On the elliptic equations Δ𝑢=𝐾(𝑥)𝑢^{𝜎} and Δ𝑢=𝐾(𝑥)𝑒^{2𝑢}

Cheng, Kuo-Shung; Lin, Jenn-Tsann

doi:10.1090/S0002-9947-1987-0911088-1

On the elliptic equations $\Delta u=K(x)u^ \sigma$ and $\Delta u=K(x)e^ {2u}$
HTML articles powered by AMS MathViewer

by Kuo-Shung Cheng and Jenn-Tsann Lin

Trans. Amer. Math. Soc. 304 (1987), 639-668

DOI: https://doi.org/10.1090/S0002-9947-1987-0911088-1

PDF | Request permission

Abstract:

We give some nonexistence results for the equations $\Delta u = K(x){u^\sigma }$ and $\Delta u = K(x){e^{2u}}$ for $K(x) \geqslant 0$.

References

Wei Yue Ding and Wei-Ming Ni, On the elliptic equation $\Delta u+Ku^{(n+2)/(n-2)}=0$ and related topics, Duke Math. J. 52 (1985), no. 2, 485–506. MR 792184, DOI 10.1215/S0012-7094-85-05224-X
R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
Nichiro Kawano, Takaŝi Kusano, and Manabu Naito, On the elliptic equation $\Delta u=\varphi (x)u^\gamma$ in $\textbf {R}^2$, Proc. Amer. Math. Soc. 93 (1985), no. 1, 73–78. MR 766530, DOI 10.1090/S0002-9939-1985-0766530-X
Nichiro Kawano and Takaŝi Kusano, On positive entire solutions of a class of second order semilinear elliptic systems, Math. Z. 186 (1984), no. 3, 287–297. MR 744820, DOI 10.1007/BF01174883
Jerry L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113–134. MR 365409
J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math. 10 (1957), 503–510. MR 91407, DOI 10.1002/cpa.3160100402
Takaŝi Kusano and Shinnosuke Oharu, Bounded entire solutions of second order semilinear elliptic equations with application to a parabolic initial value problem, Indiana Univ. Math. J. 34 (1985), no. 1, 85–95. MR 773394, DOI 10.1512/iumj.1985.34.34004
Takaŝi Kusano, Charles A. Swanson, and Hiroyuki Usami, Pairs of positive solutions of quasilinear elliptic equations in exterior domains, Pacific J. Math. 120 (1985), no. 2, 385–399. MR 810778, DOI 10.2140/pjm.1985.120.385
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
Robert C. McOwen, On the equation $\Delta u+Ke^{2u}=f$ and prescribed negative curvature in $\textbf {R}^{2}$, J. Math. Anal. Appl. 103 (1984), no. 2, 365–370. MR 762561, DOI 10.1016/0022-247X(84)90133-1

Conformal metric in

with prescribed Gaussian curvature and positive total curvature

34

Manabu Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), no. 1, 211–214. MR 750398
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
Wei Ming Ni, On the elliptic equation $\Delta u+K(x)e^{2u}=0$ and conformal metrics with prescribed Gaussian curvatures, Invent. Math. 66 (1982), no. 2, 343–352. MR 656628, DOI 10.1007/BF01389399

On the equation

33

D. H. Sattinger, Conformal metrics in $\textbf {R}^{2}$ with prescribed curvature, Indiana Univ. Math. J. 22 (1972/73), 1–4. MR 305307, DOI 10.1512/iumj.1972.22.22001