Localization of critical points in annular conical sets via the method of Nehari manifold

Andrei Stan Department of Mathematics, Babeș-Bolyai University AND Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

1. Introduction and Preliminaries

The theory of critical points has proved to be a cornerstone in the study of various problems arising from real-world mathematical models. In many cases, the solutions to such problems correspond to a critical point of some $C^{1}$ energy functional $E$ . Specifically, one seeks $u$ such that

E^{\prime}(u)=0,

where $E^{\prime}$ denotes the Fréchet derivative of $E$ . In the upcoming discussion, $H$ denotes a Hilbert space, identified with its dual, equipped with the inner product $(u,v)_{H}$ and the associated norm $|u|_{H}=\sqrt{(u,u)_{H}}$ . Also, $B_{R}$ denotes the closed ball of radius $R$ centered at the origin in $H$ .

1.1. Nehari manifold

A significant advancement in the study of critical points is attributed to the pioneering papers of Nehari (see [13, 14]). The core idea involves minimizing the functional $E$ over the so-called Nehari manifold,

\mathcal{M}=\left\{u\in H\setminus\{0\}\,:\,(E^{\prime}(u),u)_{H}=0\right\},

since in many cases the functional $E$ is bounded on $\mathcal{M}$ , even though it may be unbounded on the entire domain. Moreover, by its definition, all critical points of $E$ are contained within $\mathcal{M}$ . Typically, the Nehari method proceeds as follows,

1)

Prove that the infimum of $E$ over $\mathcal{M}$ is achieved, i.e., there exists $u_{0}\in\mathcal{M}$ such that $E(u_{0})=c=\inf_{\mathcal{N}}E(\cdot)$ .
2)

Establish that $\mathcal{M}$ is homeomorphic to the unit sphere $S^{1}$ in $H$ via a mapping $\omega$ . Then, show that $u$ is a critical point of the mapping $E\circ\omega\colon S^{1}\to\mathbb{R}$ . In the standard approach from the literature, the mapping $\omega$ is obtained as the unique maximizer of the fibering mapping $(0,\infty)\ni t\mapsto E(tu)$ (see, e.g., [6]).
3)

Show that any critical point of $E\circ\omega$ corresponds to a critical point of $E$ .

The point $u_{0}$ is usually called a ground state since it minimizes the functional over the set of all possible solutions. However, as shown in [21, Theorem 4.2], $E(u_{0})$ is just a mountain pass value given by a min-max procedure over a specific set of paths.

In cases where $\mathcal{M}$ is a $C^{1}$ manifold (e.g., when $E$ is a $C^{2}$ functional), the mapping $\omega$ is of $C^{1}$ class, allowing us to work directly on the manifold $\mathcal{M}$ . For further details, we refer the reader to the remarkable monograph by Szulkin and Weth [15]. Additionally, some excellent sources include [2, 9] and the references therein.

1.2. Localization

In addition to establishing the existence of a critical point, one may also ask for its localization or, more generally, aim to identify a critical point within a specific subset. A localization result, particularly with respect to an energetic norm, may be of interest because, for example, when modeling a real process, the parameters of the nonlinear term can be adjusted to obtain a solution whose energy lies within predefined limits. Another motivation for localization arises in nonlinear problems, where the uniqueness of solutions may fail. In such cases, it is often important to focus the analysis on a specific solution, which requires localization.

In general, to achieve localization results for nonlinear problems, one typically employs a combination of inequalities, such as Poincaré’s and Harnack’s inequalities, alongside abstract localization techniques, e.g., Krasnoselskii-type methods or approaches that use the properties of topological degree.

One of the earliest results on the localization of critical points can be found in Schechter [20, Chapter 5.3], which provides sufficient conditions to ensure the existence of a minimizing sequence of a functional within a given ball. Under the assumption of the Palais-Smale condition, this minimizing sequence has a convergent subsequence whose limit is a critical point.

Theorem 1.1 (Schechter’s Theorem).

Assume $E$ is bounded from below on $B_{R}$ ( $R>0$ ) and the following Leray-Schauder boundary condition holds

E^{\prime}(u)+\mu u\neq 0\,\,\text{ for all }u\in\partial B_{R}\text{ and }\mu% >0.

Then, there exists a sequence $\{u_{n}\}\subset B_{R}$ such that

\text{$E(u_{n})\to\inf_{B_{R}}E$ \quad and \quad$E^{\prime}(u_{n})\to 0$}.

In [17, Theorem 3.1], a result analogous to Schecter’s theorem was established for the set $\overline{B_{R}\setminus B_{r}}$ ( $0<r<R$ ), while in [16, Theorem 2.3], a similar result was obtained for the conical set

K_{r,R}=\left\{u\in K\,:\,r\leq|u|_{1}\text{ and }|u|_{2}\leq R\right\}\quad(0% <r<R<\infty),

where $|\cdot|_{i}$ ( $i=1,2$ ) are two norms on $H$ , and $K$ is a wedge, i.e., $K$ is a closed convex subset of $H$ satisfying $\lambda K\subset K$ for all $\lambda>0$ . Further results on the localization of critical points can be found in [1, 11, 19, 10].

1.3. Ekeland variational principle

The proof of our main result is essentially based on the weak form of Ekeland’s variational principle (see, e.g., [7, 8]).

Lemma 1.2 (Ekeland Principle - weak form).

Let $(X,d)$ be a complete metric space and let $\Phi:X\rightarrow\mathbb{R}\cup\{+\infty\}$ be a lower semicontinuous and bounded from below functional. Then, given any $\varepsilon>0$ , there exists $u_{\varepsilon}\in X$ such that

\Phi(u_{\varepsilon})\leq\inf_{X}\Phi+\varepsilon,

and

\Phi(u_{\varepsilon})\leq\Phi(u)+\varepsilon d(u,u_{\varepsilon}),

for all $u\in X.$

1.4. Implicit function theorem

In our analysis, we employ the following variant of the implicit function theorem. For another use of the implicit function theorem in the context of Nehari manifolds, see [3, Proposition 4.2].

Theorem 1.3.

