Four conjectures in Nonlinear Analysis

TL;DR

This paper presents four challenging conjectures in Nonlinear Analysis, covering topics like the Monge-Ampère equation, eigenvalue problems, non-local problems, and solution connectedness, aiming to stimulate further research in these areas.

Contribution

It introduces four new conjectures in Nonlinear Analysis, highlighting open problems and potential directions for future mathematical investigation.

Findings

Formulation of a conjecture on the Monge-Ampère equation

Proposal of a conjecture on an eigenvalue problem

Introduction of a conjecture on disconnectedness versus infinitely many solutions

Abstract

In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Amp\`ere equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.

Equations

$$\det (D^{2}u)=h$$

$$\nabla (\Omega )\subseteq \mathrm{conv}(\nabla (\partial \Omega )).$$

$$f(\Omega )\subseteq \overline{\mathrm{conv}}(f(\partial \Omega )),$$

$$\underset{A}{\sup }\psi \le \underset{B}{\sup }\psi .$$

$$\psi (y_{\alpha })=\psi \left(\sum _{i=1}^k{\lambda }_i{z}_i\right)\le \underset{1\le i\le k}{\max }\psi (z_i)\le \underset{B}{\sup }\psi$$

$$\psi (\tilde{y})=\underset{\alpha }{\lim }\psi (y_{\alpha })\le \underset{B}{\sup }\psi$$

$$\underset{x\in \Omega }{\sup }\psi (f(x))=\underset{x\in \partial \Omega }{\sup }\psi (f(x)).$$

$$\underset{x\to x^{\ast }}{\mathrm{limsup}}\psi (f(x))=\underset{x\in \Omega }{\sup }\psi (f(x)).$$

$$\underset{x\to x^{\ast }}{\mathrm{limsup}}\psi (f(x))=\psi (g(x^{\ast }))$$

$$\underset{x\in \partial \Omega }{\sup }\psi (g(x))=\underset{x\in \Omega }{\sup }\psi (g(x)).$$

$$\psi (g(x^{\ast }))=\underset{x\in \partial \Omega }{\sup }\psi (g(x)).$$

$$\psi (g(x^{\ast }))=\underset{x\to x^{\ast }}{\lim }\psi (f(x))$$

$$\underset{x\in \Omega }{\sup }\psi (g(x))=\underset{x\in \overline{\Omega }}{\sup }\psi (g(x))$$

$$\underset{x\to x^{\ast }}{\lim }\psi (f(x))=\underset{x\in \Omega }{\sup }\psi (f(x))$$

$$\underset{x\to z}{\mathrm{limsup}}\psi (f(x))<\underset{x\in \Omega }{\sup }\psi (f(x))$$

$$\underset{x\in U_z\cap \Omega }{\sup }\psi (f(x))<\underset{x\in \Omega }{\sup }\psi (f(x)).$$

$$\partial \Omega \subseteq \bigcup _{i=1}^k{U}_{z_i}.$$

$$U=\bigcup _{i=1}^k{U}_{z_i}.$$

$$\underset{x\in U\cap \Omega }{\sup }\psi (f(x))=\underset{1\le i\le k}{\max }\underset{x\in U_{z_i}\cap \Omega }{\sup }\psi (f(x))<\underset{x\in \Omega }{\sup }\psi (f(x)).$$

$$\underset{x\in U\cap \Omega }{\sup }\psi (f(x))<r<\underset{x\in \Omega }{\sup }\psi (f(x))$$

$$K=\{x\in \Omega :\psi (f(x))\ge r\}.$$

$$\phi (f(\bar{x}))<\underset{y\in {\psi }^{-1}(]-\infty ,r])}{\inf }\phi (y).$$

$$\phi (f(\bar{x}))<\underset{x\in \Omega \setminus K}{\inf }\phi (f(x)).$$

$$\phi (f(\bar{x}))<\rho <\underset{x\in \Omega \setminus K}{\inf }\phi (f(x))$$

$$X=\{x\in \Omega :\phi (f(x))<\rho \}.$$

$$\tilde{\lambda }=\underset{x\in X}{\inf }\frac{\phi (g(x))-{\inf }_{z\in K}\phi (g(z))}{\rho -\phi (f(x))}.$$

$$\frac{\phi (g(x_0))-{\inf }_{z\in K}\phi (g(z))}{\rho -\phi (f(x_0))}<\lambda .$$

$$\phi (g(x_0))+\lambda \phi (f(x_0))<\lambda \rho +\underset{z\in K}{\inf }\phi (g(z)).$$

Full text

Four conjectures in Nonlinear Analysis

BIAGIO RICCERI

Abstract. In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Ampère equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.

In this chapter, I intend to formulate four challenging conjectures in Nonlinear Analysis which have their roots in certain results that I have obtained in the past years.

1. A conjecture on the Monge-Ampère equation

CONJECTURE 1.1. - Let $\Omega\subset{\bf R}^{n}$ $(n\geq 2)$ be a non-empty open bounded set and let $h:\Omega\to{\bf R}$ be a non-negative continuous function.

Then, each $u\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega})$ satisfying in $\Omega$ the Monge-Ampère equation

[TABLE]

has the following property:

[TABLE]

This conjecture is motivated by [26] where I proved that it is true for $n=2$. I am going to produce such a proof here.

In what follows, $\Omega$ is a non-empty relatively compact and open set in a topological space $E$, with $\partial\Omega\neq\emptyset$, and $Y$ is a real locally convex Hausdorff topological vector space. $\overline{\Omega}$ and $\partial\Omega$ denote the closure and the boundary of $\Omega$, respectively. Since $\overline{\Omega}$ is compact, $\partial\Omega$, being closed, is compact too. Let us first recall some well-known definitions.

Let $S$ be a subset of $Y$ and let $y_{0}\in S$. As usual, we say that $S$ is supported at $y_{0}$ if there exists $\varphi\in Y^{*}\setminus\{0\}$ such that $\varphi(y_{0})\leq\varphi(y)$ for all $y\in S$. If this happens, of course $y_{0}\in\partial S$. Further, extending a maximum principle definition for real-valued functions, a continuous function $f:\overline{\Omega}\to Y$ is said to satisfy the convex hull property in $\overline{\Omega}$ (see [7], [13] and references therein) if

[TABLE]

$\overline{\hbox{\rm conv}}(f(\partial\Omega))$ being the closed convex hull of $f(\partial\Omega)$. When dim$(Y)<\infty$, since $f(\partial\Omega)$ is compact, conv$(f(\partial\Omega))$ is compact too and so $\overline{\hbox{\rm conv}}(f(\partial\Omega))=\hbox{\rm conv}(f(\partial\Omega))$.

