Hindawi 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 with Exponential Nonlinearities Qihu Zhang Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China Correspondence should be addressed to Qihu Zhang, zhangqh1999@yahoo.com.cn Received 18 August 2007; Accepted 25 November 2007 Recommended by M. Garcia-Huidobro This paper investigates the px-Laplacian equations with exponential nonlinearities − px u  e fx,u  0inΩ, ux →  ∞ as dx, ∂Ω → 0, where − px u  −div|∇u| px−2 ∇u is called px- Laplacian. The singularity of boundary blow-up solutions is discussed, and the existence of bound- ary blow-up solutions 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 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 example, 1–15. In this paper, we consider the px-Laplacian equations with exponential nonlinearities −Δ px u  e fx,u  0inΩ, ux −→  ∞ as dx, ∂Ω −→ 0, P where −Δ px u  −div|∇u| px−2 ∇u, ΩB0,R ⊂ R N is a bounded radial domain B0,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 px and fx, u satisfy that H 1  px ∈ C 1 Ω is radial and satisfies 1 <p − ≤ p  < ∞, where p −  inf Ω px,p   sup Ω px; 1.1 2 Journal of Inequalities and Applications H 2  fx, u is radial with respect to x, fx, · is increasing and fx, 00 for any x ∈ Ω; H 3  f : Ω × R → R is a continuous function and satisfies   fx, t   ≤ C 1  C 2 |t| γx , ∀x, t ∈ Ω × R, 1.2 where C 1 ,C 2 are positive constants, 0 ≤ γ ∈ CΩ. The operator −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, px-Laplacian problems are more complicated than those of p-Laplacian ones see 6; and another difficulty of this paper is that fx, u cannot be represented as hxfu. 2. Preliminary In 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   u | u is a measurable real-valued function,  Ω   ux   px dx < ∞  . 2.1 We can introduce the norm on L px Ω by |u| px  inf  λ>0 |  Ω     ux λ     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, 1.14. The space W 1,px Ω is defined by 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 is the closure of C ∞ 0 Ω in W 1,px Ω. W 1,px Ω and W 1,px 0 Ω are separable, reflexive, and uniform convex Banach spaces see 3, Theorem 2.1. If u ∈ W 1,px loc Ω ∩ CΩ, u is called a solution of P if it satisfies  Q   ∇u   px−2 ∇u∇qdx   Q fx, uqdx  0, ∀q ∈ W 1,px 0 Q, 2.5 for any domain Q  Ω,andmaxk − u, 0 ∈ W 1,px 0 Ω for any k ∈ N  . Let W 1,px 0,loc Ω  {u| there exists an open domain Q  Ω s.t. u ∈ W 1,px 0 Q}. For any u ∈ W 1,px loc Ω ∩ CΩ and ϕ ∈ W 1,px 0,loc Ω, define A : W 1,px loc Ω∩CΩ → W 1,px 0,loc Ω ∗ as Au, ϕ   Ω |∇u| px−2 ∇u∇ϕ  e fx,u ϕdx. Qihu Zhang 3 Lemma 2.1 see 5, Theorem 3.1. Let h ∈ W 1,px Ω ∩ CΩ, X  h  W 1,px 0,loc Ω ∩ CΩ. Then, A : X → W 1,px 0,loc Ω ∗ is strictly monotone. Let g ∈ W 1,px 0,loc Ω ∗ , if g,ϕ≥0, for all ϕ ∈ W 1,px 0,loc Ω,ϕ ≥ 0 a.e. in Ω, then denote 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 ∗ . Definition 2.2. Let u ∈ W 1,px loc Ω ∩ CΩ. If Au ≥ 0 Au ≤ 0 in W 1,px 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,px loc Ω ∩ CΩ 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 4, Theorem 1.1. Under the conditions (H 1 )and(H 3 ), if u ∈ W 1,px Ω is a bounded weak solution of −Δ px u  e fx,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,thenP can be transformed into  r N−1 |u  | pr−2 u     r N−1 e fr,u ,r∈ 0,R, u0u 0 ,u  00,u  r ≥ 0for0<r<R. 3.1 It means that ur is increasing. Theorem 3.1. If there exists a constant σ ∈ R/2,R such that fr, 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 dx, ∂Ω → 0), and such that, if u is a weak solution of problem P,thenux ≤ Φ 1 x. Proof. Let R 0 ∈ σ,R.Denote Θr, a, λ  R 0 r ⎡ ⎣ a  a ln  R−R 0 −λ  −1  1/s−1 s  R−R 0 −λ  ⎤ ⎦ pR o −1/pt−1   R o  N−1 t N−1 sin εt−σ  1/pt−1 dt. 3.3 Define the function gr, a on 0,R as gr, a ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  a ln R − r −1  1/s  k, R 0 ≤ r<R, k − Θr, a, 0  a ln R − R 0  −1  1/s , σ<r<R 0 , k − Θσ,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 px is a constant, R 0 ∈ σ,R,andR − R 0 is small enough, ε  π/2R 0 − σ and k 2p  /α ln R − R 0  −1  1/s Θσ,2a, 0. Obviously, for any positive constant a, gr, a ∈ C 1 0,R. When R 0 <r<R,wehave  r N−1 |g  | pr−2 g    r N−1  a 1/s s  pr−1 pr − 1 R − r pr  ln R−r −1  1/s−1pr−1  1Πr  , 3.5 where Πr 1/s − 1 ln R − r −1   r N−1 a 1/s /s pr−1   r N−1  a 1/s /s  pr−1 pr − 1 R − r  −p  r ln R − r  pr − 1  R − r 1/s − 1p  r ln ln R − r −1  pr − 1  R − r. 3.6 If R − R 0  is small enough, it is easy to see |Πr|≤1/2; from 3.5,wehave  r N−1 |g  | pr−2 g    ≤ 2r N−1  a 1/s s  pr−1  pr − 1R − r  −pr  ln R − r −1  1/s−1pr−1 ≤ r N−1  1 R − r  αa  r N−1 e αg s ≤ r N−1 e fr,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  | pr−2 g     ε  R o  N−1  a  a ln  R − R 0  −1  1/s−1 s  R − R 0   pR o −1 cos  εr − σ  ≤ r N−1 e αg s ≤ r N−1 e fr,g , σ<r<R 0 . 