Let $A,B$ be open sets in $\mathbb{R}$ , and let $x_{0}\in\text{Int }A$ , $y_{0}\in\text{Int }B$ . Suppose that $\mathcal{F}(y,x)\colon B\times A\to\mathbb{R}$ is a function that satisfies the following conditions:

(1)

$\mathcal{F}(y_{0},x_{0})=0$ ;
(2)

The function $\mathcal{F}(y,x)$ is continuously differentiable with respect to both variables in a neighborhood of $(y_{0},x_{0})$ ;
(3)

$\mathcal{F}_{y}(y_{0},x_{0})\neq 0$ ;

Then:

(a)

There exist neighborhoods $U_{0}\subset A$ of $x_{0}$ and $V_{0}\subset B$ of $y_{0}$ , as well as a unique mapping $y=\xi(x)\colon U_{0}\to V_{0}$ such that:

\xi(x_{0})=y_{0}\quad\text{and}\quad\mathcal{F}(\xi(x),x)=0\quad\text{for all % }x\in U_{0};

(b)

The mapping $\xi$ is continuously differentiable on $U_{0}$ and moreover satisfies

\xi^{\prime}(x)=-\frac{\mathcal{F}_{x}(\xi(x),x)}{\mathcal{F}_{y}(\xi(x),x)}\,% \,\text{ for all }x\in U_{0}.

2. Main Result

Let $E:H\to\mathbb{R}$ be a twice Fréchet differentiable functional, and let $K$ be a nondegenerate cone in $H$ , i.e.,

K\setminus\left\{0\right\}\neq\emptyset

\mathbb{R}_{+}K\subset K

and

K+K\subset K

The second derivative of $E$ at $x$ in the direction $y$ on the point $z$ is (see, e.g., [12]):

E^{\prime\prime}(x)(z,y)=\lim_{t\searrow 0}\frac{1}{t}\left(E^{\prime}(x+ty)-E% ^{\prime}(x),z\right)_{H}.

Throughout this paper, we assume that the operator $N\colon H\to H$ given by

N(u)=u-E^{\prime}(u)\,\,\text{ for all }u\in H,

is invariant over $K$ , i.e., $N(K)\subset K$ (recall that $H$ is identified with its duals, so $E^{\prime}:H\to H$ ).

Our aim is to determine a critical point of $E$ within the conical set

K_{r,R}=\left\{u\in K\setminus\{0\}\,:\,r\leq|u|_{H}\leq R\right\},

where $0<r<R<\infty$ are some given real numbers. The main assumption we consider on the functional $E$ is the following:

(h1)::

For each $u\in K\setminus\{0\}$ , there exists a unique $s(u)\in\left(\frac{r}{|u|_{H}},\frac{R}{|u|_{H}}\right)$ such that the mapping

\tau\mapsto\left(E^{\prime}(\tau u),u\right)_{H}\quad(\tau>0),

is strictly positive on $\left[\frac{r}{|u|_{H}},s(u)\right)$ and strictly negative on $\left(s(u),\frac{R}{|u|_{H}}\right]$ .

From this, it follows that the mapping $\alpha_{u}(\tau)=E(\tau u)$ , for some $u\in K\setminus\{0\}$ , has a unique critical point within the interval $\left(\frac{r}{|u|_{H}},\frac{R}{|u|_{H}}\right)$ at $\tau=s(u)$ , and is concave at this point. Thus, given the smoothness of the functional $E$ , these properties are characterized by

\alpha_{u}^{\prime}(s(u))=0\quad\text{and}\quad\alpha_{u}^{\prime\prime}(s(u))% \leq 0.

Moreover, we see that $s(ku)=\frac{s(u)}{k}$ for all $k>0$ .

Following the method of Nehari manifold, we look for critical points of $E$ on $K_{r,R}$ within the conical shell

\tilde{\mathcal{N}}=\left\{u\in K_{r,R}\,:\,\left(E^{\prime}(u),u\right)_{H}=0% \right\}.

Further, let $\mathcal{N}$ be the set

\mathcal{N}=\left\{s(u)u\,:\,u\in K\setminus\{0\}\right\},

and observe that $\mathcal{N}=\tilde{\mathcal{N}}$ . To see this, from (h1) one clearly has $\mathcal{N}\subset\tilde{\mathcal{N}}.$ Conversely, if $(E^{\prime}(u),u)_{H}=0$ for some $u\in K\setminus\{0\}$ , then $s(u)=1$ , which implies $u\in\mathcal{N}$ .

The main result of this paper is presented in Theorem 2.2 below, where we establish an analogue of the results presented in [16], obtained using the method of Nehari manifold.

Remark 2.1.

We mention that in [9], the existence of a ground state for a $C^{1}$ functional was established over an open cone in Banach spaces. However, our result do not require this condition, and in applications, the cones often have empty interior (as the one from Section 3).

Theorem 2.2.

Assume condition (h1) holds. In addition, we suppose that

(h2)::

The functional $E$ is bounded from below on $\mathcal{N}$ .

(h3)::

The second Fréchet derivative $E^{\prime\prime}(u)$ is bounded uniformly with respect to $u\in\mathcal{N}$ , i.e., there exists $C_{1}>0$ such that

\sup_{\begin{subarray}{c}w_{1},w_{2}\in H\\ |w_{1}|_{H}=|w_{2}|_{H}=1\end{subarray}}\left|E^{\prime\prime}(u)(w_{1},w_{2})% \right|\leq C_{1}\,\,\text{ for all }u\in\mathcal{N}.

(h4)::

There is a positive constant $C_{2}>0$ satisfying

|E^{\prime\prime}(u)(u,u)|\geq C_{2}>0\,\,\text{ for all }u\in\mathcal{N}.

Then, there exists a sequence $\{u_{n}\}\subset\mathcal{N}$ such that

E(u_{n})\to\inf_{\mathcal{N}}E\quad\text{ and }\quad E^{\prime}(u_{n})\to 0.

The following auxiliary result will be of great importance in proving Theorem 2.2.

Lemma 2.3.

Let $u\in\mathcal{N}$ , $v\in H$ and let $\varepsilon>0$ be such that

u+tv\in K\,\,\text{ for all $t\in[0,\varepsilon]$. }

If assumption (h4) holds true, then the limit

\lim_{t\searrow 0}\frac{s(u+tv)-s(u)}{t}

exists and has the value

-\frac{E^{\prime\prime}(u)(u,v)+(E^{\prime}(u),v)_{H}}{E^{\prime\prime}(u)(u,u% )}.

Proof.

Let us consider the mapping $g\colon(0,\infty)\times(-\varepsilon,\varepsilon)\to\mathbb{R}$ given by

g(\tau,t)=\left(E^{\prime}(\tau(u+tv)),u+tv\right)_{H}.