A function $\psi:Y\to{\bf R}$ is said to be quasi-convex if, for each $r\in{\bf R}$, the set $\psi^{-1}(]-\infty,r])$ is convex.

Notice the following proposition:

PROPOSITION 1.1. - For each pair $A,B$ of non-empty subsets of $Y$, the following assertions are equivalent:

$(a_{1})$*  *$A\subseteq\overline{\hbox{\rm conv}}(B)\ .$

$(a_{2})$*  For every continuous and quasi-convex function $\psi:Y\to{\bf R}$, one has*

[TABLE]

PROOF. Let $(a_{1})$ hold. Fix any continuous and quasi-convex function $\psi:Y\to{\bf R}$. Fix $\tilde{y}\in A$. Then, there is a net $\{y_{\alpha}\}$ in conv$(B)$ converging to $\tilde{y}$. So, for each $\alpha$, we have $y_{\alpha}=\sum_{i=1}^{k}\lambda_{i}z_{i}$, where $z_{i}\in B$, $\lambda_{i}\in[0,1]$ and $\sum_{i=1}^{k}\lambda_{i}=1$. By quasi-convexity, we have

[TABLE]

and so, by continuity,

[TABLE]

which yields $(a_{2})$.

Now, let $(a_{2})$ hold. Let $x_{0}\in A$. If $x_{0}\not\in\overline{\hbox{\rm conv}}(B)$, by the standard separation theorem, there would be $\psi\in Y^{*}\setminus\{0\}$ such that $\sup_{\overline{\hbox{\rm conv}}(B)}\psi<\psi(x_{0})$, against $(a_{2})$. So, $(a_{1})$ holds. $\bigtriangleup$

Clearly, applying Proposition 1.1, we obtain the following one:

PROPOSITION 1.2. - For any continuous function $f:\overline{\Omega}\to Y$, the following assertions are equivalent:

$(b_{1})$*  $f$ satisfies the convex hull property in $\overline{\Omega}$ .*

$(b_{2})$*  For every continuous and quasi-convex function $\psi:Y\to{\bf R}$, one has*

[TABLE]

In view of Proposition 1.2, we now introduce the notion of convex hull-like property for functions defined in $\Omega$ only.

DEFINITION 1.1. - A continuous function $f:\Omega\to Y$ is said to satisfy the convex hull-like property in $\Omega$ if, for every continuous and quasi-convex function $\psi:Y\to{\bf R}$, there exists $x^{*}\in\partial\Omega$ such that

[TABLE]

We have

PROPOSITION 1.3. - Let $g:\overline{\Omega}\to Y$ be a continuous function and let $f=g_{|\Omega}$.

Then, the following assertions are equivalent:

$(c_{1})$*  $f$ satisfies the convex hull-like property in $\Omega$ .*

$(c_{2})$*  $g$ satisfies the convex hull property in $\overline{\Omega}$ .*

PROOF. Let $(c_{1})$ hold. Let $\psi:Y\to{\bf R}$ be any continuous and quasi-convex function. Then, by Definition 1.1, there exists $x^{*}\in\partial\Omega$ such that

[TABLE]

But

[TABLE]

and hence

[TABLE]

So, by Proposition 1.2, $(c_{2})$ holds.

Now, let $(c_{2})$ hold. Let $\psi:Y\to{\bf R}$ be any continuous and quasi-convex function. Then, by Proposition 1.2, one has

[TABLE]

Since $\partial\Omega$ is compact and $\psi\circ g$ is continuous, there exists $x^{*}\in\partial\Omega$ such that

[TABLE]

But

[TABLE]

and, by continuity again,

[TABLE]

and so

[TABLE]

which yields $(c_{1})$. $\bigtriangleup$

The central result is as follows:

THEOREM 1.1. - For any continuous function $f:\Omega\to Y$, at least one of the following assertions holds:

$(i)$*  $f$ satisfies the convex hull-like property in $\Omega$ .*

$(ii)$*  There exists a non-empty open set $X\subseteq\Omega$, with $\overline{X}\subseteq\Omega$, satisfying the following property: for every continuous function $g:\Omega\to Y$, there exists $\tilde{\lambda}\geq 0$ such that, for each $\lambda>\tilde{\lambda}$, the set $(g+\lambda f)(X)$ is supported at one of its points.*

PROOF. Assume that $(i)$ does not hold. So, we are assuming that there exists a continuous and quasi-convex function $\psi:Y\to{\bf R}$ such that

[TABLE]

for all $z\in\partial\Omega$.

In view of $(1.1)$, for each $z\in\partial\Omega$, there exists an open neighbourhood $U_{z}$ of $z$ such that

[TABLE]

Since $\partial\Omega$ is compact, there are finitely many $z_{1},...,z_{k}\in\partial\Omega$ such that

[TABLE]

Put

[TABLE]

Hence

[TABLE]

Now, fix a number $r$ so that

[TABLE]

and set

[TABLE]

Since $f,\psi$ are continuous, $K$ is closed in $\Omega$. But, since $K\cap U=\emptyset$ and $U$ is open, in view of $(1.2)$, $K$ is closed in $E$. Hence, $K$ is compact since $\overline{\Omega}$ is so. By $(1.3)$, we can fix $\bar{x}\in\Omega$ such that $\psi(f(\bar{x}))>r$. Notice that the set $\psi^{-1}(]-\infty,r])$ is closed and convex. So, thanks to the standard separation theorem, there exists a non-zero continuous linear functional $\varphi:Y\to{\bf R}$ such that

[TABLE]

Then, from $(1.4)$, it follows

[TABLE]

Now, choose $\rho$ so that

[TABLE]

and set

[TABLE]

Clearly, $X$ is a non-empty open set contained in $K$. Now, let $g:\Omega\to Y$ be any continuous function. Set

[TABLE]

Fix $\lambda>\tilde{\lambda}$. So, there is $x_{0}\in X$ such that

[TABLE]

From this, we get

[TABLE]

By continuity and compactness, there exists $\hat{x}\in K$ such that

[TABLE]

for all $x\in K$. Let us prove that $\hat{x}\in X$. Arguing by contradiction, assume that $\varphi(f(\hat{x}))\geq\rho$. Then, taking $(1.5)$ into account, we would have

[TABLE]

contradicting $(1.6)$. So, it is true that $\hat{x}\in X$, and, by $(1.6)$, the set $(g+\lambda f)(X)$ is supported at its point $g(\hat{x})+\lambda f(\hat{x})$. $\bigtriangleup$

An application of Theorem 1.1 shows a strongly bifurcating behaviour of certain equations in ${\bf R}^{n}$.

THEOREM 1.2. - Let $\Omega$ be a non-empty bounded open subset of ${\bf R}^{n}$ and let $f:\Omega\to{\bf R}^{n}$ a continuous function.

