Remarks on the Clark theorem

TL;DR

This paper investigates the convergence behavior of critical points in the Clark theorem, providing new characterizations and examples that extend previous abstract results in critical point theory.

Contribution

It offers improved theoretical insights into the convergence of critical points for even functionals related to the Clark theorem, including new examples and characterizations.

Findings

Critical points with negative critical values can converge to non-zero critical points.

The paper provides a characterization of accumulation points of critical points.

Results extend and improve upon previous abstract results by Kajikiya and Liu-Wang.

Abstract

The Clark theorem is important in critical point theory. For a class of even functionals it ensures the existence of infinitely many negative critical values converging to $0$ and it has important applications to sublinear elliptic problems. We study the convergence of the corresponding critical points and we give a characterization of accumulation points of critical points together with examples, in which critical points with negative critical values converges to non-zero critical point. Our results improve the abstract results in Kajikiya [Ka1] and Liu-Wang [LW].

Equations

$$\gamma (A)\ge kand\underset{u\in A}{\sup }I(u)<0.$$

$$\eqalign{&c_{j}<0\quad\hbox{for all}\ j\in\mathop{\bf N}\nolimits,\cr&c_{j}\to 0\quad\hbox{as}\ j\to\infty.\cr}$$

$$c_j=\underset{A\in \mathcal{E},\gamma (A)\ge j}{\inf }\underset{u\in A}{\sup }I(u).$$

$$I^{\prime }(u_j)=0,I(u_j)<0and{u}_j\to 0asj\to \infty .$$

$$I^{\prime }(u_j)=0,I(u_j)=0,u_j\ne 0and{u}_j\to 0asj\to \infty$$

$$I^{\prime }(v_j)=0,I(v_j)<0and{v}_{j}converges\; to\; a\; non-zero\; limit.$$

$$\sup \{I(u);u\in X^k,\Vert u{\Vert }_X={\rho }_k\}<0$$

$$\Vert u{\Vert }_X=aandI(u)=0.$$

$${\hat{K}}_0\cap \{u\in X;\Vert u{\Vert }_X=r\}\ne \varnothing .$$

$$\eqalignno{&I(u_{j})<0\quad\hbox{for all}\ j\in\mathop{\bf N}\nolimits,&{{\rm({1.2}{})}}\cr&I(u_{j})\to 0\quad\hbox{as}\ j\to\infty,&{{\rm({1.3}{})}}\cr&\mathop{\rm dist}\,(u_{j},\widehat{K}_{0})\,\left(\equiv\inf\{\|{u_{j}-v}\|_{X};\,v\in\widehat{K}_{0}\}\right)\to 0\quad\hbox{as}\ j\to\infty.\cr}$$

$$u_j\to 0asj\to \infty .$$

$$I^{\prime }(u)=0andI(u)=c_{j}imply\Vert u{\Vert }_X\ge r_0.$$

$$\eqalign{B_{\delta}(u)&=\{x\in X;\,\|{x-u}\|_{X}<\delta\}\quad\hbox{for}\ u\in X,\cr N_{\delta}(D)&=\{x\in X;\,\mathop{\rm dist}\,(x,D)<\delta\}\quad\hbox{for}\ D\subset X,\cr}$$

$$\mathrm{dist}(x,D)=\underset{y\in D}{\inf }\Vert x-y{\Vert }_X.$$

$$\bigcap _{\delta >0}\overline{O_{\delta }}=\hat{D},$$

$$\hat{D}\subset \bigcap _{\delta >0}{O}_{\delta }\subset \bigcap _{\delta >0}\overline{O_{\delta }}.$$

$$A=\bigcap _{\delta >0}\overline{O_{\delta }}\subset D.$$

$$\eqalignno{&A=\bigcap_{\delta>0}D_{\delta},&{{\rm({2.1}{})}}\cr&O_{\delta}=\bigcup_{u\in D_{\delta}}B_{\delta}(u),&{{\rm({2.2}{})}}\cr&\hbox{$\delta_{1}<\delta_{2}$ implies $D_{\delta_{1}}\subset D_{\delta_{2}}$.}&{{\rm({2.3}{})}}\cr}$$

