Proceedings of the Edinburgh Mathematical Society (2016) 59, 655–669 DOI:10.1017/S001309151500036X

NEW CONDITIONS FOR THE EXISTENCE OF INFINITELY MANY SOLUTIONS FOR A QUASI-LINEAR PROBLEM FRANCESCA FARACI1 AND CSABA FARKAS2 1

Department of Mathematics and Computer Science, University of Catania, Viale A. Doria, 95125 Catania, Italy (ff[email protected]) 2 Fcaultatea de Matematic˘ a ¸si Informatic˘ a, Universitatea Babe¸s-Bolyai Cluj-Napoca, Str. Mihail Kogalniceanu nr. 1, 400084, Cluj-Napoca, Romania ([email protected]) (Received 3 May 2014)

Abstract In this paper we study a quasi-linear elliptic problem coupled with Dirichlet boundary conditions. We propose a new set of assumptions ensuring the existence of infinitely many solutions. Keywords: p-Laplacian; infinitely many solutions; Dirichlet problem; variational methods 2010 Mathematics subject classification: Primary 35J20; 35J92; 47J10

1. Introduction In this paper we deal with the following Dirichlet problem  −Δp u = h(x)f (u) in Ω, u=0

on ∂Ω,

(P)

where Ω ⊆ RN is a bounded domain with smooth boundary, p > 1, Δp is the p-Laplacian operator, i.e, Δp u = div(|∇u|p−2 ∇u), f : R → R is a continuous function, h : Ω → R is a bounded non-negative function. The existence of infinitely many solutions for such types of problem has been intensively studied under different assumptions on the nonlinearity. Since the pioneering work of Ambrosetti and Rabinowitz [1] it has been well known that symmetry assumptions on f can yield infinitely many solutions for (P). The same result also holds when the symmetry of the energy is broken by the presence of a perturbation (see, for example, the contributions of Hirano and Zou [8] and of Tehrani [16]). A different approach to the problem was proposed by Ricceri [12, 13] by means of suitable variational arguments, by Omari and Zanolin [10, 11] who exploited subsupersolution methods, and by Saint Raymond in [15] (see also [2,6,9]), where a sequence of local minima of the energy functional in suitable convex sets is exhibited. All these contributions prove the existence of infinitely many solutions when the nonlinearity exhibits c 2015 The Edinburgh Mathematical Society 

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F. Faraci and C. Farkas

a suitable oscillatory behaviour either at zero or at infinity. It is worth mentioning that the oscillation of f in itself is not enough to guarantee multiple solutions: indeed, de Figueiredo [5] proved the uniqueness of positive solutions of (P) when f (u) = λ sin u. In the aforementioned results, if F is the primitive of f , the oscillation of f is ensured by the assumption −∞ < lim inf t→

F (t) F (t) < lim sup p  +∞, p |t| |t| t→

where  = ±∞, 0± . In this paper we develop a variant of a recent existence and localization theorem by Ricceri [14] in order to prove the existence of infinitely many solutions for (P) under new conditions on the nonlinearity. First of all, our result can be applied when lim

t→

F (t) ∈R |t|p

(see Example 4.1). This is not the sole novelty of our result. We point out that the result of [14] is a consequence of the variational methods contained in [12]. Applicability of Ricceri’s variational principle (see [12]) in the framework of infinitely many weak solutions for quasi-linear problems is only known in low dimension, i.e. for p > N (see, for example, [3, 4, 7]). We give a positive contribution also when p  N , as our paper seems to provide the very first example in this direction (see Example 4.2). In conclusion, our result represents a step forward in the research of new conditions for finding infinitely many weak solutions for (P). The structure of the paper is the following: in § 2 we introduce the preliminaries, while § 3 contains our main result. To conclude the paper, § 4 is devoted to some examples. 2. Notation and preliminaries Let us introduce the notation we will use. If N  p, A denotes the class of continuous functions f : R → R such that sup t∈R

|f (t)| < +∞, 1 + |t|γ

where 0 < γ < p∗ − 1 if p < N (here p∗ = pN/(N − p)) and 0 < γ < +∞ if p = N , while if N < p, A is the class of continuous functions f : R → R. Denote by F the primitive of f , i.e.  t

F (t) =

f (s) ds. 0

Problem (P) has a variational structure. More precisely, if f ∈ A, the functional E : W01,p (Ω) → R defined by  1 E (u) = up − h(x)F (u(x)) dx, p Ω

