ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 3 (2007) No. 4, pp. 286-288
On positive weak solutions for a class of semipositone equations G. A. Afrouzi∗ , M. Baghery, S. H. Rasouli Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran (Received September 25 2006, Accepted February 2 2007) Abstract. In this paper, we study existence of positive weak solution for the semipositone problem −∆p u = λuα − c, with zero Dirichlet boundary conditions in bounded domain Ω ⊂ RN where ∆p denotes the pLaplacian operator defined by ∆p z = div(|∇z|p−2 ∇z); p > 1, α < p − 1 and c > 0 is a parameter. We establish positive constants c0 (Ω) and λ∗ (Ω, c) such that the above equation has a positive solution when c ≤ c0 and λ ≤ λ∗ . Keywords: semipositone problem, p-Laplacian, positive solution, Method of sub- and supersolution
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Introduction In this paper, we consider the existence of positive weak solution of the equation −∆p u = λuα − c, x ∈ Ω, u = 0, x ∈ ∂Ω,
(1)
where ∆p z = div(|∇z|p−2 ∇z); p > 1, α < p − 1 and c > 0 is a parameter. We also consider the equation in the bounded domain Ω ⊂ RN (N ≥ 2) with ∂Ω of class C 2 and connected. Problems involving the “p-Laplacian” arise from many branches of pure mathematics as in the theory of quasiregular and quasiconformal mapping (see [7]) as well as from various problems in mathematical physics notably the flow of non-Newtonian fluids. Semipositone problems have been of great interest during the past two decades, and continue to pose mathematically difficult problems in the study of positive solutions (see [3, 5]). We refer to [4, 6] for additional results in p-laplacian problems.
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Existence results
Let W01,p = W01,p (Ω) denote the usual Sobolev space. We first give the definition of weak subsolution and supersolution of (1). Our approach is based on the method of sub-super solutions, see [1, 2]. Definition 1. We say that ψ ∈ W01,p (Ω) (φ ∈ W01,p (Ω)) is a subsolution (a supersolution) to (1) if for any v ∈ W where W = {v ∈ C0∞ (Ω) | v ≥ 0 ∈ Ω}, we have Z Z p−2 |∇ψ| ∇ψ.∇v dx − [ψ p−1 − c]v dx ≤ 0(≥ 0). (2) Ω
Ω
Now if there exists sub and super solutions ψ and φ respectively such that 0 ≤ ψ ≤ φ for x ∈ Ω, then (1) has a positive solution u ∈ W01,p (Ω) such that ψ ≤ u ≤ φ. This follows from a result in [2]. ∗
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World Journal of Modelling and Simulation, Vol. 3 (2007) No. 4, pp. 286-288
¯ be a Let λ1 be the first eigenvalue of the ∆p with Dirichlet boundary conditions and φ1 ∈ C 1 (Ω) corresponding eigenfunction such that φ1 > 0 in Ω and kφ1 k∞ = 1. Let m > 0, δ > 0 and n > 0 be such that |∇φ1 |p − λ1 φp1 ≥ m, x ∈ Ω¯δ , (3) ¯ φ1 ≥ n, x ∈ Ω \ Ωδ , ¯δ = {x ∈ Ω; d(x, ∂Ω) ≤ δ}. where Ω Now we shall establish: Theorem 1. There exist positive constants c0 = c0 (Ω) and λ∗ = λ∗ (Ω, c) such that (1) has a positive solution for c ≤ c0 and λ ≥ λ∗ . Proof. Let λ1 , φ1 , m, δ, n and Ωδ be as describe in section 1. We establish the above theorem by the method of sub- and supersolution. p/(p−1) We shall varify that ψ = ((p − 1)/p)φ1 is a subsolution of (1). A calculation shows that ∇ψ = 1/(p−1) φ1 ∇φ1 and Z Z |∇φ1 |p−2 φ1 ∇φ1 · ∇w dx |∇ψ|p−2 ∇ψ.∇w dx = Ω ZΩ Z p−2 |∇φ1 | ∇φ1 ∇φ1 w dx − |∇φ1 |p w dx = Ω ZΩ p = (λ1 φ1 − |∇φ1 |p )w dx Ω
¯δ we have [λ1 φp − |∇φ1 |p )] ≤ −m. Hence if c ≤ c0 = m then [λ1 φp − |∇φ1 |p )] ≤ [λψ α − c] Now, in Ω 1 1 ¯δ . in Ω Next, in Ω − Ω¯δ we have [λ1 φp1 − |∇φ1 |p ] ≤ λ1 while λφα − c ≥ λnα − c
Hence if λ ≥ λ∗ = (λ1 + c)/nα then [λ1 φp1 − |∇φ1 |p )] ≤ λψ α − c. Hence if c ≤ c0 and λ ≤ λ∗ then (3) is satisfy and ψ is a subsolution. Next, we construct a supersolution z of (1). Let e1 (x) be the positive solution of the problems −∆p e1 = 1, x ∈ Ω, (4) e1 = 0, x ∈ ∂Ω. We denote z = Ae1 (x) where the constant A > 0 is sufficiently large and to be chosen later. Let w ∈ W01,p with w ≥ 0. Then we have Z Z p−2 p−1 |∇z| ∇z.∇w dx = A w dx, Ω
Ω
Let l = ke1 k∞ . Since α < p − 1, it is easy that there exists positive large constant A such that Ap−1−α ≥ λlα .
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G. Afrouzi & M. Baghery & S. Rasouli: On positive weak solutions
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Ap−1 ≥ λz α x ∈ Ω,
then (3) is satisfy and z is a supersolution. Obviously, we have ψ(x) ≤ z(x) in Ω with large A. thus, by comparison principle, there exists a solution u of (1) with ψ ≤ u ≤ z. This completes the proof of Theorem 1.
References [1] P. Drabek and J. Hernandez, Existence and uniqueness of positive solutions for some quasilinear elliptic problem, Nonl. Anal, 44(2), 2001, 189-204. [2] P. Drabek, P. Kerjci, P. Takac, Nonlinear Differential Equations, Chapman and Hall/CRC, 1999. [3] D. D. Hai. On a class of sublinear quasilinear elliptic problems, Proc. Amer. Math. Soc, 131(8), 2003. [4] D. D. Hai and R. Shivaji, An existence result on positive solutions for a class of p-Laplacian systems, Nonl. Anal, 56(7), 2004, 1007-1010. [5] S. Oruganti, and R. Shivaji, Existence results for classes of p-Laplacian semipositone equations, Boundary Value Problems, 2006, 1-7. [6] S. Oruganti, J. Shi, and R. Shivaji, Logistic equation with the p-Laplacian and constant yeild harvesting, Abstract. Applied. Anal, 9, 2004, 723-727. [7] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eqns, 51, 1984, 126-150.
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