SECOND-ORDER ESTIMATES FOR BOUNDARY BLOWUP SOLUTIONS OF SPECIAL ELLIPTIC EQUATIONS CLAUDIA ANEDDA, ANNA BUTTU, AND GIOVANNI PORRU Received 20 October 2005; Accepted November 2005 We find a second-order approximation of the boundary blowup solution of the equation β−1 Δu = eu|u| , with β > 0, in a bounded smooth domain Ω ⊂ RN Furthermore, we conu sider the equation Δu = eu+e In both cases, we underline the effect of the geometry of the domain in the asymptotic expansion of the solutions near the boundary ∂Ω Copyright © 2006 Claudia Anedda et al 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 Introduction Let Ω ⊂ RN be a bounded smooth domain In 1916, Bieberbach [10] has investigated the problem Δu = eu in Ω, u(x) −→ ∞ as x −→ ∂Ω, (1.1) and has proved the existence of a classical solution called a boundary blowup (explosive, large) solution Moreover, if δ = δ(x) denotes the distance from x to ∂Ω, we have [10] u(x) − log(2/δ (x)) → as x → ∂Ω Recently, Bandle [4] has improved the previous estimate finding the expansion u(x) = log + (N − 1)K(x)δ(x) + o δ(x) , δ (x) (1.2) where K(x) denotes the mean curvature of ∂Ω at the point x nearest to x, and o(δ) has the usual meaning Boundary estimates for various nonlinearities have been discussed in several papers, see for example [1, 3, 5, 8, 13–16] In Section of the present paper we investigate boundary blowup solutions of the β−1 equation Δu = eu|u| , with β > 0, β = We prove the estimate u(x) = Φ(δ) + β−1 (N − 1)K(x)δ Φ(δ) Hindawi Publishing Corporation Boundary Value Problems Volume 2006, Article ID 45859, Pages 1–12 DOI 10.1155/BVP/2006/45859 1−β + O(1)δ Φ(δ) 1−2β , (1.3) Second-order estimates where Φ(δ) is defined by the equation ∞ 2F(t) Φ(s) −1/2 = s, F(t) = t β−1 −∞ eτ |τ | dτ, (1.4) K(x) is the mean curvature of the surface {x ∈ Ω : δ(x) = constant}, and O(1) denotes a bounded quantity u In Section we consider boundary blowup solutions of the equation Δu = eu+e We find the estimate u(x) = Ψ(δ) + (N − 1)K(x)e−Ψ(δ) δ + O(1)e−2Ψ(δ) δ, (1.5) where Ψ is defined by the equation ∞ Ψ(s) −1/2 t 2ee − dt = s (1.6) In this paper, the distance function δ = δ(x) plays an important role Recall that if Ω is smooth then also δ(x) is smooth for x near to ∂Ω, and [12] N N δxi δxi = 1, δxi xi = (N − 1)K = H, − i =1 (1.7) i=1 where K = K(x) is the mean curvature of the surface {x ∈ Ω : δ(x) = constant} The effect of the geometry of the domain in the behaviour of boundary blowup solutions for special equations has been observed in various papers, see for example, [2, 7, 9, 11] β−1 The equation Δu = eu|u| In what follows we denote with O(1) a bounded