Definition 3.1 ([, Definition A.3])
On the space , we define the norm
On the space , we define the norm
Lemma 3.2 ([, Theorem A.4])
Assume , and , then for every , , the operator
be the mapping given by
then is bounded and continuous.
Proof By the definition of ,
Therefore is bounded. If in , by (3.1), then . Hence is continuous. □
Lemma 3.4 Suppose the nonlinearity f satisfies (f1)-(f3), then the functional , and
where , here is compact. In addition, each critical point of J is a weak solution of problem (1.1).
Proof By Lemma 3.3, we only need to prove is continuous. By Hölder inequality
where , . If in , then in . It follows from (f3) and Lemma 3.2 that
Assume in . Since is compact, then in . By Lemma 3.2 and (3.2), is compact. □
Lemma 3.5 ([, Lemma 2.1])
There exist constants and , such that for all , , we have
Lemma 3.6 Let be defined in Lemma 3.3. If in and , then in .
Proof If in , then is bounded in .
If , by Lemma 3.5,
so in .
If , by Lemma 3.5 and the Hölder inequality,
Lemma 3.7 Assume that f satisfies (f1)-(f3). Let be a sequence such that
then has a subsequence which converges to a critical point of the functional J.
Proof First we show that each sequence satisfying , , as , is bounded. By (f2) and (f3),
where in the assumption (f2), so is bounded in .
Since E is reflexive, is reflexive, then has a weakly convergent subsequence. Going if necessary to a subsequence, let . By Lemma 3.4 , and by the definition of , , then converges. So we assume . Observe that
By Lemma 3.6, .
Next we want to show that is a critical point of J, i.e. . By Lemma 3.3, ,
By (3.3), is a critical point of J. □
Now we give the proof of Theorems 2.1 and 2.2 by applying the fountain theorem and the principle of symmetric criticality. First we recall some properties of Banach space.
According to the results in , there exists a Schauder basis for E. Let , then is a Schauder basis for . Since is reflexive, there are , which are characterized by the relations , forming a basis for .
and define a group action of .
Lemma 3.8 If , then
Proof It is clear that , so we assume for , , as . For every , there exists such that and . By the definition of , in . Since the imbedding is compact, then in . Thus we get . □
Proof of Theorem 2.1 Note that J is -invariant, by the principle of symmetric criticality [, Theorem 5.4], any critical point of is a solution of problem (1.1). J is invariant with respect to the action .
Now we claim that satisfies the assumptions of the fountain theorem.
By the assumptions (f1) and (f3), for ,
where is the lower bound of . Choose , by Lemma 3.8, for , , , and as ,
This proves (A2).
Now we want to show that the condition (A1) is satisfied. By integrating, we obtain from (f2) and (f3) that, there exist two constants , such that for any , . Hence,
Since is finite dimensional, all norms are equivalent on . Therefore, and imply that
So there exists such that (A1) is satisfied.
condition is proved above. By Theorem 2.8, we find, for , that
are critical values of the functional J. So we can get an unbounded sequence of solutions of (1.1), and the solutions are radial. □
Proof of Theorem 2.2 In this proof, we will show that it suffices to find the critical points of J restricted to a subspace of invariant functions. The proof is similar to Theorem 1.31 in .
Let be a fixed integer different from . The action of on E is defined by . For is compatible with , the embedding () is compact (or see  for details).
Let be the involution defined on by
The action of on is defined by
It is clear that 0 is the only radial function on the set
Moreover, the embedding is compact. As in the proof of Theorem 2.1, we obtain a sequence of non-radial solutions of (1.1). □