Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Question

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv &0&&\text{in }\left\{0,R\right\}\end{matrix}\right.$$ Must $u$ be positive?

Answer 1

Multiplying both sides of the ODE by the integrating factor $r^{n-1}$ and integrating with respect to $r$ yields $b^{n-1} u'(b) - a^{n-1} u'(a) = -\int_a^b r^{n-1} u^p dr$. The boundary conditions imply $0 = \int_0^R r^{n-1} u^p dr$ so $u$ cannot be strictly positive on $(0,R)$.

In particular, if $p$ is even then $u=0$ is the only radial solution. This can also be shown by noting that when $p$ is even $u$ is superharmonic and a minimum principle applies.