$$\beta =\frac{1}{2}\mathrm{dist}(A_1,A_2)>0.$$

$$O_{\delta }\cap \{x\in X;\mathrm{dist}(x,A_1)=\beta \}\ne \varnothing .$$

$$D_{\delta }\cap \{x\in X;\mathrm{dist}(x,A_1)\in [\beta -\delta ,\beta +\delta ]\}\ne \varnothing .$$

$$\eqalign{&A\cap\{x\in X;\,\mathop{\rm dist}\,(x,A_{1})=\beta\}\cr=&\bigcap_{\delta>0}\Bigl{(}D_{\delta}\cap\{x\in X;\,\mathop{\rm dist}\,(x,A_{1})\in[\beta-\delta,\beta+\delta]\}\Bigr{)}\not=\emptyset,\cr}$$

$$\eqalign{K&=\{u\in X;\,I^{\prime}(u)=0\},\cr K_{0}&=\{u\in K;\,I(u)=0\},\cr K_{-}&=\{u\in X;\,\hbox{there exists $(v_{j})_{j=1}^{\infty}\subset K$ such that}\cr&\qquad\qquad\ \ I(v_{j})<0\ \hbox{for all $j$ and $I(v_{j})\to 0$, $v_{j}\to u$ as $j\to\infty$}\}.\cr}$$

$$\eqalign{&[I\leq a]=\{u\in X;\,I(u)\leq a\},\cr&[a\leq I\leq b]=\{u\in X;\,a\leq I(u)\leq b\}.\cr}$$

$$K_{-}\cap {\hat{K}}_0\ne \varnothing .$$

$${\hat{K}}_0=\bigcap _{\delta >0}\overline{O_{\delta }}.$$

$$\overline{O_{\delta }}\cap K_{-}\ne \varnothing for\; all\delta >0.$$

$$\overline{O_{{\delta }_0}}\cap K_{-}=\varnothing .$$

$$K_{0,i}=\overline{O_{{\delta }_0}}\cap K_0,K_{0,e}=K_0\setminus O_{{\delta }_0}.$$

$$\eqalignno{&\mathop{\rm dist}\,(K_{0,i},K_{0,e})\geq 2\delta_{0},&{{\rm({2.6}{})}}\cr&K_{-}\subset K_{0,e}.&{{\rm({2.7}{})}}\cr}$$

Full text

(A1) $I(0)=0$. $I(u)$ is even in $u$ and bounded from below;

(A2) $I(u)$ satisfies $(PS)_{c}$ for all $c<0$;

(A3) For any $k\in\mathop{\bf N}\nolimits$, there exists $A\in{\cal E}$ such that

[TABLE]

Then $I(u)$ has a sequence $(c_{j})_{j=1}^{\infty}$ of critical values of $I(u)$ such that

[TABLE]

Here

[TABLE]

**Remark 1.2. **In [H], it was assumed that

(A2’) $I(u)$ satisfies $(PS)_{c}$ for all $c\in\mathop{\bf R}\nolimits$.

From its proof, we can easily see that $(PS)_{c}$ just for $c<0$ is enough for the existence of critical values.

By Theorem 1.1, there exists a sequence $(u_{j})_{j=1}^{\infty}$ of critical points of $I(u)$ such that $I(u_{j})=c_{j}\to-0$ as $j\to\infty$. Thus it is natural to ask whether $u_{j}\to 0$ holds or not. More generally, the existence of a sequence of non-zero critical points $(u_{j})_{j=1}^{\infty}$ (or critical points with negative critical values) satisfying $u_{j}\to 0$ is of interest. This question has been studied by Kajikiya [Ka1] and Liu-Wang [LW] together with applications to sublinear elliptic problems. We note that Liu-Wang [LW] also studied periodic solutions of Hamiltonian systems. More precisely, under the assumptions of (A1), (A2’) and (A3), Kajikiya [Ka1] showed either

(C1) There exists a sequence $(u_{j})_{j=1}^{\infty}$ such that

[TABLE]

or

(C2) There exists two sequences $(u_{j})_{j=1}^{\infty}$ and $(v_{j})_{j=1}^{\infty}$ such that

[TABLE]

and

[TABLE]

holds.

