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 | References | Similar Articles | Additional Information

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