Note that $g$ is continuously differentiable, with partial derivatives

(2.1)		$\displaystyle g^{\prime}_{\tau}(\tau,t)$	$\displaystyle=\lim_{\delta\searrow 0}\frac{1}{\delta}\left(E^{\prime}(\tau(u+% tv)+\delta(u+tv))-E^{\prime}(\tau(u+tv)),u+tv\right)_{H}$
		$\displaystyle=E^{\prime\prime}(\tau(u+tv))(u+tv,u+tv),$

and

(2.2)	$\displaystyle g^{\prime}_{t}(\tau,t)$	$\displaystyle=\lim_{\delta\searrow 0}\left(\frac{1}{\delta}\left(E^{\prime}(% \tau(u+tv)+\delta\tau v),u+tv+\delta v\right)_{H}-\left(E^{\prime}(\tau(u+tv))% ,u+tv\right)_{H}\right)$
	$\displaystyle=\left(E^{\prime}(\tau(u+tv),v\right)_{H}+\lim_{\delta\searrow 0}% \frac{1}{\delta}\left(E^{\prime}(\tau(u+tv)+\delta\tau v)-E^{\prime}(\tau(u+tv% )),u+tv\right)_{H}$
	$\displaystyle=\left(E^{\prime}(\tau(u+tv),v\right)_{H}+E^{\prime\prime}(\tau(u% +tv))(u+tv,\tau v).$

Additionally, we have

g(1,0)=0\quad\text{ and }\quad g^{\prime}_{\tau}(1,0)<0.

Indeed, the first relation follows immediately since $s(u)=1$ , while the second one follows from (2.1) and (h4).

Employing the implicit function theorem, there exists $\varepsilon_{0}>0$ and a unique continuously differentiable mapping $\xi\colon(-\varepsilon_{0},\varepsilon_{0})\to\mathbb{R}$ such that $\xi(0)=1$ ,

g(\xi(t),t)=0\quad\text{and}\quad\xi^{\prime}(t)=-\frac{g^{\prime}_{t}(\xi(t),% t)}{g^{\prime}_{\tau}(\xi(t),t)}\,\,\text{ for all }t\in(-\varepsilon_{0},% \varepsilon_{0}).

On the other hand, for each $t\in[0,\varepsilon]$ , since $u+tv\in K$ , assumption $(h1)$ guarantees the existence of a unique value $s(u+tv)$ within the interval $\left(\frac{r}{|u+tv|_{H}},\frac{R}{|u+tv|_{H}}\right)$ satisfying

(E^{\prime}(s(u+tv)(u+tv)),u+tv)_{H}=0,\quad\text{ i.e., }\quad g(s(u+tv),t)=0.

Since $\xi$ is smooth and $\xi(0)=1\in\left(\frac{r}{|u|_{H}},\frac{R}{|u|_{H}}\right)$ , there exists $\varepsilon_{1}>0$ such that

\xi(t)\in\left(\frac{r}{|u+tv|_{H}},\frac{R}{|u+tv|_{H}}\right)\,\,\text{ for % all $t\in(-\varepsilon_{1},\varepsilon_{1})$. }

Consequently, we have

\xi(t)=s(u+tv)\,\,\text{ for all }t\in[0,\varepsilon_{2}),

where $\varepsilon_{2}=\min\{\varepsilon,\varepsilon_{0},\varepsilon_{1}\}$ . Whence

\lim_{t\searrow 0}\frac{s(u+tv)-s(u)}{t}=\xi^{\prime}(0)=-\frac{g^{\prime}_{t}% (1,0)}{g^{\prime}_{\tau}(1,0)}.

Now, taking $\tau=1$ and $t=0$ in (2.1) and (2.2), we derive the conclusion

\lim_{t\searrow 0}\frac{s(u+tv)-s(u)}{t}=-\frac{E^{\prime\prime}(u)(u,v)+(E^{% \prime}(u),v)_{H}}{E^{\prime\prime}(u)(u,u)}.

∎

Proof of Theorem 2.2.

From assumption (h2), $E$ is bounded from below on $\mathcal{N}$ . Moreover, since $\mathcal{N}$ is closed, we may apply Ekeland’s variational principle to the functional $E$ on the set $\mathcal{N}$ . This guarantees the existence of a sequence $\{u_{n}\}\subset\mathcal{N}$ such that

(2.3)		$\displaystyle E(u_{n})$	$\displaystyle\leq\inf_{\mathcal{N}}E+\frac{1}{n},$
(2.4)		$\displaystyle E(u_{n})$	$\displaystyle\leq E(u)+\frac{1}{n}\|u_{n}-u\|_{H}\,\,\text{for all }u\in\mathcal% {N}.$

For each $n\in\mathbb{N}$ , consider the mapping $\varphi\colon\mathbb{R}\to H$ given by

\varphi(t)=u_{n}-tE^{\prime}(u_{n}).

Clearly, $\varphi$ is continuously differentiable with $\varphi^{\prime}(0)=-E^{\prime}(u_{n})$ . Moreover, we have $\varphi(t)\in K$ for all $t\in[0,1]$ . Indeed, since both $u_{n}\in K$ and $N(u_{n})\in K$ , one has

u_{n}-tE^{\prime}(u_{n})=(1-t)u_{n}+t\left(u_{n}-E^{\prime}(u_{n})\right)=(1-t% )u_{n}+tN(u_{n})\in K.

Next, we define

\psi(t)=s(\varphi(t))\varphi(t)\in\mathcal{N}\,\,\text{ for all }t\in[0,1].

Choosing $u=\psi(t)$ in (2.4) yields

(2.5)

E(\psi(0))\leq E(\psi(t))+\frac{1}{n}|\psi(t)-\psi(0)|_{H}\,\,\text{ for all }% t\in[0,1].

By Lemma 2.3, the mapping $t\mapsto s(\varphi(t))$ has a right derivative at zero given by

z_{n}=\frac{E^{\prime\prime}(u_{n})(u_{n},-E^{\prime}(u_{n}))-|E^{\prime}(u_{n% })|_{H}^{2}}{E^{\prime\prime}(u_{n})(u_{n},u_{n})}.

Thus, since $\varphi(0)=u_{n}$ , when differentiating, we obtain

(2.6)

\psi^{\prime}_{+}(0)=\lim_{t\searrow 0}\frac{\psi(t)-\psi(0)}{t}=z_{n}\,u_{n}-% E^{\prime}(u_{n}).

