.1. If X is a Banach space, E

, E and D are nonempty closed subset of X, such that the pair {E

, E } is linking with D in X, ϕ : X −→ R is locally Lipschitz,

sup

E0

ϕ 6 inf

D

ϕ, Γ = 

γ ∈ C(E; X) : γ|

= id|

, c = inf

γ∈Γ

sup

v∈E

ϕ(γ(v))

and ϕ satisfies the C

-condition, then c > inf

ϕ and c is a critical value of ϕ. Moreover, if c = inf

ϕ, then there exists a critical point of ϕ in D.

Definition 2.2. Let C be a subset of the Banach space X. A continuous

deformation of C is a continuous map h : [0, 1]×C −→ C, such that h(0, ·) =

id

. If B ⊆ C, then we say that B is a weak deformation retract (resp. a

strong deformation retract ) of C, if there exists a continuous deformation

h : [0, 1] × C −→ C, such that h([0, 1] × B) ⊆ B (resp. h(t, b) = b for all

t ∈ [0, 1] and all b ∈ B) and h(1, C) ⊆ B.

For a given locally Lipschitz functional ϕ : X −→ R, we introduce the fol- lowing sets:

˙ ϕ

= 

x ∈ X : ϕ(x) < c

(the strict sublevel set of ϕ at c);

K

= 

x ∈ X : 0 ∈ ∂ϕ(x)

(the critical set of ϕ);

K

= 

x ∈ X : ϕ(x) = c

(the critical set of ϕ at level c ∈ R).

The next theorem is a nonsmooth version of the so-called “second deforma- tion theorem” (see Chang [4, p. 23] and Gasi´ nski-Papageorgiou [11, p. 628]).

Theorem 2.3. If X is a Banach space, ϕ : X −→ R is locally Lipschitz and satisfies the C-condition, a ∈ R, a < b 6 +∞, ϕ has no critical points in ϕ

(a, b) and K

is a finite set consisting of only local minima, then there exists a continuous deformation h : [0, 1] × ˙ ϕ

−→ ˙ ϕ

, such that

(a) h(t, ·)|

= id|

for all t ∈ [0, 1];

(b) h(1, ˙ ϕ

) ⊆ ˙ ϕ

∪ K

;

(c) ϕ(h(t, x)) 6 ϕ(x) for all t ∈ [0, 1] and all x ∈ ˙ ϕ

.

In particular, the set ˙ ϕ

∪ K

is a weak deformation retract of ˙ ϕ

. Next we recall some basic facts about the spectrum of the negative p-Laplacian with Dirichlet boundary condition. So, let m ∈ L

(Z)

, m 6= 0 and consider the following weighted (with weight m) nonlinear eigenvalue problem:

(2.1)

( −div k∇u(z)k

∇u(z)

= b λm(z)|u(z)|

u(z) for a.a. z ∈ Z, u|

= 0,

with 1 < p < +∞, b λ ∈ R. In what follows we use the notation −∆

u =

−div k∇u(z)k

∇u(z)

. By an eigenvalue of (−∆

, W

(Z), m), we mean a number b λ (m) ∈ R, such that (2.1) has a nontrivial solution u ∈ W

(Z).

Nonlinear regularity theory implies that u ∈ C

(Z) (see for example Gasi´ nski-

Papageorgiou [11, p. 737–738]). The least b λ ∈ R for which problem (2.1)

has a nontrivial solution, is the first eigenvalue of (−∆

, W

(Z), m) and

it is denoted by b λ

(m). We know that: b λ

(m) > 0; b λ

(m) is isolated (i.e.,

there exists ε > 0, such that (b λ

(m), b λ

(m) + ε) contains no eigenvalues

of (−∆

, W

(Z), m)); b λ

(m) is simple (i.e., the corresponding eigenspace

is one-dimensional). The first eigenvalue b λ

(m) > 0 admits the following

variational characterization:

(2.2) bλ

(m) = min

 k∇xk

R

Z

m|x|

dz : x ∈ W

(Z), x 6= 0

 .

The minimum in (2.2) is attained on the one-dimensional eigenspace of b

λ

(m). By u

we denote the L

-normalized eigenfunction for b λ

(m). We already know that u

∈ C

( b Z) and from (2.2) it is clear that u

does not change sign, so we may say that u

(z) > 0 for all z ∈ Z. Note that the Banach space C

(Z) is an ordered Banach space with order cone

C

= 

x ∈ C

(Z) : x(z) > 0 for all z ∈ Z . This cone has a nonempty interior and in fact

int C

= 

x ∈ C

: x(z) > 0 for all z ∈ Z and ∂x

∂n (z) < 0 for all z ∈ ∂Z . Here by n we denote the unit outward normal on ∂Z. By virtue of the nonlinear strong maximum principal of Vazquez [19], we have that u