Liu-Wang [LW] assumed (A1), (A2’) and the following (A3’), which is stronger than (A3),

(A3’) For any $k\in\mathop{\bf N}\nolimits$ there exists a $k$-dimensional subspace $X^{k}$ of $X$ and $\rho_{k}>0$ such that

[TABLE]

and they showed either (C1) above or

(C3) There exists $r>0$ such that for any $0<a<r$ there exists a critical point $u$ such that

[TABLE]

In what follows, we denote by $\widehat{K}_{0}$ the connected component of $K_{0}=\{u\in X;\,I^{\prime}(u)=0,\,I(u)=0\}$ including [math].

**Remark 1.3. **From their proof of their main result, Liu-Wang [LW] claimed that (C3) can be strengthened as

(C3’) There exists $r>0$ such that

[TABLE]

The aim of this paper is to show the following Theorem 1.4 and Theorem 1.6; In Theorem 1.4, we give a new characterization of accumulation points of critical points with negative critical values and unifies the results of Kajikiya and Liu-Wang. On the other hand, in Theorem 1.6 we answer a natural question concerning (C1), which is stated below. We believe that Theorems 1.4 and 1.6 give us a better understanding of the Clark theorem.

First we give our Theorem 1.4.

(A2”) $I(u)$ satisfies $(PS)_{c}$ for all $c\leq 0$.

Then there exists a sequence $(u_{j})_{j=1}^{\infty}\subset X$ of critical points of $I(u)$ such that

[TABLE]

As an immediate corollary to our Theorem 1.4, we have

Since $\widehat{K}_{0}\not=\{0\}$ implies (C2) and (C3), Corollary 1.5 covers the results of Kajikiya [Ka1] and Liu-Wang [LW].

Next we study a question concerning (C1). In many applications of the Clark theorem to sublinear elliptic equations, there exist sequences $(u_{j})_{j=1}^{\infty}$ of solutions with (1.2), (1.3) and

[TABLE]

So (C1) may be expected under the assumption of Theorem 1.4 and a natural question is to ask whether (C1) always takes place under the assumption of Theorem 1.4 or not. Our Theorem 1.6 answers this question negatively.

**Remark 1.7. **An example related to our Theorem 1.6 was given in Example 1.3 of [Ka1] (c.f. [Ka2]). It shows that there exists a functional $I\in C^{1}(X,\mathop{\bf R}\nolimits)$ which satisfies (A1), (A2”), (A3) and the following property:

There exists an $r_{0}>0$ independent of $j$ such that

[TABLE]

Here $c_{j}$ is given in (1.1) and $c_{j}$ satisfies $c_{j}<0$ and $c_{j}\to 0$ as $j\to\infty$. Thus a special case of (C1) does not hold for $I$. In Section 3.1 we give another example $I\in C^{1}(\ell^{2},\mathop{\bf R}\nolimits)$ for which we give an explicit description of all critical points of $I(u)$ and no critical points with negative critical values do not exist in a neighborhood of [math]. Especially (C1) does not hold for our $I(u)$. Our example also shows a typical situation of our Theorem 1.4.

Finally we remark that in our Theorem 1.4, (A2”), especially $(PS)_{0}$ is important. In fact, we have

In the following Section 2, we give a proof to our Theorem 1.4. Here estimates of $I^{\prime}(u)$ play important roles. In Section 3, we give two examples which show Theorems 1.6 and 1.8.

2. Proof of Theorem 1.4In what follows, we use the following notation for $\delta>0$

[TABLE]

where

[TABLE]

We note that $N_{\delta}(D)=\bigcup_{y\in D}B_{\delta}(y)$.

2.1. A fundamental fact from topologyTo show our Theorem 1.4, we need the following characterization of connected components of compact sets.

[TABLE]

where $\widehat{D}$ is the connected component of $D$ including 0.

**Proof. **By the definition of $O_{\delta}$ and $\widehat{D}$, it is clear that $\widehat{D}\subset O_{\delta}$ for all $\delta>0$. Thus

[TABLE]

By the compactness of $D$, we also have $D=\bigcap_{\delta>0}\overline{N_{\delta}(D)}$.

