In this section, we will prove that problems (1.1)-(1.2) have three distinct solutions by using the variational principle of Ricceri and a local mountain pass lemma.

Theorem 3.1.

Assume that (f1) holds. Suppose further that

(f4) there exists
such that

(f5) there exists
such that
.

Then there exist
and
such that, for every
, problem (1.1) admits at least three distinct solutions which belong to
.

Proof.

By Lemma 2.2, condition (f1) implies that the functional

is coercive. Since

is sequentially weakly lower semicontinuous (see [

16, Propositions 2.5 and 2.6]),

has a global minimizer

. By (f5), we obtain

. Let

. Since

is coercive, we can choose a large enough

such that

Now we prove that

has a strict local minimum at

. By the compact embedding of

into

, there exists a constant

such that

Choosing

, it results that

Therefore, for every

, it follows from (f4) that

which implies that
is a strict local minimum of
in
with
.

At this point, we can apply Lemma 2.5 taking

and

as perturbing terms. Then, for

small enough and any

,

, we can obtain the following.

- (i)

(ii)
(see [16, Theorem 3.6])

Let

be such that

and put

. Owing to the coerciveness of

, there exists

such that

. Since

is continuous, then

Choosing

, hence, for every

with

, one has

and when

Further, from (3.6), we have that

. Since

is arbitrary, letting

, we can obtain that

Therefore, by (3.6) and (3.11),

can be chosen small enough that

and (3.9)-(3.10) hold, for every
.

For a given

in the interval above, define the set of paths going from

to

and consider the real number
. Since
and each path
goes through
, one has
.

By (3.9) and (3.10), in the definition of

, there is no need to consider the paths going through

. Hence, there exists a sequence of paths

such that

and

Applying a general mountain pass lemma without the (PS) condition (see [17, Theorem 2.8]), there exists a sequence
such that
and
as
. Hence
is a bounded
sequence and, taking into account the fact that
is an
type mapping, admits a convergent subsequence to some
. So, such
turns to be a critical point of
, with
, hence different from
and
and
. This completes the proof.

Taking
in Theorem 3.1, we can obtain the existence of three distinct solutions for the Duffing-type equation without perturbation (1.2) as following.

Theorem 3.2.

Assume that (f1), (f4), and (f5) hold; then problem (1.2) admits at least three distinct solutions.

Together with Lemma 2.3 and Lemma 2.4, we can easily show that the following corollary.

Corollary 3.3.

Assume that (f2), (f4), and (f5) hold; then there exist
and
such that, for every
, problem (1.1) admits at least three distinct solutions which belong to
. Furthermore, problem (1.2) admits at least three distinct solutions.

Corollary 3.4.

Assume that (f3), (f4), and (f5); hold, then there exist
and
such that, for every
, problem (1.1) admits at least three distinct solutions which belong to
. Furthermore, problem (1.2) admits at least three distinct solutions.