In this article, we shall obtain solutions of problem (1.3) using the linking-type theorem. Its different definitions can be seen in [1, 7, 8] and the references therein.

**Definition 3.1** Let

*H* be a real Hilbert space and

*A* a closed set in

*H*. Let

*B* be an Hilbert manifold with boundary ∂

*B*, we state that

*A* and ∂

*B* link if

- (i)

- (ii)
If *ϕ* is a continuous map of *H* into itself s.t. *ϕ*(*u*) = *u*, ∀*u* ∈ ∂*B*, then *ϕ*(*B*) ∩ *A* ≠ ∅.

There are some typical examples as following, cf. [1, 7, 9].

**Example 3.1** Let

*H*_{1} *and H*_{2} be two closed subspaces of

*H* such that

$H={H}_{1}\oplus {H}_{2},\phantom{\rule{1em}{0ex}}dim{H}_{2}<\infty .$

Hence, if *A* = *H*_{1}, *B* = *B*_{
R
} ∩ *H*_{2}, *then*, *A* and ∂*B* link.

**Example 3.2** Let

*H*_{1} *and H*_{2} be two closed subspaces of

*H* such that

*H* =

*H*_{1} ⊕

*H*_{2}, dim

*H*_{2} *<* ∞, and consider

*e* ∈

*H*_{1}, ||

*e*|| = 1, 0

*< ρ < R*_{1},

*R*_{2}, set

$A={H}_{1}\cap {S}_{\rho},\phantom{\rule{1em}{0ex}}B=\left\{u=v+te:v\in {H}_{2}\cap {B}_{{R}_{2}},0\le t\le {R}_{1}\right\}.$

Then, *A* and ∂*B* link.

Let *X* ⊂ *H* be a Banach space densely embedded in *H*. Assume that *H* has a closed convex cone *P*_{
H
} and that *P* := *P*_{
H
} ∩ *X* has interior points in *X*. Let *J* ∈ *C*^{1}(*H*, ℝ). In the article [10], those authors construct the pseudo-gradient flow *σ* for *J*, and have the same definition as [11].

**Definition 3.1** Let *W* ⊂ *X* be an invariant set under *σ*. *W* is said to be an admissible invariant set for *J* if (a) *W* is the closure of an open set in *X*; (b) if *u*_{
n
} = *σ*(*t*_{
n
}, *v*) → *u* in *H* as *t*_{
n
} → ∞ for some *v* ∉ *W* and *u* ∈ *K*, then *u*_{
n
} → *u* in *X*; (c) If *u*_{
n
} ∈ *K* ∩ *W* is such that *u*_{
n
} → *u* in *H*, then *u*_{
n
} → *u* in *X*; (d) For any *u* ∈ *∂W*\*K*, we have $\sigma \left(t,u\right)\in \ddot{W}$ for *t >* 0.

Now let

*S* =

*X*\

*W*,

*W* =

*P* ∪ (-

*P*). Similar to the proof described in the article [

10], the

*W* is an admissible invariant set for

*J* in the following section 4. We define

$\begin{array}{c}{\varphi}^{*}=\left\{\Gamma \right|\Gamma (t,x):[0,1]\}X\to X\phantom{\rule{0.25em}{0ex}}\text{is}\phantom{\rule{0.25em}{0ex}}\text{continuous}\phantom{\rule{0.25em}{0ex}}\text{in}\phantom{\rule{0.25em}{0ex}}\text{the}\phantom{\rule{0.25em}{0ex}}X\text{-topology}\phantom{\rule{0.25em}{0ex}}\text{and}\\ \Gamma (t,W)\subset W\}.\end{array}$

In the article [7], a new linking theorem is given under the condition (PS). Since the deformation still holds under the condition (C) (see [1]), the following theorem also holds.

