Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms

Thomas Bartsch () and Qianqiao Guo ()
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Thomas Bartsch: Justus-Liebig-Universität Giessen
Qianqiao Guo: Northwestern Polytechnical University

Partial Differential Equations and Applications, 2020, vol. 1, issue 5, 1-21

Abstract: Abstract We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-\mu \frac{u}{|x|^2} = |u|^{2^{*}-2-\varepsilon }u &{} \text {in } \Omega , \\ u = 0&{} \text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$ - Δ u - μ u | x | 2 = | u | 2 ∗ - 2 - ε u in Ω , u = 0 on ∂ Ω , in a bounded domain $$\Omega \subset {\mathbb {R}}^N (N\ge 7)$$ Ω ⊂ R N ( N ≥ 7 ) with $$0\in \Omega$$ 0 ∈ Ω , as $$\mu ,\epsilon \rightarrow 0^+$$ μ , ϵ → 0 + . In [6], we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as $$\mu =O(\epsilon ^\alpha )$$ μ = O ( ϵ α ) with $$\alpha >\frac{N-4}{N-2}$$ α > N - 4 N - 2 , $$\epsilon \rightarrow 0^+$$ ϵ → 0 + . Here we prove the existence of nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as $$\mu =O(\epsilon )$$ μ = O ( ϵ ) , $$\epsilon \rightarrow 0^+$$ ϵ → 0 + .

Keywords: Hardy term; Critical exponent; Slightly subcritical problems; Nodal solutions; Bubble towers; Singular perturbation methods.; 35B44; 35B33; 35J60 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-020-00029-9

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