We set

[TABLE]

It suffices to show that $A$ is connected. For $\delta>0$ we also set $D_{\delta}=\overline{O_{\delta}}\cap D$. Then we have

[TABLE]

Arguing indirectly, we suppose that $A$ is not connected. Then there exist two compact sets $A_{1}$, $A_{2}\subset X$ such that $A_{1}\cap A_{2}=\emptyset$, $A_{1}\cup A_{2}=A$. We set

[TABLE]

For each $\delta>0$, since $O_{\delta}$ is a connected set including $A_{1}\cup A_{2}$, we have

[TABLE]

By (2.2), we can see that for any $x\in O_{\delta}$ there exists $u\in D_{\delta}$ such that $x\in B_{\delta}(u)$. Thus

[TABLE]

Since \Bigl{(}D_{\delta}\cap\{x\in X;\,\mathop{\rm dist}\,(x,A_{1})\in[\beta-\delta,\beta+\delta]\}\Bigr{)}_{\delta>0} has the finite intersection property by (2.3), we have

[TABLE]

which contradicts with the choice of $\beta>0$. Thus $A$ is a connected set.

2.2. A gradient estimateSuppose that $I(u)\in C^{1}(X,\mathop{\bf R}\nolimits)$ satisfies the assumptions of Theorem 1.4. We use the following notation:

[TABLE]

By $(PS)_{0}$, we have $K_{-}\subset K_{0}$. We also use notation for $a<b$

[TABLE]

It is clear that $0\in K_{0}$. We denote by $\widehat{K}_{0}$ the connected component of $K_{0}$ including [math]. To show our Theorem 1.4 it suffices to prove

[TABLE]

For $\delta>0$, let $O_{\delta}$ be the connected component of $N_{\delta}(K_{0})$ including [math]. By Lemma 2.2, we have

[TABLE]

Thus to prove (2.4) it suffices to show

[TABLE]

We argue indirectly and suppose for some $\delta_{0}>0$

[TABLE]

Under the assumption (2.5), we set

[TABLE]

Then $K_{0,i}$ and $K_{0,e}$ are disjoint compact sets such that $K_{0}=K_{0,i}\cup K_{0,e}$ and

[TABLE]

We note that (2.7) follows from (2.5).

First we have

[TABLE]

Moreover for any $\varepsilon\in(0,\rho)$ there exists $\nu_{\varepsilon}\in(0,\nu]$ such that

[TABLE]

**Proof. **Using $(PS)_{0}$ and the definition of $K_{-}$, we can check (2.8)–(2.10) easily for small $\rho$ and $\nu>0$. We show (2.11). Suppose that for $r$, $\rho$, $\nu>0$, (2.8)–(2.10) hold. If (2.11) does not hold, we can find $\varepsilon\in(0,\rho)$ and a sequence $(u_{j})_{j=1}^{\infty}$ such that

[TABLE]

By $(PS)$, we can extract a subsequence $(u_{j_{k}})$ such that $u_{j_{k}}\to u_{0}$ for some $u_{0}\in[-\rho\leq I\leq-\varepsilon]\cap K$. By (2.9), we have $u_{j_{k}}\not\in N_{r}(K_{0,e})$ for large $k$, which is a contradiction. Thus we have (2.11).

2.3. Deformation argumentThe aim of this section is the following

[TABLE]

**Proof. **First we define an ODE in $X$ to define $\eta_{\varepsilon}$. For a given $r\in(0,\delta_{0}/3]$, let $\rho$, $\nu>0$ be constants given in Lemma 2.2. We set

[TABLE]

Then again by Lemma 2.2, for any given $\varepsilon\in(0,d]$ there exists $\nu_{\varepsilon}>0$ with the property (2.11).

