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  for more information on Morse theory.
Let H be a Hilbert space and be a functional satisfying the (PS) condition or (C) condition, and be the q th singular relative homology group with integer coefficients. Let be an isolated critical point of I with , , and U be a neighborhood of . The group
is said to be the q th critical group of I at , where .
Let be the set of critical points of I and , the critical groups of I at infinity are formally defined by 
From the deformation theorem, we see that the above definition is independent of the particular choice of . If then
For the convenience of our proof, we first recall two interesting results and prove two important propositions.
Proposition 3.1 
Under (H2), if is an isolated critical point of I, then .
Proposition 3.2 
If and for some integer we have and , then either or .
Proposition 3.3 If the assumptions of Theorem 2.1 hold, then
Proof Under the guidance of  and , we begin to prove this result. Let . Indeed, it follows from above Proposition 3.1 that I and have same critical set. Since is dense in E, invoking Proposition 16 of Palais , we have
From equations (20) and (21), we see that in order to prove the proposition, it suffices to show that
In order to prove equation (22), we proceed as follows. We define the sets
Consider the map defined by
Clearly, is a continuous homotopy and for all . Therefore, is contractible in itself.
By equation (3) in (H4), given any , we can find such that
Similarly, from condition (H3), and by choosing even bigger if necessary, we observe that there is a number such that
Moreover, by condition (H2), we have
for some .
Let . By inequalities (23), (24), and (25), for all we have
Recalling that is arbitrary, from (26), we have
Using formula (4) in condition (H4), we see that there exist constants and such that
By (H2) and formula (2) in condition (H4), we have
for some . By inequalities (28) and (29), for any we have
where C is a positive constant. Let be the continuous embedding map. Let denote the duality brackets for the pair . We let , and so
Then, from equation (27), we obtain
From conditions (H2) and (H3), we see that given , we can find such that
Using inequality (33), we have
for , where is defined as
and is a positive constant. So is coercive, thus we find such that . We pick
Then inequality (32) implies that we can find such that
Moreover, the implicit function theorem implies that .
By the choice of a, we have
We define the set . The map defined by
is a continuous deformation of , and for all (see equations (34) and (35)). Therefore, is a strong deformation retract of . Hence we have
Recalling that in the first part of the proof, we established that is contractible. This yields
Combining with equation (36) leads to equation (22), which completes the proof. □
Proposition 3.4 If the assumptions of Theorem 2.1 hold, then
where (V being defined in Lemma 2.3).
Proof By condition (H5), given , we can find such that
Since V is finite dimensional, all norms are equivalent. Thus we can find small such that
for all . Taking inequalities (37) and (38) into account, for all with we have
Similar to the proof of Lemma 2.3, there exists such that
for all and .
On the other hand, for given , it follows from (H2) and (H5) that
for all and . By (41) and Lemma 2.3, we have
for all . From inequality (42), we infer that for ρ small enough we have
From inequalities (40) and (43), we know that I has a local linking at 0. Then invoking Proposition 2.3 of Bartsch and Li , we obtain . □