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 .
As always, let us start with a radial solution, denoted by , of the equation such that and . It is well-known that is globally defined and blows up at infinity. Moreover, the following estimate holds
where . Again, a non-radial solution that we are going to construct is of the following form
For the sake of simplicity, we denote by the function . That is, we seek for solving
which is equivalent to solving
in where .
Since is concave, is always a subsolution of the preceding equation. Let . Then a direct calculation shows
Using the l’Hopital rule,
In view of the inequalities for , we obtain
for all . Using the l’Hopital rule again, we deduce that
Hence, . Hence, we have shown that
Thus, is a super-solution provided is chosen sufficiently small; hence showing the existence of a solution of our PDE with
In case , clearly when and . In case , we may assume that , hence upon a change of variable , if necessary, we know that
In conclusion, cannot be radial about any point in .