Existence of infinitely many solutions for a quasi-linear problem

657

y ψ

α (ϕ ,ψ ,b)

x

inf ψ

ψ – λϕ ,λ = b

M(ϕ ,ψ,b):={global minima of ψ – λϕ} Figure 1. Geometrical meaning of α.

where  ·  is the classical norm in the Sobolev space W01,p (Ω), i.e.  u =

1/p |∇u(x)| dx , p

Ω

is of class C 1 in W01,p (Ω), and its critical points are precisely the weak solutions of problem (P). Denote by B(0, ) the open ball in W01,p (Ω) centred at zero and of radius . Also, let 0  a < b  +∞. For a pair of functions ϕ, ψ : R → R, if λ ∈ [a, b], we denote by M (ϕ, ψ, λ) the set of all global minima of the function λψ − ϕ or the empty set according to whether λ < +∞ or λ = +∞. We adopt the conventions sup ∅ = −∞, inf ∅ = +∞. We also put   α(ϕ, ψ, b) = max inf ψ, sup ψ R

and

 β(ϕ, ψ, a) = min sup ψ, R

M (ϕ,ψ,b)

inf

 ψ .

M (ϕ,ψ,a)

See Figures 1 and 2 for the geometrical meaning of α and β. Furthermore, let ⎧ ⎨]0, p∗ ] if N > p, q∈ ⎩]0, +∞[ if N  p,

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F. Faraci and C. Farkas y ψ

sup ψ

x

β (ϕ,ψ ,a)

ψ – λϕ ,λ = a

M(ϕ ,ψ,a):={global minima of ψ – λϕ} Figure 2. Geometrical meaning of β.



and cq =

|u(x)|q dx  Ω . p q/p u∈W 1,p (Ω)\{0} ( Ω |∇u(x)| dx) sup

0

Denote by Fq the family of all lower semi-continuous functions ψ : R → R, with supR ψ > 0, such that inf

t∈R

ψ(t) > −∞ 1 + |t|q

and γψ :=

ψ(t) < +∞. q t∈R\{0} |t| sup

In [14], Ricceri proved the following result. Theorem 2.1 (Ricceri [14]). Let f ∈ A and let h ∈ L∞ (Ω) \ {0} with h  0. Moreover, assume that there exists ψ ∈ Fq such that, for each λ ∈ ]a, b[, the function λψ − F is coercive and has a unique global minimum in R. Finally, suppose that there exists a number r > 0 satisfying α(F, ψ, b) < r < β(F, ψ, a) and sup F < ψ −1 (r)

rp/q  . p(γψ ess supΩ hcq )p/q ( Ω h(x) dx)(q−p)/q

(2.1)

Existence of infinitely many solutions for a quasi-linear problem Put

 d=



h(x) dx γψ ess supΩ hcq Ω

659

p/q .

Then problem (P) has a weak solution u with u < rp/q . More precisely, u is a global minimum of the restriction of E to B(0, drp/q ). Remark 2.2. Notice that, from [14, Proposition A], for any r ∈ ]α(F, ψ, b), β(F, ψ, a)[ there exists λr ∈ ]a, b[ such that the unique global minimum of λr ψ − F lies in ψ −1 (r). In particular, ψ −1 (r) = ∅. 3. Main results 3.1. The zero case We deal with the existence of infinitely many solutions tending to zero in the norm of W01,p (Ω). Theorem 3.1. Let f ∈ A and let h ∈ L∞ (Ω) \ {0} with h  0. Assume that there exists ψ ∈ Fq such that, for each λ ∈ ]a, b[, the function λψ − F is coercive and has a unique global minimum in R. Finally, suppose that α(F, ψ, b)  0 < β(F, ψ, a), lim inf +

supψ−1 (r) F

r→0

rp/q

<

1  , p(γψ ess supΩ hcq )p/q ( Ω h(x) dx)(q−p)/q

(3.1)

and that 0 is not a local minimum of E . Under such hypotheses, problem (P) has a sequence of non-zero weak solutions {un } with lim un  = 0. n→∞

Also, E (un ) < 0 for any n ∈ N, and {E (un )} is increasing. Proof . From condition (3.1) we can fix a decreasing sequence {rn } of positive numbers such that lim rn = 0 n

and p/q

sup F < ψ −1 (rn )

rn  p(γψ ess supΩ hcq )p/q ( Ω h(x) dx)(q−p)/q

∀n ∈ N.