∈ int C

. In addition to b λ

(m) > 0, we obtain a whole strictly increasing sequence {b λ

(m)}

⊆ R

of eigenvalues of (−∆

, W

(Z), m), such that b λ

(m) −→ +∞ as k → +∞. These are the so called Lusternik-Schnirelmann eigenvalues (LS-eigenvalues for short) of (−∆

, W

(Z), m). If p = 2 (linear eigenvalue problem), then these are all the eigenvalues. If p 6= 2 (nonlinear eigenvalue problem), we do not know if this is true. Nevertheless, since b λ

(m) > 0 is isolated and the set of eigenvalues of (−∆

, W

(Z), m) is closed, we can define

bλ

(m) = inf bλ : bλ is an eigenvalue of (−∆

, W

(Z), m), b λ > b λ

(m)

> b λ

(m),

which is an eigenvalue of (−∆

, W

(Z), m). In fact we have b λ

(m) = b λ

(m), i.e., the second eigenvalue and the second LS-eigenvalue coincide.

If m ≡ 1, then we write b λ

(m) = λ

for all k > 1 and (−∆

, W

(Z), m)

= (−∆

, W

(Z)). Since λ

> 0 is also the second LS-eigenvalue, it admits

a variational characterization provided by the Lusternik-Schnirelmann the-

ory. However, for our purposes that characterization is not convenient. In-

stead, we will use the following characterization of λ

> 0, due to Cuesta-de

Figueiredo-Gossez [6]. Let

∂B

= 

v ∈ L

(Z) : kvk

= 1 (2.3) ,

S = W

(Z) ∩ ∂B

, (2.4)

Γ

= 

γ

∈ C([−1, 1]; S) : γ

(−1) = −u

, γ

(1) = u

(2.5) .

Then

(2.6) λ

= inf

γ₀∈Γ0

sup

x∈γ0([−1,1])

k∇xk

.

Our approach also uses truncation techniques coupled with the method of upper-lower solutions. So let us recall the definition of upper and lower solutions for problem (1.1).

We say that x ∈ W

(Z) is an “upper solution” for problem (1.1), if x|

> 0 and

Z

Z

k∇xk

(∇x, ∇v)

dz >

Z

Z

uv dz

for all v ∈ W

(Z), v > 0 and some u ∈ L

(Z), with u(z) ∈ ∂j(z, x(z)) for a.a. z ∈ Z. We say that x is a “strict upper solution”, if in addition it is not a solution of (1.1). We say that x ∈ W

(Z) is a “lower solution” for problem (1.1), if

x|

6 0 and Z

Z

k∇xk

(∇x, ∇v)

dz 6 Z

Z

uv dz

for all v ∈ W

(Z), v > 0 and some u ∈ L

(Z), with u(z) ∈ ∂j(z, x(z)) for a.a. z ∈ Z. We say that x is a “strict lower solution”, if in addition it is not a solution of (1.1).

If X is a reflexive Banach space and A : X −→ X

is a map, we say that the map A is of type (S)

, if for every sequence {x

}

⊆ X, such that x

−→ x weakly in X and

lim sup

n→+∞

hA(x

), x

− xi 6 0,

one has that x

−→ x in X as n → +∞. If A : X −→ X

is of type (S)

and B : X −→ X

is compact, then A + B is of type (S)

.

If X is an ordered Banach space with order cone K, int K 6= ∅ and

e ∈ int K, then for every x ∈ X, we can find ϑ(x) > 0, such that x 6 ϑ(x)e.

In the sequel we use the notation r

= max{±r, 0} and kxk = k∇xk

for all x ∈ W

(Z).

To formulate the hypotheses on the potential j, we need to introduce the following notion. Suppose f : Z × R −→ R is a measurable function, such that for every r > 0 there exists a

∈ L

(Z)

, such that

|f (z, ζ)| 6 a

(z) for a.a. z ∈ Z and all |ζ| 6 r.

We permit f (z, ·) to have jump discontinuities and in order to be able to guarantee a solution, we fill in the discontinuities gaps. For this purpose we define

(2.7) f

(z, ζ) = lim inf

ζ⁰→ζ

f (z, ζ

) and f

(z, ζ) = lim sup

ζ⁰→ζ

f (z, ζ

).

Note that for almost all z ∈ Z, both limits are finite. We assume that f

and f

are superpositionally measurable, meaning that, if u : Z −→ R is a measurable function, then so are the functions z 7−→ f

(z, u(z)) and z 7−→ f

(z, u(z)). We set

j(z, ζ) = Z

0

f (z, r) dr.