Then, at least one of the following assertions holds:

$(d_{1})$*  $f$ satisfies the convex hull-like property in $\Omega$ .*

$(d_{2})$*  There exists a non-empty open set $X\subseteq\Omega$, with $\overline{X}\subseteq\Omega$, satisfying the following property: for every continuous function $g:\Omega\to{\bf R}^{n}$, there exists $\tilde{\lambda}\geq 0$ such that, for each $\lambda>\tilde{\lambda}$, there exist $\hat{x}\in X$ and two sequences $\{y_{k}\}$, $\{z_{k}\}$ in ${\bf R}^{n}$, with*

[TABLE]

such that, for each $k\in{\bf N}$, one has

$(j)$*  the equation*

[TABLE]

has no solution in $X$ ;

$(jj)$*  the equation*

[TABLE]

has two distinct solutions $u_{k},v_{k}$ in $X$ such that

[TABLE]

PROOF. Apply Theorem 1.1 with $E=Y={\bf R}^{n}$. Assume that $(d_{1})$ does not hold. Let $X\subseteq\Omega$ be an open set as in $(ii)$ of Theorem 1.1. Fix any continuous function $g:\Omega\to{\bf R}^{n}$. Then, there is some $\tilde{\lambda}\geq 0$ such that, for each $\lambda>\tilde{\lambda}$, there exists $\hat{x}\in X$ such that the set $(g+\lambda f)(X)$ is supported at $g(\hat{x})+\lambda f(\hat{x})$. As we observed at the beginning, this implies that $g(\hat{x})+\lambda f(\hat{x})$ lies in the boundary of $(g+\lambda f)(X)$. Therefore, we can find a sequence $\{y_{k}\}$ in ${\bf R}^{n}\setminus(g+\lambda f)(X)$ converging to $g(\hat{x})+\lambda f(\hat{x})$. So, such a sequence satisfies $(j)$. For each $k\in{\bf N}$, denote by $B_{k}$ the open ball of radius ${{1}\over{k}}$ centered at $\hat{x}$. Let $k$ be such that $B_{k}\subseteq X$. The set $(g+\lambda f)(B_{k})$ is not open since its boundary contains the point $g(\hat{x})+\lambda f(\hat{x})$. Consequently, by the invariance of domain theorem ([29], p. 705), the function $g+\lambda f$ is not injective in $B_{k}$. So, there are $u_{k},v_{k}\in B_{k}$, with $u_{k}\neq v_{k}$ such that

[TABLE]

Hence, if we take

[TABLE]

the sequences $\{u_{k}\},\{v_{k}\},\{z_{k}\}$ satisfy $(jj)$ and the proof is complete. $\bigtriangleup$

REMARK 1.1. - Notice that, in general, Theorem 1.2 is no longer true when $f:\Omega\to{\bf R}^{m}$ with $m>n$. In this connection, consider the case $n=1$, $m=2$, $\Omega=]0,\pi[$ and $f(\theta)=(\cos\theta,\sin\theta)$ for $\theta\in[0,\pi]$. So, for each $\lambda>0$, on the one hand, the function $\lambda f$ is injective, while, on the other hand, $\lambda f(]0,\pi[)$ is not contained in conv$(\{f(0),f(\pi)\})$.

If $S\subseteq{\bf R}^{n}$ is a non-empty open set, $x\in S$ and $h:S\to{\bf R}^{n}$ is a $C^{1}$ function, we denote by det$(J_{h}(x))$ the Jacobian determinant of $h$ at $x$.

A very recent and important result by J. Saint Raymond ([27]) states what follows (for anything concerning the topological dimension we refer to [8]):

THEOREM 1.A ([27], Theorem 10). - Let $A\subseteq{\bf R}^{n}$ be a non-empty open set and $\varphi:A\to{\bf R}^{n}$ a $C^{1}$ function such that the topological dimension of the set

[TABLE]

is not positive.

Then, the function $\varphi$ is open.

A joint application of Theorem 1.1 and Theorem 1.A gives THEOREM 1.3. - Let $f:\Omega\to{\bf R}^{n}$ be a $C^{1}$ function.

Then, at least one of the following assertions holds:

$(a_{1})$*  $f$ satisfies the convex hull-like property in $\Omega$ .*

$(a_{2})$*  There exists a non-empty open set $X\subseteq\Omega$, with $\overline{X}\subseteq\Omega$, satisfying the following property: for every continuous function $g:\Omega\to{\bf R}^{n}$ which is $C^{1}$ in $X$, there exists $\tilde{\lambda}\geq 0$ such that, for each $\lambda>\tilde{\lambda}$, the topological dimension of the set*

[TABLE]

is greater than or equal $1$.

PROOF. Assume that $(a_{1})$ does not hold. Let $X$ be an open set as in $(ii)$ of Theorem 1.1. Let $g:\Omega\to{\bf R}^{n}$ be a continuous function which is $C^{1}$ in $X$. Then, there is some $\tilde{\lambda}\geq 0$ such that, for each $\lambda>\tilde{\lambda}$, there exists $\hat{x}\in X$ such that the set $(g+\lambda f)(X)$ is supported at $g(\hat{x})+\lambda f(\hat{x})$. As already remarked, this implies that $g(\hat{x})+\lambda f(\hat{x})\in\partial(g+\lambda f)(X)$ and so $(g+\lambda f)(X)$ is not open. Now, $(a_{2})$ is a direct consequence of Theorem 1.A. $\bigtriangleup$

In turn, here is a consequence of Theorem 1.3 when $n=2$.

THEOREM 1.4. - Let $\Omega$ be a non-empty bounded open set of ${\bf R}^{2}$, let $h:\Omega\to{\bf R}$ be a continuous function and let $\alpha,\beta:\Omega\to{\bf R}$ be two $C^{1}$ functions such that $|\alpha_{x}\beta_{y}-\alpha_{y}\beta_{x}|+|h|>0$ and $(\alpha_{x}\beta_{y}-\alpha_{y}\beta_{x})h\geq 0$ in $\Omega$.

Then, any $C^{1}$ solution $(u,v)$ in $\Omega$ of the system

[TABLE]

satisfies the convex hull-like property in $\Omega$.

PROOF. Arguing by contradiction, assume that $(u,v)$ does not satisfy the convex hull-like property in $\Omega$. Then, by Theorem 1.3, applied taking $f=(u,v)$ and $g=(\alpha,\beta)$, there exist $\lambda>0$ and $(\hat{x},\hat{y})\in\Omega$ such that

[TABLE]

On the other hand, for each $(x,y)\in\Omega$, we have

[TABLE]

and hence

[TABLE]

which is impossible in view of our assumptions. $\bigtriangleup$

Finally, taking Proposition 1.3 in mind, here is the proof of Conjecture 1.1 when $n=2$:

THEOREM 1.5. - Let $\Omega$ be a non-empty bounded open subset of ${\bf R}^{2}$, let $h:\Omega\to{\bf R}$ be a continuous non-negative function and let $w\in C^{2}(\Omega)$ be a function satisfying in $\Omega$ the Monge-Ampère equation

[TABLE]

Then, the gradient of $w$ satisfies the convex hull-like property in $\Omega$.