**Theorem 3.1** Suppose that *W* is an admissible invariant set of *J* and *J* ∈ *C*^{1}(*H*, ℝ) such that

(*J*_{1})*J* satisfies condition (C) in ]0, +∞[;

(

*J*_{2}) There exists a closed subset

*A* ⊂

*H* and a Hilbert manifold

*B* ⊂

*H* with boundary ∂

*B* satisfying

- (a)
there exist two constants

*β > α* ≥ 0 s.t.

$J\left(u\right)\le \alpha ,\forall u\in \partial B;\phantom{\rule{1em}{0ex}}J\left(u\right)\ge \beta ,\forall u\in A$

i.e.,

${a}_{0}:=\underset{\partial B}{sup}J\le {b}_{0}:=\underset{A}{inf}J$.

- (b)

- (c)
$\underset{u\in B}{sup}J\left(u\right)<+\infty $.

Then,

*a** defines below is a critical value of

*J*${a}^{*}=\underset{\Gamma \in {\varphi}^{*}}{inf}\underset{\Gamma \left(\left[0,1\right],A\right)\cap S}{sup}J\left(u\right).$

Furthermore, assume 0 ∉ *K*_{
a
}*, then *K*_{
a
}_{*} ∩ *S* ≠ ∅, if *a** *> b*_{0} and *K*_{
a
}_{*} ∩ *A* ≠ ∅, if *a** = *b*_{0}.

In this article, we shall consider the symmetry given by a ℤ_{2} action, more precisely even functionals.

**Theorem 3.2** Suppose *J* ∈ *C*^{1}(*H*, ℝ) and the positive cone *P* is an admissible invariant for *J*, *K*_{
c
} ∩ *∂P* = ∅, for *c >* 0, such that

(*J*_{1}) *J* satisfies condition (C) in ]0, +∞[, and *J*(0) ≥ 0;

(

*J*_{2}) There exist two closed subspace

*H*^{+},

*H*^{-} of

*H*, with codim

*H*^{+} *<* +∞ and two constants

*c*_{∞} *> c*_{0} *> J*(0) satisfying

$J\left(u\right)\ge {c}_{0},\forall u\in {S}_{\rho}\cap {H}^{+};\phantom{\rule{1em}{0ex}}J\left(u\right)<{c}_{\infty},\forall u\in {H}^{-}.$

(*J*_{3}) *J* is even.

Hence, if dim *H*^{-}*>* codim *H*^{+}+1, *then J* possesses at least *m* := dim *H*^{-} -codim *H*^{+} - 1 (*m* := dim *H*^{-} -1 resp.) distinct pairs of critical points in *X*\*P* ∪ (-*P*) with critical values belong to [*c*_{0}, *c*_{∞}].

**Remark 3.1** The above theorem locates the critical points more precisely than Theorem 3.3 in [10].

We shall use pseudo-index theory to prove Theorem 3.2. First, we need the notation of genus and its properties, see [

10,

12]. Let

${\Sigma}_{X}=\left\{A\subset X:A\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{is}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{closed}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}X,A=-A\right\};$

with more preciseness, we denote *i*_{
X
}(*A*) to be the genus of *A* in *X*.

**Proposition 3.2** Assume that

*A*,

*B* ∈ ∑

_{
X
},

*h* ∈

*C*(

*X*,

*X*) is an odd homeomorphism, then

- (i)
*i*_{
X
}(*A*) = 0 if and only if *A* = ∅;

- (ii)
*A* ⊂ *B* ⇒ *i*_{
X
}(*A*) ≤ *i*_{
X
}(*B*) (monotonicity);

- (iii)
*i*_{
X
}(*A* ∪ *B*) ≤ *i*_{
X
}(*A*) + *i*_{
X
}(*B*) (subadditivity);

- (iv)
${i}_{X}\left(A\right)\le {i}_{X}\left(\overline{h\left(A\right)}\right)$ (supervariancy);

- (v)
if *A* is a compact set, then *i*_{
X
}(*A*) *<* +∞ and there exists *δ >* 0 s.t. *i*_{
X
}(*N*_{
δ
}(*A*)) = *i*_{
X
}(*A*), where *N*_{
δ
}(*A*) denotes the closed *δ* - neighborhood of *A* (continuity);

- (vi)
if *i*_{
X
}(*A*) *> k*, *V* is a *k*-dimensional subspace of *X*, then *A* ∩ *V*^{⊥} ≠ ∅;

- (vii)
if *W* is a finite dimensional subspace of *X*, then *i*_{
X
}(*h*(*S*_{
ρ
}) ∩ *W* ) = dim *W*.