By (2.9), we have $I^{\prime}(u)\not=0$ for all $u\in[-3d\leq I<0]\setminus N_{r}(K_{0,e})$. Thus there exists a locally Lipschitz odd vector field $V(u):\,[-3d\leq I<0]\setminus N_{r}(K_{0,e})\to X$ such that

[TABLE]

Let $\phi_{1}(u)$, $\phi_{2}(u):\,X\to[0,1]$ be even Lipschitz continuous functions such that

[TABLE]

We set

[TABLE]

and we note that $\widetilde{V}(u)$ is well-defined on $[I<0]$. For $u\in[I<0]$ we consider

[TABLE]

We have for all $(t,u)$

[TABLE]

Since

[TABLE]

it follows from (2.13)–(2.15) that

[TABLE]

By (2.17) and (2.18), we note that for any $u\in[I<0]$, $\eta(t,u)$ exists globally, that is, $\eta(t,u):\,[0,\infty)\times[I<0]\to[I<0]$ is well-defined. For a latter use, we note that

[TABLE]

Thus by (2.19)

[TABLE]

Next we claim that

**Claim. **Let $T_{\varepsilon}={2d\over\nu_{\varepsilon}}$. Then

[TABLE]

To prove (2.22), it suffices to show that if $u\in[I\leq-\varepsilon]$ satisfies

[TABLE]

then

[TABLE]

We note that under the condition (2.23)

[TABLE]

Step 1: Assume (2.23), i.e., (2.25). Then

[TABLE]

In fact, if (2.26) does not hold, it follows from (2.20) that

[TABLE]

Thus, by the definition of $T_{\varepsilon}$,

[TABLE]

which is in contradiction with (2.23).

Step 2: Assume (2.23), i.e., (2.25). Then (2.24) holds.

Assume (2.24) does not hold. Then $\eta(T_{\varepsilon},u)\not\in N_{3r}(K_{0,e})$ and by (2.26) the orbit $\eta(t,u)$ enters in $N_{2r}(K_{0,e})$ for some $t\in[0,T_{\varepsilon}]$. Thus there exists an interval $[t_{0},t_{1}]\subset[0,T_{\varepsilon}]$ such that

[TABLE]

By (2.17), we have

[TABLE]

Thus by (2.21),

[TABLE]

This is a contradiction to (2.23). Thus we have $\eta(T_{\varepsilon},u)\in N_{3r}(K_{0,e})$ and the conclusion of Step 2 holds.

Setting $\eta_{\varepsilon}(u)=\eta(T_{\varepsilon},u)$, we have the desired deformation.

2.4. End of the proof of Theorem 1.4

**Proof of Theorem 1.4. **Since $K_{0,e}\in{\cal E}$ is compact, we can see

[TABLE]

Moreover for small $r\in(0,\delta_{0}/3]$

[TABLE]

We fix such an $r$ and we choose $d>0$ by Proposition 2.3.

By Clark’s theorem [Cl], we have

[TABLE]

Thus there exists $k_{0}$ such that $c_{k_{0}}>-d$. That is, $\gamma([I\leq-d])<k_{0}$. Thus

[TABLE]

By the assumption of Theorem 1.4, there exists $A\in{\cal E}$ such that

[TABLE]

Choosing $\varepsilon\in(0,d)$ such that $\sup_{u\in A}I(u)<-\varepsilon$, we have

[TABLE]

On the other hand, by Proposition 2.3, there exists a continuous odd map $\eta_{\varepsilon}:\,[I<0]\to[I<0]$ such that

[TABLE]

Thus by (2.27)

[TABLE]

which is in contradiction with (2.28). Thus (2.5) cannot take place and we complete the proof of our Theorem 1.4.

3. Some examplesIn this section we give two examples which show Theorems 1.6 and 1.8.

3.1. An example which shows Theorem 1.6We give an example which shows that (A1), (A2”), (A3’) do not imply (C1). We work in the space $X=\ell^{2}$, that is,

[TABLE]

Since the first component has a special role in our argument, we use notation $(t,x_{1},x_{2},\cdots)$ for elements of $X$.

