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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A remark on positive radial solutions of the elliptic equation $ \Delta u+K(\vert x\vert )u\sp {(n+2)/(n-2)}=0$ in $ {\bf R}\sp n$


Authors: Yasuhiro Sasahara and Kazunaga Tanaka
Journal: Proc. Amer. Math. Soc. 123 (1995), 527-531
MSC: Primary 35J65; Secondary 35B40, 35J20
MathSciNet review: 1249893
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Abstract: We consider the following semilinear elliptic equation involving critical Sobolev exponents:

\begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u = K(\vert x\vert){u^{(n ... ... 0\quad {\text{as}}\;\vert x\vert \to \infty ,} \\ \end{array} \end{displaymath}

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/\vert x{\vert^{n - 2}}$ at $ \vert x\vert = \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$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1249893-7
PII: S 0002-9939(1995)1249893-7
Keywords: Critical Sobolev exponents, positive radial solutions, variational methods
Article copyright: © Copyright 1995 American Mathematical Society