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.


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


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