Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities
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Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities Hui Yang1 Received: 27 November 2019 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation
Δ2 u = up
in B1 �{0}
with an isolated singularity, where the punctured ball B1 �{0} ⊂ ℝn with n ≥ 5 and n n+4 < p < n−4 . This equation is relevant for the Q-curvature problem in conformal geomn−4 etry. We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for −Δu . We also give a new method to prove removable singularity theorem for nonlinear higher order equations. Mathematics Subject Classification 35J30 · 35B40 · 35B65
1 Introduction and main results In this paper, we study the asymptotic behavior of positive solutions of the biharmonic equation
Δ2 u = up
in B1 �{0}
(1.1)
with an isolated singularity, where the punctured ball B1 �{0} ⊂ ℝn with n ≥ 5 and n n+4 < p < n−4 . Here the unit ball B1 can be replaced by any bounded domain Ω ⊂ ℝn conn−4 taining 0. This equation serves as a basic model of nonlinear fourth-order equations and is also related to the Q-curvature problem in conformal geometry. Equation (1.1) and related equations arise in several models describing various phenomena in the applied sciences see, for instance, Gazzola et al. [13]. For an introduction to the Q-curvature problem see, for instance, Hang and Yang [20].
Communicated by A. Malchiodi. * Hui Yang [email protected]; hui‑[email protected] 1
Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
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H. Yang
n n−2
We first recall that the corresponding second order equation (when n ≥ 3 and n+2 < p < n−2 )
−Δu = up
(1.2)
in B1 �{0}
was studied by Gidas–Spruck [15] and Caffarelli–Gidas–Spruck [3]. More specifically, the following classification result is obtained.
Theorem A ([3, 15]) Let n ≥ 3 and u ∈ C2 (B1 �{0}) be a positive solution of (1.2). Assume
n n+2 . n+2 . Hence the isolated singularin−2 n−2 ties of positive solutions for the second order equation (1.2) have been very well understood. The asymptotic n+2 behavior of positive solutions for a more general second order equation −Δu = K(x)u n−2 with isolated singularity was studied by Chen–Lin [7, 8] and Taliaferro–Zhang [30]. See also González [16], Li [23] and Han–Li–Teixeira [19] for a fully nonlinear equation of second order. In the fundamental paper [24], Lin classified all positive smooth entire solutions of (1.1) in ℝn via the moving plane method. We refer to Chang–Yang [6], Martiwith 1 < p ≤ n+4 n−4 nazzi [26] and Wei–Xu [31] for the classification of smooth solutions of the higher-order , the positive smooth radial equations in ℝn . For the supercritical case, that is for p > n+4 n−4 symmetric solutions of (1.1) in ℝn were studied by Gazzola–Grunau [12], Guo–Wei [17] and Winkler
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