PROOF. It is enough to observe that $(w_{x},w_{y})$ is a $C^{1}$ solution in $\Omega$ of the system $(1.7)$ with $\alpha(x,y)=-y$ and $\beta(x,y)=x$ and that such $\alpha,\beta$ satisfy the assumptions of Theorem 1.4. $\bigtriangleup$

2. A conjecture on an eigenvalue problem

CONJECTURE 2.1. - Let $n\geq 2$ and let $\Omega=\{x\in{\bf R}^{n}:a<|x|<b\}$, with $0<a<b$.

Then, there exists $\lambda>0$ such that the problem

[TABLE]

has at least one non-zero classical solution.

The above conjecture has its roots in Pohozaev identity ([19]). Let me recall it.

So, let $\Omega\subset{\bf R}^{n}$ be a smooth bounded domain, and let $f:{\bf R}\to{\bf R}$ be a continuous function. Put

[TABLE]

for all $\xi\in{\bf R}$. For $\lambda>0$, consider the problem

[TABLE]

In the sequel, a classical solution of problem $(P_{\lambda f})$ is any $u\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega})$, zero on $\partial\Omega$, satisfying the equation pointwise in $\Omega$. Set

[TABLE]

When $n\geq 2$, the Pohozaev identity tells us that, if $u$ is a classical solution of $(P_{\lambda f})$, then one has

[TABLE]

where $\nu$ denotes the unit outward normal to $\partial\Omega$.

From $(2.1)$, in particular, it follows that, if $\Omega$ is star-shaped with respect to [math] (so $x\cdot\nu(x)\geq 0$ on $\partial\Omega$), then the set $\Lambda_{f}$ is empty in the two following cases:

$(a)$  $f(\xi)=|\xi|^{p-2}\xi$ with $n\geq 3$ and $p\geq{{2n}\over{n-2}}$ ;

$(b)$  $\sup_{\xi\in{\bf R}}F(\xi)=0$ .

A natural question arises: what about problem $(P_{\lambda f})$ in cases $(a)$ and $(b)$ when $\Omega$ is not star-shaped ?

It is very surprising to realize that, while a great amount of research has been produced on case $(a)$ (see, for instance, [1]-[5], [12], [14], [17], [18]), apparently the only papers dealing with case $(b)$ are [9]-[11], [23].

In [11], the following result has been pointed out:

THEOREM 2.1. - Let $n\geq 2$ and $\Omega=\{x\in{\bf R}^{n}:a<|x|<b\}$ with $0<a<b$.

Then for every continuous function $f:{\bf R}\to{\bf R}$, with $\sup_{\xi\in{\bf R}}F(\xi)=0$, and every $\lambda>0$, problem $(P_{\lambda f})$ has no radially symmetric non-zero classical solutions.

PROOF. Let $f:{\bf R}\to{\bf R}$ be a continuous function, with $\sup_{\xi\in{\bf R}}F(\xi)=0$, let $\lambda>0$, and let $u$ be a radially symmetric classical solution of $(P_{\lambda f})$. Then

[TABLE]

that is

[TABLE]

Multiplying both sides of the equation in (2.2) by $u^{\prime}$, we have

[TABLE]

for all $r\in(a,b)$. Let $r_{1}\in(a,b)$ be such that $u^{\prime}(r_{1})=0$. Define

[TABLE]

for all $r\in[a,b]$. Then (2.3) shows that $I_{r_{1}}^{\prime}(r)=0$ for all $r\in(a,b)$ and so, for some $c\in{\bf R}$, one has

[TABLE]

for all $r\in[a,b]$. Since

[TABLE]

we have $c\leq 0$. On the other hand, since

[TABLE]

have $c\geq 0$, and so $c=0$. In particular $I_{r_{1}}(b)=0$, which implies $u^{\prime}(b)=0$, and consequently $u(r)=0$ for all $r\in[a,b]$, as claimed. $\bigtriangleup$

REMARK 2.1. - It is important to note the drastic difference between cases $(a)$ and $(b)$ enlighted by Theorem 2.1 when $\Omega$ is an annulus. Actually, in this case, it was remarked in [13] that the problem

[TABLE]

has radially symmetric non-zero classical solutions for $p\geq{{2n}\over{n-2}}$ ($n\geq 3$), and $\lambda>0$.

Now, I recall a very general result proved in [23].

For any real Hilbert space $X$, denote by ${\cal A}_{X}$ the set of all $C^{1}$ functionals $I:X\to{\bf R}$ such that [math] is a global maximum of $I$ and $I^{\prime}$ is Lipschitzian with Lipschitz constant less than $1$. Set

[TABLE]

We have:

THEOREM 2.2. - For any real Hilbert space $(X,\langle\cdot,\cdot\rangle)$, with $X\neq\{0\}$, one has

[TABLE]

We first prove PROPOSITION 2.1. - One has

[TABLE]

PROOF. Let $I_{0}\in{\cal A}_{\bf R}$ and let $L<1$ be the Lipschitz constant of $I_{0}^{\prime}$. Set

[TABLE]

Fix $\lambda\in]0,3]$. Let us prove that [math] is the only solution of the equation

[TABLE]

Arguing by contradiction, assume that

[TABLE]

for some $x_{0}\neq 0$. It is not restrictive to assume that $x_{0}>0$ (otherwise, we would work with $I^{\prime}(-x)$). Consider now the function $g:{\bf R}\to{\bf R}$ defined by

[TABLE]

Clearly, $g\in C^{1}({\bf R})$. Let $x>0$ with $x\neq x_{0}$. Let us prove that

[TABLE]

We distinguish two cases. If $0<x\leq{{x_{0}}\over{3}}$, We have

[TABLE]

If $x>{{x_{0}}\over{3}}$, We have

[TABLE]

So, in particular, we get

[TABLE]

which contradicts the fact that the function $I$ is non-positive, since [math] is a global maximum of $I_{0}$. From what we have just proven, it clearly follows that

[TABLE]

Now, fix any $\mu>1$. Continue to consider the function $g$ defined above (for a fixed $x_{0}>0$). Clearly, the function ${{1}\over{\mu}}g$ belongs to ${\cal A}_{\bf R}$ and

[TABLE]

Of course, from this we infer that

[TABLE]

and the conclusion clearly follows. $\bigtriangleup$

Proof of Theorem 2.2. First, let us prove that

[TABLE]

To this end, fix any $\varphi\in{\cal A}_{\bf R}$ and any $\lambda>0$ such that

[TABLE]

for some $t\neq 0$. Fix also $u\in X$, with $\|u\|=1$, and consider the functional $I$ defined by putting

[TABLE]

for all $x\in X$. It is readily seen that $I\in{\cal A}_{X}$. In particular, note that

[TABLE]

Finally, set

[TABLE]

Of course, $\hat{x}\neq 0$. Since

[TABLE]

we also have

[TABLE]

and so

[TABLE]

From this, it clearly follows that

[TABLE]

and so $(2.4)$ follows now from Proposition 2.1.

