RESEARC H Open Access Sub-super solutions for (p-q) Laplacian systems Somayeh Haghaiegh 1* and Ghasem Alizadeh Afrouzi 2 * Correspondence: Haghaieghi_ch86@yahoo.com 1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Full list of author information is available at the end of the article Abstract In this work, we consider the system: ⎧ ⎨ ⎩ − p u = λ[g(x)a(u)+f (v)] in  − q v = λ[g(x)b(v)+h(u)] in  u = v =0 on∂ , where Ω is a bounded region in R N with smooth boundary ∂Ω, Δ p is the p-Laplacian operator defined by Δ p u = div (|∇u| p-2 ∇u), p, q > 1 and g (x)isaC 1 sign-changing the weight function, that maybe negative near the boundary. f, h, a, b are C 1 non- decreasing functions satisfying a(0) ≥ 0, b(0) ≥ 0. Using the method of sub-super solutions, we prove the existence of weak solution. 1 Content In this paper, we study the existence of positive weak solution for the following system: ⎧ ⎨ ⎩ − p u = λ[g(x)a(u)+f (v)] in  − q v = λ[g(x)b(v)+h(u)] in  u = v =0 on∂ , (1) where Ω is a bounded region in R N with smooth boundary ∂Ω, Δ p is the p-Laplacian operator defined by Δ p u = div(|∇u | p-2 ∇u), p, q >1andg(x)isaC 1 sign-changing the weigh t functi on, that maybe negative near the boundary . f, h, a, b are C 1 non-decreas- ing functions satisfying a(0) ≥ 0, b(0) ≥ 0. Thi s paper is motivated by results in [1-5]. We shall show the system (1) with sign- changing weight functions has at least one solution. 2 Preliminaries In this article, we use the following hypotheses: (Al) lim f ⎛ ⎝ M(h(s)) 1 q−1 ⎞ ⎠ s p−1 =0 as s ® ∞, ∀M >0 (A2) lim f (s) = lim h (s)=∞ as s ® ∞. (A3) lim a(s) s p−1 = lim b(s) s q−1 =0 as s ® ∞. Let l p , l q be the first eigenvalue of -Δ p ,-Δ q with Dirichlet boundary conditions and  p ,  q be the corresponding positive eigenfunctions with || p || ∞ =|| q || ∞ =1. Haghaiegh and Afrouzi Boundary Value Problems 2011, 2011:52 http://www.boundaryvalueproblems.com/content/2011/1/52 © 2011 Haghaiegh and Afrouzi; licensee Springer. This is an Open Access a rticle distributed under the terms of the Creative Co mmons Attribution License (htt p://creativecommons.org/licenses /by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let m, δ, g, μ p , μ q > 0 be such that  |∇ϕ p | p − λ p ϕ p ≥ m in  δ ϕ p ≥ μ p on  −  δ (2) and  |∇ϕ q | q − λ q ϕ q ≥ m in  δ ϕ p ≥ μ p on  −  δ . (3)  δ = {x ∈ ; d(x, ∂) ≤ δ}. We assume that the weight function g(x) take negative values in Ω δ , but it requires to be strictly positive in Ω-Ω δ . To be precise, we assume that there exist positive co n- stants b and h such that g(x) ≥-b on  δ and g(x) ≥ h on Ω-Ω δ .Lets 0 ≥ 0suchthat ha(s)+f (s)>0,hb(s)+h(s) > 0 for s >s 0 and f 0 =max{0, −f (0)}, h 0 =max{0, −h(0)}. For g such that g r-1 t>s 0 ; t = min {a p , a q }, r = min{p, q} we define A =max ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ γλ p ηa ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ γ 1 p − 1 α p ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ +f ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ γ 1 q − 1 α q ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , γλ q ηb ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ γ 1 q − 1 α q ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ +h ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ γ 1 p − 1 α p ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ B = min ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ mγ βa ⎛ ⎜ ⎜ ⎝ γ 1 p − 1 ⎞ ⎟ ⎟ ⎠ +f 0 , mγ βb ⎛ ⎜ ⎜ ⎝ γ 1 q − 1 ⎞ ⎟ ⎟ ⎠ +h 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ where α p = p−1 p μ p p p−1 and α q = q−1 q μ q q q−1 . We use the following lemma to prove our main results. Lemma 1.1 [6]. Suppose there exist sub and supersolutions (ψ 1 , ψ 2 ) and (z 1 , z 2 ) respectively of (1) s uch that (ψ 1 , ψ 2 ) ≤ ( z 1 , z 2 ). then (1) has a solution (u, v) such that (u, v) Î [(ψ 1 , ψ 2 ), (z 1 , z 2 )]. 3 Main result Theorem 3.1Suppose t hat (A1)-(A3) hold, then for every l Î [A, B], system (1) has at least one positive solution. Proof of Theorem 3.1 We shall verify that (ψ 1 , ψ 2 ) is a sub solution of (1.1) where ψ 1 = γ 1 p−1 p−1 p ϕ p p p−1 ψ 2 = γ 1 q−1 q−1 q ϕ q q q−1 . Haghaiegh and Afrouzi Boundary Value Problems 2011, 2011:52 http://www.boundaryvalueproblems.com/content/2011/1/52 Page 2 of 5 Let W ∈ H 0 1 () with w ≥ 0. Then   |∇ψ 1 | p−2 ∇ψ 1 ∇wdx = γ   (λ p ϕ p p −|∇ϕ p | p )wdx (4) Now, on  δ by (2),(3) we have γ (λ p ϕ p p −|∇ϕ p | p ) ≤−mγ Since l ≤ B then λ ≤ mγ βa(γ 1 p−1 )+f 0 . thus γ (λ p ϕ p p −|∇ϕ p | p ) ≤−mγ ≤ λ  −βa (γ 1 p−1 ) − f 0  ≤ λ  g(x)a (γ 1 p−1 ) − f 0  λ  g(x) a  p−1 p γ 1 p−1 ϕ p 1 p−1  + f  q−1 q γ 1 q−1 ϕ q 1 q−1  then by (4)   δ |∇ψ 1 | p−2 ∇ψ 1 ∇wdx ≤   δ λ  g(x) a  p−1 p γ 1 p−1 ϕ p p p−1  + f  q−1 q γ 1 q−1 ϕ q q q−1  wdx A similar argument shows that   δ |∇ψ 2 | q−2 ∇ψ 2 ∇wdx ≤   δ λ  g(x) b  q−1 q γ 1 q−1 ϕ q 1 q−1  + h  p−1 p γ 1 p−1 ϕ p 1 p−1  wdx Next, on  −  δ . Since l ≥ A, then λ ≥ γλ p ηa  γ 1 p−1 α p  + f  γ 1 q−1 α q  so we have γ (λ p ϕ p p −|∇ϕ p | p ) ≤ γλ p ≤ λ  ηa  γ 1 p−1 α p  + f  γ 1 q−1 α q  ≤ λ[g(x)a(ψ 1 )+f (ψ 2 )],  −  δ Haghaiegh and Afrouzi Boundary Value Problems 2011, 2011:52 http://www.boundaryvalueproblems.com/content/2011/1/52 Page 3 of 5 Then by (4) on we have − p ψ 1 ≤ λ[g(x)a(ψ 1 )+f (ψ 2 )] on  −  δ A similar argument shows that − q ψ 2 ≤ λ[g(x)b(ψ 2 )+h(ψ 1 )] We suppose that  p and  q be solutions of  − p κ p =1 in κ p =0 on∂  − q κ q =1 in κ q =0 on∂ respectively, and μ’ p =|| p ||  ,|| q ||  = μ’ q . Let (z 1 , z 2 )= ⎛ ⎝ c μ  p λ 1 p−1 κ p ,  2h  cλ 1 q−1  1 q−1 λ 1 q−1 κ q ⎞ ⎠ . Let W ∈ H 0 1 () with w ≥ 0. For sufficient C large μ  p p−1 ⎡ ⎣ ||g|| ∞ a  Cλ 1 p−1  + f  2h(Cλ 1 p−1  1 q−1 λ 1 q−1 μ  q  ⎤ ⎦ C p−1 ≤ 1 then  |∇z 1 | p−2 ∇z 1 ∇wdx = λ  C μ  p  p−1  wdx ≥ λ  ⎡ ⎣ ||g|| ∞ a (Cλ 1 p−1 )+f ⎛ ⎝ (2h (Cλ 1 p−1 )) 1 q−1 λ 1 q−1 μ  q ⎞ ⎠ ⎤ ⎦ dx ≥ λ  ⎡ ⎣ g(x) a (Cλ 1 