Evidently for almost all z ∈ Z, the function j(z, ·) is locally Lipschitz and

∂j(z, ζ) = [f

(z, ζ), f

(z, ζ)]

(see Clarke [5]). Clearly j(z, 0) = 0 for almost all z ∈ Z and if f (z, ·) is continuous at ζ = 0, then ∂j(z, 0) = {0}. The set of hypotheses on the nonsmooth potential j is the following.

H (j) j : Z × R −→ R is a function, such that j(z, ζ) = R

0

f(z, r) dr where f : Z × R −→ R is a function satisfying

(i) the function (z, ζ) 7−→ f (z, ζ) is measurable; the function f

, f

, defined by (2.7) are superpositionally measurable;

(ii) for almost all z ∈ Z, the function ζ 7−→ f (z, ζ) is continuous at ζ = 0;

(iii) for almost all z ∈ Z, all ζ ∈ R, we have |f(z, ζ)| 6 a(z) + c|ζ|

with

a ∈ L

(Z)

, c > 0;

(iv) lim

|ζ|→+∞

pj(z, ζ)

|ζ|

= λ

uniformly for almost all z ∈ Z and

ζ→−∞

lim f

(z, ζ)ζ − pj(z, ζ)

= −∞, uniformly for almost all z ∈ Z;

(v) there exists b η ∈ L

(Z)

, such that λ

< inf

z∈Z

lim inf

ζ→0

f

(z, ζ)

|ζ|

ζ 6 lim sup

ζ→0

f

(z, ζ)

|ζ|

ζ 6 η(z) b uniformly for almost all z ∈ Z;

(vi) there exist a

< 0 < a

, such that 0 ∈ ∂j(z, a

) and 0 ∈ ∂j(z, a

) for almost all z ∈ Z and there is k > 0, such that

0 6 u

6 k(a

− ζ)

for almost all z ∈ Z, all ζ ∈ [0, a

) and all u

∈ ∂j(z, ζ) and

−k(ζ − a

)

6 u

6 0

for almost all z ∈ Z, all ζ ∈ (a

, 0] and all u

∈ ∂j(z, ζ).

Remark 2.4. The first limit in hypothesis H(j)(iv) implies that the prob- lem is resonant at infinity with respect to λ

. For this reason, we need to introduce an additional asymptotic condition at infinity.

Example 2.5. The following function satisfies hypotheses H(j) (for sim- plicity we drop the z-dependence):

j(ζ) = Z

0

f (r) dr ∀ζ ∈ R, with f : R −→ R, defined by

f (ζ) =

 

 

 



−c(1 + ζ)

ln(|ζ|

+ 1) if ζ ∈ [−1, 0],

c(1 − ζ)

ln(|ζ|

+ 1) if ζ ∈ [0, 1],

λ

|ζ|

ζ − ln |ζ| if |ζ| > 1,

with c > λ

. Note that at ζ = −1, the function f exhibits a downward jump discontinuity of −λ

and at ζ = 1, an upward jump discontinuity of λ

.

3. Constant sign solutions

In this section we produce four nontrivial smooth solutions of constant sign (two positive and two negative) for problem (1.1).

We consider the following truncations of the nonsmooth potential j(z, ζ):

bj

(z, ζ) =

 



 

0 if ζ 6 0,

j(z, ζ) if ζ ∈ [0, a

], j(z, a

) if a

6 ζ, bj

(z, ζ) =

 



 

j(z, a

) if ζ 6 a

, j(z, ζ) if ζ ∈ [a

, 0], 0 if ζ > 0.

Note that for all ζ ∈ R, the functions z 7−→ bj

(z, ζ) are measurable and for almost all z ∈ Z, functions ζ 7−→ b j

(z, ζ) are locally Lipschitz.

From the nonsmooth chain rule (see Clarke [5, p. 42]), we know that

∂b j

(z, ζ) =

 

 

 

 

 

 

 



{0} if ζ < 0,

{τ ∂j(z, 0) : τ ∈ [0, 1]} = {0} if ζ = 0,

∂j(z, ζ) if ζ ∈ (0, a

),

⊆ ∂j(z, a

) if ζ = a

,

{0} if a

< ζ,

(3.1)

∂b j

(z, ζ) =

 

 

 

 

 

 

 



{0} if ζ < a

,

⊆ ∂j(z, a

) if ζ = a

,

∂j(z, ζ) if ζ ∈ (a

, 0),

{τ ∂j(z, 0) : τ ∈ [0, 1]} = {0} if ζ = 0,

{0} if 0 < ζ.

(3.2)

We consider the functionals ϕ, b ϕ

: W

(Z) −→ R, defined by b

ϕ

(x) = 1

p k∇xk

− Z

Z

bj

(z, x(z)) dz ∀x ∈ W

(Z), (3.3)

ϕ(x) = 1

p k∇xk

− Z

Z

j(z, x(z)) dz ∀x ∈ W

(Z).