Now, let us prove that

[TABLE]

To this end, fix $I\in{\cal A}_{X}$, $\lambda>0$ and $x\in X\setminus\{0\}$ such that

[TABLE]

Then, consider the function $\varphi:{\bf R}\to{\bf R}$ defined by

[TABLE]

for all $t\in{\bf R}$. Clearly, [math] is a global maximum for $\varphi$. Moreover, $\varphi\in C^{1}({\bf R})$ and one has

[TABLE]

Therefore, if $L$ is the Lipschitz constant of $I^{\prime}$, for each $t,s\in{\bf R}$, we have

[TABLE]

[TABLE]

This shows that $\varphi^{\prime}$ is a contraction, and so $\varphi\in{\cal A}_{\bf R}$. Now, from $(2.6)$, we get

[TABLE]

that is

[TABLE]

From this, we infer that

[TABLE]

So $(2.5)$ follows from Proposition 2.1, and the proof is complete. $\bigtriangleup$

Now, for each $L>0$, denote by ${\cal C}_{L}$ the class of all Lipschitzian functions $f:{\bf R}\to{\bf R}$, with Lipschitz constant $L$, such that $f(0)=0$ and $\sup_{\xi\in{\bf R}}F(\xi)=0$. Also denote by $\lambda_{1}$ the first eigenvalue of the problem

[TABLE]

From Theorem 2.2, it directly follows that

[TABLE]

In [10], X. L. Fan obtained the finer inequality

[TABLE]

Conjecture 2.1 says that $\Lambda_{f}\neq\emptyset$ for $f(\xi)=-\sin\xi$, $\Omega$ being an annulus. Due to what precedes, if Conjecture 2.1 is true, then $\lambda$ must necessarily be larger than $3\lambda_{1}$.

3. A conjecture on a non-local problem

CONJECTURE 3.1. - Let $a\geq 0$, $b>0$ and let $\Omega\subset{\bf R}^{n}$ be a smooth bounded domain, with $n>4$.

Then, for each $\lambda>0$ large enough and for each convex set $C\subseteq L^{2}(\Omega)$ whose closure in $L^{2}(\Omega)$ contains $H^{1}_{0}(\Omega)$, there exists $v^{*}\in C$ such that the problem

[TABLE]

has at least three weak solutions, two of which are global minima in $H^{1}_{0}(\Omega)$ of the functional

[TABLE]

Conjecture 3.1 comes from the results I have obtained in [25]. I am going to reproduce them here.

Let $a,b,\Omega$ be as in Conjecture 3.1.

On the Sobolev space $H^{1}_{0}(\Omega)$, we consider the scalar product

[TABLE]

and the induced norm

[TABLE]

Denote by ${\cal A}$ the class of all Carathéodory functions $f:\Omega\times{\bf R}\to{\bf R}$ such that

[TABLE]

for some $p\in\left]0,{{n+2}\over{n-2}}\right[$.

Moreover, denote by $\tilde{\cal A}$ the class of all Carathéodory functions $g:\Omega\times{\bf R}\to{\bf R}$ such that

[TABLE]

for some $q\in\left]0,{{2}\over{n-2}}\right[$. Furthermore, denote by $\hat{\cal A}$ the class of all functions $h:\Omega\times{\bf R}\to{\bf R}$ of the type

[TABLE]

with $f\in{\cal A},g\in\tilde{\cal A}$ and $\alpha\in L^{2}(\Omega)$. For each $h\in\hat{\cal A}$, define the functional $I_{h}:H^{1}_{0}(\Omega)\to{\bf R}$, by putting

[TABLE]

for all $u\in H^{1}_{0}(\Omega)$, where

[TABLE]

for all $(x,\xi)\in\Omega\times{\bf R}$.

By classical results (involving the Sobolev embedding theorem), the functional $I_{h}$ turns out to be sequentially weakly continuous, of class $C^{1}$, with compact derivative given by

[TABLE]

for all $u,w\in H^{1}_{0}(\Omega)$.

Now, recall that, given $h\in\hat{\cal A}$, a weak solution of the problem

[TABLE]

is any $u\in H^{1}_{0}(\Omega)$ such that

[TABLE]

for all $w\in H^{1}_{0}(\Omega)$. Let $\Phi:H^{1}_{0}(\Omega)\to{\bf R}$ be the functional defined by

[TABLE]

for all $u\in H^{1}_{0}(\Omega)$.

Hence, the weak solutions of the problem are precisely the critical points in $H^{1}_{0}(\Omega)$ of the functional $\Phi-I_{h}$ which is said to be the energy functional of the problem.

The central result is as follows:

THEOREM 3.1. - Let $n\geq 4$, let $f\in{\cal A}$ and let $g\in\tilde{\cal A}$ be such that the set

[TABLE]

has a positive measure.

Then, there exist $\lambda^{*}\geq 0$ such that, for each $\lambda>\lambda^{*}$ and each convex set $C\subseteq L^{2}(\Omega)$ whose closure in $L^{2}(\Omega)$ contains the set $\{G(\cdot,u(\cdot)):u\in H^{1}_{0}(\Omega)\}$, there exists $v^{*}\in C$ such that the problem

[TABLE]

has at least three weak solutions, two of which are global minima in $H^{1}_{0}(\Omega)$ of the functional

[TABLE]

If, in addition, the functional

[TABLE]

has at least two global minima in $H^{1}_{0}(\Omega)$ and the function $G(x,\cdot)$ is strictly monotone for all $x\in\Omega$, then $\lambda^{*}=0$.

The main tool we use to prove Theorem 3.1 is Theorem 3.C below which, in turn, is a direct consequence of two other results recently established in [24].

To state Theorem 3.C in a compact form, we now introduce some notations.

Here and in what follows, $X$ is a non-empty set, $V,Y$ are two topological spaces, $y_{0}$ is a point in $Y$.

