# Ngô Quốc Anh

## September 11, 2014

### Construction of non-radial solutions for the equation Δu=u^q with 0<q<1 in the whole space

Filed under: Uncategorized — Ngô Quốc Anh @ 3:01

Given $q \in (0,1)$ and $N \geqslant 2$, in this note, we are interested in construction of non-radial solutions for the following equation

$\displaystyle \Delta u = u^q$

in the whole space $\mathbb R^N$. The construction is basically due to Louis Dupaigne and mainly depends on the unique radial solution of the equation.

To start our construction, let us recall that there is a unique radial solution, denoted by $u_0$, of the equation $\Delta u = u^q$ such that $u_0 (0)=1$ and $u'_0(0)=0$. Moreover, $u_0$ is globally defined and blows up at infinity at a fixed rate

$\displaystyle \lim_{r \to +\infty} \frac{u_0(r)}{r^\alpha} = L$

where $\alpha = \frac{2}{1-q}$ and $L=[\alpha (\alpha + N-2)]^{-1/(q-1)}>0$, see a paper by Yang and Guo published in J. Partial Diff. Eqns. in 2005.

Notice that

$\displaystyle \Delta u_0 = r^{1-N} (r^{N-1} u'_0)'.$

Hence, integrating both sides of the equation for $u_0$ gives

$\displaystyle\frac{du_0}{dr} = r^{1-N} \int_0^r t^{N-1} u_0^q dt$