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Exact boundary behavior of the unique positive solution to some singular elliptic problems. (English) Zbl 1281.31006
Summary: We give an exact asymptotic of the unique solution to the following singular boundary value problem $$-\varDelta u =a(x)g(u)$$, $$x \in \varOmega$$, $$u > 0$$, in $$\varOmega$$, $$u|_{\partial \varOmega} = 0$$. Here $$\varOmega$$ is a $$C^2$$-bounded domain in $$\mathbb R^n$$ $$(n \geq 2)$$, $$g \in C^1\big((0,\infty), (0,\infty)\big)$$ is nonincreasing on $$(0,\infty)$$ with
$\lim_{t\rightarrow 0} g'(t)\int^t_0\frac{ds}{g(s)} = -C_g \leq 0,$
and the function $$a$$ is in $$C^\alpha_{\text{loc}} (\varOmega)$$, $$0 < \alpha < 1$$, satisfying
$0 < a_1 = \liminf_{ d(x)\rightarrow 0}\frac{a(x)}{ h(d(x))} \leq\limsup_{d(x)\rightarrow 0}\frac{a(x)} {h(d(x))} = a_2 < \infty,$
where $h(t) = c t^{-\lambda} \exp(\int^\eta_t\frac{z(s)}s ds),$ $$\lambda \leq 2$$, $$c > 0$$, and $$z$$ is continuous on $$[0, \eta]$$ for some $$\eta > 0$$ such that $$z(0) = 0$$. Two applications of this result are also given. The first concerns the boundary behavior of the unique solution of $$-\varDelta u +\frac\beta u |\nabla u|^2 = a(x)g(u)$$ that vanishes on the boundary, and the second concerns the behavior of $$u$$ in the case where the open set $$\varOmega$$ is annular and the behaviors of the function $$a$$ on the interior boundary and the exterior boundary may be different.

##### MSC:
 31C15 Potentials and capacities on other spaces 34B27 Green’s functions for ordinary differential equations 35K10 Second-order parabolic equations
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