On the Neumann problem for elliptic equations involving the p-Laplacian

J. Math. Anal. Appl. 358 (2009) 223–228

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

On the Neumann problem for elliptic equations involving the p-Laplacian Gabriele Bonanno a,∗ , Giuseppina D’Aguì b a b

Department of Science for Engineering and Architecture (Mathematics Section), Engineering Faculty, University of Messina, 98166 - Messina, Italy Department of Mathematics, University of Messina, 98166 - Messina, Italy

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 7 February 2009 Available online 3 May 2009 Submitted by P.J. McKenna

The aim of this paper is to establish the existence of an unbounded sequence of weak solutions to a Neumann problem for elliptic equations involving the p-Laplacian. © 2009 Elsevier Inc. All rights reserved.

Keywords: Neumann problem Multiple solutions

Here and in the sequel, Ω ⊂ Rk is a bounded open set, with a boundary of class C 1 , q ∈ L ∞ (Ω) with ess infΩ q > 0, p > k, f : Ω × R → R is a Carathéodory function. In the very interesting paper [11], Ricceri, by using an infinitely many critical points theorem [10, Theorem 2.5], established the existence of an unbounded sequence of weak solutions under an appropriate oscillating behaviour of f for a Neumann problem of the type:



− p u + q(x)|u | p −2 u = f (x, u ) in Ω, ∂ u /∂ ν = 0 on ∂Ω,

(P )

where  p = div(|∇ u | p −2 ∇ u ) is the p-Laplacian and ν is the outer unit normal to ∂Ω . Subsequently, starting from the fruitful papers of Ricceri, cited above, several authors gave further results on infinitely many solutions for Neumann elliptic problems (see, for instance, [3–6,9]). The aim of this paper is to establish the same conclusion of the result of Ricceri in [11], but under a completely different assumption. For reader’s convenience, Theorem 3 of [11] is here recalled. Theorem 1. Let h : R → R be a continuous function. Assume that the following conditions hold: (i) let {an } and {bn } be two sequences in R+ satisfying

an < bn ,

∀n ∈ N,

 max

lim bn = +∞,

n→+∞

ξ sup

ξ ∈[an ,bn ]

lim



ξ h(t ) dt ,

an

sup

ξ ∈[−bn ,−an ]

an

n→+∞

h(t ) dt

bn

 0,

= 0, ∀n ∈ N,

−an

and

*

Corresponding author. E-mail addresses: [email protected], [email protected] (G. Bonanno), [email protected] (G. D’Aguì).

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.04.055

©

2009 Elsevier Inc. All rights reserved.

224

G. Bonanno, G. D’Aguì / J. Math. Anal. Appl. 358 (2009) 223–228

ξ lim sup

(ii)

|ξ |→+∞

0

h(t ) dt = +∞. |ξ | p

Then, for every α , β ∈ L 1 (Ω), with min{α (x), β(x)}  0 almost everywhere in Ω and with function g : R → R satisfying

α = 0, and for every continuous

ξ g (t ) dt  0

sup ξ ∈R

0

and

ξ lim inf

g (t ) > −∞, |ξ | p

0

|ξ |→+∞

the problem



− p u + q(x)|u | p −2 u = α (x)h(u ) + β(x) g (u ) in Ω, ∂ u /∂ ν = 0 on ∂Ω

( P α ,β )

admits an unbounded sequence of weak solutions in W 1, p (Ω). Owing to our principal result (Theorem 3), we can restate Theorem 1, replacing the assumption (i) with the following (i∗ )

lim inf

max|t |ξ

t

0 ξp

ξ →+∞

h( z) dz

=0

(see Theorem 5 and Remark 1).

ξ

h( z) dz

Clearly, when h is nonnegative, (i∗ ) becomes lim infξ →+∞ 0 ξ p = 0. An example of elliptic Neumann problem which cannot be studied by Ricceri’s results, but admits, owing to our result, infinitely many solutions, is pointed out (see Example 1 and Remark 2). We observe that, when the nonlinear term is independent of x and nonnegative, Theorem 3 assumes a simpler form (see Corollary 1). Moreover, these results hold again substituting ξ → +∞ with ξ → 0+ (Theorem 4). We also remark that infinitely many solutions to elliptic Neumann problems have been obtained by [7] and [8] by using a different approach. It is very easy to verify that the results obtained in these papers (where sign assumptions are made) and ours, are mutually independent. In order to prove Theorem 3, we use an infinitely many critical points result (see Theorem 2). Let X be a reflexive real Banach space, Φ : X → R is a (strongly) continuous, coercive sequentially weakly lower semicontinuous and Gâteaux differentiable functional, Ψ : X → R is a sequentially weakly upper semicontinuous and Gâteaux differentiable functional. For all r > inf X Φ , put

ϕ (r ) =

inf

(supu ∈(Φ −1 ]−∞,r [) Ψ (u )) − Ψ (u ) r − Φ(u )

u ∈Φ −1 (]−∞,r [)

and

γ = lim inf ϕ (r ), r →+∞

δ=

lim inf

r →(inf X Φ)+

ϕ (r ).

