Multiplicity of positive solutions for critical singular elliptic systems with sign-changing weight function ∗

In this paper, the existence and multiplicity of positive solutions for a critical singular elliptic system with concave and convex nonlinearity and sign-changing weight function, are established. With the help of the Nehari manifold, we prove that the system has at least two positive solutions via variational methods.

The corresponding energy functional of problem (1.1) is defined by It follows from (H1)-(H2) that the functional J λ,µ is of class C 1 (E, R).Moreover, the critical points of J λ,µ are the week solutions of problem (1.1).Existence and multiplicity of solutions for elliptic problems with concave-convex nonlinearities in bounded domain Ω ⊂ R N are studied extensively.
In whole space, Ambrosetti, Garcia and Peral [4] considered the problem (1.2) for t = 0, p = 2 * and proved the existence of Λ > 0 such that problem (1.2) admits at least two non-negative solutions for λ ∈ (0, Λ), provided that f ∈ L 1 (R N ) ∪ L ∞ (R N ) and f + ≡ 0.More recently, under the proper hypothesis, Miotto [3] studied the same problem above and obtained the similar results.
Catrina and Wang [11] proved that S t is achieved by the function where From [10], we have .

By Hardy inequality
we can derive that • t defines an equivalent norm in E. We define the Palais-Smale (PS) sequence and (PS)-condition in E for J λ,µ as follows.
(ii) J λ,µ satisfies the (P S) c -condition in E, if any (P S) c -sequence {z n } in E for J λ,µ has a convergent subsequence.
As consequence of the assumptions (H1)-(H2), we have EJQTDE, 2012 No. 20, p. 4 The same conclusion still holds if f is replaced by g in Lemma 2.1.
Proof.Let z n = (u n , v n ).On the contrary, assume that z n t → ∞.
By passing to a subsequence, we can assume that It easily follows from (2.2) and 1 < q < 2, that z n t → 0 as n → ∞, which is a contradiction.
q ) for some positive constant C 0 only depending on f, g, N, q, t.
We will see that θ λ,µ > −∞.In fact, let z ∈ N λ,µ , then from the proof the Lemma 2.3, we have As in Tarantello [12], we divide N λ,µ in three parts where C f , C g are from Lemma 2.3.
Similar to Lemma 2.6 in [15], we have the following result.
Lemma 3.5 Assume that z is a local minimizer for J λ,µ on N λ,µ and z ∈ N 0 λ,µ , then Proof.The proof is almost the same as that of Theorem 2.3 in [14] and is omitted here.
The following lemma provides a precise description of the (P S) c -sequence for J λ,µ .Lemma 3.6 If 0 < λ has a convergent subsequence.
Proof.By Lemma 2.2, we have {z n } ⊂ E is bounded and there exists z = (u, v) ∈ E. We can assume, by passing to a subsequence if necessary, that z n ⇀ z in E and z n → z a.e. in R N .Now we will show that z ∈ N λ,µ .First, we prove that z = 0. On the contrary, suppose that z = 0. Then by Lemma 2.1, we have Moreover, because {z n } is a (P S) c -sequence, we have It is obvious that c > 0. Thus z n 2 t ≥ c for large n.Then by (3.4) and the definition of S t α,β , we obtain that 2 ) .
and by Lemma 2.1 we have So we can get that where a is a nonnegative constant.
EJQTDE, 2012 No. 20, p. 9 If a = 0, the proof is completed.Assume that a > 0, it follows from (3.5), that which is a contradiction.So the proof is completed.
Similar to the proof of Theorem 4. Next, we establish the existence of a positive solution of the system (1.1) on N − λ,µ .First, we consider which is an extremal function for S t , where U is defined in (2.1).Since f + , g + are continuous functions in R N and Σ = Σ f ∩ Σ g = φ.Following the method of [17], without loss of generality, we may assume the Σ is a domain of positive measure.