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?

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