The following result, as given in [2], is a more precise version of [10, Theorem 2.5]. Theorem 2. Under the above assumptions on X , Φ , and Ψ , the following conditions hold: (1) If

γ < +∞ then, for each λ ∈ ]0, γ1 [, the following alternative holds: either the functional Φ − λΨ has a global minimum, or

there exists a sequence {un } of critical points (local minima) of Φ − λΨ such that limn→+∞ Φ(un ) = +∞. (2) If δ < +∞ then, for each λ ∈ ]0, 1δ [, the following alternative holds: either there exists a global minimum of Φ which is a local minimum of Φ − λΨ , or there exists a sequence {un } of pairwise distinct critical points (local minima) of Φ − λΨ , with limn→+∞ Φ(un ) = inf X Φ , which weakly converges to a global minimum of Φ . After introducing the questions we are interested in and the approach we intend to follow, we consider the following problem

G. Bonanno, G. D’Aguì / J. Math. Anal. Appl. 358 (2009) 223–228



225

− p u + q(x)|u | p −2 u = λ f (x, u ) in Ω, ∂ u /∂ ν = 0 on ∂Ω,

( P λ)

where λ is a real positive parameter. We recall that a weak solution of the problem ( P λ ) is any u ∈ W 1, p (Ω), such that





  ∇ u (x) p −2 ∇ u (x)∇ v (x) dx +

Ω





 p −2

q(x)u (x)





u (x) v (x) dx − λ

Ω





f x, u (x) v (x) dx = 0,

∀ v ∈ W 1, p (Ω).

Ω

Put

c=

sup u ∈ W 1, p (Ω)\{0}

supx∈Ω |u (x)| . 1   ( Ω |∇ u (x)| p dx + Ω q(x)|u (x)| p dx) p

(1)

If Ω is convex, an explicit upper bound for the constant c is

c2

p −1 p





max

 1p

1

,

Ω q(x) dx

d



1

np

 p−p 1

q ∞  , m(Ω) p −n Ω q(x) dx

p−1

where d = diam(Ω)  t (see [1, Remark 1]). Put F (x, t ) = 0 f (x, ξ ) dξ for all (x, t ) ∈ Ω × R,



A = lim inf

Ω max|t |ξ F (x, t ) dx

ξ →+∞



ξp



F (x, ξ ) dx B = lim sup Ω , p

,

ξ

ξ →+∞

where, as usual, q 1 = Ω q(x) dx, moreover

λ1 =

q 1 pB

,

λ2 =

1 pc p A

(2)

.

Our main result is the following. Theorem 3. Assume that



Ω max|t |ξ F (x, t ) dx

lim inf

ξp

ξ →+∞

<



F (x, ξ ) dx lim sup Ω . p

1

c p q 1 ξ →+∞

(3)

ξ

Then, for each λ ∈ ]λ1 , λ2 [, where λ1 , λ2 are given in (2), problem ( P λ ) possesses an unbounded sequence of weak solutions in W 1, p (Ω). Proof. Our aim is to apply part (1) of Theorem 2. Take as X the Sobolev space W 1, p (Ω) endowed with the norm



u =

  ∇ u (x) p dx +

Ω





p

 1p

q(x)u (x) dx

.

Ω

We observe that the above norm is equivalent to the usual one. On the space C 0 (Ω), we consider the norm u ∞ := supu ∈Ω |u (x)|. Since p > k, X is compactly embedded in C 0 (Ω), and taking into account (1), we have

u ∞  c u .

(4)

For each u ∈ X , put

Φ(u ) :=

1 p



u p ,

Ψ (u ) :=





F x, u (x) dx. Ω

It is well known that the critical points in X of the functional Φ − λΨ are exactly the weak solutions of the problem ( P λ ). Pick λ ∈ ]λ1 , λ2 [. Let {ρn } be a real sequence such that limn→∞ ρn = +∞ and



lim

Ω max|t |ρn F (x, t ) dx

ρnp

n→∞

Put rn = 1p ( Therefore,

= A.

ρn p ) for all n ∈ N. Taking into account v p < prn and v ∞  c v , one has | v (x)|  c

ϕ (rn ) =



inf

u p < prn





ρn , for every x ∈ Ω .

sup v p < prn Ω F (x, v (x)) dx − Ω F (x, u (x)) dx sup v p < prn Ω F (x, v (x)) dx  .

u p rn rn − p

226

G. Bonanno, G. D’Aguì / J. Math. Anal. Appl. 358 (2009) 223–228

Hence,



ϕ (rn ) 

Ω max|t |ρn F (x, t ) dx

rn

 = pc p

Ω max|t |ρn F (x, t ) dx

ρnp

∀n ∈ N.