We denote by ${\cal G}$ the family of all lower semicontinuous functions $\varphi:Y\to[0,+\infty[$, with $\varphi^{-1}(0)=\{y_{0}\}$, such that, for each neighbourhood $U$ of $y_{0}$, one has

[TABLE]

Moreover, denote by ${\cal H}$ the family of all functions $\Psi:X\times V\to Y$ such that, for each $x\in X$, $\Psi(x,\cdot)$ is continuous, injective, open, takes the value $y_{0}$ at a point $v_{x}$ and the function $x\to v_{x}$ is not constant. Furthermore, denote by ${\cal M}$ the family of all functions $J:X\to{\bf R}$ whose set of all global minima (noted by $M_{J}$) is non-empty.

Finally, for each $\varphi\in{\cal G}$, $\Psi\in{\cal H}$ and $J\in{\cal M}$, put

[TABLE]

When $X$ is a topological space, a function $\psi:X\to{\bf R}$ is said to be inf-compact if $\psi^{-1}(]-\infty,r])$ is compact for all $r\in{\bf R}$. THEOREM 3.A ([24], Theorem 3.1). - Let $\varphi\in{\cal G}$, $\Psi\in{\cal H}$ and $J\in{\cal M}$.

Then, for each $\lambda>\theta(\varphi,\Psi,J)$, one has

[TABLE]

THEOREM 3.B ([24], Theorem 3.2). - Let $X$ be a topological space, $E$ a real Hausdorff topological vector space, $C\subseteq E$ a convex set, $f:X\times C\to{\bf R}$ a function which is lower semicontinuous and inf-compact in $X$, and upper semicontinuous and concave in $C$. Assume also that

[TABLE]

Then, there exists $v^{*}\in C$ such that the function $f(\cdot,v^{*})$ has at least two global minima.

THEOREM 3.C. - Let $\varphi\in{\cal G}$, $\Psi\in{\cal H}$ and $J\in{\cal M}$. Moreover, assume that $X$ is a topological space, that $V$ is a real Hausdorff topological vector space and that $\varphi(\Psi(x,\cdot))$ is convex and continuous for each $x\in X$. Finally, let $\lambda>\theta(\varphi,\Psi,J)$ and let $C\subseteq V$ a convex set, with $\{v_{x}:x\in X\}\subseteq\overline{C}$, such that the function $x\to J(x)-\lambda\varphi(\Psi(x,v))$ is lower semicontinuous and inf-compact in $X$ for all $v\in C$.

Under such hypotheses, there exist $v^{*}\in C$ such that the function $x\to J(x)-\lambda\varphi(\Psi(x,v^{*}))$ has at least two global minima in $X$.

PROOF. Set

[TABLE]

and, for each $(x,v)\in X\times V$, put

[TABLE]

Theorem 3.A ensures that

[TABLE]

But, since $f(x,\cdot)$ is continuous and $D\subseteq\overline{C}$, we have

[TABLE]

for all $x\in X$, and hence, from $(3.3)$, it follows that

[TABLE]

At this point, the conclusion follows applying Theorem 3.B to the restriction of the function $f$ to $X\times C$. $\bigtriangleup$

Proof of Theorem 3.1. For each $\lambda\geq 0$, $v\in L^{2}(\Omega)$, consider the function $h_{\lambda,v}:\Omega\times{\bf R}\to{\bf R}$ defined by

[TABLE]

for all $(x,\xi)\in\Omega\times{\bf R}$. Clearly, the function $h_{\lambda,v}$ lies in $\hat{\cal A}$ and

[TABLE]

So, the weak solutions of the problem are precisely the critical points in $H^{1}_{0}(\Omega)$ of the functional $\Phi-I_{h_{\lambda,v}}$. Moreover, if $p\in\left]0,{{n+2}\over{n-2}}\right[$ and $q\in\left]0,{{2}\over{n-2}}\right[$ are such that $(3.1)$ and $(3.2)$ hold, for some constant $c_{\lambda,v}$, we have

[TABLE]

for all $u\in H^{1}_{0}(\Omega)$. Therefore, by the Sobolev embedding theorem, for a constant $\tilde{c}_{\lambda,v}$, we have

[TABLE]

for all $u\in H^{1}_{0}(\Omega)$. On the other hand, since $n\geq 4$, one has

[TABLE]

Consequently, from $(3.4)$, we infer that

[TABLE]

Since the functional $\Phi-I_{h_{\lambda,v}}$ is sequentially weakly lower semicontinuous, by the Eberlein-Smulyan theorem and by $(3.5)$, it follows that it is inf-weakly compact.

Now, we are going to apply Theorem 3.C taking $X=H^{1}_{0}(\Omega)$ with the weak topology and $V=Y=L^{2}(\Omega)$ with the strong topology, and $y_{0}=0$. Also, we take

[TABLE]

for all $w\in L^{2}(\Omega)$. Clearly, $\varphi\in{\cal G}$. Furthermore, we take

[TABLE]

for all $u\in H^{1}_{0}(\Omega),v\in L^{2}(\Omega),x\in\Omega$. Clearly, $\Psi(u,v)\in L^{2}(\Omega)$, $\Psi(u,\cdot)$ is a homeomorphism, and we have

[TABLE]

We show that the map $u\to v_{u}$ is not constant in $H^{1}_{0}(\Omega)$. For each $x\in\Omega$, set

[TABLE]

and

[TABLE]

Since $G$ is a Carathéodory is continuous, we have

[TABLE]

and

[TABLE]

and so the functions $\alpha,\beta$ are measurable. Set

[TABLE]

Clearly, we have

[TABLE]

Hence, by assumption, meas$(A)>0$. Then, by the classical Scorza-Dragoni theorem ([6], Theorem 2.5.19), there exists a compact set $K\subset A$, of positive measure, such that the restriction of $G$ to $K\times{\bf R}$ is continuous. Fix a point $\tilde{x}\in K$ such that the intersection of $K$ and any ball centered at $\tilde{x}$ has a positive measure. Next, fix $\xi_{1},\xi_{2}\in{\bf R}$ such that

[TABLE]

By continuity, there is a closed ball $B(\tilde{x},r)$ such that

[TABLE]

for all $x\in K\cap B(\tilde{x},r)$. Finally, consider two functions $u_{1},u_{2}\in H^{1}_{0}(\Omega)$ which are constant in $K\cap B(\tilde{x},r)$. So, we have

[TABLE]

for all $x\in K\cap B(\tilde{x},r)$. Hence, as meas($K\cap B(\tilde{x},r))>0$, we infer that $v_{u_{1}}\neq v_{u_{2}}$, as claimed. As a consequence, $\Psi\in{\cal H}$. Of course, $\varphi(\Psi(u,\cdot))$ is continuous and convex for all $u\in X$. Finally, take

[TABLE]

Clearly, $J\in{\cal M}$. So, for what seen above, all the assumptions of Theorem 3.C are satisfied. Consequently, if we take

