Let us now state the main results.

Theorem 2.1.

Assume that conditions
and
hold,
, and
for some
; then problem (1.1) has at least three nontrivial solutions.

Theorem 2.2.

Assume that conditions
)–(
hold,
and
for some
; then problem (1.1) has at least three nontrivial solutions.

Consider the following problem:

Define a functional

by

where
and then

Lemma 2.3.

satisfies the (PS) condition.

Proof.

Let

be a sequence such that

as

Note that

for all

Assume that

is bounded, taking

in (2.4). By

, there exists

such that

a.e.

So

is bounded in

. If

as

set

, and then

. Taking

in (2.4), it follows that

is bounded. Without loss of generality, we assume that

in

, and then

in

. Hence,

a.e. in

. Dividing both sides of (2.4) by

, we get

Then for a.e.

, we deduce that

as

where

. In fact, when

by

we have

When

, we have

When

, we have

Since

, by (2.5) and the Lebesgue dominated convergence theorem, we arrive at

Choosing

, we deduce that

where

Now we show that there is a contradiction in both cases of
and

Case 1.

Suppose

then

a.e. in

By

we have

Thus (2.11) implies that

which contradicts to

Case 2.

Suppose

then

and

It follows from (2.11) that

which contradicts to
if
and contradicts to
if

Lemma 2.4.

Let
be the eigenfunction corresponding to
with
. If
, then

(a) there exist
such that
for all
with
;

(b)
as
.

Proof.

By

and

, if

, for any

, there exist

and

such that for all

,

where
if

Choose

such that

By (2.14), the Poincaré inequality, and the Sobolev inequality, we get

So, part (a) holds if we choose
small enough.

On the other hand, if

take

such that

. By (2.15), we have

Since

and

, it is easy to see that

and part (b) is proved.

Lemma 2.5.

Let
, where
. If
satisfies
)–(
then

(i) the functional

is coercive on

, that is,

and bounded from below on
;

(ii) the functional
is anticoercive on
.

Proof.

For

, by

, for any

, there exists

such that for all

,

Choose

such that

This proves (i).

- (ii)
We firstly consider the case

. Write

. Then

and

imply that

It follows from (2.22) that for every

, there exists a constant

such that

For

we have

Integrating (2.25) over

, we deduce that

Let

and use (2.23); we see that

for

a.e.

A similar argument shows that

for

a.e.

. Hence

for
with
, where

In the case of
, we do not need the assumption
and it is easy to see that the conclusion also holds.

Lemma 2.6.

If
, then
satisfies the (PS) condition.

Proof.

Let

be a sequence such that

. One has

for all

If

is bounded, we can take

. By

, there exists a constant

such that

a.e.

So

is bounded in

. If

, as

set

, and then

. Taking

in (2.29), it follows that

is bounded. Without loss of generality, we assume

in

, and then

in

. Hence,

a.e. in

. Dividing both sides of (2.29) by

, we get

Then for a.e.

, we have

as

In fact, if

by

, we have

If

, we have

Since

, by (2.30) and the Lebesgue dominated convergence theorem, we arrive at

It is easy to see that
. In fact, if
, then
contradicts to
. Hence,
is an eigenvalue of
. This contradicts our assumption.

Lemma 2.7.

Suppose that
and
satisfies
. Then the functional
satisfies the (C) condition which is stated in [13].

Proof.

Suppose

satisfies

In view of

, it suffices to prove that

is bounded in

. Similar to the proof of Lemma 2.6, we have

Therefore

is an eigenfunction of

, then

for a.e.

. It follows from

that

holds uniformly in

, which implies that

On the other hand, (2.34) implies that

which contradicts to (2.37). Hence
is bounded.

It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the book [14] for more information on Morse theory.

Let

be a Hilbert space, let

be a functional satisfying the (PS) condition or (C) condition, let

be the

th singular relative homology group with integer coefficients. Let

be an isolated critical point of

with

and let

be a neighborhood of

. The group

is said to be the
th critical group of
at
, where

Let

be the set of critical points of

and

; the critical groups of

at infinity are formally defined by (see [

15])

The following result comes from [14, 15] and will be used to prove the results in this paper.

Proposition 2.8 (see [15]).

Assume that

is bounded from below on

and

as

with

. Then