The existence of $$L^2$$ L 2 -constrained solutions for Kirchhoff equations with $$L^2$$ L 2 -critical general nonlinearity

Hongyu Ye () and Jiahui Yin
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Hongyu Ye: Wuhan University of Science and Technology
Jiahui Yin: Wuhan University of Science and Technology

Partial Differential Equations and Applications, 2025, vol. 6, issue 3, 1-25

Abstract: Abstract In this paper, for given $$c>0$$ c > 0 , we study the existence of a couple of solution $$(u_c,\lambda _c)\in H^1({\mathbb R}^N)\times {\mathbb R}_+$$ ( u c , λ c ) ∈ H 1 ( R N ) × R + to the following Kirchhoff type problem: $$ \left\{ \begin{array}{ll} -\left( a+b \int _{{\mathbb R}^N}|\nabla u|^2\right) \Delta u+\lambda u=f(u),\,\,\, & ~x\in {\mathbb R}^N, \\ (\int _{{\mathbb R}^N}|u|^2)^{\frac{1}{2}}=c,\,\,\, & \end{array} \right. $$ - a + b ∫ R N | ∇ u | 2 Δ u + λ u = f ( u ) , x ∈ R N , ( ∫ R N | u | 2 ) 1 2 = c , where $$N\le 3$$ N ≤ 3 , a, $$b>0$$ b > 0 are constants, $$f(u)\sim |u|^{\frac{8}{N}}u$$ f ( u ) ∼ | u | 8 N u is a $$L^2$$ L 2 -critical general nonlinearity. By using the scaling method and a new version of global compactness lemma, we prove that there exists $$c_*>0$$ c ∗ > 0 such that the problem admits no solution for $$0 c_*$$ c > c ∗ . Our main results can be viewed as an extension of [He et. al. JDE, 356:375–406, (2023)] concerning the $$L^2$$ L 2 -supercritical nonlinearity.

Keywords: Constrained minimization; Subadditivity inequality; Nehari–Pohozaev indentity; Global $$L^2$$ L 2 -constrained minimizers; $$L^2$$ L 2 -critical general nonlinearity; 35J60; 35A15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00328-z

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