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.


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