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Asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth. (English) Zbl 1307.35063

Summary: This paper concerns the blow-up behavior of large radial solutions of polyharmonic equations with power nonlinearities and positive radial weights. Specifically, we consider radially symmetric solutions of \(\operatorname{\Delta}^m u = c(| x |) | u |^p\) on an annulus \(\{x \in \mathbb{R}^n | \sigma \leq | x | < \rho \}\), with \(\rho \in(0, \infty)\) and \(\sigma \in [0, \rho)\), that diverge to infinity as \(| x | \to \rho\). Here \(n, m \in \mathbb{N}\), \(p \in(1, \infty)\), and \(c\) is a positive continuous function on the interval \([\sigma, \rho]\). Letting \(\phi_\rho(r) : = Q C_\rho /(\rho - r)^q\) for \(r \in [\sigma, \rho)\), with \(q : = 2 m /(p - 1)\), \(Q : = (q(q + 1) \cdots(q + 2 m - 1))^{1 /(p - 1)}\), and \(C_\rho : = c(\rho)^{- 1 /(p - 1)}\), we show that, as \(| x | \to \rho\), the ratio \(u(x) / \phi_\rho(| x |)\) remains between positive constants that depend only on \(m\) and \(p\). Extending well-known results for the second-order problem, we prove in the fourth-order case that \(u(x) / \phi_\rho(| x |) \to 1\) as \(| x | \to \rho\) and obtain precise asymptotic expansions if \(c\) is sufficiently smooth at \(\rho\). In certain higher-order cases, we find solutions for which the ratio \(u(x) / \phi_\rho(| x |)\) does not converge, but oscillates about 1 with non-vanishing amplitude.

MSC:

35B44 Blow-up in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
35J61 Semilinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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