quantity β−1 Lemma 2.1 Let β > 0, f (s) = es|s| , F(s) = F(s) f (s) f (s) s f (t)dt Then −2 = + O(1)s−β −∞ (2.1) Proof For s > we have F(s) f (s) f (s) −2 = f (s) f (s) −2 F(0) + f (s) f (s) = βe−s sβ−1 F(0) + e−s β β s 0 f (t)dt et βt β−1 dt + βe−s β = βe−s sβ−1 F(0) + − e−s + βe−s β s −2 β β s β s et sβ−1 − t β−1 dt β et sβ−1 − t β−1 dt β (2.2) We have lim sβ βe−s sβ−1 F(0) = 0, β s→∞ lim sβ e−s = β s→∞ (2.3) Claudia Anedda et al ˆ Moreover, using de l’Hopital’s rule we find s β et s2β−1 − sβ t β−1 dt = lim s→∞ s→∞ e sβ s tβ 0e (2β − 1)sβ−1 − βt β−1 dt e sβ β β −1 s β (β − 1)es s + et (2β − 1)(β − 1)sβ−2 dt = lim s→∞ βesβ sβ−1 β lim s β β−1 et dt = + (2β − 1)(β − 1) lim sβ s→∞ βe s β β−1 β−1 = + (2β − 1)(β − 1) lim = s→∞ β + βsβ β β (2.4) The lemma follows Remark 2.2 If β = 1, we have F(s) f (s)( f (s))−2 = We not care of this special case because it has been discussed in [2] Lemma 2.3 Let Φ = Φ(δ) be defined by ∞ Φ(δ) 2F(t) −1/2 dt = δ, F(t) = t β−1 −∞ f (τ)dτ, f (τ) = eτ |τ | (2.5) Then −Φ (δ) = + O(1) Φ(δ) −β δ f Φ(δ) (2.6) Proof By the (trivial) relation −1 + + O(1)s−β = + O(1)s−β , (2.7) using (2.1) we have −1 + 2F(s) f (s) f (s) −2 = + O(1)s−β (2.8) Multiplying by (2F(s))−1/2 we find − 2F(s) −1/2 − + 2F(s) 2F(s) 1/2 1/2 f (s) f (s) −1 f (s) −2 = 2F(s) = 2F(s) −1/2 −1/2 + O(1) 2F(s) + O(1) 2F(s) −1/2 −β −1/2 −β s , (2.9) s Integrating on (s, ∞) we get 2F(s) 1/2 f (s) −1 = ∞ s 2F(t) −1/2 ∞ dt + O(1) s 2F(t) −1/2 −β t dt (2.10) Second-order estimates Using de l’Hˆ pital’s rule we find o lim s−β s→∞ ∞ s ∞ s −1/2 2F(t) 2F(t) dt −1/2 −β t dt −1/2 −β s 2F(s) = lim s→∞ + βs−β−1 = + lim β s→∞ = + lim s→∞ −1/2 2F(t) s s 2F(s) −β 2F(t) −1/2 dt −1/2 −β s 2F(s) ∞ ∞ s dt (2.11) −1/2 −1 − s 2F(s) f (s) = In the last step we have used the limit lim s→∞ s f (s) = ∞, F(s) (2.12) which can be proved easily with de l’Hˆ pital’s rule Using (2.11), (2.10) can be rewritten o as 2F(s) 1/2 f (s) −1 ∞ = 2F(t) s −1/2 dt + O(1)s−β ∞ s 2F(t) −1/2 dt (2.13) Putting s = Φ(δ) and using the equation −Φ (δ) = (2F(Φ(δ)))1/2 , the lemma follows Theorem 2.4 Let Ω be a bounded smooth domain in RN , N ≥ 2, and let β > 0, β = If β−1 u(x) is a boundary blowup solution of Δu = eu|u| in Ω, then u(x) = Φ(δ) + β−1 Hδ Φ(δ) 1−β + O(1)δ Φ(δ) 1−2β , (2.14) where Φ(δ) is defined as in (2.5), δ = δ(x) is the distance from x to ∂Ω and H is defined by (1.7) Proof We look for a super-solution of the form w(x) = Φ(δ) + β−1 Hδ Φ(δ) 1−β + αδ Φ(δ) 1−2β , (2.15) where α is a positive constant to be determined Denoting by differentiation with respect to δ, we have wxi = Φ (δ)δxi + β−1 Hxi δ Φ(δ) 1−β + β−1 H δ Φ(δ) 1−β δxi + α δ Φ(δ) 