- (viii)
Let

*V*,

*W* be two closed subspaces of

*X* with codim

*V <* +∞, dim

*W <* +∞. Hence, if

*h* is bounded odd homeomorphism on

*X*, then we have

${i}_{X}\left(W\cap h\left({S}_{\rho}\cap V\right)\right)\ge dimW-\mathsf{\text{codim}}\phantom{\rule{2.77695pt}{0ex}}V.$

The proposition is still true when we replace ∑_{
X
} by ∑_{
H
} with obvious modification.

**Proposition 3.3** [10, 11] If *A* ∈ ∑_{
X
} with 2 ≤ *i*_{
X
}(*A*) *<* ∞, then *A* ∩ *S* ≠ ∅.

**Proposition 3.4** Let *A* ∈ ∑_{
H
}, then *A* ∩ *X* ∈ ∑_{
X
} and *i*_{
H
}(*A*) ≥ *i*_{
X
}(*A* ∩ *X*).

Now, we shall discuss about the notion of pseudo-index.

**Definition 3.2** [

1] Let

$I=\left(\Sigma ,\phantom{\rule{2.77695pt}{0ex}}\mathscr{H},\phantom{\rule{2.77695pt}{0ex}}i\right)$ be an index theory on

*H* related to a group

*G*, and

*B* ∈ ∑. We call a pseudo-index theory (related to

*B* and

*I*) a triplet

${I}^{*}=\left(B,{\mathscr{H}}^{*},{i}^{*}\right)$

where

${\mathscr{H}}^{*}\subset \mathscr{H}$ is a group of homeomorphism on

*H*, and

*i** : ∑ → ℕ ∪ {+∞} is the map defined by

${i}^{*}\left(A\right)=\underset{h\in {\mathscr{H}}^{*}}{min}i\left(h\left(A\right)\cap B\right).$

**Proof of Theorem 3.2** Consider the genus

$I=\left(\Sigma ,\phantom{\rule{2.77695pt}{0ex}}\mathscr{H},\phantom{\rule{2.77695pt}{0ex}}i\right)$ and the pseudo-index theory relate to

*I* and

*B* =

*S*_{
ρ
} ∩

*H*^{+},

${I}^{*}=\left({S}_{\rho}\cap {H}^{+},{\mathscr{H}}^{*},{i}^{*}\right)$, where

$\begin{array}{c}{\mathscr{H}}^{*}=\left\{h\right|h\phantom{\rule{0.25em}{0ex}}\text{is}\phantom{\rule{0.25em}{0ex}}\text{an}\phantom{\rule{0.25em}{0ex}}\text{odd}-\text{bounded}\phantom{\rule{0.25em}{0ex}}\text{homeomorphism}\phantom{\rule{0.25em}{0ex}}\text{on}\phantom{\rule{0.25em}{0ex}}H\phantom{\rule{0.25em}{0ex}}\text{and}\phantom{\rule{0.25em}{0ex}}h(u)=u\phantom{\rule{0.25em}{0ex}}\text{if}\\ u\overline{)\in}{J}^{-1}(]0,+\infty [)\}.\end{array}$

Obviously, conditions (

*a*_{1})(

*a*_{2}) of Theorem 2.9 [

1] are satisfied with

*a* = 0,

*b* = +∞ and

*b* =

*S*_{
ρ
} ∩

*H*^{+}. Now, we prove the condition that (

*a*_{3}) is satisfied with

$\u0100={H}^{-}$. It is obvious that

$\overline{A}\subset {J}^{-1}(]-\infty ,{c}_{\infty}])$, and by property (iv) of genus, we have

