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A remark on the existence of entire solutions of a singular semilinear elliptic problem. (English) Zbl 0891.35042

Summary: It is proved that the singular semilinear elliptic equation \(-\Delta u= p(x)g(u)\), \(0\leq p(x)\), \(x\in\mathbb{R}^n\), \(2\leq n\), \(\lim_{s\to 0^+} g(s)=+\infty\), and \(g\in C^1((0,\infty), (0,\infty))\) which is strictly decreasing in \((0,\infty)\), has a unique positive \(C^{2+\alpha}_{\text{loc}}(\mathbb{R}^n)\) solution that decays to zero near \(\infty\) provided \(\int^\infty_0 t\varphi(t)dt< \infty\), where \(\varphi(t)= \max_{|x|= t}g(x)\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
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References:

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