# A note on spherical maxima sharing the same Lagrange multiplier

- Biagio Ricceri
^{1}Email author

**2014**:25

https://doi.org/10.1186/1687-1812-2014-25

© Ricceri; licensee Springer. 2014

**Received: **14 October 2013

**Accepted: **10 January 2014

**Published: **31 January 2014

## 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: Let *X* be a real Hilbert space. For each $r>0$, let ${S}_{r}=\{x\in X:{\parallel x\parallel}^{2}=r\}$. Let $J:X\to \mathbf{R}$ be a sequentially weakly upper semicontinuous functional which is Gâteaux differentiable in $X\setminus \{0\}$. Assume that ${lim\hspace{0.17em}sup}_{x\to 0}\frac{J(x)}{{\parallel x\parallel}^{2}}=+\mathrm{\infty}$. Then, for each $\rho >0$, there exists an open interval $I\subseteq \phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[$ and an increasing function $\phi :I\to \phantom{\rule{0.2em}{0ex}}]0,\rho [$ such that, for each $\lambda \in I$, one has $\mathrm{\varnothing}\ne \{x\in {S}_{\phi (\lambda )}:J(x)={sup}_{{S}_{\phi (\lambda )}}J\}\subseteq \{x\in X:x=\lambda {J}^{\prime}(x)\}$.

*X*is a real Hilbert space and $J:X\to \mathbf{R}$ is a functional, with $J(0)=0$. For each $r>0$, set

*J*. Assuming that

*J*is ${C}^{1}$, spherical maxima are important in connection with the eigenvalue problem

*J*, by the classical Lagrange multiplier theorem, there exists ${\mu}_{\stackrel{\u02c6}{x}}\in \mathbf{R}$ such that

More specifically, one could be interested in the multiplicity of solutions for (1), in the sense of finding some $\mu \in \mathbf{R}$ for which there are more points *x* satisfying (1). In this connection, however, just because of dependence of ${\mu}_{\stackrel{\u02c6}{x}}$ on $\stackrel{\u02c6}{x}$, the existence of more spherical maxima in ${S}_{r}$ does not imply automatically the existence of some $\mu \in \mathbf{R}$ for which (1) has more solutions. So, in order to the multiplicity of solutions of (1), it is important to know when, at least for some $r>0$, the spherical maxima in ${S}_{r}$ share 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 1**

*For some*$\rho >0$,

*assume that*

*J*

*is Gâteaux differentiable in*$int({B}_{\rho})\setminus \{0\}$

*and that*

*where*

*and*

*Assume also that*,

*for some*$a>0$,

*with*

*if* ${\delta}_{\rho}<+\mathrm{\infty}$, *the restriction of the functional* ${\parallel \cdot \parallel}^{2}-aJ(\cdot )$ *to* ${B}_{\rho}$ *is sequentially weakly lower semicontinuous*.

*For each*$r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho},+\mathrm{\infty}[$,

*put*

*and*

*Then the following assertions hold*:

