A priori estimates for a nonlinear elliptic problem singular on the boundary

Question

Let us consider the following elliptic problem $$ \begin{cases} -\Delta u = \frac{u^p}{|x|^2} \mbox{ in } \Omega \\ u >0 \mbox{ in } \Omega \\ u = 0 \mbox{ on } \partial \Omega. \end{cases} $$ with $N \geq 3$, $\Omega$ a bounded domain in $\mathbb{R}^N$ with smooth boundary,$1 < p < \frac{N+2}{N-2}$ and $0 \in \partial \Omega$.

We say that $u$ is an energy solution of the preceding problem if $u \in H^1_0(\Omega)$, $u >0$ such that $$ \int_\Omega \nabla u \nabla \varphi \, dx = \int_\Omega \frac{u^p}{|x|^2} \varphi \, dx \quad \forall \varphi \in \mathcal{C}^\infty_0(\Omega). $$

The authors proved the following Lemma.

If $u$ is an energy solution of the preceding problem then $u \in L^\infty(\Omega)$.

The proof, they said, is just a consequence of the a priori estimates of Gidas and Spruck, "A priori bounds for positive solutions of nonlinear elliptic equations", Comm. Partial Differential Equations 6 (1981), after suitable scaling. Moreover they gave this hint. Suppose $x_0 \in \Omega$, $x\neq 0$. Let $r:= \frac{|x_0|}{2}$ and define the function $$ v(y) := u(x_0 + ry) \quad \mbox{in } \frac{\Omega - x_0}{r} $$ Then $v$ satisfies the equation $$ - \Delta v = \frac{r^2}{|x_0 + ry|^2}v^p \quad \mbox{in } \frac{\Omega - x_0}{r} $$ Now restrict the equation to $\frac{\Omega - x_0}{r} \cap B_1(0)$. In this region the weight $\frac{r^2}{|x_0 + ry|^2}$ is bounded and smooth. Now using the a priori estimates of Gidas and Spruck we deduce that $v(0) \leq C$ for some universal $C$. Finally since $u(x_0) = v(0)$ we have the thesis, since $C$ is not depending on $x_0$.

Howewer I don't understand how the result of Gidas and Spruck could be useful in this setting. Indeed although in the region $\frac{\Omega - x_0}{r} \cap B_1(0)$ the weight is bounded and smooth, the boundary conditions of the problem for the function $v$ are "free" ( at least in some part of the boundary of that region). So, applying the Gidas and Spruck results we have that there exists a constant, but I don't get how can this constant be universal, since it seems, to me, depending on the boundary conditions and hence on the point $x_0$.

Answer 1

I don't see either that one can apply the result of Gidas and Spruck in this situation. Foremost, this is because the argument given by Davila and Peral does not take into account what happens near the singular point $x=0$.

Still that $u\in L^\infty(\Omega)$ (and I guess this is what you're interested in) seems to be correct. Clearly, any solution $u$ satisfies $u\in \mathscr C^\infty(\overline\Omega\setminus\{0\})$. So, the question whether $u$ is bounded or not comes down to answering it near $x=0$. Then the trick is to regard $x=0$ as a conic point and to invoke something what is called singular analysis (here, flatten the boundary near $x=0$ by introducing local coordinates $y=(y_1,\dots,y_N)$ centered at $x=0$ such that $\Omega$ is locally given by $y_N>0$ and then rewrite everything in polar coordinates $\displaystyle (r,\theta) = \left(|y|,\frac{y}{|y|}\right)$, where $r>0$ and $\theta\in\mathbb S_+^{N-1}$). Good references are P. Grisvard, Elliptic Problems in Nonsmooth Domains, Chapman and Hall, 1985 and also M. Borsuk and V. Kondratiev, Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains, North Holland, 2006. Repeatedly using elliptic regularity for the (cone-degenerate elliptic) differential operator $\Delta$, one eventually arrives at $$ u(x) \sim \sum_{k\in\mathbb N_0} r^k u_k(\theta) \enspace\text{as $x\to0$} \tag{1} $$ for some functions $u_k\in\mathscr C^\infty(\mathbb S_+^{N-1})$ which satisfy $u_k\bigr|_{\partial\mathbb S_+^{N-1}}=0$. (The right-hand side in (1) possibly contains log terms for $k\geq1$ - I haven't checked - but this does not prevent $u$ from being bounded near $x=0$.)

Roughly, without going into the details, the general theory tells one that the solution $u$ admits an asymptotic expansion like the one shown in (1) containing terms of the form $r^\alpha\log^m\!r\,\varphi(\theta)$, where $\alpha\in \mathbb R$, $m\in\mathbb N_0$. Now, (1) constitutes indeed the only choice of the $\alpha$ which is consistent with the problem under study. For instance, if $u(x)$ behaved like $r^{-\gamma}u_{-\gamma}(\theta)$ (to the leading order as $x\to 0$), where $\gamma>0$, then $\displaystyle \frac{u^p}{|x|^2}$ would behave like $r^{-p\gamma-2}\psi(\theta)$, for some $\psi$. But then, by elliptic regularity, $u(x)$ would behave like $r^{-p\gamma}\phi(\theta)$, for some $\phi$, which leads to the contradiction $-\,p\gamma<-\,\gamma$ (because of $p>1$).