我們研究一維半正狄利克雷邊界問題的正解數與分枝曲線的演化與結構。
u''(x)+λg(u)-μ=0,u(-1)=u(1)=0,
這裡的λ,μ>0 為兩個分枝參數。假設非線性項g滿足g(0)=g(1)=0≥0, g(u)>0在(0,1)且g是凹函數在(0,1)或(是凹-凸函數在(0,1)且滿足一些特定條件。我們證明在(λ,‖u‖∞)平面上，對於固定μ>0, 分枝曲線包含一條⊂型曲線和其後我們研究對於改變μ>0,分枝曲線的結構與演化。(我們亦證明在(μ,‖u‖∞)平面上， 對於固定λ>π²/(4g'(0)),分枝曲線包含一條反向⊂型曲線且其後我們研究對於改變λ>π²/(4g'(0)) 分枝曲線的結構與演化。)我們亦研究在(μ,λ,‖u‖∞)空間上分枝曲面的結構與形狀。

We study the structures and evolution of bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional semipositone Dirichlet problem
u''(x)+λg(u)-μ=0,u(-1)=u(1)=0,
where λ,μ>0 are two bifurcation parameter. We assume that nonlinearity g satisfies g(0)=g(1)=0≥0, g(u)>0 on (0,1) and g either is concave on (0,1) or (is concave-convex on (0,1) and satisfies a certain condition). We prove that, for any fixed μ>0, the bifurcation diagram always consists of a ⊂-shaped curve on the (λ,‖u‖_∞)-plane and then we study the structures and evolution of bifurcation diagrams for varying μ>0. (We also prove that, for any fixed λ>π²/(4g'(0)), the bifurcation diagram always consists of a reversed ⊂-shaped curve on the (μ,‖u‖∞)-plane and then we study the structures and evolution of bifurcation diagrams for varying λ>π²/(4g'(0)).) We also study, in the (μ,λ,‖u‖∞)-space, the shape and structure of the bifurcation surface.

摘要
目錄
1.Introduction---------------2
2.Main results --------------10
3.Lemmas --------------------16
4.Proof of the main results -28
5.References-----------------30

1. J. Ali, R. Shivaji, K. Wampler, Population models with diffusion, strong Allee effect and constant yield harvesting, J. Math. Anal. Appl. 352 (2009) 907--913.
2. F. Brauer, C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.
3. A. Castro, S. Gadam, R. Shivaji, Positive solution curves of semipositone problems with concave nonlinearities, Proc. Roy. Soc. Edinburgh 127A (1997) 921--934.
4. A. Castro, S. Gadam, R. Shivaji, Evolution of positive solution curves in semipositone problems with concave nonlinearities, J. Math. Anal. Appl. 245 (2000) 282--293.
5. A. Castro, R. Shivaji, Nonnegative solutions for a class of nonpositone problems, Proc. Roy. Soc. Edinburgh 108A (1988) 291--302.
6. N. Chafee, E.F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Appl. Anal. 4 (1974) 17--37.
7. C.W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources, John Wiley & Sons, Inc., New York, 1990.
8. A. Collins, M. Gilliland, C. Henderson, S. Koone, L. McFerrin, E.K. Wampler, Population models with diffusion and constant yield harvesting, Rose Hulman Undergrad. Math. J., 5 (2004), 19 pp.
9. M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal. 52 (1973) 161--180.
10. S. Gadam, J.A. Iaia, Exact multiplicity of positive solutions in semipositone problems with concave-convex type nonlinearities, Electron. J. Qual. Theory Differ. Equ. (2001), 9 pp.
11. P. Girão, H. Tehrani, Positive solutions to logistic type equations with harvesting, J. Differential Equations 247 (2009) 574--595.
12. J. Goddard II, E.K. Lee, R. Shivaji, Population models with nonlinear boundary conditions, Electron. J. Differential Equations, Conference 19 (2010) 135--149.
13. J. Goddard II, E.K. Lee, R. Shivaji, Diffusive logistic equation with non-linear boundary conditions, J. Math. Anal. Appl. 375 (2011) 365--370.
14. J. Goddard II, E.K. Lee, R. Shivaji, Population models with diffusion, strong Allee effect, and nonlinear boundary conditions, Nonlinear Anal. 74 (2011) 6202--6208.
15. J. Goddard II, R. Shivaji, A population model with nonlinear boundary conditions and constant yield harvesting, Dynamic Systems and Applications, Vol. 6, 150--157, Dynamic, Atlanta, GA, 2012.
16. M. Guedda, L. Veron, Bifurcation phenomena associated to the p-Laplace operator, Trans. Amer. Math. Soc. 310 (1988) 419--431.
17. I.N. Herstein, Topics in Algebra, second ed., John Wiley & Sons Inc., New York, 1975.
18. K.-C. Hung, Exact multiplicity of positive solutions of a semipositone problem with concave-convex nonlinearity, J. Differential Equations 255 (2013) 3811--3831.
19. K.-C. Hung, Y.-N. Suen, S.-H. Wang, Structures and evolution of bifurcation diagrams for a diffusive generalized logistic problem with constant yield harvesting in one space variable, preprint, 2019.
20. P. Korman, Families of solution curves for some non-autonomous problems, Acta Appl. Math. 143 (2016) 165--178.
21. P. Korman, Y. Li, Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities, J. Differential Equations 257 (2014) 3730--3737.
22. P. Korman, Y. Li, T. Ouyang, Exact multiplicity results for boundary-value problems with nonlinearities generalising cubic, Proc. Roy. Soc. Edinburgh, Ser. A 126 (1996) 599--616.
23. T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J. 20 (1970) 1--13.
24. S.-Y. Lee, S.-H. Wang, C.-P. Ye, Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-Laplacian steady-state reaction-diffusion problem, Discrete Contin. Dyn. Syst., Supplement Volume (2005) 587--596.
25. S. Oruganti, J. Shi, R. Shivaji, Diffusive logistic equation with constant effort harvesting, I: Steady states, Trans. Amer. Math. Soc. 354 (2002) 3601--3619.
26. S. Oruganti, J. Shi, R. Shivaji, Logistic equation with the p-Laplacian and constant yield harvesting, Abstr. Appl. Anal. 2004 (2004) 723--727.
27. T. Ouyang, J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems: II, J. Differential Equations 158 (1999) 94--151.
28. H.L. Royden, Real Analysis, Macmillan, New York, 1988.
29. J. Shi, R. Shivaji, Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity, Discrete Contin. Dyn. Syst. 7 (2001) 559--571.
30. S. Takeuchi, Multiplicity result for a degenerate elliptic equation with logistic reaction. J. Differential Equations 173 (2001) 138--144.
31. S. Takeuchi, Y. Yamada, Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian, Nonlinear Anal. 42 (2000) 41--61.
32. S.-H. Wang, Bifurcation of positive solutions of generalized nonlinear undamped pendulum problems, Bull. Inst. Math. Acad. Sinica 21 (1993) 211--227.
33. S.-H. Wang, Positive solutions for a class of nonpositone problems with concave nonlinearities, Proc. Roy. Soc. Edinburgh 124A (1994) 507--515.
34. R.L. Wheeden, A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Marcel Dekker, New York, 1977.