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On the least-energy solutions of the pure Neumann Lane–Emden equation

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Abstract

We study the pure Neumann Lane–Emden problem in a bounded domain

$$\begin{aligned} -\Delta u = |u|^{p-1} u \text { in }\Omega , \qquad \partial _\nu u=0 \text { on }\partial \Omega , \end{aligned}$$

in the subcritical, critical, and supercritical regimes. We show existence and convergence of least-energy (nodal) solutions (l.e.n.s.). In particular, we prove that l.e.n.s. converge to a l.e.n.s. of a problem with sign nonlinearity as \(p\searrow 0\); to a l.e.n.s. of the critical problem as \(p\nearrow 2^*-1\) (in particular, pure Neumann problems exhibit no blowup phenomena at the critical Sobolev exponent); and we show that the limit as \(p\rightarrow 1\) depends on the domain. Our proofs rely on different variational characterizations of solutions including a dual approach and a nonlinear eigenvalue problem. Finally, we also provide a qualitative analysis of l.e.n.s., including symmetry, symmetry-breaking, and monotonicity results for radial solutions.

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Acknowledgements

A. Saldaña was supported by UNAM-DGAPA-PAPIIT grant IA101721, Mexico; H. Tavares was partially supported by the Portuguese government through FCT-Fundação para a Ciência e a Tecnologia, I.P., under the projects UID/MAT/04459/2020, PTDC/MAT-PUR/28686/2017, and PTDC/MAT-PUR/1788/2020.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510, Coyoacán, Ciudad de México, Mexico

    Alberto Saldaña

  2. CAMGSD and Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Hugo Tavares

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  1. Alberto Saldaña
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  2. Hugo Tavares
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Correspondence to Alberto Saldaña.

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Saldaña, A., Tavares, H. On the least-energy solutions of the pure Neumann Lane–Emden equation. Nonlinear Differ. Equ. Appl. 29, 30 (2022). https://doi.org/10.1007/s00030-022-00762-7

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