- (i)
*the function**η**is convex and decreasing in*$]{\beta}_{\rho},+\mathrm{\infty}[$,*with*${lim}_{r\to +\mathrm{\infty}}\eta (r)=0$; - (ii)
*for each*$r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$,*the set*$\mathrm{\Gamma}(r)$*is non*-*empty and*,*for every*$\stackrel{\u02c6}{x}\in \mathrm{\Gamma}(r)$,*one has*$0<{\parallel \stackrel{\u02c6}{x}\parallel}^{2}<\rho $*and*$\begin{array}{rcl}\stackrel{\u02c6}{x}& \in & \{x\in {S}_{{\parallel \stackrel{\u02c6}{x}\parallel}^{2}}:J(x)=\underset{{S}_{{\parallel \stackrel{\u02c6}{x}\parallel}^{2}}}{sup}J\}\\ \subseteq & \{x\in int({B}_{\rho}):{\parallel x\parallel}^{2}-\eta (r)J(x)=\underset{y\in {B}_{\rho}}{inf}({\parallel y\parallel}^{2}-\eta (r)J(y))\}\\ \subseteq & \{x\in X:x=\frac{\eta (r)}{2}{J}^{\prime}(x)\};\end{array}$ - (iii)
*for each*${r}_{1},{r}_{2}\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$,*with*${r}_{1}<{r}_{2}$,*and each*$\stackrel{\u02c6}{x}\in \mathrm{\Gamma}({r}_{1})$, $\stackrel{\u02c6}{y}\in \mathrm{\Gamma}({r}_{2})$,*one has*$\parallel \stackrel{\u02c6}{y}\parallel <\parallel \stackrel{\u02c6}{x}\parallel ;$ - (iv)
*if**A**denotes the set of all*$r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$*such that*$\mathrm{\Gamma}(r)$*is a singleton*,*then the function*$r\to \mathrm{\Gamma}(r)$ ($r\in A$)*is continuous with respect to the weak topology*;*if*,*in addition*,*J**is sequentially weakly upper semicontinuous in*${B}_{\rho}$,*then*${\mathrm{\Gamma}}_{|A}$*is continuous with respect to the strong topology*.

Before proving Theorem 1, let us recall a proposition from [1] that will be used in the proof.

**Proposition 1** *Let* *Y* *be a non*-*empty set*, $f,g:Y\to \mathbf{R}$ *two functions*, *and* *a*, *b* *two real numbers*, *with* $a<b$. *Let* ${y}_{a}$ *be a global minimum of the function* $f+ag$ *and* ${y}_{b}$ *a global minimum of the function* $f+bg$.

*Then one has* $g({y}_{b})\le g({y}_{a})$.

*Proof of Theorem 1*By definition, the function

*η*is the upper envelope of a family of functions which are decreasing and convex in $]{\beta}_{\rho},+\mathrm{\infty}[$. So,

*η*is convex and non-increasing. We also have

*η*is decreasing as it never vanishes. Now, fix $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$. So, we have

*ϵ*is arbitrary, we have

*u*be any global maximum of ${J}_{|{S}_{{\parallel \stackrel{\u02c6}{x}\parallel}^{2}}}$. Then we have

*u*is a local minimum of the functional ${\parallel \cdot \parallel}^{2}-\eta (r)J(\cdot )$. Consequently, we have

*A*converging to

*r*. Up to a subsequence, $\{\mathrm{\Gamma}({r}_{k})\}$ converges weakly to some $\tilde{x}\in {B}_{\rho}$. Moreover, for each $k\in \mathbf{N}$, $x\in {B}_{\rho}$, one has

*r*with respect to the weak topology. Now, assuming also that

*J*is sequentially weakly upper semicontinuous, in view of the continuity of

*η*in $]{\beta}_{\rho},+\mathrm{\infty}[$, we have

*X*is a Hilbert space and $\{\mathrm{\Gamma}({r}_{k})\}$ converges weakly to $\mathrm{\Gamma}(r)$, this implies that

which shows the continuity of ${\mathrm{\Gamma}}_{|A}$ at *r* in the strong topology. □

**Remark 1** Clearly, when *J* is sequentially weakly upper semicontinuous in ${B}_{\rho}$, the assertions of Theorem 1 hold in the whole interval $]{\beta}_{\rho},\rho {\delta}_{\rho}[$, since *a* can be any positive number.

**Remark 2**The simplest way to satisfy condition (2) is, of course, to assume that

Another reasonable way is provided by the following proposition.

**Proposition 2**

*For some*$s>0$,

*assume that*

*J*

*is Gâteaux differentiable in*${B}_{s}\setminus \{0\}$

*and that there exists a global maximum*$\stackrel{\u02c6}{x}$

*of*${J}_{|{B}_{s}}$

*such that*

*Then* (2) *holds with* $\rho ={\parallel \stackrel{\u02c6}{x}\parallel}^{2}$.

