Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 19349, 9 pages doi:10.1155/2007/19349 Research Article Existence and Asymptotic Behavior of Positive Solutions to p(x)-Laplacian Equations with Singular Nonlinearities Qihu Zhang Received 17 July 2007; Accepted 27 August 2007 Recommended by M. Garc ´ ıa-Huidobro This paper investigates the p(x)-Laplacian equations with singular nonlinearities −Δ p(x) u = λ/u γ(x) in Ω, u(x) = 0on∂Ω,where−Δ p(x) u =−div(|∇u| p(x)−2 ∇u)iscalledp(x)- Laplacian. The existence of positive solutions is given, and the asymptotic behavior of solutions near boundary is discussed. Copyright © 2007 Qihu Zhang. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work i s properly cited. 1. Introduction The study of differential equations and variational problems with nonstandard p(x)- growth conditions is a new and interesting topic. We refer to [1, 2], the background of these problems. Many results have been obtained on this kind of problems, for exam- ple, [2–13]. In [4, 7], Fan and Zhao give the regularity of weak solutions for differential equations with nonstandard p(x)-growth conditions. On the existence of solutions for p(x)-Laplacian problems in bounded domain, we refer to [5, 11, 12]. In this paper, we consider the p(x)-Laplacian equations with singular nonlinearities: − p(x) u = λ u γ(x) in Ω, u(x) = 0on∂Ω, (P) where − p(x) u =−div(|∇u| p(x)−2 ∇u)iscalledp(x)-Laplacian, Ω ⊂ R N is a bounded domain with C 2 boundary ∂Ω.Ifp(x) ≡ p (a constant), then (P)isthewell-known p-Laplacian problem. There are many results on the existence of positive solutions for p-Laplacian problems with singular nonlinearities (see [14–18]), but the results on the existence of positive solutions for p(x)-Laplacian problems with singular nonlinearities 2 Journal of Inequalities and Applications are rare. Our aim is to give the existence of positive solutions for problem (P), and give the asy mptotic behavior of positive solutions near boundary. Throughout the paper, we assume that 0 <γ(x) ∈ C(Ω)andp(x) satisfy (H 1 ) p(x) ∈ C 1 (Ω), 1 <p − ≤ p + < +∞,wherep − = inf Ω p(x), p + = sup Ω p(x). Because of the nonhomogeneity of p(x)-Laplacian, p(x)-Laplacian problems are more complicated than those of p-Laplacian ones, many results and methods for p-Laplacian problems are invalid for p(x)-Laplacian problems (see [6]), and another difficulty of this paper is that f (x,u) = 1/u γ(x) cannot be represented as h(x) f (u). Our results partially generalized the results of [18]. 