(3.2)

Fix k1 ∈ N such that α(F, ψ, b) < rk1 < β(F, ψ, a). From Theorem 2.1 we deduce the existence of u1 ∈ W01,p (Ω), which is a weak solution of (P) and satisfies p/q u1 p < drk1 .

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Since u1 is a local minimum of E , u1 = 0. Now, choose k2 ∈ N, k2 > k1 , such that rk2  d−q/p u1 q . As α(F, ψ, b) < rk2 < rk1 < β(F, ψ, a), p/q

there exists a weak solution of (P) u2 = 0 such that u2 p < drk2 . Hence, by the choice of rk2 , we also deduce that u2  < u1 . Therefore, we can construct a sequence {un } of non-zero solutions of (P) such that, for every n ∈ N, p/q

un p < drkn and

un  < un−1 . In particular, limn un  = 0 and un are pairwise distinct. It is also clear that E (un ) < 0 for all n ∈ N and that {E (un )} is increasing.  Remark 3.2. From the proof, it is clear that Theorem 3.1 holds with (3.2) in place of (3.1). We now propose some sufficient conditions ensuring that 0 is not a local minimum of E . Lemma 3.3. Assume one of the following conditions: (i0+ ) −∞ < lim inf +

F (t) F (t)  lim sup p = +∞, tp t t→0+

(i0− ) −∞ < lim inf −

F (t) F (t)  lim sup p = +∞. p |t| |t| t→0−

t→0

t→0

Then 0 is not a local minimum of E . Proof . Let us prove (i0+ ). Since h ∈ L∞ (Ω) \ {0}, h  0, it is possible to find x0 ∈ Ω and θ > 0 such that B(x0 , θ) ⊂ Ω, and h(x) > (1/p)h∞ almost everywhere in B(x0 , θ). For every s ∈ R, s = 0, define ⎧ ⎪ 0 in Ω \ B(x0 , θ), ⎪ ⎪ ⎪   ⎪ ⎨ 2s θ (θ − |x − x , |) in B(x , θ) \ B x , 0 0 0 ws (x) = θ 2 ⎪   ⎪ ⎪ θ ⎪ ⎪ ⎩s in B x0 , . 2 Then ws ∈ W01,p (Ω) and from direct calculation we have that  N −p θ ws  = (2N − 1)ωN |s|p 2 p

(3.3)

Existence of infinitely many solutions for a quasi-linear problem

661

(where ωN is the measure of the unit ball in RN ). From the left-hand inequality of our assumption, we deduce the existence of l > 0 and δ > 0 such that F (t) > −ltp Fix L such that

for 0 < t < δ.

L > (2N − 1) lp +

 2p . h∞ θp

(3.4)

Using the right-hand inequality of (i0+ ), there exists a sequence {sn } ⊂ R+ , sn → 0+ , such that F (sn ) > Lspn

for every n ∈ N.

Then we have, for n ∈ N,  N −p  1 θ N p E (wsn ) = (2 − 1)ωN sn − h(x)F (sn ) p 2 B(x0 ,θ/2)  − h(x)F (wsn ) B(x0 ,θ)\B(x0 ,θ/2)

 N −p  N θ 1 1 θ N p (2 − 1)ωN sn − h∞ ωN Lspn  p 2 p 2  N θ + h∞ l (2N − 1)ωN spn 2

 −p   N θ 1 p 1 θ N N = ωN sn (2 − 1) + lh∞ (2 − 1) − Lh∞ 2 p 2 p <0 as it follows from (3.4). Therefore, E (wsn ) < 0 = E (0).  Remark 3.4. From the proof of Lemma 3.3, we can weaken condition (i0+ ) by assuming that 

F (t) F (t) F (t) 2p N . > −∞ and lim sup p > (2 − 1) p lim inf + (j0+ ) lim inf tp t tp h∞ θp t→0+ t→0+ t→0+ Analogously, we can replace (i0− ) with



F (t) F (t) F (t) 2p N (j0− ) lim inf p > −∞ and lim sup p > (2 − 1) p lim inf p + . t t t h∞ θp t→0− t→0− t→0−

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F. Faraci and C. Farkas

Remark 3.5. The conclusion of Lemma 3.3 is also valid under the conditions F (t) 1 (k0+ ) ess inf h > 0 and lim inf p > , Ω t pcp ess inf Ω h t→0+ (k0− ) ess inf h > 0 and lim inf − Ω

t→0

F (t) 1 . > p |t| pcp ess inf Ω h 

Proof . See [14]. Corollary 3.6 easily follows from Theorem 3.1.