$\begin{array}{lll}\hfill {i}^{*}\left(\u0100\right)={i}^{*}\left({H}^{-}\right)& =\underset{h\in {\mathscr{H}}^{*}}{min}i\left(h\left({H}^{-}\right)\cap {S}_{\rho}\cap {H}^{+}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\underset{h\in {\mathscr{H}}^{*}}{min}i\left({H}^{-}\cap {h}^{-1}\left({S}_{\rho}\cap {H}^{+}\right)\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$

Now, by (viii) of Proposition 3.2, we have

$i\left({H}^{-}\cap {h}^{-1}\left({S}_{\rho}\cap {H}^{+}\right)\right)\ge dim{H}^{-}-codim{H}^{+}.$

Therefore we get

${i}^{*}\left(\u0100\right)\ge dim{H}^{-}-\mathsf{\text{codim}}\phantom{\rule{2.77695pt}{0ex}}{H}^{+}.$

Then, by Theorem 2.9 in [

11] and Proposition 3.3 above, the numbers

${c}_{k}=\underset{A\in {\Sigma}_{k}}{inf}\underset{u\in A\cap S}{sup}J\left(u\right),\phantom{\rule{1em}{0ex}}k=2,\dots ,dim{H}^{-}-\mathsf{\text{codim}}\phantom{\rule{2.77695pt}{0ex}}{H}^{+}.$

are critical values of

*J* and

$J\left(0\right)<{c}_{0}\le {c}_{k}\le {c}_{\infty},\phantom{\rule{1em}{0ex}}k=2,\dots ,dim{H}^{-}-codim{H}^{+}.$

(3.1)

If for every

*k*,

*c*_{
k
} ≠

*c*_{
k
}_{+1}, then we get the conclusion of Theorem 3.2. Assume now that

$c={c}_{k}=\cdots ={c}_{k+r}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{with}}\phantom{\rule{2.77695pt}{0ex}}r\ge 1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}k+r\le dim{H}^{-}-\mathsf{\text{co}}dim\phantom{\rule{2.77695pt}{0ex}}{H}^{+}.$

Then, similar to the proof of Theorem 2.9 [

11], where

*K*_{
c
} is replaced by

*K*_{
c
}∩

*S* and

*A* by

*A* ∩

*S*, we have

$i\left({K}_{c}\cap S\right)\ge r+1\ge 2$

(3.2)

Now, from Proposition 3.3 and (3.1), we deduce that

$0\notin {K}_{c}\cap S.$

(3.3)

Since a finite set (not containing 0) has genus 1, we deduce from (3.2) and (3.3) that *K*_{
c
} above contains infinitely many sign-changing critical points. Therefore, *J* has at least *m* := dim *H*^{-} -codim *H*^{+} -1 distinct pairs of sign-changing critical points in *X*\*P* ∪ (-*P*) with critical values belonging to [*c*_{0}, *c*_{∞}].

If codim *H*^{+} = 0, then we consider *c*_{
j
} for *j* ≥ 2. As per the above arguments, $J\left(0\right)<{c}_{0}\le {c}_{2}\le {c}_{3}\le \cdots \le {c}_{dim{H}^{-}}\le {c}_{\infty}$ and if *c* := *c*_{
j
} = ⋯ = *c*_{
j
}_{+}_{
l
} for 2 ≤ *j* ≤ *j* + *l* ≤ dim *H*^{-} with *l* ≥ 1, then *i*(*K*_{
c
} ∩ *S*) ≥ *l* + 1 ≥ 2.

Therefore, *J* has at least dim *H*^{-} -1 pairs of sign-changing critical points with values belong to [*c*_{0}, *c*_{∞}]. ■

**Remark 3.2** Theorem 3.1 above can also be proved by the pseudo-index theory in the same way as Theorem 3.2.