2. Preliminary In order to deal with p(x)-Laplacian problems, we need some theories on the spaces L p(x) (Ω), W 1,p(x) (Ω) and properties of p(x)-Laplacian which we w ill use later (see [3, 8]). Let L p(x) (Ω) = u | u is a measurable real-valued function, Ω u(x) p(x) dx < ∞ , C + 0 (Ω) = u ∈ C(Ω) | u>0inΩ, u = 0on∂Ω . (2.1) We can introduce the norm on L p(x) (Ω)by |u| p(x) = inf μ>0 | Ω u(x) μ p(x) dx ≤ 1. . (2.2) The space (L p(x) (Ω), |·| p(x) ) becomes a Banach space. We call it generalized Lebesgue space. The space (L p(x) (Ω), |·| p(x) ) is a separable, reflexive, and uniform convex Banach space (see [3, Theorems 1.10, Theorem 1.14]). The space W 1,p(x) (Ω)isdefinedby W 1,p(x) (Ω) = u ∈ L p(x) (Ω) | ∇u ∈ L p(x) (Ω) , (2.3) and it can be equipped with the norm |u|=|u| p(x) + |∇u| p(x) , ∀u ∈ W 1,p(x) (Ω). (2.4) W 1,p(x) 0 (Ω)istheclosureofC ∞ 0 (Ω)inW 1,p(x) (Ω), W 1,p(x) (Ω)andW 1,p(x) 0 (Ω)aresep- arable, reflexive, and uniform convex Banach spaces (see [3, Theorem 2.1]). If u ∈ W 1,p(x) loc (Ω) ∩ C + 0 (Ω), u is called a positive solution of (P)ifu(x) satisfies Q |∇u| p(x)−2 ∇u∇qdx − Q λ u γ(x) qdx = 0, ∀q ∈ W 1,p(x) 0 (Q), (2.5) for any domain Q Ω. Qihu Zhang 3 Let W 1,p(x) 0,loc (Ω) ={u | there is an open domain Q Ω s.t. u ∈ W 1,p(x) 0 (Q)},anddefine A : W 1,p(x) loc (Ω) ∩ C + 0 (Ω) → (W 1,p(x) 0,loc (Ω)) ∗ as Au,ϕ= Ω |∇ u| p(x)−2 ∇u∇ϕ − λ u γ(x) ϕ dx, (2.6) where u ∈ W 1,p(x) loc (Ω) ∩ C + 0 (Ω), ϕ ∈ W 1,p(x) 0,loc (Ω); then we have the following lemma. Lemma 2.1 (see [5, Theorem 3.1]). A : W 1,p(x) loc (Ω) ∩ C + 0 (Ω) → (W 1,p(x) 0,loc (Ω)) ∗ is strictly monotone. Let g ∈ (W 1,p(x) 0,loc (Ω)) ∗ ,ifg,ϕ≥0,forallϕ ∈ W 1,p(x) 0,loc (Ω), ϕ ≥ 0 a.e. in Ω,thendenote g ≥ 0 in (W 1,p(x) 0,loc (Ω)) ∗ ; correspondingly, if −g ≥ 0 in (W 1,p(x) 0,loc (Ω)) ∗ , then denote g ≤ 0 in (W 1,p(x) 0,loc (Ω)) ∗ . Definit ion 2.2. Let u ∈ W 1,p(x) loc (Ω) ∩ C + 0 (Ω). If Au ≥ 0(Au ≤ 0) in (W 1,p(x) 0,loc (Ω)) ∗ ,thenu is called a weak supersolution (weak subsolution) of (P). Copying the proof of [10], we have the following lemma. Lemma 2.3 (comparison principle). Let u,v ∈ W 1,p(x) loc (Ω) ∩ C(Ω) be positive and satisfy Au − Av ≥ 0 in (W 1,p(x) 0,loc (Ω)) ∗ .Letϕ(x) = min{u(x) − v(x),0}.Ifϕ(x) ∈ W 1,p(x) 0,loc (Ω) (i.e., u ≥ v on ∂Ω), then u ≥ v a.e. in Ω. Lemma 2.4 (see [7]). If g(x,u) is continuous on Ω × R, u ∈ W 1,p(x) (Ω) is a bounded weak solution of − p(x) u + g(x,u) = 0 in Ω, u = w 0 on ∂Ω,wherew 0 ∈ W 1,p(x) (Ω), then u ∈ C 1,α loc (Ω),whereα ∈ (0,1) is a constant. 3. Existence of positive solutions In order to deal with the existence of positive solutions, let us consider the problem − p(x) u = λ | u| + a n γ(x) in Ω, u(x) = 0forx ∈ ∂Ω, (3.1) where {a n } is a positive strictly decreasing sequence and lim n→+∞ a n = 0. We have the following lemma. Lemma 3.1. For any n = 1,2, ,problem(3.1) possesses a weak positive s olution n ∈ C(Ω). Proof. The relative functional of (3.1)is ϕ = Ω 1 p(x) ∇ u(x) p(x) dx − Ω F n (x, u)dx, (3.2) where F n (x, u) = u 0 λ/((|t| + a n ) γ(x) )dt.Sinceϕ is coercive in W 1,p(x) 0 (Ω), then ϕ possesses a nontrivial minimum point n ,then| n | is also a nontrivial minimum point of problem (3.1), then (3.1) possesses a weak positive solution. The proof is completed. 