A note on spherical maxima sharing the same Lagrange multiplier

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A note on spherical maxima sharing the same

Lagrange multiplier

Biagio Ricceri

Dedicated to Professor Wataru Takahashi, with esteem and friendship, on the occasion of his 70th birthday

*_{Correspondence:}

[email protected] Department of Mathematics, University of Catania, Viale A. Doria 6, Catania, 95125, Italy

Abstract

In this paper, we establish a general result on spherical maxima sharing the same Lagrange multiplier of which the following is a particular consequence: LetXbe a real Hilbert space. For eachr> 0, letSr={x∈X:x2=r}. LetJ:X→Rbe a sequentially weakly upper semicontinuous functional which is Gâteaux differentiable inX\ {0}. Assume that lim sup_x→0 J(x)

x2 = +∞. Then, for each

ρ

ϕ

ρ

λ

λJ

Here and in what follows,X is a real Hilbert space andJ:X→Ris a functional, with

J() = . For eachr> , set

Sr=

x∈X:x=r,

Br=

x∈X:x≤r.

A pointxˆ∈Srsuch that

J(xˆ) =sup

Sr

J

is called a spherical maximum ofJ. Assuming thatJisC_{, spherical maxima are important}

in connection with the eigenvalue problem

J(x) =μx. ()

Actually, ifxˆis a spherical maximum ofJ, by the classical Lagrange multiplier theorem, there existsμxˆ∈Rsuch that

J(xˆ) =μxˆxˆ.

More specifically, one could be interested in the multiplicity of solutions for (), in the sense of finding someμ∈Rfor which there are more pointsxsatisfying (). In this connec-tion, however, just because of dependence ofμx_ˆonxˆ, the existence of more spherical max-ima inSrdoes not imply automatically the existence of someμ∈Rfor which () has more

solutions. So, in order to the multiplicity of solutions of (), it is important to know when, at least for somer> , the spherical maxima inSrshare the same Lagrange multiplier.

The aim of the present note is to give a contribution along such a direction. Here is our basic result.

Theorem  For someρ> ,assume that J is Gâteaux differentiable inint(Bρ)\ {}and that

βρ

ρ <δρ, ()

where

βρ=sup

Bρ

J

and

δρ= sup

x∈Bρ\{}

J(x)

x.

Assume also that,for some a> ,with

a> ρ

ρδρ–βρ

ifδρ< +∞,the restriction of the functional · –aJ(·)to Bρis sequentially weakly lower

semicontinuous.

For each r∈]βρ, +∞[,put

η(r) =sup

y∈Bρ

ρ–y

r–J(y)

and

(r) =

x∈Bρ:ρ–x



r–J(x) =η(r)

.

Then the following assertions hold:

(i) the functionηis convex and decreasing in]βρ, +∞[,withlimr→+∞η(r) = ; (ii) for eachr∈]βρ+ρ_a,ρδρ[,the set(r)is non-empty and,for everyxˆ∈(r),one has

 <ˆx<ρ

and

ˆ

x∈x∈S_ˆx:J(x) =sup

S_ˆx

J

⊆x∈int(Bρ) :x–η(r)J(x) = inf

y∈Bρ

y_–η(r)J(y)

⊆

x∈X:x=η(r)  J(x)

(iii) for eachr,r∈]βρ+ρ_a,ρδρ[,withr<r,and eachxˆ∈(r),ˆy∈(r),one has

ˆy<ˆx;

(iv) ifAdenotes the set of allr∈]βρ+ρ_a,ρδρ[such that(r)is a singleton,then the

functionr→(r)(r∈A)is continuous with respect to the weak topology;if,in

addition,Jis sequentially weakly upper semicontinuous inBρ,then_|Ais

continuous with respect to the strong topology.

Before proving Theorem , let us recall a proposition from [] that will be used in the proof.

Proposition  Let Y be a non-empty set, f,g:Y →Rtwo functions,and a,b two real numbers,with a<b.Let ya be a global minimum of the function f +ag and yb a global

minimum of the function f +bg.

Then one has g(yb)≤g(ya).

Proof of Theorem By definition, the functionηis the upper envelope of a family of

func-tions which are decreasing and convex in ]βρ, +∞[. So,ηis convex and non-increasing.

We also have

η(r)≤ ρ

r–βρ

()

for allr>βρand so

lim

r→+∞η(r) = .

In turn, this implies thatηis decreasing as it never vanishes. Now, fixr∈]βρ+ρ_a,ρδρ[. So,

we have

ρ

r–βρ

<a.

