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On positive entire solutions to singular, nonlinear \(p\)-harmonic equations in \(\mathbb{R}^N\, (N\geq 3)\). (Chinese. English summary) Zbl 1056.31005

The \(N\)-dimensional singular nonlinear \(p\)-harmonic equation of the norm \[ \Delta\bigl((\Delta u)^{p-1^*}\bigr)=f\bigl(|x|,u,|\nabla u| \bigr)u^{-\beta},\quad x\in\mathbb{R}^N\quad (N\geq 3) \] is considered, where \(p>1\), \(\beta\geq 0\), \(\xi^{\alpha^*}=|\xi|^{\alpha-1}\xi\), \(\xi\in\mathbb{R}\), \(\alpha >0\), and \(f:\overline\mathbb{R}_+ \times\mathbb{R}_+ \times\overline\mathbb{R}_+\) is a continuous function. The author obtains some sufficient conditions for the existence of infinitely many positive symmetric entire solutions which are bounded or asymptotic to positive constant multiples of \(|x|^{(2p-N)/(p-1)}\) when \(p>N/2\), and \(\log t\) when \(p=N/2\) as \(|x|\to\infty\).

MSC:

31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J30 Higher-order elliptic equations
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