4 Journal of Inequalities and Applications Here and hereafter, we will use the notation d(x,∂Ω) to denote the distance of x ∈ Ω to the boundary of Ω. Denote d(x) = d(x,∂Ω)and∂Ω ={x ∈ Ω | d(x) < } .Since∂Ω is C 2 regularly, then there exists a positive constant σ such that d(x) ∈ C 2 (∂Ω 2σ ), and |∇d(x)|≡1. Let δ ∈ (0,(1/3)σ) be a small enough constant. Denote v 1 (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d(x), d(x) <δ, δ + d(x) δ 2δ − t δ 2/(p − −1) dt, δ ≤ d(x) < 2δ, δ + 2δ δ (2δ − t) δ 2/(p − −1) dt,2δ ≤ d(x). (3.3) Obviously, v 1 (x) ∈ C 1 (Ω) ∩ C + 0 (Ω). Lemma 3.2. If λ>0 is large enough, then v 1 (x) is a subsolution of (P). Proof. Since |∇d(x)|≡1, when λ>0 is large enough, we have − p(x) v 1 =−d(x) ≤ λ v 1 (x) γ(x) , ∀x ∈ Ω, d(x) <δ. (3.4) By computation, when δ<d(x) < 2δ,wehave − p(x) v 1 =−div 2δ − d(x) δ 2/(p − −1) p(x)−1 ∇d(x) =− 2δ − d(x) δ 2/(p − −1) p(x)−1 d(x) − 2δ − d(x) δ 2/(p − −1) p(x)−1 ∇ d(x)∇p(x) ln 2δ − d(x) δ 2/(p − −1) + 2 δ p(x) − 1 p − − 1 2δ − d(x) δ (2(p(x)−1)/(p − −1))−1 . (3.5) When λ>0 is large enough, it is easy to see that − p(x) v 1 ≤ λ v 1 (x) γ(x) , ∀x ∈ Ω, δ<d(x) < 2δ, − p(x) v 1 = 0 ≤ λ v 1 (x) γ(x) , ∀x ∈ Ω,2δ<d(x). (3.6) From (3.4)and(3.6), we can conclude that v 1 (x)isasubsolutionof(P). Qihu Zhang 5 Theorem 3.3. If λ>0 is a large enough constant, then problem ( P) possesses only one posi- tive solution u λ ,andu λ is increasing with respect to λ. Proof. Denote u n = n + a n ,where n is a solution of (3.1). Since {u n } is a sequence of positive solutions of − p(x) u = λ u γ(x) in Ω, u(x) = a n for x ∈ ∂Ω, (II) then every u n is subsolution and supersolution of − p(x) u = λ/u γ(x) in Ω.Accordingto comparison principle, we have that u n ≥ u n+1 for n = 1,2, Sincev 1 (x)isasubsolution of (P)andv 1 (x) = 0on∂Ω,thenu n ≥ u n+1 ≥ v 1 for n = 1,2, AccordingtoLemma 2.4, we have that {u n } has uniform C 1,α local regularity property, and hence we can choose a subsequence, which we denoted by {u 1 n },suchthatu 1 n → w and ∇u 1 n → h in Ω.Infact, h =∇w in Ω. For any domain D Ω,foranyϕ ∈ W 1,p(x) 0 (D). The C 1,α regularity result implies that the sequences {u n } and {∇u n } are equicontinuous in D;fromtheC 1,α estimate we con- clude that ∇w ∈ C α (D)forsome0<α<1. Thus w ∈ W 1,p(x) (D) ∩ C 1,α (D). From the C 1,α regularity result, we see that |∇u 1 n | p−1 |∇ϕ|≤C|∇ϕ| on D, and since the function ξ →|ξ| p−2 ξ is continuous on R n , it follows that |∇u 1 n (x)| p−2 ∇u 1 n (x) ·∇ϕ(x) → |∇ w(x)| p−2 ∇w(x) ·∇ϕ(x)forx ∈ D. Thus, by the dominated convergence theorem, for any ϕ ∈ W 1,p(x) 0 (D), we can see that D ∇ u 1 n (x) p−2 ∇u 