(3.4)

We know that b ϕ

and ϕ are Lipschitz continuous on bounded sets, hence locally Lipschitz (see Clarke [5, p. 83]).

In what follows by h·, ·i we denote the duality brackets for the pair of spaces W

(Z), W

(Z)

(where

+

= 1). Let A : W

(Z) −→

W

(Z) be the nonlinear operator defined by (3.5) hA(x), yi =

Z

Z

k∇xk

(∇x, ∇y)

dz ∀x, y ∈ W

(Z).

Lemma 3.1. If A : W

(Z) −→ W

(Z) is the nonlinear operator defined by (3.5), then A is of type (S)

.

We know that

(3.6) ∂ ϕ b

(x) ⊆ A(x) − b N

(x) and

∂ϕ(x) ⊆ A(x) − b N (x) ∀x ∈ W

(Z), where

N b

(x) = 

u ∈ L

(Z) : u(z) ∈ ∂b j

(z, x(z)) a.e. on Z , N b (x) = 

u ∈ L

(Z) : u(z) ∈ ∂j(z, x(z)) a.e. on Z (see Clarke [5, p. 83]).

Proposition 3.2. If hypotheses H(j) hold, then problem (1.1) has at least two solutions x

∈ int C

and v

∈ −int C

, x

is a minimizer of ϕ b

, v

is a minimizer of ϕ b

and both are local minimizers of ϕ.

P roof. From (3.3), we see that b ϕ

is coercive. Exploiting the compactness of the embedding W

(Z) ⊆ L

(Z), we can check that b ϕ

is weakly lower semicontinuous. So by the Weierstrass theorem, we can find x

∈ W

(Z), such that

(3.7) ϕ b

(x

) = b m

= inf  b

ϕ

(x) : x ∈ W

(Z) .

We show that without any loss of generality, we can assume that b m

< 0.

Let u

∈ int C

be the L

-normalized principal eigenfunction. Using hy- pothesis H(j)(v) we can find δ > 0 such that

λ

6 f (z, ζ)

|ζ|

ζ for all |ζ| ≤ δ and a.a. z ∈ Z,

so also

(3.8) λ

p |ζ|

6 j(z, ζ) for all |ζ| ≤ δ and a.a. z ∈ Z.

Next we can find ε

> 0 small enough and such that 0 6 ε

u

(z) 6 β < min{δ, a

} ∀z ∈ Z,

where a

is as in hypothesis H(j)(vi). Then from (3.8), for all ε ∈ (0, ε

], we have

(3.9) λ

p ε

u

(z)

6 j(z, εu

(z)) = b j

(z, εu

(z)) for a.a. z ∈ Z.

Hence from (3.3), (2.2), (3.9) and recalling that ku

k = 1, we have b

ϕ

(εu

) = ε

p k∇u

k

− Z

Z

bj

(z, εu

) dz 6 ε

p λ

− λ

p ε

= 0, so b m

6 0.

If b m

= 0, then b ϕ

(εu

) = 0 for all ε ∈ (0, ε

]. So 0 ∈ ∂ b ϕ

(εu

) and A(εu

) = u

, with u

∈ b N

(εu

), so

(3.10) −∆

(εu

)(z) = u

(z) for a.a. z ∈ Z.

Since 0 < εu

(z) 6 β < a

for all z ∈ Z, using also (3.1), we have

∂b j

(z, εu

(z)) = ∂j(z, εu

(z)) for a.a. z ∈ Z

and so from (3.10) we deduce that εu

is a solution of (1.1) and a minimizer of b ϕ

for any ε ∈ (0, ε

]. Moreover, since εu

∈ int C

and εu

(z) 6 β < a

for all z ∈ Z, we can find r > 0 small, such that

b ϕ

|

BC10 r (εu1)

= ϕ

|

BC10 r (εu1)

,

so εu

is a local C

(Z)-minimizer of ϕ, thus also a local W

(Z)-minimizer

of ϕ (see Gasi´ nski-Papageorgiou [10, p. 655]). Thus, if b m

= 0, then we

have a whole continuum of functions (the functions {εu

}

) satisfying

the properties claimed by the proposition.

So without any loss of generality we may assume that b m

< 0, so from (3.7), we have b ϕ

(x

) < 0 = b ϕ

(0), thus x

6= 0. We have 0 ∈ ∂ b ϕ

(x

), so

(3.11) A(x

) = u

,

with u

∈ b N

(x

). Acting on (3.11) with −x

∈ W

(Z) and using (3.1), we obtain k∇x

k

= 0, so x

= 0 and thus x

> 0, x

6= 0. From (3.11), we have

(3.12)

( −∆

x

(z) = u

(z) ∈ ∂b j

(z, x

(z)) for a.a. z ∈ Z x

|

= 0.