We consider a functional $I(t,x_{1},x_{2},\cdots):\,X\to\mathop{\bf R}\nolimits$ in a form

[TABLE]

where $x_{+}=\max\{x,0\}$, $x_{-}=\max\{-x,0\}$ and $a_{+}(t)$, $a_{-}(t)\in C^{1}(\mathop{\bf R}\nolimits,\mathop{\bf R}\nolimits)$ are given by

[TABLE]

Here $\mu(t)\in C^{1}(\mathop{\bf R}\nolimits,\mathop{\bf R}\nolimits)$ satisfies

[TABLE]

It follows from (3.1)–(3.5) that

[TABLE]

Finally we define $\varphi(t)\in C^{1}(\mathop{\bf R}\nolimits,\mathop{\bf R}\nolimits)$ by

[TABLE]

We can see that $I(t,x_{1},x_{2},\cdots)$ has the following properties.

Proposition 3.1.

(i) $I(t,x_{1},x_{2},\cdots)\in C^{1}(X,\mathop{\bf R}\nolimits)$;

(ii) $I(t,x_{1},x_{2},\cdots)$ is bounded from below and coercive on $X$;

(iii) $I(t,x_{1},x_{2},\cdots)$ satisfies $(PS)_{c}$ for all $c\in\mathop{\bf R}\nolimits$;

(iv) $I(t,x_{1},x_{2},\cdots)$ is even in $(t,x_{1},x_{2},\cdots)$, that is, $I(-t,-x_{1},-x_{2},\cdots)=I(t,x_{1},x_{2},\cdots)$;

(v) $I(t,x_{1},x_{2},\cdots)$ satisfies (A3’).

**Proof. **It follows from Hölder inequality that

[TABLE]

from which we can see that $I(t,x_{1},x_{2},\cdots)$ is well-defined as a functional on $X$. Using (3.9), we can also see (i)–(iii). (iv) follows from (3.7).

For $k\in\mathop{\bf N}\nolimits$, setting $X^{k}=\{(0,x_{1},x_{2},\cdots,x_{k},0,0,\cdots);\,(x_{1},x_{2},\cdots,x_{k})\in\mathop{\bf R}\nolimits^{k}\}$, we can easily find $\rho_{k}>0$ such that

[TABLE]

Thus (v) holds.

By Proposition 3.1, we can apply the Clark Theorem to $I(t,x_{1},x_{2},\cdots)$. On the other hand, we have

[TABLE]

where

[TABLE]

Moreover we have

[TABLE]

**Proof. **Since

[TABLE]

we see that $I^{\prime}(t,x_{1},x_{2},\cdots)=0$ implies $t\in[-1,1]$.

In what follows, we assume that $(t,x_{1},x_{2},\cdots)$ ($t\in[-1,1]$) is a critical point of $I$ and give more precise description. First we show

Step 1: For any $j\in\mathop{\bf N}\nolimits$,

[TABLE]

In particular, we have

(i) if $x_{j}\not=0$,

[TABLE]

(ii) $K\cap\{t=1\}=N$, $K\cap\{t=-1\}=-N$, where $N$ is given in (3.11).

In fact, it follows from $\partial_{x_{j}}I(t,x_{1},x_{2},\cdots)=0$ that

[TABLE]

From which we can get (3.15). (i) and (ii) follow from the property (3.6) and $a_{+}(1)=a_{-}(-1)=3$, $a_{+}(-1)=a_{-}(1)=1$.

Step 2: When $t\in(-1,1)$, it holds that $x_{j}=0$ for all $j$.

It follows from $\partial_{t}I(t,x_{1},x_{2},\cdots)=0$ that

[TABLE]

By (3.8),

[TABLE]

Arguing indirectly, we assume that $x_{j}\not=0$ for some $j$ and let $j_{0}$ be the smallest integer such that $x_{j}\not=0$. Then we have

[TABLE]

By (3.16),

[TABLE]

Again by (3.16),

[TABLE]

which is in contradiction with (3.17). Thus we have $x_{j}=0$ for all $j\in\mathop{\bf N}\nolimits$.

Step 3: Conclusion.

(3.10) follows from Steps 1–2. We can also verify (3.12)–(3.13) easily.

As an immediate corollary to Proposition 3.2, we have

(ii) $(1,0,0,\cdots)$, $(-1,0,0,\cdots)\in X$ are accumulation points of critical points with negative critical values.

