We assume that

. Let

denote the usual bases in

and set

Then

is the topological direct sum of subspaces

,

,

and

, where

and

are finite dimensional subspaces. We also set

We have the following two pairs of the sphere-torus variational linking inequalities.

Lemma 3.1. (First Sphere-Torus Variational Linking).

Assume that
satisfies the conditions
,
,
, and the condition

(

*H* 2)

^{'}
suppose that there exist

and

such that

and

Then there exist

,

,

, and

such that

, and for any

and

with

and

Proof.

Let

. By

we have

for some

. Since

, there exists

such that if

, then

. Thus,

. Moreover, if

, then

, so we have

. Next, we will show that there exist

,

and

such that if

, then

. Let

with

,

,

, where

is a small number. Let

for some

and

. Then

and

. By

, there exists

such that

Since
,
, and
, there exist a small number
and
with
and
such that if
and
, then
. Thus, we have
. Moreover, if
and
, then we have
. Thus,
. Thus, we prove the lemma.

Lemma 3.2.

Let
be the number introduced in Lemma 3.1. Then for any
and
with
and
, if
is a critical point for
, then
.

Proof.

We notice that from Lemma 3.1, for fixed

, the functional

is weakly convex in

, while, for fixed

, the functional

is strictly concave in

. Moreover,

is the critical point in

with

. So if

is another critical point for

, then we have

So we have
.

Let

be the orthogonal projection from

onto

and

Then

is the smooth manifold with boundary. Let

. Let us define a functional

by

Let us define the functional

by

Then

. We note that if

is the critical point of

and lies in the interior of

, then

is the critical point of

. We also note that

We note that
and
have the same topological structure as
,
,
and
, respectively.

Lemma 3.3.

satisfies the

condition with respect to

for every real number

such that

Proof.

Let

be a sequence such that

,

be a sequence in

such that

, for all

,

and

. Set

(and hence

) and

. We first consider the case in which

, for all

. Since for

large

, we have

In the first case, the claim follows from the limit Palais-Smale condition for

. In the second case,

. We claim that

is bounded. By contradiction, we suppose that

and set

. Up to a subsequence

weakly for some

. By the asymptotically linearity of

, we have

where

. Passing to the limit we, get

Since

and

are bounded and

in

,

. On the other hand, we have

Since

converges to 0 weakly and

is bounded,

. Since

,

converges to 0 strongly, which is a contradiction. Hence,

is bounded. Up to a subsequence, we can suppose that

converges to

for some

. We claim that

converges to

strongly. We have

By

and the boundedness of

,

That is,

converges. Since

,

converges, so

converges to

strongly. Therefore, we have

So we proved the first case.

We consider the case
, that is,
. Then
, for all
. In this case,
and
. Thus, by the same argument as the first case, we obtain the conclusion. So we prove the lemma.

Proposition 3.4.

Assume that

satisfies the conditions

,

,

,

. Then there exists a number

such that for any

and

with

and

, there exist at least two nontrivial critical points

,

, in

for the functional

such that

where
,
, and
are introduced in Lemma 3.1.

Proof.

First, we will find two nontrivial critical points for

. By Lemma 3.1,

satisfies the torus-sphere variational linking inequality, that is, there exist

,

,

, and

such that

, and for any

and

with

and

By Lemma 3.3,

satisfies the

condition with respect to

for every real number

such that

Thus by Theorem 2.5, there exist two critical points

,

for the functional

such that

Setting

,

, we have

We claim that

, that is

, which implies that

are the critical points for

in

, so

are the critical points for

in

. For this we assume by contradiction that

. From (3.12),

, namely,

,

, are the critical points for

. By Lemma 3.2,

, which is a contradiction for the fact that

Lemma 3.2 implies that there is no critical point

such that

Hence,
,
. This proves Proposition 3.4.

Lemma 3.5. (Second Sphere-Torus Variational Linking).

Assume that
satisfies the conditions
,
,
, and the condition

(

*H* 2)

^{''}
suppose that there exist

and

such that

and

Then there exist

,

,

, and

such that

, and for any

and

with

and

,

Proof.

Let

. By

we have

for some

. Since

, there exists

such that if

, then

. Thus we have

. Moreover, if

, then

, so we have

. Next, let

with

, where

is a small number. We also let

and

. Then

and

. By

, there exists

such that

Since
and
, there exist a small number
and
with
and
such that if
and
, then
. Thus we have
.

Moreover, if
, then
. Thus we have
. Thus we prove the lemma.

Lemma 3.6.

For any
there exists a constant
such that for any
and
with
and
, if
is a critical point for
with
, then
.

Proof.

By contradiction, we can suppose that there exist

, a sequence

,

such that

,

with

, and a sequence

in

such that

and

. We claim that

is bounded. If we do not suppose that

, let us set

. We have up to a subsequence, that

weakly for some

. Furthermore,

Adding (3.37) and (3.39), we have

Dividing by

and going to the limit, we have

which is a contradiction since

. So

is bounded and we can suppose that

for

. From (3.42), we have

Thus,

converges to

strongly. We claim that

. Assume that

. By (

*H* 1)

, for some

and

. If

with

for

and

,

If

,

, and

which is absurd because of
and
. Thus
. We proved the lemma.

Let

be the orthogonal projection from

onto

and

Then

is the smooth manifold with boundary. Let

. Let us define a functional

by

Let us define the functional

by

Then

. We note that if

is the critical point of

and lies in the interior of

, then

is the critical point of

. We also note that

We note that
and
have the same topological structure as
,
,
, and
, respectively.

We have the following lemma whose proof has the same arguments as that of Lemma 3.5 except the space
,
,
instead of the space
,
,
.

Lemma 3.7.

satisfies the

condition with respect to

for every real number

such that

where
,
, and
are introduced in Lemma 3.5.

Proposition 3.8.

Assume that

satisfies the conditions

,

,

, and

. Then there exists a small number

such that for any

and

with

and

, there exist at least two nontrivial critical points

,

, in

for the functional

such that

where
,
, and
are introduced in Lemma 3.5.

Proof.

It suffices to find the critical points for

. By Lemma 3.5,

satisfies the torus-sphere variational linking inequality, that is, there exist

,

,

, and

such that

, and for any

and

with

By Lemma 3.7,

satisfies the

condition with respect to

for every real number

such that

Then by Theorem 2.5, there exist two critical points

,

for the functional

such that

Setting

,

, we have

We claim that

, that is

, which implies that

are the critical points for

, so

are the critical points for

. For this we assume by contradiction that

. From (3.54),

, namely,

,

, are the critical points for

. By Lemma 3.6,

, which is a contradiction for the fact that

It follows from Lemma 3.6 that there is no critical point

such that

Hence,
,
. This proves Proposition 3.8.

Proof of Theorem 1.1.

Assume that

satisfies conditions

–

. By Proposition 3.4, there exist

,

,

, and

such that for any

and

with

,(1.1) has at least two nontrivial solutions

,

, in

for the functional

such that

By Proposition 3.8, there exist

,

,

, and

such that for any

and

with

and

, (1.1) has at least two nontrivial solutions

,

, in

for the functional

such that

Then for any
and
with
and
, (1.1) has at least four nontrivial solutions, two of which are in
and two of which are in
.