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Proceedings of the American Mathematical Society

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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 $\textbf {R}^ n$


Authors: Yasuhiro Sasahara and Kazunaga Tanaka
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
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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 $r > 0$.


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Keywords: Critical Sobolev exponents, positive radial solutions, variational methods
Article copyright: © Copyright 1995 American Mathematical Society