[TABLE]

and fix $\lambda>\lambda^{*}$ and a convex set $C\subseteq L^{2}(\Omega)$ whose closure in $L^{2}(\Omega)$ contains the set $\{G(\cdot,u(\cdot)):u\in H^{1}_{0}(\Omega)\}$, there exists $v^{*}\in C$ such that the functional $\Phi-I_{h_{\lambda,v^{*}}}$ has at least two global minima in $H^{1}_{0}(\Omega)$ which are, therefore, weak solutions of the problem. To guarantee the existence of a third solution, denote by $k$ the inverse of the restriction of the function $at+bt^{3}$ to $[0,+\infty[$. Let $T:X\to X$ be the operator defined by

[TABLE]

Since $k$ is continuous and $k(0)=0$, the operator $T$ is continuous in $X$. For each $u\in X\setminus\{0\}$, we have

[TABLE]

In other words, $T$ is a continuous inverse of $\Phi^{\prime}$. Then, since $I_{h_{\lambda,v^{*}}}^{\prime}$ is compact, the functional $\Phi-I_{h_{\lambda,v^{*}}}$ satisfies the Palais-Smale condition ([30], Example 38.25) and hence the existence of a third critical point of the same functional is assured by Corollary 1 of [20].

Finally, assume that the functional $\Phi-I_{f}$ has at least two global minima, say $\hat{u}_{1},\hat{u}_{2}$. Then, the set $D:=\{x\in\Omega:\hat{u}_{1}(x)\neq\hat{u}_{2}(x)\}$ has a positive measure. By assumption, we have

[TABLE]

for all $x\in D$, and so $v_{\hat{u}_{1}}\neq v_{\hat{u}_{2}}$. Then, by definition, we have

[TABLE]

and so $\lambda^{*}=0$ in view of $(3.6)$. $\bigtriangleup$

Notice the following corollary of Theorem 3.1:

COROLLARY 3.1. - Let $n\geq 4$, let $\nu\in{\bf R}$ and let $p\in\left]0,{{n+2}\over{n-2}}\right[$.

Then, for each $\lambda>0$ large enough and for each convex set $C\subseteq L^{2}(\Omega)$ whose closure in $L^{2}(\Omega)$ contains $H^{1}_{0}(\Omega)$, there exists $v^{*}\in C$ such that the problem

[TABLE]

has at least three solutions, two of which are global minima in $H^{1}_{0}(\Omega)$ of the functional

[TABLE]

PROOF. Apply Theorem 3.1 taking $f(x,\xi)=|\xi|^{p-1}\xi$ and $g(x,\xi)=1$. $\bigtriangleup$

REMARK 3.1. - In Theorem 3.1, the assumption made on $g$ (besides $g\in\tilde{\cal A}$) is essential. Indeed, if $g=0$, for $f=0$ (which is an allowed choice), the problem would have the zero solution only.

REMARK 3.2. - The assumption $n\geq 4$ is likewise essential. Indeed, Corollary 3.1 does not hold if $n=3$. To see this, take $p=4$ (which, when $n=3$, is compatible with the condition $p<{{n+2}\over{n-2}}$) and observe that the corresponding energy functional is unbounded below.

Besides Corollary 3.1, among the consequences of Theorem 3.1, we highlight the following

THEOREM 3.2. - Let $n\geq 4$, let $f\in{\cal A}$ and let $g\in\tilde{\cal A}$ be such the set

[TABLE]

has a positive measure. Moreover, assume that, for each $x\in\Omega$, $f(x,\cdot)$ is odd, $g(x,\cdot)$ is even and $G(x,\cdot)$ is strictly monotone.

Then, for each $\lambda>0$, there exists $\mu^{*}>0$ such that, for each $\mu>\mu^{*}$ and for each convex set $C\subseteq L^{2}(\Omega)$ whose closure in $L^{2}(\Omega)$ contains the set $\{G(\cdot,u(\cdot)):u\in H^{1}_{0}(\Omega)\}$, there exists $v^{*}\in C$ such that the problem

[TABLE]

has at least three weak solutions, two of which are global minima in $H^{1}_{0}(\Omega)$ of the functional

[TABLE]

PROOF. Set

[TABLE]

By assumption, meas$(D)>0$. Then, by the Scorza-Dragoni theorem, there exists a compact set $K\subset D$, of positive measure, such that the restriction of $F$ to $K\times{\bf R}$ is continuous. Fix a point $\hat{x}\in K$ such that the intersection of $K$ and any ball centered at $\hat{x}$ has a positive measure. Choose $\hat{\xi}\in{\bf R}$ so that $F(\hat{x},\hat{\xi})>0$. By continuity, there is $r>0$ such that

[TABLE]

for all $x\in K\cap B(\hat{x},r)$. Set

[TABLE]

Since $f\in{\cal A}$, we have $M<+\infty$. Next, choose an open set $\tilde{\Omega}$ such that

[TABLE]

and

[TABLE]

Finally, choose a function $\tilde{u}\in H^{1}_{0}(\Omega)$ such that

[TABLE]

for all $x\in K\cap B(x,r)$,

[TABLE]

for all $x\in\Omega\setminus\tilde{\Omega}$ and

[TABLE]

for all $x\in\Omega$. Thus, we have

[TABLE]

[TABLE]

Now, fix any $\lambda>0$ and set

[TABLE]

Fix $\mu>\mu^{*}$. Hence

[TABLE]

From this, we infer that the functional $\Phi-\mu I_{f}+{{\lambda}\over{2}}I_{Gg}$ possesses at least to global minima since it is even. At this point, we can apply Theorem 3.1 to the functions $g$ and $\mu f-\lambda Gg$. Our current conclusion follows from the one of Theorem 3.1 since we have $\lambda^{*}=0$ and hence we can take the same fixed $\lambda>0$. $\bigtriangleup$

4. A conjecture on disconnectedness versus infinitely many solutions

CONJECTURE 4.1. - Let $\Omega\subset{\bf R}^{n}$ be a smooth bounded domain, with $n\geq 3$. Let $\tau$ be the strongest vector topology on $H^{1}_{0}(\Omega)$.

Then, there exists a continuous function $f:{\bf R}\to{\bf R}$, with

[TABLE]

such that the set

[TABLE]

is disconnected in $(H^{1}_{0}(\Omega),\tau)\times(H^{1}_{0}(\Omega),\tau)$.

The importance of Conjecture 4.1 is shown by Proposition 4.3 below. But, first the relevant theory should be fixed.

The central abstract result, obtained in [21], is as follows (see also [16]):

THEOREM 4.1. - Let $X$ be a connected topological space, let $E$ be a real topological vector space, with topological dual $E^{*}$, and let $A:X\rightarrow E^{*}$ be an operator such that the set

[TABLE]

is dense in $E$ and the set

[TABLE]

is disconnected.

