Ngô Quốc Anh

January 8, 2010

A trick on finding sup- and super-solutions for some elliptic equation involing critical exponents

Let us consider the following equation

\displaystyle\Delta u + a{u^{ - \frac{{3n - 2}}{{n - 2}}}} - b{u^{\frac{{n + 2}}{{n - 2}}}} = 0

in \mathbb R^n. We assume that both a \geqslant 0 and b \geqslant 0. We are interested in finding positive solutions for the above equation.

By sup- and super-solutions to the above equation we mean functions u_\star and u^\star such that

\displaystyle \Delta {u^\star} + a({u^\star})^{ - \frac{{3n - 2}}{{n - 2}}} - b({u^\star})^{\frac{{n + 2}}{{n - 2}}} \leqslant 0 \leqslant \Delta {u_ \star } + au_ \star ^{ - \frac{{3n - 2}}{{n - 2}}} - bu_ \star ^{\frac{{n + 2}}{{n - 2}}}

and

\displaystyle {u_\star } \leqslant {u^\star }.

The key point of sup- and super-solutions method is to tell us that having the existence of u_\star and u^\star we can prove the existence of one solution u satisfying

\displaystyle {u_\star } \leqslant u \leqslant {u^\star }.

We also note that this method says nothing about the uniqueness. We will try to construct u_\star such that u_\star \geqslant 0.

Construction of sup-solution. We first consider equation (when b=0)

\displaystyle\Delta u + a{u^{ - \frac{{3n - 2}}{{n - 2}}}}= 0.

This equation admits a constant sub-solution u_-=1 but no finite constant super-solution. However, it admits a non-constant super-solution, namely, the function

\displaystyle u_+=\varphi_+ +1

with \varphi_+ a solution of the linear equation

\displaystyle \Delta \varphi_+=-a.

Indeed, the maximum principle shows that \varphi_+ \geqslant 0, hence u_+ \geqslant 1 and

\displaystyle\Delta {u_ + } + au_ + ^{ - \frac{{3n - 2}}{{n - 2}}} = \Delta {\varphi _ + } + au_ + ^{ - \frac{{3n - 2}}{{n - 2}}} = - a + au_ + ^{ - \frac{{3n - 2}}{{n - 2}}} \leqslant 0.

Thus, we can prove the existence of a super-solution u^\star.

Construction of super-solution. We next consider equation (when a=0)

\displaystyle\Delta u - b{u^{\frac{{n + 2}}{{n - 2}}}} = 0.

This equation admits the sub-solution u_-=0 and the super-solution u_+=1. And thus, there exists u_\star a solution to the equation. In fact we can prove u_\star>0 (see the proof of Brill-Canton theorem).

The proof completed. We now consider the given equation, clear u_\star and u^\star are sup- and super-solutions which proves the existence of solution to the equation.

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