Regarding $z_{n}$ , from assumption (h3) and (h4), one has

(2.7)

|z_{n}|\leq\frac{C_{1}}{C_{2}}|u_{n}|_{H}|E^{\prime}(u_{n})|_{H}+\frac{1}{C_{2% }}|E^{\prime}(u_{n})|^{2}_{H}.

Now, dividing relation (2.5) by $t>0$ gives

(2.8)

\displaystyle\frac{E(\psi(0))-E(\psi(t))}{t}\leq\frac{1}{n}\left|\frac{\psi(t)% -\psi(0)}{t}\right|_{H}.

Taking the limit as $t\searrow 0$ , from (2.6) and $\left(E^{\prime}(u_{n}),u_{n}\right)_{H}=0$ , the left-hand side of (2.8) becomes

(2.9)	$\displaystyle\lim_{t\searrow 0}\frac{E(\psi(0))-E(\psi(t))}{t}$	$\displaystyle=-\left(E^{\prime}(\psi(0)),\psi^{\prime}_{+}(0)\right)_{H}$
	$\displaystyle=-\left(E^{\prime}(u_{n}),z_{n}u_{n}-E^{\prime}(u_{n})\right)_{H}$
	$\displaystyle=\|E^{\prime}(u_{n})\|^{2}_{H}.$

For the right-hand side of (2.8), using (2.6) and (2.7), we obtain

(2.10)

\displaystyle\lim_{t\searrow 0}\frac{1}{n}\left|\frac{\psi(t)-\psi(0)}{t}% \right|_{H}\leq\frac{C}{n}\left(|u_{n}|_{H}|E^{\prime}(u_{n})|_{H}+|E^{\prime}% (u_{n})|^{2}_{H}\right).

Consequently, from (2.8), (2.9) and (2.10), we obtain

|E^{\prime}(u_{n})|_{H}\leq\frac{C}{n}\left(|u_{n}|_{H}+|E^{\prime}(u_{n})|_{H% }\right)\leq\frac{C}{n}\left(R+|E^{\prime}(u_{n})|_{H}\right),

which ensures that $E^{\prime}(u_{n})\to 0$ as $n\to\infty$ .

∎

If we further assume a compactness condition, Theorem 2.2 leads to the following critical point principle.

Theorem 2.4.

Assume (h1)-(h4) hold true. If in addition the operator $N$ given in (2) is completely continuous, then there exists $u^{\ast}\in\mathcal{N}$ such that

E(u^{\ast})=\inf_{\mathcal{N}}E\quad\text{ and }\quad E^{\prime}(u^{\ast})=0.

Proof.

From assumptions (h1)-(h4), Theorem 2.2 guarantees the existence of a sequence $\{u_{n}\}\subset\mathcal{N}$ such that

(2.11)

E(u_{n})\leq\inf_{\mathcal{N}}E+\frac{1}{n}\quad\text{ and }\quad E^{\prime}(u% _{n})\to 0.

Since $N$ is completely continuous and $\{u_{n}\}$ is bounded (recall that $\mathcal{N}\subset K_{r,R}$ ), it follows that, after possibly passing to a subsequence, the sequence $\{N(u_{n})\}$ is convergent to some $u^{\ast}\in H$ . Now, taking the limit in (2.11), one has that $u_{n}\to u^{\ast}$ , hence $u^{\ast}\in\mathcal{N}$ , $E(u^{\ast})=\inf_{\mathcal{N}}E$ and $E^{\prime}(u^{\ast})=0$ . ∎

Finally, we note that Theorem 2.4 immediately yields multiplicity results if the hypotheses are satisfied for several finite or infinitely many pairs of numbers.

Remark 2.5 (Multiplicity).

Let $\{r_{i}\}_{1\leq i\leq m}$ and $\{R_{i}\}_{1\leq i\leq m}$ be two sequences of positive real numbers satisfying

0<r_{1}<R_{1}<r_{2}<R_{2}<\dots<r_{m}<R_{m}<\infty.

For each pair $(r_{i},R_{i})$ , we assume that condition $(h1)$ holds with $s_{i}:=s$ . Additionally, suppose that conditions $(h2)$ – $(h4)$ hold for each Nehari manifold

\mathcal{N}_{i}=\left\{u\in K\setminus\{0\}\,:\,s_{i}(u)u\right\},\quad i=1,% \dots,m,

and that the operator $N$ is completely continuous. Then, for each $i=1,\dots,m$ , there exists $u^{\ast}_{i}\in\mathcal{N}_{i}$ such that

E(u^{\ast}_{i})=\inf_{u\in\mathcal{N}_{i}}E(u)\quad\text{and}\quad E^{\prime}(% u^{\ast}_{i})=0.

3. Application

In this section, we present an application of Theorem 2.4. We aim to obtain a symmetric and positive solution for the Dirichlet problem

(3.1)

\begin{cases}-u^{\prime\prime}\left(t\right)=g(t)f\left(u\left(t\right)\right)% \,\,\text{ on }\left[0,1\right],\\ u\left(0\right)=u\left(1\right)=0,\end{cases}

where $f\colon\mathbb{R}\to\mathbb{R}$ is continuously differentiable. In addition, we suppose that $f$ is nondecreasing and positive on $\mathbb{R}_{+}$ . The function $g\colon[0,1]\to\mathbb{R}_{+}$ is assumed to be bounded, nondecreasing on $[0,1/2]$ and symmetric with respect to the middle of the interval, i.e.,

g(t)=g(1-t)\quad\text{ for all }t\in[0,1/2].

Note that, since $g$ is nondecreasing and bounded on $[0,1/2]$ , it is measurable on this interval, and by symmetry, the same holds on the entire interval $[0,1]$ .
Consider the Sobolev space $H=H_{0}^{1}(0,1)$ endowed with the inner product

(u,v)_{H_{0}^{1}}=\int_{0}^{1}u^{\prime}(t)v^{\prime}(t)dt,

and the energetic norm

|u|_{H_{0}^{1}}=\sqrt{\left(u,u\right)_{H_{0}^{1}}}.

We identify the space $H_{0}^{1}(0,1)$ with its dual $H^{-1}(0,1)$ via the mapping

J\colon H_{0}^{1}(0,1)\to H^{-1}(0,1),\quad Ju=-u^{\prime\prime}.

