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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A remark on positive radial solutions of the elliptic equation $\Delta u+K(\vert x\vert )u^ {(n+2)/(n-2)}=0$ in $\textbf {R}^ n$
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by Yasuhiro Sasahara and Kazunaga Tanaka PDF
Proc. Amer. Math. Soc. 123 (1995), 527-531 Request permission

Abstract:

We consider the following semilinear elliptic equation involving critical Sobolev exponents: \[ \begin {array}{*{20}{c}} { - \Delta u = K(|x|){u^{(n + 2)/(n - 2)}}\quad {\text {in}}\;{{\mathbf {R}}^n},} \\ {u(x) \to 0\quad {\text {as}}\;|x| \to \infty ,} \\ \end {array} \] where $n \geq 3,K(r) \in C([0,\infty ),{\mathbf {R}})$. We prove the existence of a positive radial solution with asymptotic behavior $C/|x{|^{n - 2}}$ at $|x| = \infty$ under the conditions (i) $K(r) > 0$ for all $r > 0$, (ii) $K(0) = K(\infty )$, and (iii) there exist C, $\delta > 0$ such that $K(r) \geq K(0) - C{r^\delta }$ for small $r > 0$ and $K(r) \geq K(0) - C{r^{ - \delta }}$ for large $r > 0$.
References
  • Abbas Bahri and Yan Yan Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\textbf {R}^N$, Rev. Mat. Iberoamericana 6 (1990), no.Β 1-2, 1–15. MR 1086148, DOI 10.4171/RMI/92
  • A. Bahri and P. L. Lions, On the existence of a positive solution of a semilinear elliptic equations in unbounded domains, preprint.
  • Gabriele Bianchi and Henrik Egnell, A variational approach to the equation $\Delta u+Ku^{(n+2)/(n-2)}=0$ in $\mathbf R^n$, Arch. Rational Mech. Anal. 122 (1993), no.Β 2, 159–182. MR 1217589, DOI 10.1007/BF00378166
  • HaΓ―m BrΓ©zis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no.Β 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
  • W.-Y. Ding and W.-M. Ni, On the elliptic equations $\Delta u + K{u^{(n + 2)/(n - 2)}} = 0$ and related topics, Duke Math. J. 52 (1985), 485-506.
  • Chang Shou Lin and Song-Sun Lin, Positive radial solutions for $\Delta u+K(r)u^{(n+2)/(n-2)}=0$ in $\textbf {R}^n$ and related topics, Appl. Anal. 38 (1990), no.Β 3, 121–159. MR 1116178, DOI 10.1080/00036819008839959
  • Yi Li and Wei-Ming Ni, On conformal scalar curvature equations in $\textbf {R}^n$, Duke Math. J. 57 (1988), no.Β 3, 895–924. MR 975127, DOI 10.1215/S0012-7094-88-05740-7
  • 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.Β 4, 493–529. MR 662915, DOI 10.1512/iumj.1982.31.31040
  • E. Yanagida and S. Yotsutani, Existence of positive radial solutions to $\Delta u + K(|x|){u^{(n + 2)/(n - 2)}} = 0$ in ${{\mathbf {R}}^n}$, preprint. β€”, Classification of the structure of positive radial solutions to $\Delta u + K(r){u^{(n + 2)/(n - 2)}} = 0$ in ${{\mathbf {R}}^n}$, preprint.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 527-531
  • MSC: Primary 35J65; Secondary 35B40, 35J20
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1249893-7
  • MathSciNet review: 1249893