In this section, we always assume that and are real symmetric positive definite matrices for all . Let
and for , let
Then E is a Hilbert space with the above inner product, and the corresponding norm is
As usual, for , set
and their norms are defined by
Lemma 2.1 
where and .
), there exist two constants and such that
Lemma 2.2 Suppose that L satisfies (L
). Then, for , E is compactly embedded in ; moreover,
Proof Let . Then . For and , it follows from (2.2), (2.5) and the Hölder inequality that
This shows that (2.3) holds. Hence, from (2.2), (2.6) and the Hölder inequality, one has
This shows that (2.4) holds.
Finally, we prove that E is compactly embedded in . Let be a bounded sequence. Then by (2.1), there exists a constant such that
Since E is reflexive, possesses a weakly convergent subsequence in E. Passing to a subsequence if necessary, it can be assumed that in E. It is easy to verify that
For any given number , we can choose such that
It follows from (2.8) that there exists such that
On the other hand, it follows from (2.3), (2.7) and (2.9) that
Since ε is arbitrary, combining (2.10) with (2.11), we get
This shows that possesses a convergent subsequence in . Therefore, E is compactly embedded in for . □
Lemma 2.3 Suppose that L and W satisfy (L
) and (W1′). Then, for ,
Proof For , it follows from (2.4) and (W1′) that
This shows (2.12) holds. □
Lemma 2.4 Assume that L and W satisfy (L
), (W1′) and (W2′). Then the functional defined by
is well defined and of class
Furthermore, the critical points of f in E are solutions of (1.1) with .
Proof Lemma 2.3 implies that f defined by (2.16) is well defined on E. Next, we prove that (2.17) holds. By (W2′), one can choose an such that
For any , there exists an integer such that for . Then, for any sequence with for and any number , by (W2′), (2.3) and Lemma 2.2, we have
where . Then by (2.16), (2.19) and Lebesgue’s dominated convergence theorem, we have
This shows that (2.17) holds. Observe that for ,
It follows from (2.17) and (2.20) that for all if and only if
So, the critical points of f in E are the solutions of system (1.1) with .
Let us prove now that is continuous. Let in E. Then there exists a constant such that
It follows from (2.1) that
By (W2′), one can choose an such that
From (2.3), (2.17), (2.21), (2.22), (2.23), (W2′) and the Hölder inequality, we have
which implies the continuity of . The proof is complete. □
Lemma 2.5 
Let E be a real Banach space and let satisfy the (PS)-condition. If f is bounded from below, then is a critical value of f.
Lemma 2.6 
Let E be a real Banach space, let with f even, bounded from below, and satisfying the (PS)-condition. Suppose that , there is a set such that K is homeomorphic to by an odd map, and . Then f possesses at least k distinct pairs of critical points.