From Theorem 7.1, p. 286 of Ladyzhenskaya-Uraltseva [13], we have that x

∈ L

(Z). Then invoking Theorem 1 of Lieberman [14], we have that x

∈ C

(Z).

(3.13) u

(z) = 0 for a.a. z ∈ {x > a

}

(see (3.1)). So if we act on (3.11) with (x

− a

)

∈ W

(Z) ∩ C(Z), we

obtain Z

{x0>a+}

k∇x

k

dz = 0,

so |{x

> a

}|

= 0, and thus 0 6 x

(z) 6 a

for almost all z ∈ Z. Note that (3.12) and hypothesis H(j)(vi) implies that ∆

x

(z) 6 0 for almost all z ∈ Z. Invoking the strong maximum principle of Vazquez [19], we conclude that x

∈ int C

. Moreover, using hypothesis H(j)(vi), we have

−∆

(a

− x

)(z) = ∆

x

(z) = −u

(z) > −k(a

− x

)(z)

. Thus

(3.14) ∆

(a

− x

)(z) 6 k(a

− x

)(z)

for a.a. z ∈ Z.

Invoking once more the nonlinear strong maximum principle of Vazquez [19], we obtain from (3.14), x

(z) < a

for all z ∈ Z. Recalling the definition of bj

(z, ζ), we see that, if we choose r > 0 small, we get

b ϕ

|

BC10 r (x0)

= ϕ|

BC10 r (x0)

.

Therefore, x

is a local C

(Z)-minimizer of ϕ thus also a local W

(Z)-

minimizer of ϕ (see Gasi´ nski-Papageorgiou [10, p. 655]).

Similarly, working with b ϕ

and using this time (3.2), we obtain v

∈ −int C

, a minimizer of b ϕ

, which is also a local minimizer of ϕ and v

solves problem (1.1).

Next we can produce two more nontrivial smooth solutions of constant sign.

Theorem 3.3. If hypotheses H(j) hold, then problem (1.1) has at least four nontrivial solution x

, b x ∈ int C

, bv, v

∈ −int C

, such that

x

6 x, x b

6= b x and bv 6 v

, bv 6= v

.

P roof. From Proposition 3.2, we already have two solutions x

∈ int C

and v

∈ −int C

. First we will show that we can find b x ∈ int C

, the solution of (1.1), such that x

6 x, x b

6= b x. For this purpose we introduce the functional ψ

: W

(Z) −→ R defined by

(3.15) ψ

(y) = 1

p k∇(x

+ y)k

− Z

Z

j(z, x

+ y

) dz + Z

Z

u

y

dz + ξ,

where u

∈ L

(Z), u

(z) ∈ ∂j(z, x

(z)) for almost all z ∈ Z and satisfies

−∆

x

(z) = u

(z) for a.a. z ∈ Z (see Proposition 3.2) and ξ = R

Z

j(z, x

(z)) dz. Evidently, ψ

is locally Lipschitz on W

(Z). Note that

(3.16) k∇(x

+ y)k

= k∇(x

+ y

)k

− k∇x

k

+ k∇(x

− y

)k

. Using (3.16) in (3.15), we obtain

(3.17) ψ

(y) = ϕ(x

+ y

) − ϕ(x

) + 1

p k∇(x

− y

)k

+ Z

Z

u

y

dz.

Since u

(z) ∈ ∂j(z, x

(z)) for almost all z ∈ Z and x

(z) ∈ [0, a

] for all z ∈ Z, from hypothesis H(j)(vi), we have u

(z) > 0 for almost all z ∈ Z. Also recall that x

∈ int C

is a local minimizer of ϕ. So from (3.17), it can be shown that the origin is a local minimizer of ψ

(cf.

Gasi´ nski-Papageorgiou [10, p. 675]). Next following the ideas of Gasi´ nski-

Papageorgiou [10, Chapter 4], we can show the following:

(1) there exists % > 0 small, such that inf{ψ

(y) : kyk = %} = β

> 1

p k∇x

k

= ψ

(0).

(2) ψ

satisfies the C-condition.

(3) there exists y ∈ W

(Z) with kyk > %, such that ψ

(y) < ψ

(0) 6 β

. Then the three above facts permit the use of the nonsmooth mountain pass theorem (with relaxed boundary condition; see Gasi´ nski-Papageorgiou [10, p. 140] and Theorem 2.1). Then, there exists y

∈ W

(Z), y

6= 0, such that

(3.18) β

6 ψ

(y

) and 0 ∈ ∂ψ

(y

).