Corollary 3.6. Let f ∈ A and ψ ∈ Fq such that, for each λ ∈ ]a, b[, the function λψ − F is coercive, has a unique global minimum in R and one of the conditions (i0+ ), (i0− ) hold. Finally, suppose that α(F, ψ, b)  0 < β(F, ψ, a) and lim inf

supψ−1 (r) F rp/q

r→0+

< +∞.

(3.5)

Under such hypotheses, there exists μ > 0 such that for every μ ∈ ]0, μ ], the problem −Δp u = μf (u) in Ω, u=0

on ∂Ω,

has a sequence of non-zero weak solutions {un } with lim un  = 0.

n→∞

When lim inf t→0+ F (t)/tp = 0 in condition (j0+ ) we can easily obtain (3.1) of Theorem 3.1 as in the following theorem. Theorem 3.7. Let f ∈ A with f (0)  0, and let h ∈ L∞ (Ω) \ {0} with h  0. Assume for each λ ∈ ]0, +∞[ that lim (λtq − F (t)) = +∞ t→+∞

and that the function t → λt − F (t) has a unique global minimum in [0, +∞[. Finally, suppose that condition (j0+ ) holds with lim inf t→0+ F (t)/tp = 0. Under such hypotheses, problem (P) has a sequence of non-zero non-negative weak solutions {un } with lim un  = 0. q

n→∞

Proof . We apply Theorem 3.1 to the functions  f (0) if t  0, f˜(t) = f (t) if t > 0, ψ(t) = |t|q .

Existence of infinitely many solutions for a quasi-linear problem It is clear that f˜ ∈ A and ψ ∈ Fq . Since   t f (0)t ˜ ˜ f (s) ds = F (t) = F (t) 0

663

if t  0, if t > 0,

we obtain the coercivity of the function λψ − F˜ for every λ > 0. Also, as f (0)  0, λψ(t) − F˜ (t) = λ|t|q − f (0)t > 0 for every t < 0. From the assumptions there exists a unique tλ  0 such that λψ(tλ ) − F (tλ )  λψ(t) − F (t) for every t  0. Therefore, λψ(tλ ) − F˜ (tλ )  0 < λψ(t) − F˜ (t) for every t < 0, which implies the uniqueness of the global minimum of the function λψ − F˜ in R. In this framework, given that a = 0 and b = +∞, it follows that α(F˜ , ψ, b) = 0

and β(F˜ , ψ, a) > 0.

From our assumptions, the existence of a sequence {tn } with tn → 0+ such that F (tn ) →0 tpn also follows. Put rn = ψ(tn ) = tqn . It is clear that rn → 0+ and that ψ −1 (rn ) = {tn , −tn }. Therefore, lim sup

supψ−1 (rn ) F˜ p/q rn

n→∞

= lim sup n→∞

sup{F (tn ), −f (0)tn } F (tn )  lim = 0, n→∞ tpn tpn

which implies condition (3.1) for F˜ . All the assumptions of Theorem 3.1 are fulfilled, so the functional E˜ : W01,p (Ω) → R,  1 E˜ (u) = up − h(x)F˜ (u(x)) dx, p Ω has a sequence of non-zero critical points {un } with limn→∞ un  = 0. In order to conclude the proof it is enough to show that each critical point of E˜ is non-negative, that is, it is a solution of (P). Indeed, if u is a critical point of E˜ , then   |∇u(x)|p−2 ∇u(x)∇v(x) dx = h(x)f˜(u(x))v(x) dx Ω

for every v ∈

W01,p (Ω),

Ω

and by choosing as test function v = −u− = min{0, u} we obtain  h(x)f (0)u(x) dx  0, u− p = {u0}

that is, u  0 a.e. in Ω. The claim follows.



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3.2. The ∞-case The existence of infinitely many solutions at infinity is possible by ensuring that β = +∞. Theorem 3.8. Let f ∈ A and let h ∈ L∞ (Ω) \ {0} with h  0. Assume that there exists ψ ∈ Fq such that, for each λ ∈ ]a, b[, the function λψ − F is coercive and has a unique global minimum in R. Finally, suppose that α(F, ψ, b) < +∞ lim inf r→+∞

supψ−1 (r) F rp/q

and β(F, ψ, a) = +∞, 1

<

p/q

p(γψ ess supΩ hcq )

,  ( Ω h(x) dx)(q−p)/q

(3.6)

and E is unbounded from below. Under such hypotheses, problem (P) has a sequence of weak solutions {un } with lim un  = +∞.

n→∞

Also, E (un ) < 0 for any n ∈ N and {E (un )} is decreasing. Proof . From (3.6) we can fix an increasing sequence {rn } of positive numbers such that limn rn = +∞ and p/q

sup F < ψ −1 (rn )

rn  p(γψ ess supΩ hcq )p/q ( Ω h(x) dx)(q−p)/q

∀n ∈ N.

