# Ngô Quốc Anh

## March 9, 2013

### On the integrability of the inverse of functions in the Sobolev space H_0^1

Filed under: PDEs — Ngô Quốc Anh @ 9:05

Yesterday, I read a recent paper by S. Yijing and Z. Duanzhi published in the journal Calculus of Variations in 2013. In one of their results, they proved the following

Theorem 2. Let $\Omega \subset\mathbb R^N$, $N \geq 3$ be bounded open set with smooth boundary. If $p\geq 3$, then

$\displaystyle \int_\Omega |u|^{1-p}dx =+\infty, \quad \forall u \in H_0^1(\Omega).$

Since $p \geq 3$, it is clear that $1-p \leq -2$. By definition, $\int_\Omega |u|^2dx <+\infty$ which gives us some comparison between $u$ and its inverse power.

In order to prove the above theorem, the authors first proved the following

Theorem 1. Let $\Omega \subset\mathbb R^N$, $N \geq 3$ be bounded open set with smooth boundary, the function $h\in L^1(\Omega)$ is positive a.e. in $\Omega$, and $p>1$. Then the equation

$\displaystyle\begin{array}{rcl} \Delta u + h(x){u^{ - p}} &=& 0 \quad \text{ in } \Omega \hfill \\ u &>& 0 \quad \text{ in } \Omega \\ u &=& 0 \quad \text{ on }\partial \Omega \end{array}$

admits a unique $H_0^1$-solution if and only if there exists $u_0 \in H_0^1(\Omega)$ such that

$\displaystyle \int_\Omega h(x) |u_0|^{1-p}dx <+\infty.$

The proof of Theorem 1 is variational which makes use of the Nehari manifold and the fibering map. Basically, the authors tried to minimize the functional

$\displaystyle I(u) = \frac{1}{2}\int_\Omega {|\nabla u{|^2}dx} + \frac{1}{{p - 1}}\int_\Omega {h(x)|u{|^{1 - p}}dx}$

over the closed set (in $H_0^1$-topology)

$\displaystyle N = \left\{ {u \in H_0^1(\Omega ):\int_\Omega {|\nabla u{|^2}dx} - \int_\Omega {h(x)|u{|^{1 - p}}dx} \geqslant 0} \right\}.$

To examine the best minimizing sequence for $\inf_N I$, the authors used the Ekeland variational principle. Making use of the fibering map, the authors was successfully locate the critical point which gives nothing but a solution to the problem. In view of this setting, if the integral $\int_\Omega h(x)|u|^{1-p}dx <+\infty$, the set $N$ is non-empty. Conversely, the solution of the problem gives the existence of such an $u_0 \in H_0^1(\Omega)$. It turns out that the difficulty is when the limiting function, say $u^\star$, lies in the boundary of $N$, say

$\displaystyle N^\star = \left\{ {u \in H_0^1(\Omega ):\int_\Omega {|\nabla u{|^2}dx} - \int_\Omega {h(x)|u{|^{1 - p}}dx} =0} \right\}.$

It is worth noticing that by $H_0^1$-solution we mean that

$\displaystyle\int_\Omega {\nabla u \cdot \nabla \varphi dx} - \int_\Omega {h(x){u^{ - p}}\varphi dx} = 0, \quad \forall \varphi \in H_0^1(\Omega ).$

Since in general it is not true that

$\displaystyle\int_\Omega {h(x){u^{ - p}}{\varphi _n}dx} \to \int_\Omega {h(x){u^{ - p}}\varphi dx}$

where $\{\varphi_n\}\subset C_0^1(\Omega)$ with $\varphi_n \to \varphi$ in $H_0^1(\Omega)$, the validity of

$\displaystyle\int_\Omega {\nabla u \cdot \nabla \varphi dx} - \int_\Omega {h(x){u^{ - p}}\varphi dx} = 0, \quad \forall \varphi \in C_0^1(\Omega )$

is preferred to $H_{\rm loc}^1(\Omega)$ solutions.

In order to prove Theorem 2, the authors made use of a result of Lazer and McKenna which says that the problem

$\displaystyle\begin{array}{rcl} \Delta u +{u^{ - p}} &=& 0 \quad \text{ in } \Omega \hfill \\ u &>& 0 \quad \text{ in } \Omega \\u &=& 0 \quad \text{ on }\partial \Omega \end{array}$

has a $H_0^1$-solution if and only if $p<3$.

We are now in a position to prove Theorem 2. Indeed, we argue by contradiction that there exists some $u_0 \in H_0^1(\Omega)$ satisfying

$\displaystyle \int_\Omega |u_0|^{1-p}dx < \infty$

with $p \geqslant 3$. Then in view of Theorem 1, one can obtain some $u^\star \in H_0^1(\Omega)$ solving the following

$\displaystyle\begin{array}{rcl} \Delta u +{u^{ - p}} &=& 0 \quad \text{ in } \Omega \hfill \\ u &>& 0 \quad \text{ in } \Omega \\u &=& 0 \quad \text{ on }\partial \Omega \end{array}$

while by the Lazer-McKenna result this problem has $u^\star$ if and only if $p<3$. The proof follows.

In my opinion, the result in Theorem 2 is interesting which provides a necessary and sufficient condition for the question: why $3$ becomes such an obstruction to solvability in the study of elliptic equations with negative exponents?