Positive solutions of a logistic equation on unbounded intervals

Ma, Li; Xu, Xingwang

doi:10.1090/S0002-9939-02-06405-5

Positive solutions of a logistic equation on unbounded intervals
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by Li Ma and Xingwang Xu

Proc. Amer. Math. Soc. 130 (2002), 2947-2958

DOI: https://doi.org/10.1090/S0002-9939-02-06405-5

Published electronically: April 22, 2002

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Abstract:

In this paper, we study the existence of positive solutions of a one-parameter family of logistic equations on $R_+$ or on $R$. These equations are stationary versions of the Fisher equations and the KPP equations. We also study the blow-up region of a sequence of the solutions when the parameter approaches a critical value and the non-existence of positive solutions beyond the critical value. We use the direct method and the sub and super solution method.

References

Giuseppe Buttazzo, Mariano Giaquinta, and Stefan Hildebrandt, One-dimensional variational problems, Oxford Lecture Series in Mathematics and its Applications, vol. 15, The Clarendon Press, Oxford University Press, New York, 1998. An introduction. MR 1694383
Zhiren Jin, Principal eigenvalues with indefinite weight functions, Trans. Amer. Math. Soc. 349 (1997), no. 5, 1945–1959. MR 1389781, DOI 10.1090/S0002-9947-97-01797-2
J. López-Gómez and J. C. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. Differential Equations 148 (1998), no. 1, 47–64. MR 1637517, DOI 10.1006/jdeq.1998.3456
K. J. Brown, C. Cosner, and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\textbf {R}^n$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 147–155. MR 1007489, DOI 10.1090/S0002-9939-1990-1007489-1
W. Allegretto, Principal eigenvalues for indefinite-weight elliptic problems in $\textbf {R}^n$, Proc. Amer. Math. Soc. 116 (1992), no. 3, 701–706. MR 1098396, DOI 10.1090/S0002-9939-1992-1098396-9
Jerry L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113–134. MR 365409
Tiancheng Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$ on the compact manifolds, Trans. Amer. Math. Soc. 331 (1992), no. 2, 503–527. MR 1055810, DOI 10.1090/S0002-9947-1992-1055810-7
Manuel A. del Pino, Positive solutions of a semilinear elliptic equation on a compact manifold, Nonlinear Anal. 22 (1994), no. 11, 1423–1430. MR 1280207, DOI 10.1016/0362-546X(94)90121-X
Allan L. Edelson and Adolfo J. Rumbos, Linear and semilinear eigenvalue problems in $\textbf {R}^n$, Comm. Partial Differential Equations 18 (1993), no. 1-2, 215–240. MR 1211732, DOI 10.1080/03605309308820928
Yihong Du and Qingguang Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal. 31 (1999), no. 1, 1–18. MR 1720128, DOI 10.1137/S0036141099352844
Li Ma, Conformal deformations on a noncompact Riemannian manifold, Math. Ann. 295 (1993), no. 1, 75–80. MR 1198842, DOI 10.1007/BF01444877
Peter Li, Luen-Fai Tam, and DaGang Yang, On the elliptic equation $\Delta u+ku-Ku^p=0$ on complete Riemannian manifolds and their geometric applications, Trans. Amer. Math. Soc. 350 (1998), no. 3, 1045–1078. MR 1407497, DOI 10.1090/S0002-9947-98-01886-8
Wei Ming Ni, On the elliptic equation $\Delta u+K(x)u^{(n+2)/(n-2)}=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493–529. MR 662915, DOI 10.1512/iumj.1982.31.31040
Patricio Aviles and Robert C. McOwen, Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds, J. Differential Geom. 27 (1988), no. 2, 225–239. MR 925121
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
Fang-Hua Lin, On the elliptic equation $D_i[a_{ij}(x)D_jU]-k(x)U+K(x)U^p=0$, Proc. Amer. Math. Soc. 95 (1985), no. 2, 219–226. MR 801327, DOI 10.1090/S0002-9939-1985-0801327-3
M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 14 (1997), no. 2, 237–274 (English, with English and French summaries). MR 1441394, DOI 10.1016/S0294-1449(97)80146-1
Y.H.Du and L. Ma, Positive solutions of an elliptic partial differential equation on $R^n$, preprint, 2000.
Y.H.Du and L.Ma, Logistic type equations on $R^n$ by a squeezing method involving boundary blow-up solutions, Journal of London Mathematical Society, 64(2001)107-124.
G. A. Afrouzi and K. J. Brown, On a diffusive logistic equation, J. Math. Anal. Appl. 225 (1998), no. 1, 326–339. MR 1639260, DOI 10.1006/jmaa.1998.6044
E.N.Dancer and S.P.Hasting, On the global bifurcation diagram for the one dimensional Ginzburg-Landau model of superconductivity, European J. Applied Math., 11(2000)271-291.