我們研究 p-Laplacian 邊界爆炸值問題解存在之充分與必要條件.我們也研究分歧曲線 λ(ρ) 的漸近行為，其中ρ代表解u(x)在(0,1)的最小值。特別的是，我們探討分歧曲線 λ(ρ)在ρ=0的連續性。我們研究的結果有助於去決定對於給定的參數λ>0，p-Laplacian 邊界爆炸值問題解的個數，且我們也提供了幾個有趣的例子。

We investigate the necessary and sufficient conditions for the existence of solutions of the p-Laplacian boundary blow-up problem .We also study asymptotic behaviors of the bifurcation curve λ(ρ) , where ρ:=min{x in (0,1)}u(x). In particular, we study the continuity of λ(ρ) at ρ=0. Our results help to determine the number of solutions for any λ>0. Some interesting examples are given.

1.A. Aftalion and W. Reichel, Existence of two boundary blow-up solutions for semilinear elliptic equations, J. Differential Equations 141 (1997), 400-421.
2.V. Anuradha, C. Brown and R. Shivaji, Explosive nonnegative solutions to two point boundary value problems, Nonlinear Analysis 26(1996), 613-630.
3.C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Analyse Math.58 (1992), 9-24.
4.C. Bandle and M. Marcus, Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary, Ann. Inst. H. Poincare Anal. Non Lineaire} 12(1995), 155--171.
5.C. Bandle and M. Marcus, On second-order effects in the boundary behaviour of large solutions of semilinear elliptic problems, Differential Integral Equations 11(1998), 23--34.
6.L. Bieberbach, △u=e^(u) and die automorphen Funktionen,Math. Annln 77 (1916), 173-212.
7.Y. J. Cheng, Some surprising results on a one-dimensional elliptic boundary value blow-up problem, Z. Anal. Anwendungen 18 (1999), 525--537.
8.G. Diaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear Analysis 20 (1993), 97-125.
9.J. B. Keller, On solutions of △u=f(u), Comm. Pure Appl. Math.10 (1957), 503-510.
10.A. V. Lair and A. W. Wood, Large solutions of semilinear elliptic problems, Nonlinear Analysis 37 (1999), 805-812.
11.A. C. Lazer and P. J. McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Analysis 21 (1993), 327-335.
12.C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis (A Collection of Paper Dedicated to Lipman Bers, (1974) pp. 245-272, Academic Press, New York.
13.M. Marcus and L. Veron, Uniqueness of solutions with blowup at the boundary for a class of nonlinear elliptic equations, C. R. Acad. Sci. Paris 317(1993), 559-563.
14.J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Analyse Math.69 (1996), 229-247.
15.P. J. McKenna, W. Reichel, and W. Walter, Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, Nonlinear Analysis 28 (1997), 1213-1225.
16.R. Osserman, On the inequality △u≧f(u),Pacific J. Math.7 (1957), 1641-1647.
17.H. Rademacher, Einige besondere problem partieller Differential gleichungen, in Die Differential- and Integralgleichungen, der Mechanik and Physik I, 2nd edition (1943), pp. 838-845, Rosenberg, New York.
18.S.-H. Wang,\ Existence and multiplicity of boundary blow-up nonnegative solutions to two point boundary value problems,Nonlinear Analysis 42 (2000), 139-162.
19.S.-H. Wang, Y.-T. Liu and I-A Cho, An explicit formula of the bifurcation curve for a boundary blow-up problem, to appear in Dynamics of Continuous, Discrete and Impulsive Systems.
20.S.-H. Wang and M.-T. Shieh, Necessary and sufficient conditions for the existence of solutions of a boundary blow-up problem, submitted.