# Ngô Quốc Anh

## October 12, 2014

### Construction of non-radial solutions for a Lichnerowicz type equation in the whole space

Filed under: Uncategorized — Ngô Quốc Anh @ 1:23

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

$\displaystyle -\Delta u = -u^q + u^{-q-2}$

in the whole space $\mathbb R^N$.

In the previous post, we showed how to construct non-radial solutions of the following equation

$\displaystyle -\Delta u = -u^q.$

Clearly, this equation comes from the Lichnerowicz type equation by writing off the term with a negative exponent.

To start our construction and for simplicity, let us denote by $f$ the following

$\displaystyle f(t) = t^q - t^{-q-2},$

then a simple calculation shows $f'(t)=qt^{q-1} + (q+2)t^{-q-3}$ and $f''(t)=q(q-1)t^{q-2} - (q+2)(q+3)t^{-q-4}$. Hence, the function $f$ is monotone increasing in $[0,+\infty)$. Moreover, there exists a real number $a>0$ sufficiently large such that $f>0$ and $f$ is concave in $[a,+\infty)$. In addition, we can choose the number $a$ even large in such a way that

$\displaystyle \frac 1C f''(t) \leqslant f''(2t) \leqslant C f''(t)$

for some constant $C>0$.