1 n (x) ·∇ϕ(x)dx −→ D ∇ w(x) p−2 ∇w(x) ·∇ϕ(x)dx. (3.7) Furthermore, since 0 ≤ λ/([u 1 n (x)] γ(x) ) ≤ λ/([u 1 n+1 (x)] γ(x) ), and λ/([u 1 n (x)] γ(x) ) → λ/([w(x)] γ(x) )foreachx ∈ D, by the monotone convergence theorem we obtain D λ u 1 n (x) γ(x) ϕdx −→ D λ w(x) γ(x) ϕdx, ∀ϕ ∈ W 1,p(x) 0 (D). (3.8) Therefore, it follows that D ∇ w(x) p−2 ∇w(x) ·∇ϕ(x)dx − D λ w(x) γ(x) ϕdx = 0, ∀ϕ ∈ W 1,p(x) 0 (D), (3.9) and hence w is a weak solution of −Δ p(x) w = λ/([w(x)] γ(x) )onD. Obviously, w is a solution of (P), and satisfies w ≥ v 1 . According to comparison prin- ciple, it is easy to see that (P) possesses only one positive solution, and u λ is increasing with respect to λ. 6 Journal of Inequalities and Applications 4. Asymptotic behavior of positive solutions In the following, we will use C i to denote positive constants. Theorem 4.1. If u is a positive weak solution of problem (P), then C 2 d(x) ≤ u(x) as x → ∂Ω. Proof. Similar to the proof of Lemma 3.2, there exists a positive constant C 2 such that when δ>0 is small enoug h, then v 2 (x) = C 2 d(x)isasubsolutionof(P)on∂Ω δ .Thus u(x) ≥ v 2 (x) = C 2 d(x)on∂Ω δ . The proof is completed. Denote γ ∗ = max x∈∂Ω 2σ γ(x)andγ ∗ = min x∈∂Ω 2σ γ(x). Theorem 4.2. If 1 ≤ γ ∗ <γ ∗ , for any weak solution u of problem (P), we have C 3 d(x) θ 1 ≤ u(x) ≤ C 4 d(x) θ 2 as x −→ ∂Ω, (4.1) where θ 1 = max d(x)≤σ (p(x)/(p(x) − 1+γ(x))), θ 2 = min d(x)≤σ (p(x)/(p(x) − 1+γ(x))). Proof. From Theorem 4.1 we only consider (P) in the case of 1 <γ ∗ <γ ∗ . Denote v 3 (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a d(x) θ , d(x) <δ, aδ θ + d(x) δ aθδ θ−1 2δ − t δ 2/(p − −1) dt, δ ≤ d(x) < 2δ, aδ θ + 2δ δ aθδ θ−1 2δ − t δ 2/(p − −1) dt,2δ ≤ d(x), (4.2) where a and θ are positive constants and satisfy θ ∈ (0,1), 0 <δis smal l enough. Obviously, v 3 (x) ∈ C 1 (Ω) ∩ C + 0 (Ω). By computation, − p(x) v 3 (x) =−(aθ) p(x)−1 (θ − 1) p(x) − 1 d(x) (θ−1)(p(x)−1)−1 1+Π(x) , d(x) <δ, (4.3) where Π(x) = d ( ∇p∇d)lnaθ (θ − 1) p(x) − 1 + d ( ∇p∇d)lnd p(x) − 1 + d d (θ − 1) p(x) − 1 . (4.4) Obviously |Π(x)|≤1/2, when δ>0 is small enough. Let θ = θ 1 and a ∈ (0,1) is small enough, when δ ∈ (0,a) is small enough, we can conclude that − p(x) v 3 (x) ≤ λ v 2 (x) γ(x) , d(x) <δ. (4.5) Qihu Zhang 7 By computation, when δ<d(x) < 2δ,wehave − p(x) v 3 =−div aθδ θ−1 2δ − d(x) δ 2/(p − −1) p(x)−1 ∇d(x) =− aθδ θ−1 2δ − d(x) δ 2/(p − −1) p(x)−1 × ∇d(x)∇p(x) lnaθδ θ−1 2δ − d(x) δ 2/(p − −1) − aθδ θ−1 2δ − d(x) δ 2/(p − −1) p(x)−1 d(x) + 2 δ p(x) − 1 p − − 1 aθδ θ−1 p(x)−1 2δ − d(x) δ (2(p(x)−1)/(p − −1))−1 . (4.6) Thus, there exists a positive constant C ∗ such that − p(x) v 3 ≤ C ∗ δ (θ−1)(p(x)−1)−1 , δ<d(x) < 2δ. (4.7) Obviously, v 3 (x) ≤ a(θ +1)δ θ , δ<d(x) < 2δ. (4.8) Let θ = θ 1 ,whena ∈ (0,1) is small enough, δ ∈ (0,a) is small enough, then − p(x) v 3 (x) ≤ λ v 2 (x) γ(x) , δ<d(x) < 2δ. (4.9) It is easy to see that − p(x) v 3 (x) = 0 ≤ λ v 2 (x) γ(x) ,2δ<d(x). (4.10) Combining (4.5), (4.9), and (4.10), it is easy to see that when θ = θ 1 , a ∈ (0,1) is small enough and δ ∈ (0, a) is small enough, then v(x)isasubsolutionof(P), then u(x) ≥ C 3 [d(x)] θ 1 on ∂Ω δ . Similarly, when δ>0 is small enough, θ = θ 2 ,anda ≥ max x∈∂Ω δ (u(x)/δ θ )islarge enough, we can see that v(x)isasupersolutionof(P)on ∂Ω δ ,andu(x) ≤ a[d(x)] θ 2 on ∂Ω δ . The proof is completed. Theorem 4.3. If lim d(x)→0 p(x) = p 0 and lim d(x)→0 p(x)/(p(x) − 1+γ(x)) = s,wheres ≤ 1 is a positive constant, u is a solution of (P), then lim d(x)→0 u(x) C d(x) s = 1, C = lim d(x)→0 λ θ p(x)−1 (1 − θ) p(x) − 1 1/(p(x)−1+γ(x)) . (4.11) 8 Journal of Inequalities and Applications Proof. It can be obtained easily from Theorem 4.2. Theorem 4.4. If 1 ≥ γ ∗ , for any positive constant θ ∈ (0,1), u is a weak solution of problem (P), then there exists a positive constant C 5 such that C 1 d(x) ≤ u(x) ≤ C 5 (d(x)) θ as x → ∂Ω. Proof. According to Theorem 4.1,itonlyneedstoproveu(x) ≤ C 5 (d(x)) θ as x → ∂Ω. Define a function on ∂Ω δ as v 4 (x) = C 5 (d(x)) θ ,whereC 5 ≥ (1/δ θ )max x∈∂Ω δ u(x). Similar to the proof of Theorem 4.2,whenδ>0 is small enough, then v 4 (x)isasupersolutionof (P)on ∂Ω δ ,thenu(x) ≤ v 4 (x) = C 5 (d(x)) θ on ∂Ω δ .Theproofiscompleted. Theorem 4.5. If γ ∗ < 1 <γ ∗ , u is a weak solution of problem (P), then there exists a positive constant C 6 such that C 1 d(x) ≤ u(x) ≤ C 6 (d(x)) θ as x → ∂Ω,whereθ = min d(x)≤δ (p(x)/ (p(x) − 1+γ(x))). Proof. According to Theorem 4.1,itonlyneedstoproveu(x) ≤ C 6 (d(x)) θ as x → ∂Ω. Similar to the proof of Theorem 4.2,whenδ>0 is small enough, then v 5 (x) = C 6 (d(x)) θ is a supersolution of (P)on∂Ω δ ,thenu(x) ≤ v 5 (x) = C 6 (d(x)) θ on ∂Ω δ .Theproofis completed. Acknowledgments This work was partially supported by the National Science Foundation of China (no. 10701066 and no. 10671084) and the Natural Science Foundation of Henan Education Committee (no. 2007110037). References [1] Y. Chen, S. Levine, and M. 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Corporation Journal of Inequalities and Applications Volume 2007, Article ID 19349, 9 pages doi:10.1155/2007/19349 Research Article Existence and Asymptotic Behavior of Positive Solutions to p(x)-Laplacian Equations. the existence of positive solutions for p-Laplacian problems with singular nonlinearities (see [14–18]), but the results on the existence of positive solutions for p(x)-Laplacian problems with singular. nonlinearities 2 Journal of Inequalities and Applications are rare. Our aim is to give the existence of positive solutions for problem (P), and give the asy mptotic behavior of positive solutions near