On the elliptic equation 𝐷ᵢ[𝑎ᵢⱼ(𝑥)𝐷ⱼ𝑈]-𝑘(𝑥)𝑈+𝐾(𝑥)𝑈^{𝑝}=0

Lin, Fang-Hua

doi:10.1090/S0002-9939-1985-0801327-3

On the elliptic equation $D_ i[a_ {ij}(x)D_ jU]-k(x)U+K(x)U^ p=0$
HTML articles powered by AMS MathViewer

by Fang-Hua Lin

Proc. Amer. Math. Soc. 95 (1985), 219-226

DOI: https://doi.org/10.1090/S0002-9939-1985-0801327-3

PDF | Request permission

Abstract:

The problem of the existence and nonexistence of entire, positive solutions to the uniformly elliptic, semilinear equation ${D_i}[{a_{ij}}(x){D_j}U] - k(x)U + K(x){U^p} = 0$ in ${{\mathbf {R}}^n}$, where $p > 1$, is studied. A limiting case when $K(x)$ is negative and has quadratic decay at infinity is also treated.

References

E. Fabes, D. Jerison, and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vi, 151–182 (English, with French summary). MR 688024, DOI 10.5802/aif.883
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
Carlos E. Kenig and Wei-Ming Ni, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math. 106 (1984), no. 3, 689–702. MR 745147, DOI 10.2307/2374291

Conformal metrics with prescribed curvatures

4

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
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019

Semi-linear equations in

without conditions at infinity