Given and , in this note, we are interested in construction of non-radial solutions for the following equation
in the whole space . 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 , of the equation such that and . Moreover, is globally defined and blows up at infinity at a fixed rate
where and , see a paper by Yang and Guo published in J. Partial Diff. Eqns. in 2005.
Hence, integrating both sides of the equation for gives
from which it follows that
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 . Clearly, solves the linearized equation
in . Since , we know that . Hence, we can conclude that
as . Thanks to , the function needs to satisfy the following
Using the Taylor expansion, we further obtain
in for some . Hence, our aim is to solve the preceding equation for . To this purpose, we shall use the method of sub-super solutions. Clearly, is always a sub-solution. A direct calculation shows which solves
could be a super-solution provided is sufficiently small.
Hence, for fixed , we can always find a function in . By elliptic regularity, up to a subsequence, converges in to a solution of the previous PDE. Moreover, there holds
Observe that, when and , there holds ; hence cannot be radial about any point in since the term cannot be radial.