Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 279306, 8 pages doi:10.1155/2008/279306 ResearchArticleBoundaryBlow-UpSolutionsto px-Laplacian EquationswithExponentialNonlinearitiesQihu Zhang Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China Correspondence should be addressed toQihu Zhang, zhangqh1999@yahoo.com.cn Received 18 August 2007; Accepted 25 November 2007 Recommended by M. Garcia-Huidobro This paper investigates the px-Laplacian equationswithexponentialnonlinearities − px u e fx,u 0inΩ, ux → ∞ as dx, ∂Ω → 0, where − px u −div|∇u| px−2 ∇u is called px- Laplacian. The singularity of boundaryblow-upsolutions is discussed, and the existence of bound- ary blow-upsolutions is given. Copyright q 2008 Qihu Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The study of differential equations and variational problems with nonstandard px-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 example, 1–15. In this paper, we consider the px-Laplacian equationswithexponentialnonlinearities −Δ px u e fx,u 0inΩ, ux −→ ∞ as dx, ∂Ω −→ 0, P where −Δ px u −div|∇u| px−2 ∇u, ΩB0,R ⊂ R N is a bounded radial domain B0,R {x ∈ R N ||x| <R} . Our aim is to give the existence and asymptotic behavior of solutions for problem P. Throughout the paper, we assume that px and fx, u satisfy that H 1 px ∈ C 1 Ω is radial and satisfies 1 <p − ≤ p < ∞, where p − inf Ω px,p sup Ω px; 1.1 2 Journal of Inequalities and Applications H 2 fx, u is radial with respect to x, fx, · is increasing and fx, 00 for any x ∈ Ω; H 3 f : Ω × R → R is a continuous function and satisfies fx, t ≤ C 1 C 2 |t| γx , ∀x, t ∈ Ω × R, 1.2 where C 1 ,C 2 are positive constants, 0 ≤ γ ∈ CΩ. The operator −Δ px u −div|∇u| px−2 ∇u is called px-Laplacian. Especially, if px ≡ p a constant, P is the well-known p-Laplacian problem see 16–18. Because of the nonhomogeneity of px-Laplacian, px-Laplacian problems are more complicated than those of p-Laplacian ones see 6; and another difficulty of this paper is that fx, u cannot be represented as hxfu. 2. Preliminary In order to deal with px-Laplacian problems, we need some theories on spaces L px Ω and W 1,px Ω, and properties of px-Laplacian, which we will use later see 3, 7.Let L px Ω u | u is a measurable real-valued function, Ω ux px dx < ∞ . 2.1 We can introduce the norm on L px Ω by |u| px inf λ>0 | Ω ux λ px dx ≤ 1 . 2.2 The space L px Ω, |·| px becomes a Banach space. We call it generalized Lebesgue space. The space L px Ω, |·| px is a separable, reflexive, and uniform convex Banach space see 3, Theorems 1.10, 1.14. The space W 1,px Ω is defined by W 1,px Ω u ∈ L px Ω | ∇u ∈ L px Ω , 2.3 and it can be equipped with the norm u |u| px ∇u px , ∀u ∈ W 1,px Ω. 2.4 W 1,px 0 Ω is the closure of C ∞ 0 Ω in W 1,px Ω. W 1,px Ω and W 1,px 0 Ω are separable, reflexive, and uniform convex Banach