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Introduction to geometric potential theory. (English) Zbl 0754.31002
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