Given and
, in this note, we are interested in construction of non-radial solutions for the following Lichnerowicz type equation
in the whole space .
In the previous post, we showed how to construct non-radial solutions of the following equation
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 the following
then a simple calculation shows and
. Hence, the function
is monotone increasing in
. Moreover, there exists a real number
sufficiently large such that
and
is concave in
. In addition, we can choose the number
even large in such a way that
for some constant .