spaces see 3, Theorem 2.1. If u ∈ W 1,px loc Ω ∩ CΩ, u is called a solution of P if it satisfies Q ∇u px−2 ∇u∇qdx Q fx, uqdx 0, ∀q ∈ W 1,px 0 Q, 2.5 for any domain Q Ω,andmaxk − u, 0 ∈ W 1,px 0 Ω for any k ∈ N . Let W 1,px 0,loc Ω {u| there exists an open domain Q Ω s.t. u ∈ W 1,px 0 Q}. For any u ∈ W 1,px loc Ω ∩ CΩ and ϕ ∈ W 1,px 0,loc Ω, define A : W 1,px loc Ω∩CΩ → W 1,px 0,loc Ω ∗ as Au, ϕ Ω |∇u| px−2 ∇u∇ϕ e fx,u ϕdx. Qihu Zhang 3 Lemma 2.1 see 5, Theorem 3.1. Let h ∈ W 1,px Ω ∩ CΩ, X h W 1,px 0,loc Ω ∩ CΩ. Then, A : X → W 1,px 0,loc Ω ∗ is strictly monotone. Let g ∈ W 1,px 0,loc Ω ∗ , if g,ϕ≥0, for all ϕ ∈ W 1,px 0,loc Ω,ϕ ≥ 0 a.e. in Ω, then denote g ≥ 0 in W 1,px 0,loc Ω ∗ ; correspondingly, if −g ≥ 0 in W 1,px 0,loc Ω ∗ , then denote g ≤ 0 in W 1,px 0,loc Ω ∗ . Definition 2.2. Let u ∈ W 1,px loc Ω ∩ CΩ. If Au ≥ 0 Au ≤ 0 in W 1,px 0,loc Ω ∗ , then u is called a weak supersolution weak subsolution of P. Copying the proof of 9, we have the following lemma. Lemma 2.3 comparison principle. Let u, v ∈ W 1,px loc Ω ∩ CΩ satisfy Au − Av ≥ 0 in W 1,px 0,loc Ω ∗ .Letϕxmin {ux − vx, 0}.Ifϕx ∈ W 1,px 0,loc Ω (i.e., u ≥ v on ∂Ω), then u ≥ v a.e. in Ω. Lemma 2.4 see 4, Theorem 1.1. Under the conditions (H 1 )and(H 3 ), if u ∈ W 1,px Ω is a bounded weak solution of −Δ px u e fx,u 0 in Ω,thenu ∈ C 1,ϑ loc Ω, where ϑ ∈ 0, 1 is a constant. 3. Main results and proofs If u is a radial solution of P,thenP can be transformed into r N−1 |u | pr−2 u r N−1 e fr,u ,r∈ 0,R, u0u 0 ,u 00,u r ≥ 0for0<r<R. 3.1 It means that ur is increasing. Theorem 3.1. If there exists a constant σ ∈ R/2,R such that fr, u ≥ αu s as u −→ ∞ for r ∈ σ,R uniformly, 3.2 where α and s are positive constants, then there exists a continuous function Φ 1 x which satisfies Φ 1 x → ∞ (as dx, ∂Ω → 0), and such that, if u is a weak solution of problem P,thenux ≤ Φ 1 x. Proof. Let R 0 ∈ σ,R.Denote Θr, a, λ R 0 r ⎡ ⎣ a a ln R−R 0 −λ −1 1/s−1 s R−R 0 −λ ⎤ ⎦ pR o −1/pt−1 R o N−1 t N−1 sin εt−σ 1/pt−1 dt. 3.3 Define the function gr, a on 0,R as gr, a ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ a ln R − r −1 1/s k, R 0 ≤ r<R, k − Θr, a, 0 a ln R − R 0 −1 1/s , σ<r<R 0 , k − Θσ,a,0 a ln R − R 0 −1 1/s ,r≤ σ, 3.4 4 Journal of Inequalities and Applications where a>1/α sup |x|≥R 0 px is a constant, R 0 ∈ σ,R,andR − R 0 is small enough, ε π/2R 0 − σ and k 2p /α ln R − R 0 −1 1/s Θσ,2a, 0. Obviously, for any positive constant a, gr, a ∈ C 1 0,R. When R 0 <r<R,wehave r N−1 |g | pr−2 g r N−1 a 1/s s pr−1 pr − 1 R − r pr ln R−r −1 1/s−1pr−1 1Πr , 3.5 where Πr 1/s − 1 ln R − r −1 r N−1 a 1/s /s pr−1 r N−1 a 1/s /s pr−1 pr − 1 R − r −p r ln R − r pr − 1 R − r 1/s − 1p r ln ln R − r −1 pr − 1 R − r. 3.6 If R − R 0 is small enough, it is easy to see |Πr|≤1/2; from 3.5,wehave r N−1 |g | pr−2 g ≤ 2r N−1 a 1/s s pr−1 pr − 1R − r −pr ln R − r −1 1/s−1pr−1 ≤ r N−1 1 R − r αa r N−1 e αg s ≤ r N−1 e fr,g , ∀r ∈ R 0 ,R . 