Multiplicity of Positive Solutions for Semilinear Elliptic Systems

and Applied Analysis 3 Let Kλ,μ : E → R be the functional defined by Kλ,μ (z) = ∫ Ω (λf (x) |u| q + μg (x) |V| q ) dx ∀z = (u, V) ∈ E. (11) We know that Iλ,μ is not bounded below on E. From the following lemma, we have that Iλ,μ is bounded from below on the Nehari manifoldNλ,μ defined in (9). Lemma 3. The energy functional Iλ,μ is coercive and bounded below onNλ,μ. Proof. If z = (u, V) ∈ Nλ,μ, then by (10), the Hölder inequality, and the Sobolev embedding theorem, we get Iλ,μ (z) = 2 ∗ − 2 22 ‖z‖ 2 E − 2 ∗ − q 2q Kλ,μ (z) (12) ≥ 1 N ‖z‖ 2 E − 2 ∗ − q 2q γ∞S −q/2 |Ω| (2 ∗ −q)/2 ∗ × (λ 2/(2−q) + μ 2/(2−q) ) (2−q)/2 ‖z‖ q E . (13) Hence, we have that Iλ,μ is coercive and bounded below on Nλ,μ. Define Φλ,μ (z) = ⟨I 󸀠 λ,μ (z) , z⟩ . (14) Then, for z ∈ Nλ,μ,