Int. Journal of Math. Analysis, Vol. 2, 2008, no. 20, 987 - 991
On Critical Exponent for Existence of Positive Solutions for Some Semipositone Problems Involving the Weight Function G. A. Afrouzi a , J. Vahidi a
b
and S. H. Rasouli
a
Department of Mathematics, Faculty of Basic Sciences Mazandaran University, Babolsar, Iran
[email protected],
[email protected] b
Department of Computer Sciences Shomal University, Amol, Iran
[email protected] Abstract
In this paper, we study existence of positive solution for the semipositone problem of the form
−Δu = λm(x)uα − c, , x ∈ Ω, u(x) = 0, x ∈ ∂Ω,
where Δ denote the Laplacian operator, Ω is a smooth bounded domain in RN with ∂Ω of class C 2 , λ, c are positive parameters and the weight m(x) satisfying m(x) ∈ C(Ω) and m(x) ≥ m0 > 0 for x ∈ Ω. A critical exponent is optained for existence of positive solution by applying the method of sub-super solution.
Mathematics Subject Classification: 35J55 Keywords: Semipositone problem; Positive solutions, Method of subsuper solution
1
Introduction
In this work, we consider the existence of positive solution to boundary value problem of the form
−Δu = λm(x)uα − c, , x ∈ Ω, u(x) = 0, x ∈ ∂Ω,
(1)
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G. A. Afrouzi, J. Vahidi and S. H. Rasouli
where Δ denote the Laplacian operator, Ω is a smooth bounded domain in RN with ∂Ω of class C 2 , λ, c are positive parameters and the weight m(x) satisfying m(x) ∈ C(Ω) and m(x) ≥ m0 > 0 for x ∈ Ω. Here we consider the challenging semipositone case c > 0. 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 [6, 7]). We refer to [1, 2, 3] for additional results in semipositone problems. Our approach is based on the method of sub-super solutions, see [4, 8].
2
Existence results
We first give the definition of sub-super solution of (1). A super solution to (1) is defined as a function z ∈ C 2 (Ω) such that −Δz ≥ λg(x, z) x ∈ Ω, z ≥ 0, x ∈ ∂Ω. Sub solutions are defined similarly with the inequalities reversed and it is well known that if there exists a sub solution ψ and a super solution z to ¯ then (1) has a solution u such that (1) such that ψ(x) ≤ z(x) for x ∈ Ω, ¯ Further note that if ψ(x) ≥ 0 for x ∈ Ω then ψ(x) ≤ u(x) ≤ z(x) for x ∈ Ω. u ≥ 0 for x ∈ Ω. To precisely state our existence result we consider the eigenvalue problem
−Δφ = λ φ, x ∈ Ω, φ = 0, x ∈ ∂Ω.
(2)
Let φ1 ∈ C 1 (Ω) be the eigenfunction corresponding to the first eigenvalue λ1 of (2) such that φ1 (x) > 0 in Ω, and ||φ1 ||∞ = 1. It can be shown that ∂φ1 < 0 on ∂Ω. Here n is the outward normal. This result is well known ( ∂n see [5] ), and hence, depending on Ω, there exist positive constants k, η, μ such that ¯ η, λ1 φ21 − |∇φ1 |2 ≤ −k, x∈Ω (3) ¯ η, φ1 ≥ μ, x ∈ Ω0 = Ω \ Ω (4) ¯ η = {x ∈ Ω | d(x, ∂Ω) ≤ η}. with Ω We will also consider the unique solution, ζ ∈ C 1 (Ω), of the boundary value problem
−Δζ = 1, x ∈ Ω, ζ = 0, x ∈ ∂Ω,
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Positive solutions for semipositone problems
to discuss our existence result. It is known that ζ > 0 in Ω and
∂ζ ∂n
< 0 on ∂Ω.
Our main result is as follows: Theorem 2.1. If α < 1, then 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. To obtain the existence of positive solution to problem (1), we constructing a positive subsolution ψ and supersolution z. We shall verify that ψ = 12 φ21 is a subsolution of (1). Since ∇ψ = φ1 ∇φ1 , a calculation shows that 1 −Δψ = −Δ ( φ21 ) 2 = −(|∇φ1 |2 + φ1 Δφ1 ) = [φ1 (−Δφ1 ) − |∇φ1 |2 ] = λ1 φ21 − |∇φ1 |2 . Then ψ is a subsolution if λ1 φ21 − |∇φ1 |2 ≤ λ m(x) ψ α − c. ¯ η , and therefore Now λ1 φ21 − |∇φ1 |2 ≤ −k in Ω if c ≤ c0 = k, then λ1 φ21 − |∇φ1 |2 ) ≤ λ m(x) ψ α − c, ¯ η , also in Ω0 we have Furthermore, we note that φ1 ≥ μ > 0 in Ω0 = Ω \ Ω λ1 φ21 − |∇φ1 |2
≤ λ1 ≤ λ m(x) ψ α − c,
if λ ≥ λ∗ =
λ1 + c . μ2α m0 2α
Hence if c ≤ c0 and λ ≥ λ∗ then (3) is satisfy and ψ is a subsolution. Next, we construct a supersolution z of (1). We denote z = Aζ(x), where the constant A > 0 is large and to be chosen later. We shall verify that z is a
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G. A. Afrouzi, J. Vahidi and S. H. Rasouli
supersolution of (1). A calculation shows that −Δz = A (−Δζ) = A. Thus z is a supersolution if A ≥ λ m(x) z α − c, and therefore if A ≥ A0 where 1
A0 = (λ ||m||∞ ||ξ||α∞) 1−α , we have −Δz ≥ λ m(x) z α − c, and hence z is supersolution of (1). Since ζ > 0 and ∂ζ/∂n < 0 on ∂Ω, we can choose A large enough so that ψ ≤ z is also satisfied. Thus, by comparison principle, there exists a solution u of (1) with ψ ≤ u ≤ z. This completes the proof of Theorem 2.1.
References [1] G.A. Afrouzi and S.H. Rasouli, On positive solutions for some nonlinear semipositone elliptic boundry value problems, Nonlinear Analysis: Modeling and Control, 4 (11) (2006), 323-329. [2] V. Anuradha, D.D. Hai and R. Shivaji, Existence results for superlinear semipositone boundary value problems, Proc. AMS, 124(3) (1996), 757763. [3] A. Castro, S. Gadam and R. Shivaji, Evolution of Positive Solution Curves in Semipositone Problems with Concave Nonlinearities, Jour. Math. Anal. Appl., 245, (2000), 282-293. [4] P. Drabek and J. Hernandez, Existence and uniqueness of positive solutions for some quasilinear elliptic problem, Nonl. Anal, 44 (2001), no. 2, 189-204. [5] A. Friedman, Partial Differential Equations of Parabolic type, Prentice Hall, Inc., Englewood Cliffs, NJ, 1964.
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[6] S. Oruganti, J. Shi, and R. Shivaji, Diffusive logistic equation with constant yeild harvesting, I: steady states, Tran. Amer. Math. Soc., 354 (2002), no. 9, 3601-3619. [7] S. Oruganti, and R. Shivaji, Existence results for classes of p-Laplacian semipositone equations, Boundary Value Problems, (2006), 1-7. [8] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Received: November 27, 2007