Clearly, $J$ is invertible and its inverse $J^{-1}v$ is the weak solution of the Dirichlet problem $-u^{\prime\prime}(t)=v,\,u(0)=u(1)=0$ . If $v\in L^{2}(0,1)$ , then

(J^{-1}v)(t)=\int_{0}^{1}G(t,s)v(s)ds,

where $G$ is the Green’s function of the differential operator $J$ with respect to the boundary conditions $u(0)=u(1)=0$ (see, e.g., A. Cabada [5, Example 1.8.18]),

G(t,s)=\begin{cases}s\left(1-t\right),s\leq t\\ t\left(1-s\right),s\geq t.\end{cases}

Moreover, given the continuous embedding of $H_{0}^{1}(0,1)$ in $C[0,1]$ , one has

(3.2)

\sup_{t\in[0,1]}|u(t)|\leq|u|_{H_{0}^{1}}\,\,\text{ for all }u\in H_{0}^{1}(0,% 1).

Additionally, the Wirtinger inequality holds [4],

(3.3)

\int_{0}^{1}u^{2}(t)dt\leq\frac{1}{\pi^{2}}|u|_{H_{0}^{1}}^{2}.

The energy functional of the problem (3.1) is $E\colon H_{0}^{1}(0,1)\to\mathbb{R},$

E\left(u\right)=\frac{1}{2}\left|u\right|_{H_{0}^{1}}^{2}-\int_{0}^{1}F\left(u% (t)\right)g(t)dt,

where $F\left(\xi\right)=\int_{0}^{\xi}f\left(s\right)ds.$ Clearly, the smoothness of $f$ implies that $E$ is a $C^{2}$ functional. Its first derivative is given by

E^{\prime}(u)=u-N(u)\quad\left(u\in H_{0}^{1}(0,1)\right),

where

N(u)(t)=\int_{0}^{1}G(t,s)f(u(s))g(s)ds\quad(u\in H_{0}^{1}(0,1)),

while the second derivative is expressed by

E^{\prime\prime}(u)(w_{1},w_{2})=\left(w_{1},w_{2}\right)_{H_{0}^{1}}-\int_{0}% ^{1}f^{\prime}(u(t))g(t)w_{1}(t)w_{2}(t)dt,

for $u,w_{1},w_{2}\in H_{0}^{1}(0,1).$

In $H_{0}^{1}(0,1)$ , we consider the cone

	$\displaystyle K=\Big{\{}u\in H_{0}^{1}(0,1)\,:$	$\displaystyle u\text{ is nondecreasing on }[0,1/2],$
		$\displaystyle u(t)=u(1-t)\text{ and }u(t)\geq\phi(t)\|u\|_{H_{0}^{1}}\text{ for % all }t\in[0,1/2]\Big{\}},$

where

\phi\colon[0,1/2]\to\mathbb{R},\text{ \quad$\phi(t)=t(1-2t)$.}

Note that the function $u(t)=\sin(\pi t)$ belongs to $K$ , so the cone $K$ is nondegenerate.

We claim that operator $N$ is invariant with respect to $K$ , i.e.,

N(K)\subset K.

An important result in proving our claim is the following Harnack-type inequality obtained in [11, Lemma 3.1].

Lemma 3.1.

For every function $u\in K$ , with $Ju\in C([0,1];\mathbb{R}^{+})$ nondecreasing on $[0,1/2]$ , one has

(3.4)

u(t)\geq\phi(t)|u|_{H_{0}^{1}}\quad\text{for all }t\in[0,1/2].

Let $u\in K$ . From the definition of $N$ , one has

J(N(u))=g(\cdot)f(u)\in C[0,1].

Simple computations show that the Green’s function satisfies

G(t,s)=G(1-t,1-s)\quad\text{ and }\quad G(t,1-s)=G(1-t,s),\,

for all $t,s\in[0,1/2]$ . Thus, from the symmetry of $u$ , for any $t\in[0,1/2]$ , we obtain

	$\displaystyle N(u)(t)$	$\displaystyle=\int_{0}^{1/2}G(t,s)f(u(1-s))g(1-s)ds+\int_{0}^{1/2}G(t,1-s)f(u(% s))g(s)ds$
		$\displaystyle=\int_{1/2}^{1}G(1-t,s)f(u(s))g(s)ds+\int_{0}^{1/2}G(1-t,s)f(u(s)% )g(s)ds$
		$\displaystyle=N(u(1-t)),$

which proves that $N(u)$ is symmetric. Since $u$ takes positive values and both $f$ and $g$ are nonnegative, it follows that $N(u)$ is concave, hence $N(u)$ is nondecreasing on $[0,1/2]$ . Furthermore, the monotonicity of $f$ on $\mathbb{R}_{+}$ and of $u$ and $g$ on $[0,1/2]$ ensures that $J(N(u))=g(\cdot)f(u)$ is also nondecreasing on $[0,1/2]$ . Therefore, by Lemma 3.1, inequality (3.4) holds for $N(u)$ . Thus, $N(u)\in K$ , as desired.

For the sake of completeness, below we provide a proof of Lemma 3.1 shorter than the one given in [11].

Proof of Lemma 3.1.

From the hypothesis, it follows directly that both $u$ and $u^{\prime}$ are positive and concave on $[0,1/2]$ . Moreover, the symmetry of $u$ ensures that $u$ is increasing on $[0,1/2]$ , while $u^{\prime}$ is decreasing, with $u^{\prime}(1/2)=0$ . For any $t\in(0,1/2)$ , the monotonicity of $u^{\prime}$ implies

(3.5)

u(t)=\int_{0}^{t}u^{\prime}(s)ds\geq tu^{\prime}(t),

while its concavity yields

(3.6)		$\displaystyle u^{\prime}(t)$	$\displaystyle=u^{\prime}\left(2t\,\frac{1}{2}+(1-2t)0\right)\geq 2tu^{\prime}% \left(\frac{1}{2}\right)+(1-2t)u^{\prime}(0)$
		$\displaystyle=(1-2t)u^{\prime}(0).$

Finally, we obtain the conclusion using (3.5), (3.6) and

|u|_{H_{0}^{1}}^{2}=\int_{0}^{1}u^{\prime}(s)^{2}ds=2\int_{0}^{1/2}u^{\prime}(% s)^{2}ds\leq u^{\prime}(0)^{2}.