Then, $A$ does vanish at some point of $X$.

PROOF. Denote by $p_{X}$ the projection from $X\times E$ onto $X$. Moreover, for any $C\subseteq X\times E$, $x\in X$, put

[TABLE]

Arguing by contradiction, assume that $A(x)\neq 0$ for all $x\in X$. Denote by $\Gamma$ the set

[TABLE]

Since $\Gamma$ is disconnected, there are two open sets $\Omega_{1},\Omega_{2}\subseteq X\times E$ such that

[TABLE]

We now prove that $p_{X}(\Omega_{1}\cap\Gamma)$ is open in $X$. So, let $(x_{0},y_{0})\in\Omega_{1}\cap\Gamma$. Since $E$ is locally connected ([28], p.35), there are a neighbourhood $U_{0}$ of $x_{0}$ in $X$ and an open connected neighbourhood $V_{0}$ of $y_{0}$ in $E$ such that $U_{0}\times V_{0}\subseteq\Omega_{1}$. Since $\langle A(x_{0}),\cdot\rangle$ is a non-null continuous linear functional, it has no local extrema. Consequently, since $\langle A(x_{0}),y_{0}\rangle=1$, the sets

[TABLE]

[TABLE]

are both non-empty and open. Then, thanks to our density assumption, there are $u_{1},u_{2}\in V_{0}$ such that the set

[TABLE]

is a neighbourhood of $x_{0}$. Then, if $x$ is in this set, due to the connectedness of $V_{0}$, there is some $y\in V_{0}$ such that $\langle A(x),y\rangle=1$, and so, $x$ actually lies in $p_{X}(\Omega_{1}\cap\Gamma)$, as desired. Likewise, it is seen that $p_{X}(\Omega_{2}\cap\Gamma)$ is open. Now, observe that, for any $x\in X$, the set $\{x\}\times\Gamma_{x}$ is non-empty and connected, and so it is contained either in $\Omega_{1}$ or in $\Omega_{2}$. Summarizing, we then have that the sets $p_{X}(\Omega_{1}\cap\Gamma)$ and $p_{X}(\Omega_{2}\cap\Gamma)$ are non-empty, open, disjoint and cover $X$. Hence, $X$ would be disconnected, a contradiction. $\bigtriangleup$

Once Theorem 4.1 has been obtained, we can state the following formally more complete result:

THEOREM 4.2. - Let $X$ be a topological space, let $E$ be a real topological vector space, and let $A:X\rightarrow E^{*}$ be such that the set

[TABLE]

is dense in $E$.

Then, the following assertions are equivalent:

(i)  The set

[TABLE]

is disconnected.

(ii)  The set $X\setminus A^{-1}(0)$ is disconnected.

PROOF. Let (i) hold. Since

[TABLE]

if $X\setminus A^{-1}(0)$ were connected, we could apply Theorem 4.1 to $A_{|(X\setminus A^{-1}(0))}$, and so $A$ would vanish at some point of $X\setminus A^{-1}(0)$, which is absurd.

Conversely, if (ii) holds, then (i) follows at once observing that, with the notations of the proof of Theorem 4.1, one has $X\setminus A^{-1}(0)=p_{X}(\Gamma).$ $\bigtriangleup$

REMARK 4.1. - When $X$ is a connected topological space, $E$ is an infinite-dimensional real vector space (with algebraic dual $E^{\prime}$), and $A:X\rightarrow E^{\prime}$ is a $\sigma(E^{\prime},E)$-continuous operator, one could try to apply Theorem 4.1 endowing $E$ with the strongest vector topology ([15], p.53).

REMARK 4.2. - In Theorem 4.1, the role of the constant $1$ can actually be assumed by any continuous real function on $X$. Precisely, we have the following

PROPOSITION 4.1. - Let $X$ be a topological space, let $E$ be a real topological vector space, and let $A:X\rightarrow E^{\prime}$. Assume that, for some continuous function $\alpha:X\rightarrow{\bf R}$, the set

[TABLE]

is disconnected.

Then, either $A(x)=0$ for some $x\in X$, or the set

[TABLE]

is disconnected.

PROOF. Assume that $A^{-1}(0)=\emptyset$. So, $p_{X}(\Gamma)=X$. Consider the function $f:X\times E\rightarrow X\times E$ defined by putting $f(x,y)=(x,\alpha(x)y)$ for all $(x,y)\in X\times E$. Of course, $f$ is continuous. Arguing by contradiction, assume that $\Gamma$ is connected. Then, $f(\Gamma)$ is connected too. Now, observe that

[TABLE]

Furthermore, note that, if $x\in\alpha^{-1}(0)$, then $(x,0)\in f(\Gamma)\cap(\{x\}\times\Lambda_{x})$, and so $f(\Gamma)\cup(\{x\}\times\Lambda_{x})$ is connected. In turn, the sets $f(\Gamma)\cup(\{x\}\times\Lambda_{x})$ ($x\in\alpha^{-1}(0))$ are clearly pairwise non-disjoint, and hence $\Lambda$ is connected, a contradiction. $\bigtriangleup$

In [22], the following proposition was pointed out:

PROPOSITION 4.2 ([22], Proposition 3). - Let $E$ be an infinite-dimensional Hausdorff topological vector space and $K$ a relatively compact subset of $E$.

Then, the set $E\setminus K$ is connected.

Finally, as said, the following proposition shows the importance of Conjecture 4.1:

PROPOSITION 4.3. - Let $f$ be a function satisfying Conjecture 4.1.

Then, the problem

[TABLE]

has infinitely many weak solutions.

PROOF. Let $X=W^{1,2}_{0}(\Omega)$, with the usual norm $\|u\|=(\int_{\Omega}|\nabla u(x)|^{2}dx)^{1\over 2}$. For $0<q\leq{{n+2}\over{n-2}}$ and $f\in{\cal A}_{q}$, put

[TABLE]

for all $u\in X$.

So, the functional $J$ is of class $C^{1}$ on $X$ and one has

[TABLE]

for all $u,v\in X$. Hence, the critical points of $J$ in $X$ are exactly the weak solutions of the problem. Since $J$ is of class $C^{1}$, clearly the operator $J^{\prime}:X\to X^{*}$ is $\tau$-weakly-star continuous. Hence, by Theorem 4.2, the set $X\setminus(J^{\prime})^{-1}(0)$ is $\tau$-disconnected. Then, due to Proposition 4.2, the set $(J^{\prime})^{-1}(0)$ is not $\tau$-relatively compact, and hence is infinite. $\bigtriangleup$

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Department of Mathematics

University of Catania

Viale A. Doria 6

95125 Catania

Italy

e-mail address: ricceri@dmi.unict.it