Ngô Quốc Anh

September 11, 2014

Construction of non-radial solutions for the equation Δu=u^q with 0<q<1 in the whole space

Filed under: Uncategorized — Ngô Quốc Anh @ 3:01

Given q \in (0,1) and N \geqslant 2, in this note, we are interested in construction of non-radial solutions for the following equation

\displaystyle \Delta u = u^q

in the whole space \mathbb R^N. 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 u_0, of the equation \Delta u = u^q such that u_0 (0)=1 and u'_0(0)=0. Moreover, u_0 is globally defined and blows up at infinity at a fixed rate

\displaystyle \lim_{r \to +\infty} \frac{u_0(r)}{r^\alpha} = L

where \alpha = \frac{2}{1-q} and L=[\alpha (\alpha + N-2)]^{-1/(q-1)}>0, see a paper by Yang and Guo published in J. Partial Diff. Eqns. in 2005.

Notice that

\displaystyle \Delta u_0 = r^{1-N} (r^{N-1} u'_0)'.

Hence, integrating both sides of the equation for u_0 gives

\displaystyle\frac{du_0}{dr} = r^{1-N} \int_0^r t^{N-1} u_0^q dt


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