3.7 Obviously, if R − R 0 is small enough, then g ≥ 2p /α ln R−R 0 −1 1/s is large enough, so we have r N−1 |g | pr−2 g ε R o N−1 a a ln R − R 0 −1 1/s−1 s R − R 0 pR o −1 cos εr − σ ≤ r N−1 e αg s ≤ r N−1 e fr,g , σ<r<R 0 . 3.8 Obviously, r N−1 |g | pr−2 g 0 ≤ r N−1 e fr,g , 0 ≤ r<σ. 3.9 Since g|x|,a is a C 1 function on B0,R,if0<R− R 0 is small enough R 0 depends on R, p, s, α,from3.7, 3.8,and3.9, we can see that g|x|,a is a supersolution of P. Define the function g m r, a − on 0,R− 1/m as g m r, a − ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a − ln R − 1 m − r −1 1/s k, R 0 ≤r<R− 1 m , k − Θ r, a − , 1 m a − ln R − 1 m − R 0 −1 1/s , σ<r<R 0 , k − Θ σ,a − , 1 m a − ln R − 1 m − R 0 −1 1/s , r ≤ σ, 3.10 Qihu Zhang 5 where m is a big-enough integer such that 0 < 1/m ≤ R − R 0 /2, ε π/2R 0 − σ,0<<1, is a positive small constant such that αa − > sup |x|≥R 0 px. Obviously, g m |x|,a− is a supersolution of P on B0,R−1/m.Ifu is a solution of P, according to the comparison principle, we get that g m |x|,a− ≥ ux for any x ∈ B0,R−1/m. For any x ∈ B0,R− 1/m \ B0,R 0 , we have g m |x|,a− ≥ g m1 |x|,a− . Thus, ux ≤ lim m→∞ g m |x|,a− , ∀x ∈ B0,R \ B 0,R 0 . 3.11 When dx, ∂Ω > 0 is small enough, we have lim m→∞ g m |x|,a− < a ln R − r −1 1/s k ≤ g |x|,a . 3.12 According to the comparison principle, we obtain that g|x|,a ≥ ux, for all x ∈ B0,R, then Φ 1 xg|x|,a is an upper control function of all of the solutions of P. The proof is completed. Theorem 3.2. If there exists a σ ∈ R/2,R such that fr, u ≤ βu s as u −→ ∞ for r ∈ σ,R uniformly, 3.13 where β and s are positive constants, then there exists a continuous function Φ 2 x which satisfies Φ 2 x → ∞ (as dx, ∂Ω → 0),andsuchthat,ifux is a solution of problem P,thenux ≥ Φ 2 x. Proof. Let z 1 be a radial solution of −Δ px z 1 x−μ in Ω 1 B0,σ, z 1 0on∂Ω 1 , 3.14 where μ>2 is a positive constant. We denote z 1 z 1 rz 1 |x|,thenz 1 satisfies z 1 σ0, z 1 00, and z 1 rμ N 1/pr−1 , z 1 − σ r rμ N 1/pr−1 dr. 3.15 Denote h b r, δ on σ,R 0 as h b r, δ R 0 r R o N−1 t N−1 t − σ R 0 − σ b b ln R δ − R 0 −1 1/s−1 s R δ − R 0 pR o −1 σ N−1 t N−1 R 0 − t R 0 − σ tμ N 1/pt−1 pσ−1 1/pt−1 dt. 3.16 It is easy to see that −h b σ,0z 1 σ σμ N 1/pσ−1 , − h b R 0 , 0 b b ln R − R 0 −1 1/s−1 s R − R 0 . 3.17 6 Journal of Inequalities and Applications Define the function vr, b on B0,R as vr, b ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ b ln R − r −1 1/s − k ∗ ,R 0 ≤ r<R, b ln R − R 0 −1 1/s − k ∗ − h b r, 0, σ<r<R 0 , − σ r rμ N 1/pr−1 dr b ln R − R 0 −1 1/s − k ∗ − h b σ,0, r ≤ σ, 3.18 where b ∈ 0, 1/βinf |x|≥R 0 px is a constant, R 0 ∈ σ,R,andR − R 0 is small enough, and k ∗ 2p /β ln 2R − R 0 −1 1/s . Obviously, for any positive constant b, vr, b ∈ C 1 0,R. Similar to the proof of Theorem 3.1,whenR − R 0 is small enough, we have r N−1 |v | pr−2 v ≥ r N−1 e fr,v , ∀r ∈ R 0 ,R . 