1−2β δx i (2.16) Using (1.7) we find Δw = Φ (δ) − Φ (δ)H + β−1 ΔHδ Φ(δ) + β−1 H δ Φ(δ) + α δ Φ(δ) 1−2β 1−β 1−β + 2β−1 ∇H · ∇δ δ Φ(δ) − β−1 H δ Φ(δ) − α δ Φ(δ) 1−2β H 1−β 1−β (2.17) Claudia Anedda et al β−1 With f (τ) = eτ |τ | , by (2.5) we have Φ (δ) = f (Φ) Often we write Φ instead of Φ(δ) and Φ instead of Φ (δ) Lemma 2.3 yields −Φ = + O(1)Φ−β δ f (Φ) (2.18) Using (2.18) and the equation Φ = −(2F(Φ))1/2 we find lim δ →0 Φ(δ) δ Φ(δ) 1−β −β f (Φ) = lim δ →0 Φ = lim −Φ δ →0 s2 = lim s→∞ 2F(s) 1/2 Φ 2F(Φ) 1/2 s = lim s→∞ f (s) (2.19) 1/2 = Let us write the last result as 1−β Φ(δ) = o(1)δ Φ(δ) −β f (Φ), (2.20) where o(1) denotes a quantity which tends to zero as δ → Using (2.18) again we find lim δ →0 −β Φ(δ) −β δ Φ(δ) Φ f (Φ) = −1 (2.21) Therefore, δ Φ(δ) 1−β 1−β = Φ(δ) + (1 − β)δ Φ(δ) = o(1)δ Φ(δ) −β −β Φ (2.22) f (Φ) Further differentiation yields δ Φ(δ) 1−β = 2(1 − β) Φ(δ) −β + (1 − β)δ Φ(δ) Φ − β(1 − β)δ Φ(δ) −β −β −1 (Φ )2 (2.23) f (Φ) Moreover, recalling (2.12) we find lim δ →0 δ Φ(δ) −β −1 δ Φ(δ) −β (Φ )2 f (Φ) = lim δ →0 2F(Φ) 2F(s) = lim = Φ f (Φ) s→∞ s f (s) (2.24) Using the last result and (2.21), from (2.23) we find δ Φ(δ) 1−β = O(1)δ Φ(δ) −β δ Φ(δ) 1−2β = o(1)δ Φ(δ) −2β f (Φ) (2.25) Similarly, we find δ Φ(δ) 1−2β = O(1)δ Φ(δ) f (Φ), −2β f (Φ) (2.26) Second-order estimates Denoting by M1 a nonnegative constant independent of α and using (2.18), (2.20), (2.22), (2.25), (2.26), by (2.17) we get Δw < f (Φ) + Hδ + M1 δΦ−β + αM1 δΦ−2β (2.27) On the other side, we have f (w) = e(Φ+β = eΦ −1 HδΦ1−β +αδΦ1−2β )β β (1+β−1 HδΦ−β +αδΦ−2β )β (2.28) Let us take δ0 > and α such that for {x ∈ Ω : δ(x) < δ0 } we have − < β−1 Hδ Φ(δ) −β + αδ Φ(δ) −2β < (2.29) Then, denoting by M2 a nonnegative constant independent of α we find f (w) > eΦ β (1+HδΦ−β +αβδΦ−2β −M = f (Φ)eHδ+αβδΦ −β (δΦ −β ) −M2 (αδΦ−2β )2 ) −M2 δ Φ−β −M2 (αδ)2 Φ−3β > f (Φ) + Hδ + αβδΦ −β − M2 δ Φ −β (2.30) − M2 (αδ) Φ −3β By (2.27) and (2.30) we find that Δw < f (w) (2.31) when + Hδ + M1 δΦ−β + αM1 δΦ−2β < + Hδ + αβδΦ−β − M2 δ Φ−β − M2 (αδ)2 Φ−3β (2.32) Rearranging we find M1 + M2 δ < α β − M2 αδΦ−2β − M1 Φ−β (2.33) We can take δ0 small and α large so that (2.33) and (2.29) hold for δ(x) < δ0 β−1 Our function f (t) = et|t| is positive and increasing for all t, and F(t)t −2 is increasing t for large t Moreover, if G(t) = F(s)ds, for a and b such that < a < < b, we have a F(t) G(t) F(t) ≤ ≤b f (t) G (t) f (t) for large t (2.34) Therefore, by [7, Theorem 4(ii)] we have, for some constant C > 0, Cδ Φ (δ) + Φ(δ) ≤ u(x) ≤ Φ(δ) + CδΦ(δ) (2.35) Using the right-hand side of (2.35) we find w(x) − u(x) ≥ Φ(δ) β−1 Hδ Φ(δ) −β + αδ Φ(δ) −2β − Cδ (2.36) Claudia Anedda et al Take α and δ0 such that (2.33) holds and put αδ0 (Φ(δ0 ))−2β = q Decrease δ0 and increase α so that αδ0 (Φ(δ0 ))−β = q and β−1 Hδ Φ(δ) −β + q − Cδ > (2.37) for δ(x) = δ0 Then, w(x) ≥ u(x) on {x ∈ Ω : δ(x) = δ0 } When α is fixed, by (2.36) we get liminf x→∂Ω [w(x) − u(x)] ≥ Hence, using (2.31) we find w(x) ≥ u(x) on {x ∈ Ω : δ(x) < δ0 } We look for a subsolution of the form v(x) = Φ(δ) + β−1 Hδ Φ(δ) 1−β 1−2β − αδ Φ(δ) , (2.38) where α is a positive constant to be determined Instead of (2.27), now we find Δv > f (Φ) + Hδ − M1 δΦ−β − αM1 δΦ−2β (2.39) Of course, the constant M1 in (2.39) and the constants Mi in what follows are not necessarily the same as in the previous case Now we have f (v) = eΦ β (1+β−1 HδΦ−β −αδΦ−2β )β (2.40) Let us take δ0 > and α such that, for {x ∈ Ω : δ(x) < δ0 } we have − < β−1 Hδ Φ(δ) −β − αδ Φ(δ) −2β < (2.41) Then, f (v) < eΦ β (1+HδΦ−β −αβδΦ−2β +M = f (Φ)eHδ −αβδΦ −β (δΦ −β ) +M2 (αδΦ−2β )2 ) +M2 δ Φ−β +M2 (αδ)2 Φ−3β (2.42) In our next step, we take δ and α such that αδΦ−β < 1, Hδ − αβδΦ−β + M2 δ Φ−β + M2 (αδ)2 Φ−3β < (2.43) Then we find f (v) < f (Φ) + Hδ − αβδΦ−β + M3 δ + M3 (αδ)2 Φ−2β (2.44) By (2.39) and (2.44) we find that Δv > f (v) provided + Hδ − M1 δΦ−β − αM1 δΦ−2β > + Hδ − αβδΦ−β + M3 δ + M3 (αδ)2 Φ−2β (2.45) Rearranging we have α β − M1 Φ−β − M3 αδΦ−β > M1 + M3 δΦβ (2.46) Since δΦβ → as δ → 0, inequality (2.46) (in addition to (2.41) and (2.43)) holds for δ(x) < δ0 with suitable δ0 and α 8 Second-order estimates Using the left-hand side of (2.35) we find v(x) − u(x) ≤ β−1 Hδ Φ(δ) 1−β = Φ(δ) 1−β − αδ Φ(δ) 1−2β β−1 Hδ − αδ Φ(δ) −β − Cδ Φ (δ) − Cδ Φ (δ) Φ(δ) β −1 (2.47) Take α and δ0 such that (2.46) holds, and put αδ0 (Φ(δ0 ))−β = q Decrease δ0 and increase α so that αδ0 (Φ(δ0 ))−β = q and β−1 Hδ − q − Cδ Φ (δ) Φ(δ) β −1 eΨ + He−Ψ δ + αe−2Ψ δ , (3.19) we find f (w) = ew+e > eΨ+He w Ψ = eΨ+e e[He −Ψ −Ψ δ+αe−2Ψ δ+eΨ [1+He−Ψ δ+αe−2Ψ δ] δ+αe−2Ψ δ+Hδ+αe−Ψ δ] (3.20) > f (Ψ) − M3 e−Ψ δ + Hδ + αe−Ψ δ By (3.18) and (3.20) we have Δw < f (w) (3.21) + Hδ + M1 e−Ψ δ + αM2 e−2Ψ δ < − M3 e−Ψ δ + Hδ + αe−Ψ δ (3.22) provided Rearranging we find M1 + M3 < α − M2 e−Ψ(δ) (3.23) Inequality (3.23) holds provided δ is small and α is large enough t The function f (t) = et+e is positive and increasing for all t If F(t) is defined as in t Lemma 3.1, the function F(t)t −2 is increasing for large t Moreover, if G(t) = F(s)ds, for < a < < b we have a F(t) G(t) F(t) ≤ ≤b f (t) G (t) f (t) for large t (3.24) Therefore, by [7, Theorem 4(ii)] we have, for some constant C > 0, Cδ Ψ (δ) + Ψ(δ) ≤ u(x) ≤ Ψ(δ) + CδΨ(δ) (3.25) Claudia Anedda et al 11 