(3.7)

Without loss of generality we can assume that α(F, ψ, b) < rn < β(F, ψ, a) = +∞ for every n ∈ N. From Theorem 2.1 we deduce that for every n ∈ N there exists a solution of (P), un ∈ W01,p (Ω), with un p < drnp/q . In particular, from Theorem 2.1, un is a global minimum of the restriction of E to p/q B(0, drn ). We claim that {un } is unbounded. Assume by contradiction that there exists M > 0 such that un   M for every n ∈ N. Then, by the reflexivity of W01,p (Ω), there exists a subsequence {unk } and u ˜ ∈ W01,p (Ω) such that ˜ in W01,p (Ω). u nk  u p/q

Let u ∈ W01,p (Ω). Since rn → +∞, there exists ν ∈ N such that u ∈ B(0, drnk ) for k  ν. So, E (˜ u)  lim inf E (unk )  E (u), k→∞

which means that u ˜ is a global minimum for E , a contradiction. The thesis follows. Remark 3.9. It is clear that Theorem 3.8 holds with (3.7) in place of (3.6). Notice that it is crucial to require that E has no global minima.



Existence of infinitely many solutions for a quasi-linear problem

665

Lemma 3.10. Assume that one of the following conditions holds: (i+∞ ) −∞ < lim inf

F (t) F (t) < lim sup p = +∞, p t t t→+∞

(i−∞ ) −∞ < lim inf

F (t) F (t) < lim sup p = +∞, |t|p t→−∞ |t|

t→+∞

t→−∞

(k+∞ ) ess inf h > 0 and lim inf

F (t) 1 , > tp pcp ess inf Ω h

(k−∞ ) ess inf h > 0 and lim inf

F (t) 1 > . |t|p pcp ess inf Ω h

Ω

Ω

t→+∞

t→−∞

Then E is unbounded from below. In the same spirit as Remark 3.4, we have the following remark. Remark 3.11. Condition (i+∞ ) holds if (j+∞ ) lim inf t→+∞



F (t) F (t) F (t) 2p N . > −∞ and lim sup > (2 − 1) p lim inf + t→+∞ tp tp tp h∞ θp t→+∞

Analogously, for (i−∞ ) we need (j−∞ ) lim inf t→−∞



F (t) F (t) F (t) 2p N . > −∞ and lim sup > (2 − 1) p lim inf + t→−∞ tp tp tp h∞ θp t→−∞

Remark 3.12. Notice that actually the unboundedness of E follows from the weaker assumptions F (t) F (t) −∞ < lim inf p  lim sup p = +∞ t→+∞ t t t→+∞ and −∞ < lim inf t→−∞

F (t) F (t)  lim sup p = +∞. p |t| t→−∞ |t|

It is worth pointing out that assumption (i+∞ ) (respectively, (i−∞ )) cannot be replaced by the condition   F (t) F (t) lim = +∞ respectively, lim = +∞ . t→+∞ tp t→−∞ |t|p Indeed, if this was the case, then lim F (t) = +∞.

t→+∞

From the coercivity of λψ − F , one also has that limt→+∞ ψ(t) = +∞. From Remark 2.2, for every r ∈ ]α, +∞[ there exists tr ∈ ψ −1 (r), and so tr → +∞ as r → +∞. As ψ ∈ Fq , there exists a positive number c such that ψ(t)  c|t|q

for any t.