From the inclusion in (3.18), we obtain

(3.19) A(x

+ y

) = u

− b u

,

with u

∈ L

(Z), u

(z) ∈ ∂e j

(z, y

(z)) for almost all z ∈ Z and b u

∈ L

(Z), b u

(z) ∈ ∂i

(y

(z)) for almost all z ∈ Z, where

ej

(z, ζ) = j(z, x

(z) + ζ

) and i

(ζ) = ζ

. Acting with −y

∈ W

(Z) we obtain

hA(x

+ y

), −y

i = Z

Z

u

(−y

) dz − Z

Z

u b

(−y

) dz so hA(x

− y

), −y

i = hA(x

), −y

i and thus

(3.20) hA(x

− y

) − A(x

), −y

i = 0.

But clearly A is strictly monotone (strongly monotone if p > 2). So from (3.18) we infer that y

= 0, i.e.: y

> 0, y

6= 0. Moreover, from (3.19), we have

(3.21)

( −∆

(x

+ y

)(z) = u

(z) − b u

(z) for a.a. z ∈ Z, (x

+ y

)|

= 0.

So from nonlinear regularity theorem, we obtain x

+ y

∈ int C

. Let us

set b x = x

+y

. Then b x ∈ int C

, x

6 x b and x

6= b x. Also from Stampacchia

theorem, for almost all z ∈ {y

= 0}, we have

−∆

b x(z) = −∆

x

(z) = u

(z) ∈ ∂j(z, x

(z)) = ∂j(z, b x(z)).

On the other hand, from (3.21), we have

−∆

b x(z) = u

(z) ∈ ∂j(z, b x(z)) for a.a. z ∈ {y

> 0}.

Therefore b x ∈ int C

is a solution of problem (1.1).

In a similar way, we obtain bv ∈ W

(Z) with bv ∈ −int C

, bv 6 v

, bv 6= v

and bv is a solution of problem (1.1).

4. Nodal solution

The strategy to produce a fifth nontrivial smooth nodal solution, was in- spired by work of Dancer-Du [7] the semilinear case (p = 2) and smooth j(z, ·) ∈ C

(R). The first step in the execution of our solution plan, is to establish certain lattice-type properties of the sets of upper-lower solutions for problem (1.1). Let S ⊆ W

(Z) be a nonempty set. We say that S is downward (respectively upward) directed, if for any u

, u

∈ S, we can find u

∈ S, such that u

6 min{u

, u

} (respectively u

> max{u

, u

}). We can prove the following lemma.

Lemma 4.1. If hypotheses H(j) hold, then the set of upper solutions for problem (1.1) is downward directed. In fact, if y

, y

are two upper solutions for problem (1.1), then min{y

, y

} is an upper solution too. Similarly the set of lower solutions for problem (1.1) is upward directed. In fact, if v

, v

are two lower solutions for problem (1.1), then max{v

, v

} is a lower solution too.

Next we can show that problem (1.1) admits positive lower solutions and negative upper solutions.

Proposition 4.2. If hypotheses H(j) hold, then we can find ε

> 0, such

that for all ε ∈ (0, ε

], the function x

= εu

∈ int C

is a strict lower

solution for problem (1.1) and v

= −εu

∈ int C

is a strict upper solution

for problem (1.1).

P roof. By virtue of hypothesis H(j)(v), we can find β > λ

and δ ∈ (0, a

), such that βζ

6 f

(z, ζ) for almost all z ∈ Z and all ζ ∈ [0, δ].

Since u

∈ int C

, we can find ε

> 0 small, such that εu

(z) ∈ [0, δ] for all z ∈ Z and all ε ∈ (0, ε

]. Thus, using the fact that β > λ

, we have

−∆

(εu

)(z) = λ

ε

u

(z)

< βε

u

(z)

6 f

(z, εu

(z)) 6 u

(z)

(4.1)

for almost all z ∈ Z and all u

∈ L

(Z), u

(z) ∈ ∂j(z, εu

(z)) for almost all z ∈ Z. So εu

∈ int C

is a strict lower solution for problem (1.1). Similarly we show that −εu

∈ −int C

is a strict upper solution for problem (1.1).

Observe that x ≡ a

is an upper solution for problem (1.1) and v = a

is a lower solution for problem (1.1). Then, using the lower-upper solution pairs {x = εu

, x = a

} and {v = a

, v = −εu

}, 0 < ε 6 ε

, we define the following order intervals:

[x, x] = {x ∈ W

(Z) : x(z) 6 x(z) 6 x(z) for a.a. z ∈ Z}, [v, v] = {v ∈ W

(Z) : v(z) 6 v(z) 6 v(z) for a.a. z ∈ Z}.

In the next proposition, we produce extremal solutions of problem (1.1) on those two order intervals.

Proposition 4.3. If hypotheses H(j) hold, then problem (1.1) has the small- est solution in [x, x] and the greatest solution in [v, v].

