On the positive solutions of semilinear equations Δ𝑢+𝜆𝑢-ℎ𝑢^{𝑝}=0 on the compact manifolds

Ouyang, Tiancheng

doi:10.1090/S0002-9947-1992-1055810-7

On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^ p=0$ on the compact manifolds
HTML articles powered by AMS MathViewer

by Tiancheng Ouyang

Trans. Amer. Math. Soc. 331 (1992), 503-527

DOI: https://doi.org/10.1090/S0002-9947-1992-1055810-7

PDF | Request permission

Abstract:

In this paper, we study the existence, nonexistence, and uniqueness of positive solutions of semilinear equations $\Delta u + \lambda u - h{u^p}= 0$ on compact Riemannian manifolds as well as on bounded smooth domains in ${R^n}$ with homogeneous Dirichlet or Neumann boundary conditions.

References

Jerry L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Regional Conference Series in Mathematics, vol. 57, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 787227, DOI 10.1090/cbms/057
Kentaro Yano and Morio Obata, Conformal changes of Riemannian metrics, J. Differential Geometry 4 (1970), 53–72. MR 261500
Jerry L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113–134. MR 365409
Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. MR 125546
Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR 240748

On variations of metric

29

Thierry Aubin, Métriques riemanniennes et courbure, J. Differential Geometry 4 (1970), 383–424 (French). MR 279731
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
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
Lars Inge Hedberg, Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math. 147 (1981), no. 3-4, 237–264. MR 639040, DOI 10.1007/BF02392874
Lars Inge Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299–319. MR 328088, DOI 10.1007/BF01181619