Consequently, by (),

η(r) <a.

Observe that, for eachλ∈],a[, the restriction toBρ of the functional · –λJ(·) is sequentially weakly lower semicontinuous. In this connection, it is enough to notice that

a a–λ

x_{–λJ(x) =x}₊ λ a–λ

x_{–aJ(x) .}

Fix a sequence{xn}inBρsuch that

lim

n→∞

ρ–xn

Up to a subsequence, we can suppose that{xn}converges weakly to somexˆr∈Bρ. Fix

∈],η(r)[. For eachn∈Nlarge enough, we have

ρ–xn

r–J(xn)

>η(r) –

and so

xn+

η(r) – r–J(xn) <ρ.

But then, by sequential weak lower semicontinuity, we have

ˆxr+

η(r) – r–J(xˆr) ≤lim inf n→∞

xn+

η(r) – r–J(xn)

≤ρ.

Hence, since is arbitrary, we have

ˆxr+η(r)

r–J(xˆr) ≤ρ

and so

ρ–ˆxr

r–J(xˆr) =η(r),

that is,xˆr∈(r). Now, letxˆbe any point of(r). Let us show thatxˆ= . Indeed, since r

ρ<δρ, there existsx˜∈Bρ\ {}such that

J(x˜)

˜x > r

ρ.

Clearly, this is equivalent to

ρ

r <

ρ–˜x

r–J(x˜).

So

ρ

r <

ρ–ˆx r–J(xˆ)

and hence, sinceJ() = , we havexˆ= , as claimed. Clearly,ˆx_{<ρasη(r) > . Moreover,}

ifx∈S_ˆx, we have



r–J(x)≤ 

r–J(xˆ)

from which we get

J(xˆ) =sup

S_ˆx

Now, letube any global maximum ofJ|S_ˆx. Then we have

ρ–u r–J(u) =η(r)

and so

u_{–η(r)J(u) =ρ–rη(r)≤ x}_–η(r)J(x)

for allx∈Bρ. Hence, asu<ρ, the pointuis a local minimum of the functional · – η(r)J(·). Consequently, we have

u=η(r)  J(u),

and the proof of (ii) is complete. To prove (iii), observe that



η(r)=xinf_<ρ

r–J(x) ρ–x.

As a consequence, for each r,r∈]βρ+ ρ_a,ρδρ[, with r <r, and for each xˆ∈(r), ˆ

y∈(r), we have

r–J(xˆ) ρ–ˆx =_xinf_<ρ

r–J(x) ρ–x

and

r–J(yˆ) ρ–ˆy=_xinf_<ρ

r–J(x) ρ–x.

Therefore, in view of Proposition , we have

 ρ–ˆy≤

 ρ–ˆx

and so

ˆy ≤ ˆx.

We claim that

ˆy<ˆx.

Arguing by contradiction, assume that ˆy=ˆx. In view of (ii), this would imply that

J(yˆ) =J(xˆ) and so, at the same time,

ˆ y=η(r)

 J(yˆ)

and

ˆ y=η(r)

In turn, this would implyη(r) =η(r) and hencer=r, a contradiction. So, (iii) holds. Finally, let us prove (iv). For eachr∈A, continue to denote by(r) the unique point of(r). Letr∈Aand let{rk}be any sequence inAconverging tor. Up to a subsequence,{(rk)} converges weakly to somex˜∈Bρ. Moreover, for eachk∈N,x∈Bρ, one has

ρ–x rk–J(x) ≤

ρ–(rk)

rk–J((rk)) .

From this, after easy manipulations, we get

(rk)



–ρ–x



r–J(x)J

(rk) –

ρ–x rk–J(x)

–ρ–x



r–J(x)

J(rk)

≤ρ–ρ–x



rk–J(x)

rk. ()

Since the sequence{J((rk))}is bounded above, we have

lim sup

k→∞

ρ–x rk–J(x)

–ρ–x



r–J(x)

J(rk) ≤. ()

On the other hand, by sequential weak semicontinuity, we also have

˜x–ρ–x



r–J(x)J(x˜)≤lim infk→∞

(rk)



–ρ–x



r–J(x)J

(rk) . ()

Now, passing in () to thelim inf, in view of () and (), we obtain

˜x–ρ–x



r–J(x)J(x˜)≤ρ–

ρ–x r–J(x) r,

which is equivalent to

ρ–x r–J(x) ≤

ρ–˜x r–J(x˜) .