p−1 κ p μ  p )+f ⎛ ⎝ (2h (Cλ 1 p−1 )) 1 q−1 λ 1 q−1 κ q ⎞ ⎠ ⎤ ⎦ dx =  [g(x) a (z 1 )+f (z 2 )] wdx Similarly, choosing C large so that ||g|| ∞ ⎛ ⎝ b  2h  Cλ 1 p−1  1 q−1 λ 1 q−1 μ  q ⎞ ⎠ h  Cλ 1 p−1  ≤ 1 Haghaiegh and Afrouzi Boundary Value Problems 2011, 2011:52 http://www.boundaryvalueproblems.com/content/2011/1/52 Page 4 of 5 then  |∇z 2 | q−2 ∇z 2 ∇wdx =2λh  Cλ 1 p−1   wdx ≥ λ   ||g|| ∞ b(z 2 )+h(z 1 )  wdx. Hence by Lemma (1.1), there exist a positive solution (u, v) of (1) such that (ψ 1 , ψ 2 ) ≤ (u, v) ≤ (z 1 , z 2 ). Author details 1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran 2 Department of Mathematics, Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran Authors’ contributions SH has presented the main purpose of the article and has used GAA contribution due to reaching to conclusions. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 13 August 2011 Accepted: 2 December 2011 Published: 2 December 2011 References 1. Ali, J, Shivaji, R: Existence results for classes of Laplacian system with sign-changing weight. Appl Math Anal. 20, 558–562 (2007) 2. Rasouli, SH, Halimi, Z, Mashhadban, Z: A remark on the existence of positive weak solution for a class of (p, q)-Laplacian nonlinear system with sign-changing weight. Nonlinear Anal. 73, 385–389 (2010). doi:10.1016/j.na.2010.03.027 3. Ali, J, Shivaji, R: Positive solutions for a class of (p)-Laplacian systems with multiple parameters. J Math Anal Appl. 335, 1013–1019 (2007). doi:10.1016/j.jmaa.2007.01.067 4. Hai, DD, Shivaji, R: An existence results on positive solutions for class of semilinear elliptic systems. Proc Roy Soc Edinb A. 134, 137–141 (2004). doi:10.1017/S0308210500003115 5. Hai, DD, Shivaji, R: An Existence results on positive solutions for class of p-Laplacian systems. Nonlinear Anal. 56, 1007–1010 (2004). doi:10.1016/j.na.2003.10.024 6. Canada, A, Drabek, P, Azorero, PL, Peral, I: Existence and multiplicity results for some nonlinear elliptic equations. A survey Rend Mat Appl. 20, 167–198 (2000) doi:10.1186/1687-2770-2011-52 Cite this article as: Haghaiegh and Afrouzi: Sub-super solutions for (p-q) Laplacian systems. Boundary Value Problems 2011 2011:52. Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Haghaiegh and Afrouzi Boundary Value Problems 2011, 2011:52 http://www.boundaryvalueproblems.com/content/2011/1/52 Page 5 of 5 . RESEARC H Open Access Sub-super solutions for (p-q) Laplacian systems Somayeh Haghaiegh 1* and Ghasem Alizadeh Afrouzi 2 * Correspondence: Haghaieghi_ch86@yahoo.com 1 Department. (2000) doi:10.1186/1687-2770-2011-52 Cite this article as: Haghaiegh and Afrouzi: Sub-super solutions for (p-q) Laplacian systems. Boundary Value Problems 2011 2011:52. Submit your manuscript to. (p, q) -Laplacian nonlinear system with sign-changing weight. Nonlinear Anal. 73, 385–389 (2010). doi:10.1016/j.na.2010.03.027 3. Ali, J, Shivaji, R: Positive solutions for a class of (p)-Laplacian