Suppose $B_1=\{x\in\mathbb{R}^n:\ \|x\|<1\}$, $N\geq 2$, $p\in (1,\infty)$, $u\in W^{1,p}_{loc}(B_1)$ satisfies $$\int_{B_1}|\nabla u|^{p-2}\nabla u\nabla\phi=\int_{B_1}f\phi,\ \forall\ \phi\in C_0^\infty(B_1) $$

where $f\in L^{q}(B_1)$, $q>\frac{N}{p}$.

Consider Theorem 4.1 from this book. In trying to adpat the demonstration of this theorem to this problem, I had to impose the additional condition that $$q\geq \frac{N(p-1)}{Np+p(p-1)-2N}$$

Note that if $p=2$ this condition is redundant. My question is: Does anyone knows if this condition is necessary?

  • $\begingroup$ I dont know how to post a screen here and the link works fine here. Try it again: books.google.com.br/… $\endgroup$ – Tomás Dec 19 '12 at 18:15
  • $\begingroup$ Interesting @PavelM. They have achieved a global estimate and looks like their theorem is more general. I was reviewing my calculations and I found some $\delta>0$ such that the imposed condition is necessary if and only if $p\in (1,2-\delta)$. $\endgroup$ – Tomás Dec 20 '12 at 10:25

I expanded my comments to the point where they look like an answer, and moved them here. My assumption is that that the screen grabs embedded below are within fair use.

First the context: a form of the Harnack maximum principle for uniformly elliptic equations is stated in the book Elliptic Partial Differential Equations by Han and Lin as follows:

Theorem 4.1 from the book by Han and Lin

The question was whether it is possible to obtain such an estimate to the $p$-Laplace equation, and in a particular range of exponents. The classical reference is Serrin's paper Local behavior of solutions of quasilinear equations where a form of Harnack's inequality is proved for $1<p<n$, $q>n/p$. This paper was cited a bazillion times and the top result of this search, the book "Nonlinear potential theory of degenerate elliptic equations" by Heinonen Kilpeläinen and Martio, is probably the best book exposition of the subject that I've seen.

As an aside, I mention a global estimate from the recent (or future?) paper A new Aleksandrov–Bakelman–Pucci maximum principle for p-Laplacian operator by Tingting Wang and Lizhou Wang, in "Nonlinear Analysis: Theory, Methods & Applications", vol. 77, (2013), 171–179. They consider the equation $$-\operatorname{div} |\nabla u|^{p-2}\nabla u=-f \tag{1.1}$$ in a bounded domain $\Omega\subset \mathbb R^n$, where $1<p<\infty$ and $f\in L^q$ with $q>n/p$. If $u$ is a subsolution of (1.1) and $1/p+1/q\le 1$, then the ABP maximum principle holds: $\sup_\Omega u - \sup_{\partial \Omega} u^+$ is controlled by the $L^q$ norm of $f$, with an appropriate dimensional constant. The more refined statement is below.

Theorem by T. Wang and L. Wang


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