3.8 Obviously,  r N−1 |g  | pr−2 g     0 ≤ r N−1 e fr,g , 0 ≤ r<σ. 3.9 Since g|x|,a is a C 1 function on B0,R,if0<R− R 0 is small enough R 0 depends on R, p, s, α,from3.7, 3.8,and3.9, 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/2R 0 − σ,0<<1, is a positive small constant such that αa −  > sup |x|≥R 0 px. Obviously, g m |x|,a− is a supersolution of P on B0,R−1/m.Ifu is a solution of P, according to the comparison principle, we get that g m |x|,a− ≥ ux for any x ∈ B0,R−1/m. For any x ∈ B0,R− 1/m \ B0,R 0 , we have g m |x|,a−  ≥ g m1 |x|,a− . Thus, ux ≤ lim m→∞ g m  |x|,a−   , ∀x ∈ B0,R \ B  0,R 0  . 3.11 When dx, ∂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 ≥ ux, for all x ∈ B0,R, then Φ 1 xg|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 fr, 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 dx, ∂Ω → 0),andsuchthat,ifux is a solution of problem P,thenux ≥ Φ 2 x. Proof. Let z 1 be a radial solution of −Δ px z 1 x−μ in Ω 1  B0,σ, z 1  0on∂Ω 1 , 3.14 where μ>2 is a positive constant. We denote z 1  z 1 rz 1 |x|,thenz 1 satisfies z 1 σ0, z  1 00, and z  1      rμ N     1/pr−1 , z 1  −  σ r     rμ N     1/pr−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   pR o −1  σ N−1 t N−1 R 0 − t R 0 − σ      tμ N     1/pt−1  pσ−1  1/pt−1 dt. 3.16 It is easy to see that −h  b σ,0z  1 σ     σμ N     1/pσ−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 vr, b on B0,R as vr, 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/pr−1 dr   b ln  R − R 0  −1  1/s − k ∗ − h b σ,0, r ≤ σ, 3.18 where b ∈ 0, 1/βinf |x|≥R 0 px is a constant, R 0 ∈ σ,R,andR − R 0 is small enough, and k ∗ 2p  /β ln 2R − R 0  −1  1/s . Obviously, for any positive constant b, vr, 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  | pr−2 v    ≥ r N−1 e fr,v , ∀r ∈  R 0 ,R  . 3.19 When R − R 0 is small enough, for all r ∈ σ,R 0 , since fr, v ≤ 0, then  r N−1 |v  | pr−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   pR 0 −1 ≥ r N−1 e fr,v . 3.20 Obviously,  r N−1 |v  | pr−2 v     r N−1 μ ≥ r N−1 e fr,v , ∀r ∈ 0,σ. 3.21 Combining 3.19, 3.20,and3.21, we can see that vr, a is a subsolution of P. Define the function v m r, b   on B0,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/pr−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/βinf |x|≥R 0 px. We can see that v m r, b   ∈ C 1 0,R is a subsolution of P on BR 0 ,R, according to the comparison principle, we get that v m |x|,b   ≤ ux for any x ∈ B0,R. For any x ∈ B0,R \ B0,R 0 , we have v m |x|,b  ≤ v m1 |x|,b . Thus, ux ≥ lim m→∞ v m  |x|,b   , ∀x ∈ B0,R \ B  0,R 0  . 3.23 When dx, ∂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 ≤ ux, ∀x ∈ B0,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 px >Nand there exists a σ ∈ R/2,R such that fr, 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 boundary blow-up solutions of P, let us consider the problem −Δ px u  e fx,u  0inΩ, uxj for x ∈ ∂Ω, 3.26 where j  1, 2, Since inf x∈Ω px >N,thenW 1,px Ω → C α Ω,whereα ∈ 0, 1.The relative functional of 3.26 is ϕu  Ω 1 px   ∇ux   px dx   Ω Fx, udx, 3.27 where Fx, u  u 0 e fx,t dt. Since ϕ is coercive in X j : j  W 1,px 0 Ω, then ϕ possesses a nontrivial minimum point u j , then problem 3.26 possesses a weak solution u j . According to the comparison principle, we get u j x ≤ u j1 x for any x ∈ Ω and j  1 , 2, Since Φ 1 x defined in Theorem 3.1 is a supersolution, according to the comparison principle, we have u j x ≤ Φ 1 x on Ω for all j  1, 2, Since Φ 1 x is locally bounded, from Lemma 2.4, every weak solution of P is a locally C 1,ϑ loc function. Thus, {u j x} possesses a subsequence we still denote it by {u j x}, such that lim j→∞ u j  u is a solution of P. Acknowledgments This work was supported by the National Science Foundation of China 10701066 & 10671084and China Postdoctoral Science Foundation 20070421107 and the Natural Science Foundation of Henan Education Committee 2007110037. References 1 Y. Chen, S. Levine, and M. 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Lair, “A necessary and sufficient condition for existence of large solutions to 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-up solutions to 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