*Proof*For each $t\in \phantom{\rule{0.2em}{0ex}}]0,1]$, set

*ω*is derivable in $]0,1]$. In particular, one has

which implies the validity of (2) with $\rho ={\parallel \stackrel{\u02c6}{x}\parallel}^{2}$. □

Also, notice the following consequence of Theorem 1.

**Theorem 2** *For some* $\rho >0$, *let the assumptions of Theorem * 1 *be satisfied*.

*Then there exists an open interval*$I\subseteq \phantom{\rule{0.2em}{0ex}}]0,+\mathrm{\infty}[$

*and an increasing function*$\phi :I\to \phantom{\rule{0.2em}{0ex}}]0,\rho [$

*such that*,

*for each*$\lambda \in I$,

*one has*

*Proof*Take

*I*is an open interval since

*η*is continuous and decreasing. Now, for each $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho}+\frac{\rho}{a},\rho {\delta}_{\rho}[$, pick ${v}_{r}\in \mathrm{\Gamma}(r)$. Finally, set

for all $\lambda \in I$. Taking (iii) into account, we then realize that the function *φ* (whose range is contained in $]0,\rho [$) is the composition of two decreasing functions, and so it is increasing. Clearly, the conclusion follows directly from (ii). □

We conclude deriving from Theorem 1 the following multiplicity result.

**Theorem 3**

*For some*$\rho >0$,

*assume that*

*J*

*is sequentially weakly upper semicontinuous in*${B}_{\rho}$,

*Gâteaux differentiable in*$int({B}_{\rho})\setminus \{0\}$

*and satisfies*(2).

*Moreover*,

*assume that there exists*$\tilde{\rho}$

*satisfying*

*where*

*such that* ${J}_{|{S}_{\tilde{\rho}}}$ *has either two global maxima or a global maximum at which* ${J}^{\prime}$ *vanishes*.

*Then there exists*$\tilde{\lambda}>0$

*such that the equation*

*has at least two non*-*zero solutions which are global minima of the restriction of the functional* $\frac{1}{2}{\parallel \cdot \parallel}^{2}-\tilde{\lambda}J(\cdot )$ *to* $int({B}_{\rho})$.

*Proof*For each $r\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho},\rho {\delta}_{\rho}[$, in view of (7), we can pick ${v}_{r}\in \mathrm{\Gamma}(r)$ (recall Remark 1), so that

Two cases can occur. First, assume that $\tilde{\rho}\in \psi (]{\beta}_{\rho},\rho {\delta}_{\rho}[)$. So, $\psi (\tilde{r})=\tilde{\rho}$ for some $\tilde{r}\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho},\rho {\delta}_{\rho}[$. So, by (ii), for each global maximum *u* of ${J}_{|{S}_{\tilde{\rho}}}$, we have ${J}^{\prime}(u)\ne 0$. As a consequence, in this case, ${J}_{|{S}_{\tilde{\rho}}}$ has at least two global maxima which, by (ii) again, satisfies the conclusion with $\tilde{\lambda}=\frac{1}{2}\eta (\tilde{r})$. Now, suppose that $\tilde{\rho}\notin \psi (]{\beta}_{\rho},\rho {\delta}_{\rho}[)$. In this case, in view of (8), the function *ψ* is discontinuous and hence, in view of (iv), there exists some ${r}^{\ast}\in \phantom{\rule{0.2em}{0ex}}]{\beta}_{\rho},\rho {\delta}_{\rho}[$ such that $\mathrm{\Gamma}({r}^{\ast})$ has at least two elements which, by (ii), satisfy the conclusion with $\tilde{\lambda}=\frac{1}{2}\eta ({r}^{\ast})$. □

## Declarations

## Authors’ Affiliations

## References

- Ricceri B: Uniqueness properties of functionals with Lipschitzian derivative.
*Port. Math.*2006, 63: 393–400.MathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.