# 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.