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Then, in particular, ψ(tr )p/q  c |tr |p (notice that ψ and F are definitively positive). So supψ−1 (r) F rp/q



F (tr ) F (tr )   p c tr ψ(tr )p/q

and the latter goes to +∞. So, (3.6) is contradicted. Question 3.13. Which kind of assumptions on f ensure that E has no global minimum but is still bounded from below? We conclude the section with the following counterpart of Theorem 3.7. Theorem 3.14. Let f ∈ A with f (0)  0, let h ∈ L∞ (Ω) \ {0} with h  0. Assume that, for each λ ∈ ]0, +∞[, lim (λtq − F (t)) = +∞ t→+∞

and the function t → λt −F (t) has a unique global minimum in [0, +∞[. Finally, suppose that condition (j+∞ ) holds with lim inf t→+∞ F (t)/tp = 0. Under such hypotheses, problem (P) has a sequence of non-negative weak solutions {un } with lim un  = +∞. q

n→∞

4. Applications In this section we provide some examples and applications of the theory. In particular, we want to stress the novelty of our results. As mentioned in the introduction, our results can be applied when F (t) ∈ R, |t|→∞ |t|p lim

which represents a novelty compared with the many papers existing on the subject (see, for example, [10, 13, 15]). Example 4.1. Let h(x) ≡ 1, let |Ω| = 1 and let F (t) =

1 p |t| − G(t), pcp

where G : R → R is a C 1 function that will be specified later. Choose ψ(t) = |t|p and put   1 p 1 hλ (t) = λψ(t) − F (t) = λ|t|p − |t|p + G(t). |t| + G(t) = λ − pcp pcp In order to apply Theorem 3.8 we need the coercivity of hλ for λ ∈ ]1/pcp , +∞[ and the uniqueness of the global minimum of hλ for λ ∈ ]1/pcp , +∞[. Equivalently, we require that (g1 ) the function t → τ |t|p + G(t) is coercive for τ > 0, (g2 ) the function t → τ |t|p + G(t) has a unique global minimum in R for τ > 0.

Existence of infinitely many solutions for a quasi-linear problem y

667

F

x Figure 3. Graph of F .

We will also need that G(t) = 0; |t|→∞ |t|p

(g3 ) G(0) = 0, G has no global minima and lim

(g4 ) there exists a divergent sequence {rk } such that min{G(rk ), G(−rk )} > 0 for all k; (g5 ) the energy E (u) =

1 1 up − upp + p pcp

 G(u) dx Ω

is unbounded from below. If the above conditions hold, then α = 0, β = +∞ and (3.7) holds (see Remark 3.9). Let us propose a concrete example of a function G fulfilling the above assumptions: choose p = 2, N = 1 and  |t|3/2 sin(ln(t2 ) − 3/4), t  0, G(t) = t > 0. |t|3/2 , In order to verify (g5 ) it is enough to test E on the sequence  1 vk = μk u1 (u1 being the first eigenfunction of −Δp ) and μk = −e2kπ → −∞. Since 0 G(u1 (x)) dx < 0, we have E (vk ) → −∞. Theorem 3.8 applies. The last example emphasises the applicability of the present theory (Theorems 3.7 and 3.14) to a higher-dimensional framework (i.e. when 1 < p < N ). Example 4.2. Let p = 2, N = 3 and q = 5, and let h and θ be as in the general framework. For η > 0 to be chosen later, define F : ]0, +∞[ by putting (see Figure 3) F (t) = ηt2 (sin(ln t2 ) + 1).

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F. Faraci and C. Farkas y

λ = 100 λ = 100

x

Figure 4. Graph of λψ − F .

We extend F to the whole real line by putting F (t) = 0 for every t  0. Then f (t) = F  (t) = 2ηt[sin(ln t2 ) + cos(ln t2 ) + 1] for every t > 0 and f (t) = 0 for every t  0. Clearly, f ∈ A and for every λ > 0 lim (λt5 − F (t)) = +∞.

t→+∞

Also, for every λ > 0 it is possible to see (see Figure 4) that the function t → λt5 − F (t) has a unique global minimum in [0, +∞[. We also have that lim inf s→0+

F (s) =0 s2

and

lim sup s→0+

F (s) = 2η, s2

and

F (s) F (s) = 0 and lim sup 2 = 2η. 2 s→+∞ s s s→+∞ Theorems 3.7 and 3.14 are applicable if we choose lim inf

η>

14 . h∞ θ2

Acknowledgements. F.F. is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). C.F. was supported by the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project PN-II-ID-PCE-2011-3-0241. C.F. was also supported by ´ TAMOP (Grant 4.2.4. A/2-11-1-2012-0001) ‘National Excellence Program – Elaborating and operating an inland student and researcher personal support system’. The project was subsidized by the European Union and cofinanced by the European Social Fund. C.F. was also supported by Collegium Talentum of Hungary. C.F. thanks the Department of Mathematics and Computer Sciences of the University of Catania, where the work was initiated, for its hospitality. Both authors express their gratitude to Professor Biagio Ricceri for stimulating discussions during the preparation of the manuscript.

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