P roof. We prove the existence of the smallest solution in [x, x], the proof of the other part being similar.

Let S

be the set of solutions of problem (1.1) in the interval [x, x]. From Proposition 3.2 we know that problem (1.1) has a solution x

∈ int C

. So by choosing ε

> 0 smaller if necessary, we get x

− x ∈ int C

and so S

6= ∅. Next we can show that the set S

is downward directed and also that S

has a minimal element (we omit the details).

Let x

be the minimal element of S

. We show that this is the smallest element of S

. Indeed, let y ∈ S

. We can find b x ∈ S

, such that b x 6 min{x

, y }. If b x 6= x

, then we contradict the minimality of x

. So x

is the smallest element of S

.

Now we can establish that problem (1.1) admits the smallest positive solu-

tion and the greatest negative solution.

Proposition 4.4. If hypotheses H(j) hold, then problem (1.1) has the small- est positive solution x

∈ int C

and the greatest negative solution v

∈

−int C

.

P roof. Let ε

& 0 and set x

= ε

u

. Then by Proposition 4.3, we can find e x

∈ int C

, the smallest solution of (1.1) in the ordered interval [x

, a

]. The sequence {e x

}

⊆ W

(Z) is bounded and so we may assume that e x

−→ x

, weakly in W

(Z). Next following some ideas of Gasi´ nski-Papageorgiou [10, Chapter 4], we can show that x

is the smallest positive solution of (1.1).

To produce v

∈ −int C

, we work with the pair [v = a

, v

= ε

(−u

)]

in a similar way.

Now we are ready to produce the nodal solution.

Theorem 4.5. If hypotheses H(j) hold, then problem (1.1) has at least five nontrivial solutions x

, x, v b

, bv, y

∈ C

(Z), such that

x

∈ int C

, x

6 b x, x

6= b x, v

∈ −int C

, bv 6 v

, v

6= bv and y

is nodal.

P roof. From Theorem 3.3, we already have the four solutions of constant sign x

, x, v b

, bv ∈ C

(Z). It remains to produce the nodal solution y

∈ C

(Z). Let x

∈ int C

be the smallest positive solution and v

∈ −int C

the greatest negative solution obtained in Proposition 4.4. We have A(x

) = u

and A(v

) = u

, with u

∈ L

(Z), u

(z) ∈ ∂j(z, x

(z)) for almost all z ∈ Z and u

∈ L

(Z), u

(z) ∈ ∂j(z, v

(z)) for almost all z ∈ Z.

We introduce the following truncations of the nonlinearity f (z, ζ):

f

(z, ζ) =

 



0 if ζ < 0,

f (z, ζ) if 0 6 ζ 6 x

(z), u

(z) if x

(z) < ζ, f

(z, ζ) =

 



u

(z) if ζ < v

(z), f (z, ζ) if v

(z) 6 ζ 6 0, 0 if 0 < ζ,

f b (z, ζ) =

 



u

(z) if ζ < v

(z),

f (z, ζ) if v

(z) 6 ζ 6 x

(z),

u

(z) if x

(z) < ζ.

Then we define the corresponding potential functions by j

(z, ζ) =

Z

f

(z, r) dr and b j(z, ζ) = Z

0

f b (z, r) dr.

Finally, we introduce the corresponding functionals ϕ

, ϕ b : W

(Z) −→ R, ϕ

(x) = 1

p k∇xk

− Z

Z

j

(z, x(z)) dz ∀x ∈ W

(Z), b

ϕ(x) = 1

p k∇xk

− Z

Z

bj(z, x(z)) dz ∀x ∈ W

(Z).

We use the following order intervals in W

(Z):

I

= [0, x

], I

= [v

, 0] and I b = [v

, x

].

We can show the following:

(1) The critical points of ϕ

are in I

, of ϕ

are in I

and of b ϕ are in b I . (2) The set of critical points of ϕ

is {0, x

} and the set of the critical

points of ϕ

is {0, v

}.

(3) Both x

∈ int C

and v

∈ −int C

are local minimizers of b ϕ.

Based on the last fact, we may assume that x

and v

are isolated critical points (in fact isolated local minimizers) of b ϕ, because, if this is not the case, then we can find a sequence {x

}

⊆ W

(Z) of critical points of

b

ϕ, x

6∈ {0, v

, v

}, such that, for example x

−→ x

in W

(Z). Then x

∈ b I for all n > 1 and so x

must be nodal (due to the extremality of v

, x

). We have produced a whole sequence of distinct nodal solutions for problem (1.1) and so we are done. Assume without any loss of generality that b ϕ(v

) 6 b ϕ(x

). Similarly as in the proof of Theorem 3.3, we can find

% > 0, such that b ϕ

(x

) < inf{ b ϕ(x) : kx−x

k = %} 6 0. Let D = ∂B

(x

) and E

= {v

, x

}, E = b I. It is easily seen that the pair {E

, E} links with D in W

(Z). Since b ϕ is coercive, we can easily check that it satisfies the C- condition. Therefore, we can apply Theorem 2.1 and obtain y

∈ W

(Z), such that

(4.2) ϕ(v b

) 6 b ϕ(x

) < b ϕ(y

) = inf

γ∈Γ

t∈[0,1]

max ϕ(γ(t)), b

where Γ = {γ ∈ C([−1, 1]; W

(Z)) : γ(−1) = v

, γ(1) = x

}.