Thus Corollary 3.3 shows that (C1) does not hold in general under the conditions (A1), (A2”), (A3’).

3.2. An example which shows Theorem 1.8Next we give another example, which shows that without $(PS)_{0}$ the conclusion of Theorem 1.4 does not hold in general. Here we work in the Hilbert space $(E,\|{\cdot}\|_{E})$ given by

[TABLE]

For $p\in(0,1)$ we define $J(u)\in C^{1}(E,\mathop{\bf R}\nolimits)$ by

[TABLE]

Critical points of $J(u)$ are solutions of the following sublinear elliptic equation:

[TABLE]

and it has the following properties:

(i) $J(0)=0$, $J(u)$ is even in $u$, bounded from below and coercive;

(ii) For any $k\in\mathop{\bf N}\nolimits$, there exists a compact subset $A\subset E\setminus\{0\}$, which is symmetric with respect to [math], such that

[TABLE]

Actually, for any $k$-dimensional subspace $H\subset E$,

[TABLE]

with small $\rho>0$ gives the desired compact set.

(iii) $J(u)$ satisfies $(PS)_{c}$ for all $c\in\mathop{\bf R}\nolimits$.

We define $I(u):\,E\to\mathop{\bf R}\nolimits$ by

[TABLE]

Then we have

**Proof. **Clearly $I(u)$ is even, bounded from below and coercive. Moreover $I(u)$ also satisfies $(PS)_{c}$ for all $c<0$. In fact, if $(u_{j})_{j=1}^{\infty}$ satisfies $I(u_{j})\to c<0$ and $\|{I^{\prime}(u_{j})}\|\to 0$, then we can easily see that $(u_{j})_{j=1}^{\infty}$ is bounded as $j\to\infty$ and, after taking a subsequence, we may assume that $\|{u_{j}}\|_{E}\to d$ for some $d>1$. Using this fact, we can see that $(u_{j})_{j=1}^{\infty}$ has a strongly convergent subsequence. (Since all points on the unit sphere $S=\{x\in E;\ \|{u}\|_{E}=1\}$ are critical points of $I(u)$ with critical value [math] and $S$ is not compact, we note that $(PS)_{0}$ fails.) Thus $I(u)$ satisfies (A1) and (A2). We can see that (A3) holds easily. In fact, for any $k$-dimensional subspace $H\subset E$, choosing $\rho>0$ small,

[TABLE]

satisfies $\gamma(A)\geq k$ and $\sup_{u\in A}I(u)<0$. Thus $I(u)$ satisfies the assumptions of Theorem 1.1.

Clearly [math] is an isolated critical point of $I(u)$ and $I(u)$ does not have a sequence of critical points with (1.2)–(1.4).

Acknowledgments. This research is motivated by Professor Zhaoli Liu’s lecture in Capital Normal University, to which the first author attended. Authors would like to thank Professor Liu for suggesting them to study such an interesting problem. Authors would also like to thank Professor Ryuji Kajikiya for helpful discussions.

Authors started this research during the first and the third authors’ visit to Department of Mathematics, School of Science and Engineering, Waseda University. They would like to gratefully acknowledge Waseda University for cordial invitations and hospitality. The first author wishes to express his gratitude to the support of Graduate School of Capital Normal University. The third author is greatly indebted to China Scholarship Council for their support.

The second author is partially supported by JSPS Grants-in-Aid for Scientific Research (B) (25287025) and Waseda University Grant for Special Research Projects 2016B-120.

References

[Cl] D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972), 65–74.

[Co] C. V. Coffman, A minimum-maximum principle for a class of non-linear integral equations, J. Analyse Math. 22 (1969), 391–419 .

[H] H.-P. Heinz, Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems, J. Differential Equations 66 (1987), no. 2, 263–300.

[Ka1] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), no. 2, 352–370.

[Ka2] R. Kajikiya, private communication, 2017.

[Kr] M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations. Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book The Macmillan Co., New York 1964 xi + 395 pp.

[LW] Z. Liu, Z.-Q. Wang, On Clark’s theorem and its applications to partially sublinear problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 5, 1015–1037.

[R] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. viii+100 pp.