∎

Let $0<r<R<\infty$ be positive real numbers, $\beta\in(0,1/4)$ , and define

\tilde{A}=\left(\int_{0}^{1}g^{2}(t)dt\right)^{1/2},\quad\tilde{B}=\int_{0}^{% \beta}g(t)dt\quad\text{ and }\quad\tilde{C}=\int_{\beta}^{1/2}g(t)dt.

In what follows, we assume

(H1):

The constants $\tilde{A}$ and $\tilde{C}$ are strictly positive, and moreover

\frac{f(r)}{r}<\frac{\pi}{\tilde{A}}\quad\text{and}\quad\frac{f\left(R\phi(% \beta)\right)}{R}>\frac{1}{2\phi(\beta)\tilde{C}}.

Additionally, suppose that the function $f$ satisfies one of the following three conditions:

(H2):

There exists a continuous mapping $\theta\colon[0,R]\to\mathbb{R}$ such that $\theta(t)>0$ for $t\in(0,R]$ , and

tf^{\prime}(t)-f(t)\geq\theta(t)\,\,\text{ for all }t\in[0,R];

(H3):

There exists constants $\mu=\mu(r,R)>1$ and $\lambda=\lambda(r,R)>0$ such that

tf^{\prime}(t)-\mu f(t)\geq 0\text{ and }f^{\prime}(t)\geq\lambda\,\,\text{ % for all }t\in[r\phi(\beta),R],

and

(3.7)

\tilde{B}f(r\phi(\beta))<\lambda\tilde{C}\left(1-\frac{1}{\mu}\right)r\phi(% \beta);

(H4):

The support of the function $g$ is included in $[\beta,1/2]$ , i.e.,

(3.8)

g(t)=0\quad\text{ for all }t\in[0,\beta].

Additionally, the function $f$ is of class $C^{2}$ on $[r\phi(\beta),R]$ with a strictly positive second derivative, that is, there exists $M=M(r,R)>0$ with

f^{\prime\prime}(t)\geq M\quad\text{ for all }t\in[r\phi(\beta),R].

Furthermore, there exists a continuous positive mapping $\tilde{\theta}\colon[r\phi(\beta),R]\to\mathbb{R}$ such that

(3.9)

tf^{\prime}(t)-f(t)\geq\tilde{\theta}(t)\quad\text{ for all }t\in[r\phi(\beta)% ,R].

We show that under condition (H1) and either (H2), (H3) or (H4), conditions (h1)-(h4) are satisfied. Note that (h2) and (h3) follow directly from (3.2), Wirtinger’s inequality, and the smoothness of the function $f$ .

To prove (h1), let $u\in K\setminus\{0\}$ . The mapping $\alpha_{u}$ introduced in (h1) is given by

\alpha_{u}(\tau)=\frac{1}{2}\tau^{2}|u|_{H_{0}^{1}}^{2}-\int_{0}^{1}F(\tau u(t% ))g(t)dt,\quad\tau\in\left(\frac{r}{|u|_{H_{0}^{1}}},\frac{R}{|u|_{H_{0}^{1}}}\right)

and its derivative is

\displaystyle\alpha_{u}^{\prime}(\tau)

\displaystyle=\tau|u|_{H_{0}^{1}}^{2}-\int_{0}^{1}f(\tau u(t))g(t)u(t)dt.

First, we show that

(3.10)

\text{$\alpha_{u}^{\prime}\left(\frac{r}{|u|_{H_{0}^{1}}}\right)>0$ \quad and % \quad$\alpha_{u}^{\prime}\left(\frac{R}{|u|_{H_{0}^{1}}}\right)<0$}.

Denote

v(t)=\frac{u(t)}{|u|_{H_{0}^{1}}}\in K\setminus\{0\},

so $|v|_{H_{0}^{1}}=1$ . Then, we have

\displaystyle\alpha_{u}^{\prime}\left(\frac{\tau}{|u|_{H_{0}^{1}}}\right)=\tau% |u|_{H_{0}^{1}}\left(1-\int_{0}^{1}\frac{1}{\tau}f\left(\tau v(t)\right)g(t)v(% t)dt\right).

From (3.2), (H1) and both Wirtinger and Hölder inequalities, one has

	$\displaystyle\int_{0}^{1}\frac{1}{r}f\left(rv(t)\right)g(t)v(t)dt\leq\int_{0}^% {1}\frac{f\left(r\right)}{r}g(t)v(t)dt$
	$\displaystyle\leq\frac{f(r)}{r}\left(\int_{0}^{1}g^{2}(t)dt\right)^{1/2}\left(% \int_{0}^{1}v^{2}(t)dt\right)^{1/2}$
	$\displaystyle\leq\frac{f(r)}{r}\frac{\tilde{A}}{\pi}\|v\|_{H_{0}^{1}}=\frac{f(r)% }{r}\frac{\tilde{A}}{\pi}<1,$

which proves the first inequality in (3.10). For the second one, using the monotonicity of $v$ and the Harnack inequality, we see that

(3.11)

v(t)\geq\phi(\beta)\quad\text{ for all }t\in[\beta,1/2].

Therefore, from (H1), the symmetry of $v$ and (3.11), we obtain

	$\displaystyle\int_{0}^{1}\frac{f\left(Rv(t)\right)}{R}g(t)v(t)dt$	$\displaystyle=2\int_{0}^{1/2}\frac{f(Rv(t))}{R}g(t)v(t)dt\geq 2\int_{\beta}^{1% /2}\frac{f(Rv(t))}{R}g(t)v(t)dt$
		$\displaystyle\geq 2\int_{\beta}^{1/2}\frac{f(R\phi(t))}{R}g(t)\phi(t)dt\geq 2% \phi(\beta)\tilde{C}\frac{f\left(R\phi(\beta)\right)}{R}>1,$

whence relation (3.10) holds.

Define $\sigma=\tau|u|_{H_{0}^{1}}$ and the functions $h,\tilde{h}\colon[r,R]\to\mathbb{R}$ ,

	$\displaystyle h(\sigma)=1-\int_{0}^{1}\frac{f(\sigma v(t))}{\sigma}g(t)v(t)dt,$
	$\displaystyle\tilde{h}(\sigma)=\sigma-\int_{0}^{1}f(\sigma v(t))g(t)v(t)dt.$

Clearly, $h$ and $\tilde{h}$ are dependent on the chosen $u$ , so one should write $h_{u}$ and $\tilde{h}_{u}$ , respectively. However, for simplicity, we omit the subscript in the following analysis, as the dependence on $u$ is clear from the context.