Using the right-hand side of (3.25) we find w(x) − u(x) ≥ He−Ψ δ + αe−2Ψ δ − CδΨ(δ) (3.26) Take α and δ0 so that (3.23) holds for δ(x) = δ0 and put q = αe−2Ψ(δ0 ) δ0 Decrease δ0 and increase α so that αe−2Ψ(δ0 ) δ0 = q and He−Ψ δ + q − CδΨ(δ) > for δ(x) = δ0 Recall that δΨ(δ) → as δ → Then, w(x) ≥ u(x) on {x ∈ Ω : δ(x) = δ0 } Moreover, by (3.26) we have w(x) − u(x) ≥ on ∂Ω Hence, using (3.21) we find w(x) ≥ u(x) on {x ∈ Ω : δ(x) < δ0 } Let us prove that v = Ψ + He−Ψ δ − αe−2Ψ δ (3.27) is a subsolution provided α is a suitable positive constant By computation, instead of (3.18), now we find Δv > f (Ψ) + Hδ − M4 e−Ψ δ − αM5 e−2Ψ δ (3.28) The next step is slightly delicate Take α and δ such that eαe−Ψ δ < 1, He−Ψ δ − αe−2Ψ δ < (3.29) Then, using the second inequality in (3.29), we find ev = eΨ+He −Ψ δ −αe−2Ψ δ < eΨ + He−Ψ δ − αe−2Ψ δ + e He−Ψ δ 2 + e αe−2Ψ δ (3.30) Hence, using the first inequality in (3.29), we get f (v) = ev+e < eΨ+He v < f (Ψ)eHδ+M6 e −Ψ −Ψ δ −αe−2Ψ δ+eΨ +Hδ −αe−Ψ δ+eH e−Ψ δ +eα2 e−3Ψ δ δ −αe−Ψ δ < f (Ψ) + Hδ + M7 e−Ψ δ − αe−Ψ δ + αe−Ψ δ (3.31) Comparing the last estimate with (3.28) we have Δv > f (v) (3.32) provided + Hδ − M4 e−Ψ δ − αM5 e−2Ψ δ > + Hδ + M7 e−Ψ δ − αe−Ψ δ + αe−Ψ δ (3.33) Rearranging, this inequality reads as α − αe−Ψ δ − M5 e−Ψ > M4 + M7 (3.34) Of course, (3.34) and (3.29) hold provided α is large and δ is small enough Using the left-hand side of (3.25), decreasing δ0 and increasing α if necessary, one proves that v(x) − u(x) ≤ at all points in Ω with δ(x) = δ0 Moreover, using (3.25) again we observe that v(x) − u(x) ≤ on ∂Ω Therefore, by (3.32) it follows that v(x) is a subsolution on {x ∈ Ω : δ(x) < δ0 } The theorem is proved 12 Second-order estimates References [1] L Andersson and P T Chru´ ciel, Solutions of the constraint equations in general relativity satiss fying “hyperboloidal boundary conditions”, Dissertationes Mathematicae (Rozprawy Matematyczne) 355 (1996), 1–100 [2] C Anedda, A Buttu, and G Porru, Boundary estimates for blow-up solutions of elliptic equations with exponential growth, to appear in Proceedings Differential and Difference Equations [3] C Anedda and G Porru, Higher order boundary estimates for blow-up solutions of elliptic equations, to appear in Differential Integral Equations [4] C Bandle, Asymptotic behaviour of large solutions of quasilinear elliptic problems, Zeitschrift fă r u Angewandte Mathematik und Physik 54 (2003), no 5, 731–738 [5] C Bandle and E Giarrusso, Boundary blow up for semilinear elliptic equations with nonlinear gradient terms, Advances in Differential Equations (1996), no 1, 133–150 [6] C Bandle and M