Introduction to geometric potential theory. (English) Zbl 0754.31002

Suzuki, Takashi

Functional-analytic methods for partial differential equations, Proc. Conf. Symp., Tokyo/Jap. 1989, Lect. Notes Math. 1450, 88-103 (1990).

[For the entire collection see Zbl 0707.00017.]
The author studies blow-up sets for solutions of semilinear equations. Non-negative functions \(\{u_ k\}\) on \(\overline\Omega\), a bounded domain in \(\mathbb{R}^ 2\), are said to make blow-up if \(\| u_ k\|_ \infty\to\infty\) as \(k\to\infty\). The blow-up set \(S\) is defined as \(S=\{x_ 0\in\overline\Omega\mid\) there exists some \(\{x_ k\}\subset\Omega\) such that \(x_ k\to x_ 0\) and \(v(x_ k)\to\infty\}\). \(S_ I=S\cap\Omega\) denotes the interior blow-up set.
One of the main results is the following (called by the author Harnack principle):
Theorem: The interior blow-up set \(S_ I\) of functions \(\{u_ k\}\subset C^ 2(\Omega)\cap C(\overline\Omega)\) satisfying \[ 0\leq- \Delta u_ k\leq\lambda_ k e^{u_ k},\qquad u_ k\geq 0\qquad (\Omega\subset\mathbb{R}^ 2) \] for some positive constants \(\{\lambda_ k\}\) either has an interior point or is finite, provided that \[ \Sigma_ k=\int_ \Omega \lambda_ k e^{u_ k} dH^ 2\in O(1). \] Furthermore, if \(S_ I\) has no interior points we have \[ \# S_ I\leq \liminf[\Sigma_ k/8\pi]. \] This result applies to semilinear equations of the form \(-\Delta u=\lambda f(u)\), \(u>0\) in \(\Omega\subset\mathbb{R}^ 2\) with \(u=0\) on \(\partial\Omega\) under suitable conditions on \(f\).

Reviewer: S.Salsa (Milano)

MSC:

31A35	Connections of harmonic functions with differential equations in two dimensions
35J25	Boundary value problems for second-order elliptic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

semilinear equations; blow-up set; Harnack principle