We now show that the function $h$ is decreasing under either (H2) or (H3), while the function $\tilde{h}$ is concave under (H4).

(a) In case (H2) holds, the mapping

t\mapsto\frac{f(t)}{t}\,\text{ is increasing on }(0,R].

The conclusion now follows directly, as $\sigma v(t)\in[0,R]$ for all $t\in[0,1]$ and $\sigma\in[r,R]$ .

(b) Under condition (H3), we prove that $h^{\prime}(\sigma)<0$ for all $\sigma\in[r,R]$ . Using the symmetry of $v$ , a straightforward computation yields

h^{\prime}(\sigma)=\frac{2}{\sigma^{2}}\int_{0}^{1/2}\left(f(\sigma v(t))-f^{% \prime}(\sigma v(t))\sigma v(t)\right)g(t)v(t)dt.

Let $\mathcal{B}$ denote the set

\mathcal{B}=\{t\in[0,1/2]\,:\,\sigma v(t)\leq r\phi(\beta)\}.

From (3.11), it follows that

(3.12)

\mathcal{B}\subset[0,\beta]\quad\text{ and }\quad[\beta,1/2]\subset[0,1/2]% \setminus\mathcal{B}.

Since the derivative of $f$ is nonnegative, one has

\displaystyle\frac{2}{\sigma^{2}}\int_{\mathcal{B}}\left(f(\sigma v(t))-f^{% \prime}(\sigma v(t))\sigma v(t)\right)g(t)v(t)dt

\displaystyle\leq\frac{2}{\sigma^{2}}\int_{\mathcal{B}}f(\sigma v(t))g(t)v(t)dt.

Now, using the monotonicity of $v$ and $f$ , along with (3.11) and (3.12), we obtain

(3.13)

\frac{2}{\sigma^{2}}\int_{\mathcal{B}}f(\sigma v(t))g(t)v(t)dt=\frac{2}{\sigma% ^{3}}\int_{\mathcal{B}}f(\sigma v(t))g(t)\sigma v(t)dt\leq\frac{2\tilde{B}}{% \sigma^{3}}f(r\phi(\beta))r\phi(\beta).

On the other hand, from (H3), since $\sigma v(t)\in[r\phi(\beta),R]$ for all $t\in[0,1/2]\setminus\mathcal{B}$ , we derive

(3.14)		$\displaystyle\frac{2}{\sigma^{2}}\int_{[0,1/2]\setminus\mathcal{B}}\left(f^{% \prime}(\sigma v(t))\sigma v(t)-f(\sigma v(t))\right)g(t)v(t)dt$
	$\displaystyle\geq\frac{2}{\sigma^{3}}\left(1-\frac{1}{\mu}\right)\int_{[0,1/2]% \setminus\mathcal{B}}g(t)f^{\prime}(\sigma v(t))\sigma^{2}v(t)^{2}dt$
	$\displaystyle\geq\frac{2}{\sigma^{3}}\left(1-\frac{1}{\mu}\right)\int_{\beta}^% {1/2-\beta}g(t)f^{\prime}(\sigma v(t))\sigma^{2}v(t)^{2}dt$
	$\displaystyle\geq\frac{2}{\sigma^{3}}\lambda\left(1-\frac{1}{\mu}\right)\int_{% \beta}^{1/2-\beta}g(t)\sigma^{2}v(t)^{2}dt\geq\frac{2\tilde{C}}{\sigma^{3}}% \lambda\left(1-\frac{1}{\mu}\right)r^{2}\phi(\beta)^{2}.$

Whence, the above two inequalities (3.13) and (3.14), together with (3.7), yield

h^{\prime}(\sigma)\leq\frac{2\tilde{B}}{\sigma^{3}}f(r\phi(\beta))r\phi(\beta)% -\frac{2\tilde{C}}{\sigma^{3}}\lambda\left(1-\frac{1}{\mu}\right)r^{2}\phi(% \beta)^{2}<0,

as expected.

\widetilde{h}(\sigma)=\sigma-2\int_{\beta}^{1/2}f(\sigma v(t))g(t)v(t)\,dt.

Since $\sigma v(t)\in[r\phi(\beta),R]$ for all $t\in[\beta,1/2]$ by (3.11), it follows that $\widetilde{h}$ is of class $C^{2}[r,R]$ . From (H4), one has

\widetilde{h}^{\prime\prime}(\sigma)=-2\int_{\beta}^{1/2}f^{\prime\prime}(% \sigma v(t))g(t)v(t)^{3}\,dt\leq-2\widetilde{C}\phi(\beta)^{3}\min_{[r\phi(% \beta),R]}f^{\prime\prime}(\cdot)<0,

which shows that $\widetilde{h}$ is strictly concave on $[r,R]$ .

Consequently, in all three cases, from (3.10), it follows immediately that $\alpha_{u}^{\prime}(\tau)$ has exactly one zero within the interval $\left(\frac{r}{|u|_{H_{0}^{1}}},\frac{R}{|u|_{H_{0}^{1}}}\right)$ . Moreover, $\alpha_{u}^{\prime}$ is positive on $\left(\frac{r}{|u|_{H_{0}^{1}}},s(u)\right)$ , while $\alpha^{\prime}_{u}$ is negative on $\left(s(u),\frac{R}{|u|_{H_{0}^{1}}}\right)$ , so condition (h1) is verified.
To prove the last assertion (h4), assume $u\in\mathcal{N}$ . Thus, $r<|u|_{H_{0}^{1}}<R$ and

|u|_{H_{0}^{1}}^{2}=\int_{0}^{1}f(u(t))g(t)u(t)dt.

