Existence of non-negative solutions for semilinear elliptic systems via variational methods

. In this paper we consider a semilinear elliptic system with nonlinearities, indeﬁnite weight functions and critical growth terms in bounded domains. The existence result of nontrivial nonnegative solutions is obtained by variational methods.


Introduction
In this paper we consider the existence results of the following two coupled semilinear equation −∆u = λau − pav 2 u|u| p−2 + 2au|v| p , x ∈ Ω, −∆v = λav − pau 2 v|v| p−2 + 2av|u| p , x ∈ Ω, where α and λ are real parameters, p < 2 * − 2, for 2 * = 2N N −2 , Ω is an open bounded domain in R N , N ≥ 3 with a smooth boundary ∂Ω, and a : Ω → R is a sign changing weight function. This work is motivated by the results in the literature for the single equation case, namely the equation of the form were established for the case when a single equation is replace by a system of equations. Also we refer to [7] where ∆ is replaced by ∆ p . In this work we extend this studies to classes of Robin boundary conditions. We prove our existence results via variational methods.
The system (1) is posed in the framework of the Sobolev space H = H 1 (Ω)×H 1 (Ω) with the norm Moreover a pair of functions (u, v) ∈ H is said to be a weak solution of the system (1) if + p Ω av 2 |u| p−2 uφ 1 + au 2 |v| p−2 vφ 2 + 2 Ω au|v| p φ 1 + av|u| p φ 2 = 0 for all (φ 1 , φ 2 ) ∈ H. Thus the corresponding energy functional to the system (1) is defined by where for t = u or v, and It is well known that the weak solutions of the system (1) are the critical points of the Euler functional J λ . Let I be the Euler functional associated with an elliptic problem on a Banach space X. If I is bounded below and has a minimizer on X, thus this minimizer is a critical point of I, and so is a solution of the corresponding elliptic problem. However, the Euler functional J λ is not bounded below on the whole space H, but is bounded on an appropriate subset, and a minimizer on this set (if it exists) gives a solution to the system (1).
Then we introduce the following notation: For any functional f : In fact we have
Moreover for (u, v) ∈ H, there exists a constant k > 0 such that In fact by using the embedding of H 1 (Ω) into L 2 (∂Ω), we have where c 1 is the best Sobolev constant of the embedding of H 1 (Ω) into L 2 (∂Ω), Applying compactly embedding of H in L 2 (Ω) × L 2 (Ω) and L 2 (∂Ω) × L 2 (∂Ω), we have (u n , v n ) → (u, v) in L 2 (Ω) × L 2 (Ω) and L 2 (∂Ω) × L 2 (∂Ω), respectively. Taking (u n , v n ) λ → 0 into account, from (5), we have (u n , v n ) → (0, 0) in L 2 (Ω) × L 2 (Ω), and so u = v = 0. This implies (u n , v n ) → (0, 0) in L 2 (∂Ω) × L 2 (∂Ω), which concludes that This contradicts with the fact (u n , v n ) H = 1 for all n. Hence, . λ and . H are equivalent norms. Now we consider the Nehari minimizing problem It is clear that all critical points of J λ must lie on M λ which is well-known as the Nehari manifold (see [2,8]). We will see below that local minimizers of J λ on M λ contains every non-zero solution of the system (1). First we claim that for In fact, since the function a changes sign, we can choose non-zero function By a similar way we can define K + 0 and K − 0 . For α ∈ [0, 1], we have that λ → K + λ is a concave continuous curve on the interval [λ − α , λ + α ]. By using a similar arguments we have these facts for K − λ . To state our main result, we now present some important properties of K + λ and K − λ .
By the Sobolev embedding theorem and Hölder inequality, there exists ε 0 < 2 N −2 such that By the same argument we have Proof. Suppose that ϕ + is a positive eigenfunction of the linear problem (4) corresponding to the principal eigenvalue λ + α . Then L(ϕ + ) = 0. On the other hand, if λ = 0 be the principal eigenvalue of (4) with corresponding positive principal eigenfunction ϕ, then λ Ω aϕ p+1 > 0 (see [3]). So in this case, since λ + α > 0, we have Ω a(ϕ + ) p+1 > 0. We now let i.e., K + λ + α = 0. By using a similar way we can prove that K − which means that the claim is true.
Using a similar argument, if (u, v) ∈ M λ , then Indeed, The latest equality follows from that and so In fact we proved the following result: 3 Main results Now let (u n , v n ) = (un,vn) (un,vn) λ , then (u n , v n ) λ = 1, i.e., {(u n , v n )} is a bounded sequence in L 2 * (Ω) × L 2 * (Ω) equipped with the norm Moreover by the boundedness of {(u n , v n )} in H, and applying the inequality mentioned in Lemma 1, we derive that the sequence {(G 1 +G 2 )(u n , v n )} is bounded in H, and so the right hand side of the last equality tends to 1. This contradiction proves the lemma.

Now let
Then for λ ∈ (λ + α − δ 2 , λ + α ), if the sequence {(u n , v n )} be a minimizing sequence of J λ on M λ , we derive that This implies {(u n , v n )} is bounded in H and so there exist a subsequence, which for convenience we again denote by for some q > 0. Also we have Now by Lemma 5, we obtain (u n , v n ) λ → 0, i.e., L(u n )+L(v n ) → 0, and if Ω a(u 2 n + v 2 n ) → 0, then we get So by taking λ = 0, we have which is a contradiction. Therefore,
Lemma 6. The production of two Hilbert spaces, is a Hilbert space.
Proof. For Hilbert spaces H 1 and H 2 , define It is easy to see that, the mentioned bilinear form defines an inner product on H 1 ×H 2 .
To state our significant proposition form [9], first let B be a ball in the Hilbert space H, centered at 0 and of radial . Proposition 1. Let Φ be a C 1 -functional on a Hilbert space X = X 1 × X 2 , where X 1 and X 2 are Hilbert spaces, and let Γ be a closed subset in X such that for any (u, v) ∈ Γ with Φ (u, v) = 0 and > 0 small arbitrary, there exists Frechet differentiable function If Φ is bounded below on Γ , then for any minimizing sequence {(u n , v n )} for Φ in Γ , there exists another minimizing sequence Proof. Let η = inf γ∈Γ Φ(γ). Using Ekland variational principle, we get a minimizing sequence Apply the hypothesis on the set Γ with (u, v) = (u * n , v * n ) to find the function Then, ∈ Γ for all small enough δ ≥ 0. By the mean value theorem we have Now we pass to the limit as δ → 0, and we derive Thus, we have This completes the proof.
By using Theorem 2, we have lim n→∞ J λ (u n , v n ) λ = 0. Then {(u n , v n )} is bounded and we can find a weak limit point of the sequence in H, i.e., u n u and v n v both in H 1 (Ω), for some u, v ∈ H 1 (Ω), and so u n → u and v n → v both in L q (Ω) for q < 2N N −2 . In particular for any (h 1 , h 2 ) ∈ H, = lim n→∞ J λ (u n , v n ), (h 1 , h 2 ) ≤ lim n→∞ J λ (u n , v n ) λ (h 1 , h 2 ) = 0, that means J λ (u, v), (h 1 , h 2 ) = 0 for all (h 1 , h 2 ) ∈ Y . Therefore, (u, v) is a weak solution for the system (1).
On the other hand, J λ is weakly lower semicontinuous, and so we have which follows that J λ (u, v) = c and that (u n , v n ) λ → (u, v) λ which implies that u n → u and v n → v both in H 1 (Ω).