In this section, we give the proof of our main results. Firstly, we prove Theorem 2.1.
Proof of Theorem 2.1 Note that satisfies the following equation:
for all . Since
By Moser’s iteration, we have
for some C independent of n. To show that is a critical point of , we use some arguments in [22, 23] (see more references therein). In (5.1), we choose , , where , , , and is a constant. Substituting into (5.1), we have
Note that , are positive for M large enough. By Fatou’s lemma, the weak convergence of and the fact that is bounded, we have
Let , . We may choose , such that , and . Then we obtain
for all , .
Similarly, we may obtain an opposite inequality. Thus we have
for all . That is, is a critical point of and a solution for system (1.1). By doing approximations, we have in the place of of (5.7)
Setting in (5.1), we have
Using as , (5.8), (5.9) and lower semi-continuity, we obtain
In particular, we have
as . This completes the proof of Theorem 2.1. □
Next, we apply the mountain pass theorem to obtain the existence of critical points of . Set
Let us consider the functional
Here and in what follows, we denote . The functional satisfies condition. Similarly, we may verify that satisfies condition. By the ε-Young inequality, for any , there exists such that
for ε, ρ small. Thus we have
for and for small enough. Choose , and . Define a path by . When T is large enough, we have
for some m independent of .
From the mountain pass theorem we obtain that
is a critical value of .
Let be a critical point corresponding to . We have . Thus is a positive critical point of by the strong maximum principle. In summary, we have the following.
There exist positive constants
has a positive critical point
Finally, we give the proof of Theorem 2.2.
Proof of Theorem 2.2 For a positive solution of system (1.1), the proof follows from Proposition 5.1 and Theorem 2.1. A similar argument gives a negative solution of system (1.1). This completes the proof of Theorem 2.2. □