(a)’ In case that (H2) holds, since $u(t)\geq\phi(\beta)|u|_{H_{0}^{1}}\geq r\phi(\beta)$ by (3.11), we compute

	$\displaystyle E^{\prime\prime}(u)(u,u)$	$\displaystyle=\|u\|^{2}_{H_{0}^{1}}-\int_{0}^{1}f^{\prime}(u(t))g(t)u^{2}(t)dt$
		$\displaystyle=\int_{0}^{1}f(u(t))g(t)u(t)dt-\int_{0}^{1}f^{\prime}(u(t))g(t)u(% t)^{2}dt$
		$\displaystyle\leq-2\int_{0}^{1/2}\theta(u(t))g(t)u(t)dt$
		$\displaystyle\leq-2r\phi(\beta)\int_{\beta}^{1/2}\theta(u(t))g(t)dt$
		$\displaystyle\leq-2\tilde{C}r\phi(\beta)\min_{[r\phi(\beta),R]}\theta(\cdot)<0.$

(b)’ Assume (H3) is satisfied. Then, taking $\sigma=|u|_{H_{0}^{1}}$ and performing the same computations as in the estimates for the derivative of $h$ in (b), we have

	$\displaystyle E^{\prime\prime}(u)(u,u)$	$\displaystyle=2\int_{0}^{1/2}\left(f(u(t))-f^{\prime}(u(t))u(t)\right)g(t)u(t)dt$
		$\displaystyle\leq 2\int_{0}^{\beta}f(u(t))g(t)u(t)dt+2\int_{\beta}^{1/2}\left(% f(u(t))-f^{\prime}(u(t))u(t)\right)g(t)u(t)dt$
		$\displaystyle\leq 2\int_{0}^{\beta}f(u(t))g(t)u(t)dt-2\left(1-\frac{1}{\mu}% \right)\int_{\beta}^{1/2}f^{\prime}(u(t))g(t)u(t)^{2}dt$
		$\displaystyle\leq 2r\phi(\beta)\left(\tilde{B}f(r\phi(\beta))-\lambda\tilde{C}% \left(1-\frac{1}{\mu}\right){r\phi(\beta)}\right)<0,$

where the latter inequality follows from (3.7).
(c)’ For the last case, using (3.8) from (H4) and the symmetry of $v$ , we see that

	$\displaystyle E^{\prime\prime}(u)(u,u)$	$\displaystyle=\|u\|^{2}_{H_{0}^{1}}-2\int_{\beta}^{1/2}f^{\prime}(u(t))g(t)u(t)^% {2}dt$
		$\displaystyle=2\int_{\beta}^{1/2}f(u(t))g(t)u(t)dt-2\int_{\beta}^{1/2}f^{% \prime}(u(t))g(t)u(t)^{2}dt.$

Since $u(t)\in[r\phi(\beta),R]$ for all $t\in[\beta,1/2]$ , the conclusion follows immediately by an argument similar to that in (a)’ , that is,

E^{\prime\prime}(u)(u,u)\leq-2\tilde{C}r\phi(\beta)\min_{[r\phi(\beta),R]}% \tilde{\theta}(\cdot)<0.

Therefore, in all three cases, condition (h4) is verified. Summing up, we have the following result.

Theorem 3.2.

Assume (H1) and either (H2), (H3) or (H4) hold true. Then, problem (3.1) has a solution $u^{\ast}\in K$ such that $r<|u^{\ast}|_{H_{0}^{1}}<R$ .

Proof.

As established above, assumptions (h1)-(h4) are verified, thus Theorem 2.2 guarantees the existence of a sequence $u_{n}\in\mathcal{N}$ such that

E(u_{n})\to\inf_{\mathcal{N}}E\quad\text{ and }\quad E^{\prime}(u_{n})\to 0.

Since $L^{2}(0,1)$ embeds compactly into $H^{-1}(0,1)$ , the operator $N$ is completely continuous from $H_{0}^{1}(0,1)$ into itself (see, e.g., [18]); therefore Theorem 2.4 applies and gives the conclusion. ∎

It is worth providing some commentaries on the conditions (H2), (H3), and (H4).

(i):

The first condition, (H2), naturally extends the standard condition obtained via the Nehari manifold method over the entire domain. For instance, let $a>0$ , $p>1$ , $g\equiv 1$ and choose $0<r<R<\infty$ such that

(3.15)

a<\frac{\pi}{r^{p}}\quad\text{and}\quad a>\frac{1}{R^{p-1}\beta^{p+1}(1-2\beta% )^{p+2}},

for some $\beta\in(0,1/4)$ . Then, the function $f(t)=at^{p}$ satisfies (H1) and (H2). Indeed, conditions (3.15) ensure (H1), while (H2) follows directly since

tf^{\prime}(t)-f(t)=a(p-1)t^{p}.

However, it is not difficult to see that this setup does not lead to multiplicity. To see this, suppose (H2) holds for two pairs of points $(r_{1},R_{1})$ and $(r_{2},R_{2})$ with $r_{1}<R_{1}<r_{2}<R_{2}$ . Then, as the map $f(t)/t$ is increasing on $[0,r_{2}]$ (see (a)), we have

\frac{f(R_{1})}{R_{1}}<\frac{f(r_{2})}{r_{2}}<\frac{\pi}{\tilde{A}},

which implies $\alpha_{u}\left(\frac{R_{1}}{|u|_{H_{0}^{1}}}\right)>0$ for all $u\in K\setminus\{0\}$ , contrary to (3.10).

(ii):

Conditions (H3) and (H4) may lead to multiplicity, as they impose restrictions intrinsically linked to the values of $r$ and $R$ . Both conditions require an Ambrosetti-Rabinowitz-type assumption, which is often encountered in the study of the existence of solutions for nonlinear equations.

Additionally, (H3) includes an extra requirement on the derivative of $f$ and the associated coefficients, while (H4) assumes that the function $g$ vanishes on a small interval starting from 0 and that the function $f$ is strictly convex on the interval $[r\phi(\beta),R]$ . We also remark that the three conditions (H2), (H3), and (H4) can be applied independently to different pairs $(r_{i},R_{i})$ . However, it is important to note that if (H2) is used, it should only be applied to the first pair $(r_{1},R_{1})$ , as remark (i) shows.

Acknowledgements

The author expresses gratitude to Prof. Radu Precup for his valuable insights on the subject and thorough verification of the paper.

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Localization of critical points in annular conical sets via the method of Nehari manifold

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Localization of critical points in annular conical sets via the method of Nehari manifold

1. Introduction and Preliminaries

1.1. Nehari manifold

1.2. Localization

Theorem 1.1 (Schechter’s Theorem).

1.3. Ekeland variational principle

Lemma 1.2 (Ekeland Principle - weak form).

1.4. Implicit function theorem

Theorem 1.3.

2. Main Result

Remark 2.1.

Theorem 2.2.

Lemma 2.3.

Proof.

Proof of Theorem 2.2.

Theorem 2.4.

Proof.

Remark 2.5 (Multiplicity).

3. Application

Lemma 3.1.

Proof of Lemma 3.1.

Theorem 3.2.

Proof.

Acknowledgements

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