Since this holds for allx∈Bρ, we havex˜=(r). So,|Ais continuous atrwith respect to the weak topology. Now, assuming also thatJis sequentially weakly upper semicontinuous, in view of the continuity ofηin ]βρ, +∞[, we have

lim

k→∞

ρ–(rk)

rk–J((rk))

=ρ–(r)



r–J((r)) ,

and hence

lim inf

k→∞

ρ–(rk) =

ρ–(r) r–J((r)) lim infk→∞

rk–J

(rk)

= ρ–(r)



r–J((r))

r–lim sup

k→∞

J(rk)

≥ρ–(r) r–J((r))

from which

lim sup

k→∞

(rk)≤(r).

SinceXis a Hilbert space and{(rk)}converges weakly to(r), this implies that

lim

k→∞

(rk) –(r)= ,

which shows the continuity of|Aatrin the strong topology.

Remark  Clearly, whenJis sequentially weakly upper semicontinuous inBρ, the asser-tions of Theorem  hold in the whole interval ]βρ,ρδρ[, sinceacan be any positive number.

Remark  The simplest way to satisfy condition () is, of course, to assume that

lim sup

x→ J(x)

x= +∞.

Another reasonable way is provided by the following proposition.

Proposition  For some s> ,assume that J is Gâteaux differentiable in Bs\ {}and that

there exists a global maximumx of Jˆ |Bs such that

J(xˆ),xˆ< J(xˆ).

Then()holds withρ=ˆx_.

Proof For eacht∈], ], set

ω(t) = J(tˆx)

tˆx.

Clearly,ωis derivable in ], ]. In particular, one has

ω() =J(xˆ),xˆ – J(xˆ)

ˆx .

So, by assumption,ω() <  and hence, in a left neighborhood of , we have

ω(t) >ω(),

which implies the validity of () withρ=ˆx.

Also, notice the following consequence of Theorem .

Theorem  For someρ> ,let the assumptions of Theorembe satisfied.

Then there exists an open interval I⊆], +∞[and an increasing functionϕ:I→],ρ[

such that,for eachλ∈I,one has

∅ =x∈Sϕ(λ):J(x) =sup

Sϕ(λ)

Proof Take

I= η

βρ+

ρ

a,ρδρ

.

Clearly,Iis an open interval sinceηis continuous and decreasing. Now, for eachr∈]βρ+ ρ

a,ρδρ[, pickvr∈(r). Finally, set

ϕ(λ) =vη–(λ)

for allλ∈I. Taking (iii) into account, we then realize that the functionϕ(whose range is contained in ],ρ[) is the composition of two decreasing functions, and so it is increasing.

Clearly, the conclusion follows directly from (ii).

We conclude deriving from Theorem  the following multiplicity result.

Theorem  For someρ> ,assume that J is sequentially weakly upper semicontinuous in Bρ,Gâteaux differentiable inint(Bρ)\ {}and satisfies().Moreover,assume that there existsρ˜satisfying

inf

x∈Dx

_<ρ˜<sup

x∈Dx

_{, ()}

where

D=

r∈]βρ,ρδρ[ (r),

such that J|Sρ_˜ has either two global maxima or a global maximum at which J vanishes.

Then there existsλ˜> such that the equation

x=λ˜J(x)

has at least two non-zero solutions which are global minima of the restriction of the func-tional 

 ·

_–λ˜_{J(·)toint(Bρ).}

Proof For eachr∈]βρ,ρδρ[, in view of (), we can pickvr∈(r) (recall Remark ), so that

inf ]βρ,ρδρ[

ψ<ρ˜< sup ]βρ,ρδρ[

ψ, ()

where

ψ(r) =vr.

Two cases can occur. First, assume that ρ˜∈ ψ(]βρ,ρδρ[). So, ψ(r˜) =ρ˜ for some ˜r∈

withλ˜=_η(˜r). Now, suppose thatρ˜∈/ ψ(]βρ,ρδρ[). In this case, in view of (), the

func-tionψis discontinuous and hence, in view of (iv), there exists somer∗∈]βρ,ρδρ[ such that

(r∗) has at least two elements which, by (ii), satisfy the conclusion withλ˜=_η(r∗).

Competing interests

The author declares that he has no competing interests.

Received: 14 October 2013 Accepted: 10 January 2014 Published:31 Jan 2014

References

1. Ricceri, B: Uniqueness properties of functionals with Lipschitzian derivative. Port. Math.63, 393-400 (2006)

10.1186/1687-1812-2014-25