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 .

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