Marcus, “Large” solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour, Journal d’Analyse Math´ matique 58 (1992), 9–24 e , On second-order effects in the boundary behaviour of large solutions of semilinear elliptic [7] problems, Differential and Integral Equations 11 (1998), no 1, 23–34 , Dependence of blowup rate of large solutions of semilinear elliptic equations, on the curva[8] ture of the boundary, Complex Variables Theory and Application 49 (2004), no 7–9, 555–570 [9] S Berhanu and G Porru, Qualitative and quantitative estimates for large solutions to semilinear equations, Communications in Applied Analysis (2000), no 1, 121–131 [10] L Bieberbach, Δu = eu und die automorphen Funktionen, Mathematische Annalen 77 (1916), no 2, 173–212 [11] M del Pino and R Letelier, The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Analysis Theory, Methods & Applications Series A: Theory and Methods 48 (2002), no 6, 897–904 [12] D Gilbarg and N S Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften, vol 224, Springer, Berlin, 1977 [13] A Greco and G Porru, Asymptotic estimates and convexity of large solutions to semilinear elliptic equations, Differential and Integral Equations 10 (1997), no 2, 219–229 [14] J B Keller, On solutions of Δu = f (u), Communications on Pure and Applied Mathematics 10 (1957), 503–510 [15] A C Lazer and P J McKenna, Asymptotic behavior of solutions of boundary blowup problems, Differential and Integral Equations (1994), no 3-4, 1001–1019 [16] R Osserman, On the inequality Δu ≥ f (u), Pacific Journal of Mathematics (1957), no 4, 1641– 1647 Claudia Anedda: Dipartimento di Matematica, Universit´ di Cagliari, Via Ospedale 72, a 09124 Cagliari, Italy E-mail address: canedda@unica.it Anna Buttu: Dipartimento di Matematica, Universit´ di Cagliari, Via Ospedale 72, a 09124 Cagliari, Italy E-mail address: buttu@uncia.it Giovanni Porru: Dipartimento di Matematica, Universit´ di Cagliari, Via Ospedale 72, a 09124 Cagliari, Italy E-mail address: porru@unica.it ... in the boundary behaviour of large solutions of semilinear elliptic [7] problems, Differential and Integral Equations 11 (1998), no 1, 23–34 , Dependence of blowup rate of large solutions of semilinear... boundary estimates for blow-up solutions of elliptic equations, to appear in Differential Integral Equations [4] C Bandle, Asymptotic behaviour of large solutions of quasilinear elliptic problems,... Boundary estimates for blow-up solutions of elliptic equations with exponential growth, to appear in Proceedings Differential and Difference Equations [3] C Anedda and G Porru, Higher order boundary estimates