Existence of a Nontrivial Solution for a Class of Superquadratic Elliptic Problems ()
We prove an abstract result on the existence of a critical point for the functional 
on a Hilbert space via the local linking theorem. Different from the works in the literature, the new theorem is constructed under the 
condition instead of 
condition.
1. Introduction and Main Results
Consider the Dirichlet boundary value problem
(1)
where 
and 
is a bounded domain whose boundary is a smooth manifold.
We assume that
, where
. In [1], Li and Willem established the existence of a nontrivial solution for problem (1) under the following well-known Ambrosetti-Rabinowitz superlinearity condition: there exists 
and 
such that
(AR)
for all 
and
, which has been used extensively in the literature; see [1-4] and the references therein. It is easy to see that condition (AR) does not include some superquadratic nonlinearity like
(G0)
In [5], Qin Jiang and Chunlei Tang completed the Theorem 4 in [1], and obtained the existence of a nontrivial solution for problem (1) under a new superquadratic condition which covered the case of (G0). The conditions are as follows:
(G1)
, as 
uniformly on
(G2)
, as 
uniformly on
(G3) There are constants 
and 
such that
for all
(G4) There are constants
, 
and
such that
for all 
and
If 0 is an eigenvalue of 
(with Dirichlet boundary condition) assume also the condition that:
(G5) There exists 
such that:
1)
, for all
,
; or 2)
, for all
,
.
Note that (G4) is also (AR) type condition.
The aim of this paper is to consider the nontrivial solution of problem (1) in a more general sense. Without the Ambrosetti-Rabinowitz superlinearity condition (AR) or (G4), the superlinear problems become more complicated. We do not know in our situations whether the (PS) or 
sequence are bounded. However, we can check that any Cerami (or
) sequence is bounded. The definition of 
(or
) sequence can be found in [6].
We will obtain the same conclusion under the 
condition instead of 
condition. So we only need the following conditions instead of (G3) (G4):
(G3') Let 
satisfying 1) 
if
2) 
if
, where
,
.
It is easy to see that function
satisfies conditions of (G1) (G2) (G5) and (G3').
Our main result is the following theorem:
Theorem 1.1. Suppose that 
satisfies (G1) (G2) (G5) and (G3'). If 0 is an eigenvalue of 
(with Dirichlet boundary condition). Then problem (1) has at least one nontrivial solution.
Remark 1. There are many functions which are superlinear but it is not necessary to satisfy AmbrosettiRabinowitz condition. For example,
where
. Then it is easy to check that (AR) does not hold. On the other hand, in order to verify (AR), it usually is an annoying task to compute the primitive function of 
and sometimes it is almost impossible. For example,
where
.
Remark 2. Our condition is much weaker than (AR) type condition (cf. [6]).
2. Proof of Theorem
Define a functional 
in the space 
by
where
,
, 
is the space spanned by the eigenvectors corresponding to negative (positive) eigenvalue of
.
In this paper, we shall use the following local linking theorem (Lemma 2.1) to prove our Theorem . Let 
be a real Banach space with 
and 
such that
, 
. For every multi-index
, let
. We know that
,
. A sequence 
is admissible if for every 
there is 
such that 
. We say 
satisfies the 
condition if every sequence 
such that 
is admissible and satisfies
contains a subsequence which converges to a critical point of
.
Lemma 2.1. ([6]) Suppose that 
satisfies the following assumptions:
(f1) 
has a local linking at 0(f2) 
satisfies 
condition(f3) 
maps bounded sets into bounded sets(f4) For every
, 
as
, on
.
Then 
has at least two critical points.
Proof of Theorem 1. We shall apply Lemma 2.1 to the functional 
associated with (1), we consider the case where 0 is an eigenvalue of 
and
(2)
The other case are similar.
1) 
and 
maps bounded sets into bounded sets.
Let
,
. Choose Hilbertian basis 
for 
and 
for
, define
Assumption (G3') implies there are constants 
such that
(3)
so
where
.
Hence 
and maps bounded sets into bounded sets.
In fact,
so (f3) holds.
2) 
has a local linking at 0.
It follows from (g2) and (g3) that, for any
, there exists
, such that
(4)
we obtain, on
, for some
,
choosing
, then
,
.
Decompose 
into 
when 
,
. Also set 
. Since 
is a finitedimensional space, there exists
, such that
(5)
First we set 
and
On
, we have, by (5)
hence, by (2)
On
, we have also by (5)
hence, by (4)
and for some 
Therefore we deduce that
choosing
, then
, 
, 
Let
, then (f1) holds.
3) 
satisfies 
condition.
Consider a sequence 
such that 
is admissible and
I) 
is bounded.
For n large, from assumption (g3'), with
, for some
,
where 
So
(6)
Arguing indirectly, assume
. Set 
, Then 
and 
for all 
. In addition, using (6)
hence by Interpolation inequality for 
norms, for 
(7)
where 
or
.
Since
,
so
From (7) for some 
as
, therefore, 
, a contradiction. Hence 
sequence is bounded.
II) From (I) we see that 
is bounded in
, going if necessary to a subsequence, we can assume that 
in
. Since
, 
in
.
which implies that 
in
. Similarly, 
in
. It follows then that 
in 
and
.
4) Finally, we claim that, for every
,
Indeed, by (g1) we have, there exists 
such that
so on
,
where
. □
NOTES
#Corresponding author.