,

Then,

γ  lim inf ϕ (rn )  pc p A < +∞. n→+∞

Now, we claim that the functional Φ − λΨ is unbounded from below. Let {dn } be a real sequence such that limn→∞ dn = +∞ and



lim

F (x, dn ) dx p

n→∞

dn

Ω

= B.

(5)

For each n ∈ N, put w n (x) = dn , for all x ∈ Ω. Clearly w n ∈ W 1, p (Ω) for each n ∈ N. Hence, p

w n p = dn q 1 and

Φ( w n ) − λΨ ( w n ) =

wn p p

 −λ





p



dn q 1

F x, w n (x) dx =

p

−λ

Ω

Now, if B < +∞, let



F (x, dn ) dx. Ω

 ∈ ]0, B − pq λ1 [. From (5) there exists ν such that p

F (x, dn ) dx > ( B −  )dn ,

∀n > ν .

Ω

Therefore,



p

Φ( w n ) − λΨ ( w n ) =

dn q 1 p

−λ

p

F (x, dn ) dx <

dn q 1 p

p

p



− λdn ( B −  ) = dn

q 1 p

 − λ( B −  ) .

Ω

 , one has

 lim Φ( w n ) − λΨ ( w n ) = −∞.

From the choice of n→+∞

If B = +∞, fix M >



q 1 . From (5) there exists pλ p

νM such that

∀n > νM .

F (x, dn ) dx > Mdn , Ω

Moreover,



p

Φ( w n ) − λΨ ( w n ) =

dn q 1 p

−λ

p

F (x, dn ) dx <

dn q 1 p

p

p

− λ Mdn = dn



q 1 p

 − λM .

Ω

Taking into account the choice of M, also in this case, one has

lim

 Φ( w n ) − λΨ ( w n ) = −∞.

n→+∞

From part (1) of Theorem 2, the functional Φ − λΨ admits a sequence un of critical points, and the conclusion is proven. Now, we point out the following consequence of Theorem 3. Corollary 1. Let f : R → R be a nonnegative continuous function. Moreover assume that

lim inf ξ →+∞

F (ξ )

ξp

Then, for each λ ∈ ]



<

1

lim sup

c p q 1 ξ →+∞

q 1

p |Ω| lim supξ →+∞

F (ξ ) ξp

,

F (ξ )

ξp

.

1 F (ξ ) pc p |Ω| lim infξ →+∞ p

[, the problem

ξ

− p u + q(x)|u | p −2 u = λ f (u ) in Ω, ∂ u /∂ ν = 0 on ∂Ω

possesses an unbounded sequence of weak solutions in W 1, p (Ω).

2

G. Bonanno, G. D’Aguì / J. Math. Anal. Appl. 358 (2009) 223–228

227

Furthermore, by using (2) of Theorem 2 and arguing as in the proof of Theorem 3, put

q 1

λ∗1 :=

p lim supξ →0+



and

Ω F (x,ξ ) dx

λ∗2 :=

ξp

1 

pc p

Ω max|t |ξ F (x,t ) dx

lim infξ →0+

,

ξp

we obtain the following result. Theorem 4. Let f : Ω × R → R be a continuous function. Assume that



max|t |ξ F (x, t ) dx lim inf Ω < p ξ →0+

ξ

1



F (x, ξ ) dx lim sup Ω . p

c p q 1 ξ →0+

ξ

Then, for each λ ∈ ]λ∗1 , λ∗2 [, problem ( P λ ) possesses a sequence of non-zero weak solutions which strongly converges uniformly to 0 in W 1, p (Ω). The following ensures what was claimed at the beginning. Theorem 5. Let h : R → R be a continuous function, such that

(i )

lim inf

max|t |ξ

t

h( z) dz

0 ξp

ξ →+∞

< +∞ and (ii )

ξ lim sup

0

ξ →+∞

h(t ) dt

ξp

= +∞.

Then, for every α , β ∈ L 1 (Ω), with min{α (x), β(x)}  0 almost everywhere in Ω and with function g : R → R satisfying

α = 0, and for every continuous

ξ g (t ) dt  0

sup ξ ∈R

(6)

0

and

ξ lim inf

0

ξ →+∞

g (t ) > −∞, ξp

(7)

the problem



 − p u + q(x)|u | p −2 u = λ α (x)h(u ) + β(x) g (u ) in Ω, ∂ u /∂ ν = 0 on ∂Ω,

for every λ ∈ ]0,

1 pc p α 1 lim infξ →+∞

 max|t |ξ 0t h( z) dz ξp

( P α ,β,λ )

[, admits an unbounded sequence of weak solutions in W 1, p (Ω).

