We first establish a compactness result.
Lemma 4.1 Suppose that () holds. Then J satisfies the -condition for all
Proof Suppose that satisfies and . The standard argument shows that is bounded in .
For some , we have
Therefore, is a solution to (1.1). Then by the concentration-compactness principle [11–13] and up to a subsequence, there exist an at most countable set , a set of different points , nonnegative real numbers , , , and , , () such that the following convergence holds in the sense of measures:
By the Sobolev inequalities , we have
We claim that is finite, and for any , or .
In fact, let be small enough for any , and for , . Let be a smooth cut-off function centered at such that , for , for and . Then
Then we have
By the Sobolev inequality, ; and then we deduce that or , which implies that is finite.
Now, we consider the possibility of concentration at points (), for small enough that for all and for and , . Let be a smooth cut-off function centered at such that , for and . Then
Thus, we have
From (4.1) and (4.2) we derive that , , and then either or . On the other hand, from the above arguments, we conclude that
If for all and , then , which contradicts the assumption that . On the other hand, if there exists an such that or there exists a with , then we infer that
which contradicts our assumptions. Hence, , as in . □
First, under the assumptions (), (), we have the following notations:
where is a minimal point of , and therefore a root of the equation
Lemma 4.2 Suppose that () holds. Then we have
has the minimizers , , where are the extremal functions of defined as in (2.2).
Proof The argument is similar to that of . □
Lemma 4.3 Under the assumptions of (), we have
Proof Suppose () holds. Define the function
Note that and as t is close to 0. Thus, is attained at some finite with . Furthermore, , where and are the positive constants independent of ε. By using (1.2), we have
and and so .
From (4.3), Lemma 4.2 and Lemma 4.3, it follows that
so . Hence, , and
Proof of Theorem 1.3 Set , where
Suppose that () holds. For all , from the Young and Hardy-Sobolev inequalities, it follows that
and there exists a constant small such that
Since as , there exists such that and . By the mountain-pass theorem , there exists a sequence such that and , as .
From Lemma 4.2 it follows that
By Lemma 4.1 there exists a subsequence of , still denoted by , such that strongly in . Thus, we get a critical point of J satisfying (1.1), and c is a critical value. Set .
Replacing respectively u, ν with and in terms of the right-hand side of (1.1) and repeating the above process, we can get a nonnegative nontrivial solution of (1.1). If , we get by (1.1) and the assumption . Similarly, if , we also have . There, . From the maximum principle, it follows that in Ω. □