3.19 When R − R 0 is small enough, for all r ∈ σ,R 0 , since fr, v ≤ 0, then r N−1 |v | pr−2 v ≥ 1 2 R o N−1 R 0 − σ b b ln R − R 0 −1 1/s−1 s R − R 0 pR 0 −1 ≥ r N−1 e fr,v . 3.20 Obviously, r N−1 |v | pr−2 v r N−1 μ ≥ r N−1 e fr,v , ∀r ∈ 0,σ. 3.21 Combining 3.19, 3.20,and3.21, we can see that vr, a is a subsolution of P. Define the function v m r, b on B0,R as v m r, b ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ b ln R 1 m − r −1 1/s − k ∗ ,R 0 ≤ r<R, b ln R 1 m − R 0 −1 1/s − k ∗ − h b r, 1 m , σ<r<R 0 , − σ r μr N 1/pr−1 dr b ln R 1 m −R 0 −1 1/s −k ∗ −h b σ, 1 m ,r≤ σ, 3.22 where is a small-enough positive constant such that b < 1/βinf |x|≥R 0 px. We can see that v m r, b ∈ C 1 0,R is a subsolution of P on BR 0 ,R, according to the comparison principle, we get that v m |x|,b ≤ ux for any x ∈ B0,R. For any x ∈ B0,R \ B0,R 0 , we have v m |x|,b ≤ v m1 |x|,b . Thus, ux ≥ lim m→∞ v m |x|,b , ∀x ∈ B0,R \ B 0,R 0 . 3.23 When dx, ∂Ω is small enough, we have lim m→∞ v m |x|,b >v |x|,b . 3.24 From the comparison principle, we obtain v|x|,b ≤ ux, ∀x ∈ B0,R,thenΦ 2 x v|x|,b is a lower control function of all of the solutions of P. Qihu Zhang 7 Theorem 3.3. If inf x∈Ω px >Nand there exists a σ ∈ R/2,R such that fr, u ≥ au s as u −→ ∞ for r ∈ σ,R uniformly, 3.25 where a and s are positive constants, then P possesses a solution. Proof. In order to deal with the existence of boundaryblow-upsolutions of P, let us consider the problem −Δ px u e fx,u 0inΩ, uxj for x ∈ ∂Ω, 3.26 where j 1, 2, Since inf x∈Ω px >N,thenW 1,px Ω → C α Ω,whereα ∈ 0, 1.The relative functional of 3.26 is ϕu Ω 1 px ∇ux px dx Ω Fx, udx, 3.27 where Fx, u u 0 e fx,t dt. Since ϕ is coercive in X j : j W 1,px 0 Ω, then ϕ possesses a nontrivial minimum point u j , then problem 3.26 possesses a weak solution u j . 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Lair, “A necessary and sufficient condition for existence of large solutionsto semilinear elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 240, no. 1, pp. 205–218, 1999. 18 A. Mohammed, “Existence and asymptotic behavior of blow-upsolutionsto weighted quasilinear equations,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 621–637, 2004. . −Δ p x u −div|∇u| p x −2 ∇u is called p x -Laplacian. Especially, if p x ≡ p a constant, P is the well-known p- Laplacian problem see 16–18. Because of the nonhomogeneity of p x -Laplacian, . order to deal with p x -Laplacian problems, we need some theories on spaces L p x Ω and W 1 ,p x Ω, and properties of p x -Laplacian, which we will use later see 3, 7.Let L p x Ω. Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 279306, 8 pages doi:10.1155/2008/279306 Research Article Boundary Blow-Up Solutions to p x -Laplacian Equations