Proof. In order to prove this result, we apply Theorem 3. To this end, put f (x, z) = α (x)h( z) + β(x) g ( z), for all (x, z) ∈ Ω × R. ξ ξ One has F (x, ξ ) = 0 f (x, z) dz = α (x) H (ξ ) + β(x)G (ξ ), for all (x, ξ ) ∈ Ω × R, where G (ξ ) = 0 g ( z) dz and H (ξ ) =

ξ 0

h( z) dz, for all ξ ∈ R. From (ii ) and (7), one has



F (x, ξ ) dξ

α 1 H (ξ ) + β 1 G (ξ ) lim sup Ω = lim sup = +∞. p p

ξ

ξ →+∞

ξ

ξ →+∞

On the other hand, from (i ) and (6)



max|t |ξ F (x, t ) dx

α 1 max|t |ξ H (t ) dx lim inf Ω  lim inf < +∞. p p ξ →+∞

ξ →+∞

ξ

ξ

Hence, (3) of Theorem 3 is satisfied. Hence, since

λ2 =

1



pc p

lim infξ →+∞

the conclusion is achieved.

2

Ω max|t |ξ F (x,t ) dx

ξp



1 pc p

α 1 lim infξ →+∞

max|t |ξ H (t )

ξp

,

228

G. Bonanno, G. D’Aguì / J. Math. Anal. Appl. 358 (2009) 223–228

Remark 1. Clearly, if lim infξ →+∞

max|t |ξ

t

ξp

0

h( z) dz

= 0, (i ) is verified and problem ( P α ,β,λ ) admits an unbounded sequence

of weak solutions in W 1, p (Ω) for each λ > 0. Hence, we obtain what we stated at the beginning. Example 1. Put

an :=

2n!(n + 2)! − 1 4(n + 1)!

bn :=

,

2n!(n + 2)! + 1 4(n + 1)!

for every n ∈ N and define the positive continuous function h : R → R as follows



h(ξ ) = One

has

32(n+1)!2 [n p −1 (n+1)! p −(n−1) p −1 n! p ]

π



1 16(n+1)!2

− (ξ −

n!(n+2) 2 ) 2

+ 1 if ξ ∈

1

limn→+∞ H (ξ )



n∈N [an , bn ],

otherwise. H (bn ) p bn

= +∞

and

limn→+∞

H (an ) p an

= 0.

Therefore,

we

obtain

H (ξ )

lim infξ →+∞ ξ p

= 0

and

lim supξ →+∞ ξ p = +∞. So, from Theorem 5, for each λ > 0, the problem



− p u + q(x)|u | p −2 u = λh(u ) in Ω, ∂ u /∂ ν = 0 on ∂Ω

admits an unbounded sequence of weak solutions in W 1, p (Ω). Remark 2. We observe that Theorem 3 of [11] requires a kind of sign hypothesis on the nonlinear term. More precisely, the function cannot be positive only. Therefore, we note the problem in Example 1, given above, cannot be investigated using results in [11], seeing that the function is positive. Remark 3. We also observe that in Theorem 1 a nonlinear term behaviour at −∞ is requested (see assumption (ii)). On the contrary, in Theorem 3 and its consequences, the nonlinear term behaviour at −∞ is completely arbitrary. References [1] G. Bonanno, P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel) 80 (2003) 424–429. [2] G. Bonanno, G. Molica Bisci, Infinitely many solution for a Sturm–Liouville boundary value problem, Bound. Value Probl. 2009 (2009) 1–20. [3] P. Candito, Infinitely many solutions to a Neumann problem for elliptic equations involving the p-Laplacian and with discontinuous nonlinearities, Proc. Edinb. Math. Soc. (2) 45 (2002) 397–409. [4] G. Dai, Infinitely many solutions for a Neumann-type differential inclusion problem involving the p (x)-Laplacian, Nonlinear Anal. 70 (2009) 2297–2305. [5] X. Fan, C. Ji, Existence of infinitely many solutions for a Neumann problem involving the p (x)-Laplacian, J. Math. Anal. Appl. 334 (2007) 248–260. [6] A. Kristály, Infinitely many solutions for a differential inclusion problem in R N , J. Differential Equations 220 (2006) 511–530. [7] C. Li, The existence of Infinitely many solutions of a class of nonlinear elliptic equations with a Neumann boundary conditions for both resonance and oscillation problems, Nonlinear Anal. 54 (2003) 431–443. [8] C. Li, S. Li, Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary conditions, J. Math. Anal. Appl. 298 (2004) 14–32. [9] S. Marano, D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to the Neumann-type problem involving the p-Laplacian, J. Differential Equations 182 (2002) 108–120. [10] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410. [11] B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc. 33 (3) (2001) 331–340.