Note that (4.2) implies that y

6= v

, y

6= x

. So, if we show that y

6= 0, then y

is a nodal solution. According to (4.2), we can show the nontriviality of y

by producing a path γ

∈ Γ, such that b ϕ|

< 0. To do this we proceed as follows.

By hypothesis H(j)(v), we can find δ, δ

> 0 small enough, such that λ

+ δ <

for almost all z ∈ Z, all |ζ| 6 δ

and all u

∈ ∂j(z, ζ).

Recall that for almost all z ∈ Z, the function j(z, ·) is differentiable almost everywhere on R (Rademacher theorem) and at a point of differentiability we have

j(z, ζ) ∈ ∂j(z, ζ), so

(4.3) 1

p (λ

+ δ)|ζ|

< j(z, ζ) for a.a. z ∈ Z, all 0 6 |ζ| 6 δ

.

Let ∂B

= {x ∈ L

(Z) : kxk

= 1} and S = W

(Z) ∩ ∂B

furnished with the relative W

(Z)-topology. Similarly, let S

= W

(Z) ∩ C

(Z) ∩

∂B

furnished with the relative C

(Z)-topology. Evidently S

is dense in S and this implies that C([−1, 1]; S

) is dense in C([−1, 1]; S). Let

Γ

= 

γ

∈ C([−1, 1]; S

) : γ

(−1) = −u

, γ

(1) = u

.

Then Γ

is dense in Γ

(see (2.5)) and because of (2.6), we can find bγ

∈ Γ

, such that

(4.4) max{k∇xk

: x ∈ bγ

([−1, 1])} 6 λ

+ δ.

Since bγ

∈ Γ

and −v

, x

∈ int C

, we can find ε > 0 small enough, and such that ε|x(z)| 6 δ for all z ∈ Z, all x ∈ bγ

([−1, 1]) and εx ∈ b I for all x ∈ bγ

([−1, 1]). Therefore, from (4.3) and (4.4), if x ∈ bγ

([−1, 1]), then

b

ϕ(εx) = ε

p k∇xk

− Z

Z

j(z, εx) dz < ε

p

 (λ

+ δ) − (λ

+ δ) 

= 0.

So, if γ

= εbγ

, then b ϕ|

< 0. Next, we produce a continuous path joining εu

and x

, along which b ϕ is strictly negative. For this purpose, let

a = m

= inf ϕ

= ϕ

(x

) < 0 = ϕ

(0) = b.

Since ϕ

is coercive it satisfies the C-condition. Also K

= {x

}. So

we can apply Theorem 2.3 and obtain a continuous deformation

h : [0, 1] × ˙ ϕ

−→ ˙ ϕ

, such that:

• h(t, ·)|

a

= id|

for all t ∈ [0, 1];

• h(1, ˙ ϕ

) ⊆ ˙ ϕ

∪ K

;

• ϕ

(h(t, x)) 6 ϕ

(x) for all (t, x) ∈ [0, 1] × ˙ ϕ

.

Then we set γ

(t) = h(t, εu

) for all t ∈ [0, 1]. Since ϕ

(εu

) < 0 then γ

is well defined and of course continuous. Moreover, since h is a deformation, we have γ

(0) = εu

and since ˙ ϕ

∪ K

= K

= {x

}, we have that γ

(1) = x

. Therefore, γ

is a continuous path which joins εu

with x

.

Note that ϕ

(γ

(t)) = ϕ

(h(t, εu

)) 6 ϕ

(εu

) < 0, so ϕ

|

< 0. But due to the sign condition (see hypothesis H(j)(vi)), we have b ϕ|

6 ϕ

|

, so b ϕ|

< 0. In a similar fashion, we produce a continuous path γ

which joins −εu

with v

and such that b ϕ|

< 0. Concatenating γ

, γ

and γ

, we have a continuous path γ

∈ Γ, such that b ϕ|

< 0. From (4.2), we conclude that b ϕ(y

) < 0 = b ϕ(0), so y

6= 0 and so y

is nodal. Nonlinear regularity theory implies that y

∈ C

(Z).

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Received 24 November 2009